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1-2016
Discrete Mathematics: Chapter 2, Predicate Logic
Calvin Jongsma
Dordt College, [email protected]
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Discrete Mathematics: Chapter 2, Predicate Logic
Abstract
In this chapter we will explore Predicate Logic (PL), an extension of Sentential Logic, the system we studied
in Chapter 1. There is the potential here to get tangled up in picky details, since PL is a refinement that deals
with the inner logical structure of sentences as well as sentential connectives. We will keep our treatment fairly
informal, however, since our goal is not to master the fine points of logic but to learn the system of PL in order
to better analyze mathematical propositions and understand mathematical proof strategies.
In this chapter we will explore Predicate Logic (PL), an extension of Sentential Logic, the system we studied
in Chapter 1. There is the potential here to get tangled up in picky details, since PL is a refinement that deals
with the inner logical structure of sentences as well as sentential connectives. We will keep our treatment fairly
informal, however, since our goal is not to master the fine points of logic but to learn the system of PL in order
to better analyze mathematical propositions and understand mathematical proof strategies.
Keywords
predicate, logic, proof, analysis, algebra
Disciplines
Christianity | Computer Sciences | Mathematics
Comments
• From Discrete Mathematics: An Integrated Approach, a self-published textbook for use in Math 212
• © 2016 Calvin Jongsma
Creative Commons License
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0
License.
This book chapter is available at Digital Collections @ Dordt: http://digitalcollections.dordt.edu/faculty_work/432
z
Chapter 2 z
PREDICATE LOGIC
2.1 Symbolizing Mathematical Sentences
In this chapter we will explore Predicate Logic (PL), an extension of Sentential Logic, the
system we studied in Chapter 1. There is the potential here to get tangled up in picky details,
since PL is a refinement that deals with the inner logical structure of sentences as well as
sentential connectives. We will keep our treatment fairly informal, however, since our goal is
not to master the fine points of logic but to learn the system of PL in order to better analyze
mathematical propositions and understand mathematical proof strategies.
PL is built on a rich and fully symbolic language. If you feel at home in algebra or are
quick to learn a computer language, you may catch on to how PL symbolizes sentences fairly
well, but it takes most people a little while to get accustomed to the syntax of PL. As you are
starting to learn PL, you will probably write down some strange looking and rather confused
sentences; this seems to be universal. Funny mistakes are made by almost everybody as they
learn the intricacies of a new language. But you will eventually become familiar with PL, so
don’t get discouraged if it seems a bit perplexing at first.
Sentential Logic: Complete But Deficient
We noted at the outset that our Natural Deduction System of Sentential Logic is both
sound and complete (see Section 1.5). It is sound because if a sentence can be proved from a
set of premises, then it is a logical consequence of those premises: If P − Q, then P = Q.
It is complete because if a sentence is a logical consequence of a set of premises, then it can
be proved from them using rules of inference from our system: If P = Q, then P − Q.
We explained why SL’s Deduction System is sound as we proceeded, but we left its deductive
completeness unexplored because that is more technically complex.
You may wonder, though, whether SL’s completeness doesn’t make an extension of Sentential Logic unnecessary. This is definitely not the case. Predicate Logic is needed for mathematics
precisely because SL is still deficient. This doesn’t contradict the fact that SL is complete, but
to explain why not, we need to make a few more remarks about the completeness of SL.
There are actually two technical senses in which SL is complete and another less technical
sense in which it is incomplete. The thing to keep in mind concerning these technical notions
of completeness is that they are system-dependent. Completeness is achieved relative to the
syntax and semantics of SL and not in an absolute sense.
The deductive completeness of SL’s inference system, mentioned above, is one way in which
SL is complete. The second way relates to SL’s expressive capabilities. The logical connectives
of SL form a complete set of connectives: any sentence that can be formulated by means of
truth-functional connectives, regardless of the number of sentences combined or the types of
connectives employed, is logically equivalent to one involving only primitive connectives. In
fact, fewer connectives than what SL actually contains are sufficient. This is a pleasantly
surprising result and not too difficult to prove when approaced in the right way; we will return
to argue for it later in the text (see Section 7.5).
Taken together, SL’s two completeness results say the following. The syntactic capabilities
of SL suffice to articulate anything that can be said in a given language of SL by means of truthfunctional connectives, and the deductive capabilities of SL are sufficient to prove anything that
follows from a set of sentences by virtue of the logical structure treated by SL. In both instances,
the power of SL is relative to how sentences may be combined, to what counts as logical form
in that system; namely, to inter-sentential combinatorial structure.
Thus the completeness of SL is system-bound. And, as a matter of fact, SL is incomplete
in an absolute sense. It fails to serve as the underlying logic for mathematical argumentation
(or even everyday reasoning, for that matter). As we will see shortly, it is rather easy to find
arguments that are intuitively valid but whose conclusion cannot be proved using SL’s inference
rules. Almost any mathematical argument picked at random will demonstrate SL’s inadequacy.
2.1 -1
Doesn’t this directly contradict the deductive completeness of SL? No, for the following
reason. While everyone may recognize arguments such as the ones we’ll give below as intuitively
valid , their validity is not due to combinatorial truth-functional logical structure, which is all
that SL takes into consideration. Their validity depends rather upon the deeper internal logical
structure of the sentences involved. SL judges such arguments not to be valid based on its
criteria of validity. Not only are the conclusions thus not derivable from their premises via SL’s
deduction system; they are also not logically implied by their premises, so far as SL is aware.
And so the completeness of SL is not contradicted.
To assert that SL is inadequate for mathematics, therefore, amounts to asserting that SL’s
syntax and semantics are still deficient. This is true even though SL is expressively complete.
SL does the whole job it was designed to do, but it doesn’t do everything that needs to be done.
A fork may be fully adequate for eating solid foods, but sometimes you need a spoon: soup
slips right through a fork. While SL is able to capture the global inter-sentential combinatorial
structure of mathematical sentences and arguments, a residual internal logical structure sits
inside most mathematical sentences, something the machinery of SL is too coarse to capture.
Numerous intuitively valid argument forms used in mathematical proofs appear to be invalid
when viewed from the standpoint of SL. An argument that is valid on the basis of SL’s syntax
and semantics will, of course, remain valid when viewed from the broader, more inclusive
vantage point of PL. But an argument that is judged invalid may well be thought so simply
because SL’s point of view is too narrow. SL’s deduction system is as powerful as it needs to
be for the syntax and semantics of SL. It is the system as a whole that falls short of our needs
for deductive argumentation.
It is always important to strive to construct a complete Deduction System; otherwise we
will not have a sufficiently powerful arsenal of proof techniques at our disposal. Having attained
this, though, we should remember that a Deduction System’s completeness is always systembound. It should not be thought of in absolute terms. Only if the formal system of logic being
considered is held to be the underlying logic for all argumentation should meta-theoretical
results about that system be taken as claims about deductive reasoning in general.
The Need For Predicate Logic: Sentential Logic's Inadequacy
The cause of SL’s deficiency lies in its lofty perspective, in its failure to pay attention
to minute details. SL takes the sentence as the smallest logical unit or building block for
constructing new sentences. The syntax of SL permits us to build compound sentences out
of simpler ones, but it never allows us to probe down into the atomic level of a compound
sentence to investigate how primitive sentences themselves are constructed. Consequently, we
cannot learn from studying SL whether atomic sentences contribute anything to the overall
logical structure of a compound sentence. SL only has the macroscopic, combinatorial logical
structure of a sentence within its purview, not the microscopic, logical structure lodged within
the interior of a sentence. Thus, inferences that depend upon the logical form of the premises’
atomic constituents will be missed by SL. To substantiate this claim, we will look at two
relatively simple mathematical arguments and see how SL treats them.
z EXAMPLE 2.1 - 1
Show that the following argument is invalid when considered as an argument in SL:
• For all real numbers x and y, x < y iff there is a positive real number r such that
x + r = y,
• e < 3;
therefore,
• there is a positive real number r such that e + r = 3.
2.1 -2
Solution
Using sentence forms from SL, this is probably best written as ‘P ↔ Q, R; therefore S’.
Statements Q and S differ, even though they are closely related, so two letters are required.
This is certainly not a sound inference from the point of view of SL, even though the original
argument is perfectly valid. The conclusion logically follows from the premises because of
the internal logical structure of the atomic sentences. (We will be able to demonstrate this
later.) The structure generated by the sentential connectives also enters into the argument,
but by itself this is insufficient to account for the argument’s validity.
For our second example, we will take the following
argument, which occurs as a fragment
√
in the middle of the Proof by Contradiction that 2 is irrational.
z EXAMPLE 2.1 - 2
Show that the following argument is invalid when considered as an argument in SL.
• All rational numbers can be written in reduced form m/n for some integers m and n,
√
• 2 is a rational number;
therefore,
√
• 2 = m/n for some integers m and n.
Solution
To make the sentential structure clearer, we will rewrite these sentences in the following
way before symbolizing them:
• If any number x is rational, then x can be written in reduced form as m/n for some
integers m and n;
√
• 2 is a rational number;
therefore,
√
• 2 = m/n for some integers m and n.
Taking sentence variables to stand for the four atomic sentences here, we obtain “P → Q,
R; therefore S” – which is obviously invalid in SL. The original argument, however, is
valid. (The fact that the second premise is false is, of course, irrelevant to the argument’s
validity.) The argument’s validity depends largely upon the inner logical structure of the
sentences; upon how the objects, properties, and relations mentioned by the premises are
interrelated.
The additional logical structure embedded within such mathematical sentences arises largely
from the way in which equality and the quantifiers “some” and “all” are used. In order to
expand our logical system to deal with such structure, therefore, we will have to develop a
theory of identity and a theory of logical quantifiers. We will also need to take into account
other internal structure — the more strictly linguistic components — in order to fully symbolize
the logical structure of a sentence.
We will begin our study of Predicate Logic in this section by learning how to read and
write quantified sentences. In Section 2.2 we will discuss more systematically the syntax and
semantics of PL in preparation for developing a Natural Deduction System for PL in Sections 2.3
and 2.4.
The Symbolic Vocabulary of Predicate Logic
Compared with sentences of ordinary language, mathematical sentences are highly symbolic.
Special symbols
are used to stand for various mathematical entities. Some of these symbols, like
√
0, 1, π, 2, + , and = , are familiar to most of us. Other symbols, such as the greatest integer
R
*
function symbol bnc, the integral sign in f (x) dx, or vector notation −
v , become familiar only
2.1 -3
by studying particular mathematical topics. Such specialized notation makes mathematics
inaccessible to those who have not mastered its concepts and language. Mathematicians are
occasionally guilty of overusing mathematical notation, introducing symbols where words might
serve just as well or better, but a good mathematical notation often simplifies and clarifies ideas
and is essential to concise mathematical exposition. Think how hard it would be to do algebra
using only words, something that was done prior to the seventeenth century!
The vocabulary of Predicate Logic generalizes and extends that of mathematical theories.
PL takes to the extreme the modern mathematical tendency to use symbolic representation.
Formulating a sentence by means of PL completely transforms it into a string of special symbols.
All traces of natural language fade away; no words are present in the final formulation. This is
hardly the way in which propositions are written in other mathematics courses or textbooks.
But it will be beneficial for us to do it this way for a short time so we can gain a better
understanding of the internal logical structure of sentences and so we can explicitly formulate
the rules of inference that govern quantified sentences.
Symbols are used in PL to denote objects, operations, functions, and relations, just as in
mathematics, but they are also used to stand for properties, such as a number being odd or a
function being one-to-one, which mathematical language usually leaves in verbal form.* The
vocabulary of PL also includes logical symbolism, such as the connectives from SL, and a few
quantifier symbols. Punctuation is provided for sentences of PL by means of parentheses, just
like in SL.
The standard symbol used by mathematicians for the universal quantifier is ‘∀’, which is
read “for all”. Think of it as an upside down ‘A’, the first letter of ‘All’. The symbol ∀ is used
to formulate universal sentences in the language of mathematical logic.
The symbol ‘∃’, which looks like a backward or rotated ‘E’ (as in ‘∃xists’), is our existential
quantifier . It can be read variously as “there exists”, “there is an”, “there is some”, or “there
is at least one”. It is used in formal statements to assert the existence of an object having a
certain property.
A third quantifier, ∃! , is read “there exists a unique.” This unique existence quantifier can
be defined either in terms of ∃ and sentential connectives or in terms of both ∃ and ∀ along
with sentential connectives (see Exercise 67). We could therefore eliminate it without any loss
of expressive power if we wished to do so, but we will instead make free use of it. It greatly
simplifies symbolic formulations of uniqueness statements.
We have briefly indicated the vocabulary of PL, but we have not said anything about PL’s
syntax or grammar, the ways in which the symbols of PL can and must be combined to give
meaningful statements. We will postpone a discussion of this matter until the next section,
after you have had a chance to begin learning the language of PL by reading and writing a
number of sentences of PL. This immersion approach imitates the way young children initially
learn to speak, read, and write and the way adults often learn a second language. Think of PL
as using a specialized foreign language constructed for the purpose of serious logical analysis.
Getting Ready to Read and Write
Formulating the propositions of a mathematical theory in an artificial language like PL
requires an interpretation key to make both the overall context and the specific meaning of the
symbols clear. Without this no communication of ideas is possible using abstract symbols.
An interpretation key must first of all identify the subject matter or intended universe of
discourse. More precisely, the universe of discourse for a theory is a structure consisting of
a set of all those objects considered by the theory, together with their properties, operations,
functions, and relations.
* Predicate Logic derives its name from the fact that it deals with predicates : one-place predicates (properties) as well as
two- or n-place predicates (relations, functions, operations).
2.1 -4
Once a universe of discourse is specified, variables and quantifiers automatically obtain a
definite context and range. Universal statements are understood to be statements about all
the objects belonging to that universe of discourse; existential statements claim the existence
of an object in the given universe of discourse.
Secondly, a key must provide a lexicon (dictionary) that assigns symbols to stand for
the various mathematical entities mentioned in your sentences. For symbols with a known
standard meaning, such a lexicon is not important, but if symbols are unfamiliar or are used in
unconventional ways, their meaning must be stipulated. Logical symbols such as connectives
or quantifiers will always be interpreted in a fixed way.
Having a universe of discourse and a key, we can then proceed to combine the symbols
(our vocabulary) to make sentences according to PL’s rules of syntax or grammar. Faithful
transcription of a sentence into PL format thus assumes both a good intuitive knowledge of
how to compose well-formed sentences in PL and a knack for analyzing the underlying logical
structure of sentences expressed in informal mathematical English. Constructing PL sentences
that say exactly what we want them to say takes some practice.
It is a bit easier to move in the other direction: to translate PL sentences into informal
mathematical statements. Given a PL sentence along with an interpretation key, we can translate it into an ordinary mathematical sentence. We may have to do this in several stages, first
translating the sentence into a stilted literal formulation, and then transforming it into a more
idiomatic natural-language equivalent. This translation process is analogous to translating a
statement from a foreign language into our native tongue.
Reading Singly - Quantified Sentences
We will begin our PL language study by considering a few fairly simple examples, ones
that involve only a single quantifier. At the end of this section, we will tackle several more
complicated examples with multiple quantifiers.
Our first examples start with a PL formulation of a sentence and then provide an interpretation for it stated in mathematical English (reading PL sentences). We then give a rendition
of the sentence in mathematical English.
z EXAMPLE 2.1 - 3
Translate the PL-sentence ∀n(P n ∧ n 6= 2 → On) into good mathematical English, using
the following interpretation key for the non-standard symbols involved:
KEY :
U = {1, 2, 3, . . .} = N+
P n : n is prime
On : n is odd
Solution
As an abstract, uninterpreted symbolic statement, this sentence can be read as follows:
“For all n, if P n and n is not equal to 2, then O n.”
Given the above interpretation key, however, we can read it in a more meaningful way.
Since the universe of discourse is N+ , the set of all positive natural numbers, this sentence
should be read as a universal statement about them: ∀n should be read as “for all positive
natural numbers n.”
Two properties of positive natural numbers, being prime and being odd, are symbolized
in the sentence and interpreted in the key. Since no additional symbols are given in our
key, we assume that 2 and = (6=) are to be taken in their usual sense. The same is true of
the logical connective →.
2.1 -5
The inner sentence can thus be read, “If n is prime and n is not equal to 2, then n is
odd.”
Since this whole inner sentence is placed in parentheses behind the universal quantifier,
it is being asserted in a universal way: “For all positive natural numbers n, if n is prime
and n is not equal to 2, then n is odd.”
This still sounds rather stiff. A more idiomatic translation of the sentence would be
“All prime numbers different from 2 are odd.”
z EXAMPLE 2.1 - 4
Translate the PL-sentence ¬∃x(x 6= 0 ∧ x + x = x) into good mathematical English, using
the following interpretation key:
KEY :
U = R, the set of all real numbers
Solution
The key only tells us that the intended universe of discourse is the set of real numbers.
All symbols in the sentence will thus be taken in their conventional sense: = means “is
equal to,” 0 stands for zero, and + is ordinary addition.
This sentence is a negated existential sentence. It can be read, “It is not the case that
there exists a real number x such that x 6= 0 and x + x = x.” To indicate that the inner
sentence is governed by the existential quantifier, the words ‘such that’ are inserted after
the quantifier phrase. This wording is typical of existential sentences since the sentence
following the quantifier qualifies the object whose existence is being claimed. ‘Such that’
is not used after the universal quantifier because there nothing has been asserted of the
objects yet (see the last example).
The negation may be pushed a bit further into the sentence formulation by reading,
“There are no real numbers x such that x 6= 0 and x + x = x.” This is better mathematical
English than the original translation, but we can make it more compact and idiomatic yet
if we read it as “There are no non-zero real numbers x such that x + x = x.” Or, if we
want to put it completely into words, we can read it as “No non-zero real numbers remain
unchanged when added to themselves.”
Writing Singly - Quantified Sentences
In transcribing mathematical sentences from idiomatic, mathematical English into the formal language of Predicate Logic, you should use the following approach.
1) Determine and stipulate a universe of discourse, the meaning structure containing all
the objects under consideration, for your statements.
2) Draw up an interpretation key with symbols to stand for the various objects, properties,
operations, functions, and relations mentioned in the sentences.
3) Break down the logical structure of the sentences into its various components, so you
know what logical symbols (connectives, quantifiers, etc.) are to be incorporated into
the sentences and where.
4) If necessary, rewrite the given statements in a more stilted style of mathematical prose
that better indicates their underlying logical structure.
5) Write the sentences in the symbolism of PL.
The next two examples illustrate this procedure.
z EXAMPLE 2.1 - 5
Analyze the logico-lingual structure of the sentence “If a function is differentiable, then it
is continuous.” Then symbolize it using PL notation.
2.1 -6
Solution
This sentence is less complex than the one we looked at in the last example, but since
we are now going at it from the other direction, it may seem more difficult. We need to
analyze what this sentence says, to tease out the logical structure lying below the surface,
as it were, so that we can find a way to write it using the language of PL. Try to do it for
yourself before reading further, and compare your result with what we get below.
We begin by specifying a universe of discourse. Since this sentence is usually asserted in
an introductory calculus course dealing with real-valued functions of a real variable, we will
assume that as our context. The only objects under consideration then, are such functions,
so we will take that as our universe of discourse.
Two properties of these functions enter into the sentence: being differentiable and being
continuous. Each of these can be spelled out by definitions involving limits, but here we
will simply choose one-place predicate letters to stand for these properties.
KEY :
U = {f : f is a real-valued function of a real variable}
Df : f is differentiable
Cf : f is continuous
Although the proposition seems to refer to a single function, the obvious intent of the
sentence is to make a universal assertion. It could be formulated by “Every differentiable
function is continuous.” This makes the universal quantifier explicit, but it hides the
conditional nature of the inner sentence. A formulation making both aspects of the logical
structure apparent is the more awkward version “For every real-valued function f of a real
variable, if f is differentiable, then f is continuous.” Stilted or not, this now makes both
the quantifier and the logical connective involved clearly visible.
FORMULATION :
∀f (Df → Cf )
The key to success in transcribing sentences in the language of PL, as this example makes
clear, lies in being able to recognize the underlying logical form of the sentence. The formulation
of a proposition in ordinary mathematical English often leaves some logical structure implicit.
If you do your analysis and retranslate the sentence, verbally or mentally, into a form that
better reveals its logical deep-structure, you are more than half way done. This is an art that
improves with practice. Especially once you learn more what logico-lingual parts there are to
mathematical sentences and how they are put together (the grammar of PL), you will be in a
better position to do the necessary analysis.
z EXAMPLE 2.1 - 6
Use the language of PL to write out the second premise and the conclusion of the argument
from Example 1: “e < 3” and “there is a positive real number r such that e + r = 3.”
Solution
Writing the second premise is trivial: it’s just as it was given: e < 3. Since e, <, and 3
are used with their normal meanings, no key is necessary for these symbols.
A universe of discourse that is compatible with the conclusion is the set of positive real
numbers, R+ . Choosing U in this way, however, makes it impossible to formulate the first
premise, which refers to all real numbers, not just positive ones. So we will choose R as our
universe of discourse and restrict our attention to positive real numbers manually, as it were,
by how we formulate the inner sentence. This can be done and the rest of the conclusion
written without the use of any special symbols. Our interpretation key, therefore, consists
merely of stipulating the universe of discourse.
KEY :
U = R, the set of real numbers
2.1 -7
Analyzing the conclusion, we see that it claims the existence of a real number with
two qualifying properties: it is positive, and its sum with e is 3. Since both of these must
hold, we must conjoin them. The two sentences are thus written as follows, using standard
mathematical and logical notation.
FORMULATION :
Premise: e < 3;
Conclusion: ∃r(r > 0 ∧ e + r = 3)
Using Restricted Quantifiers: Abbreviated Notation
Working mathematicians rarely bother to specify a universe of discourse. It is sometimes
understood from the context, but often the restrictions are explicitly incorporated into the
sentence itself. This latter practice involves qualifying the variables that are used and so
without some convention makes the resulting PL sentences long and cumbersome. Rather than
restricting the variable within the sentence, as we did in the last example, we can use restricted
quantifiers. For example, instead of writing ∃r(r > 0 ∧ e + r = 3) as we did above, we could
write (∃r ∈ R+ )(e + r = 3),* where ∃r ∈ R+ is read as “there exists an r in R+ .” This practice
assumes an underlying set theory for mathematics and so has implications for foundations of
mathematics, but its use is rather widespread, so we will use it as a bona fide option in PL
formulations. Such idiomatic notation becomes even more handy when mathematical sentences
are complicated by the presence of multiple quantifiers (see Exercise 19).
Restricted variables could have been used in the preceding example, too. Keeping the same
universe of discourse and using D now to stand for the set of all differentiable real–valued
functions, we can formulate the sentence with a restricted quantifies as follows: (∀f ∈ D)Cf ,
where Cf still represents the property of being continuous.
It is obvious that your choice for a universe of discourse will affect your PL formulation.
If you are studying a particular field, the universe of discourse will usually be laid out for you
(those entities being investigated). Otherwise, the universe of discourse ought to be the set
of objects everyone naturally assumes is the one being talked about without needing to be
mentioned. You can then formulate the sentence accordingly, restricting the variables further
either within the sentence or by means of restricted quantifiers.
Let’s generalize these last two examples. A universal sentence, which asserts some property
P about all objects in a set S, is of the form (∀ x ∈ S)P x; while an existential sentence, which
claims that there is something in S with property P , is of the form (∃ x ∈ S)P x. When these
sentences are put in expanded form, using ordinary quantifiers, we see that the inner sentences
are quite different in logical form. The universal sentence becomes the quantified conditional
sentence ∀x(x ∈ S → P x), while the existential sentence becomes the quantified conjunction
∃x(x ∈ S ∧ P x).
This is typical of universal and existential sentences: Universal sentences usually contain
the connective → while existential sentences involve ∧ . It is important to keep these two
straight and not interchange the two main interior connectives → and ∧ . You should check
when your PL formulation is finished to see whether the connective you have chosen really says
what the original sentence asserted.
Restricting the quantifier does away with the problem of which connective to use: none is
present. It gets submerged in the restricted quantifier. There may be times, though, when you
wish to expand a restricted existential quantifier or restricted universal quantifier in order to
work further with a given sentence. To do so, you must know what the notation abbreviates.
Knowing how to treat these two different cases, you will be able to take advantage of the
convenient notation of restricted quantification without being ignorant of the logical structure
of the sentence in expanded form.
* Alternatively, we could write (∃r > 0)(e + r = 3).
2.1 -8
Reading and Writing Multiply-Quantified Sentences
The examples we have seen so far contain only a single quantifier, but mathematical sentences typically have multiple quantifiers. We will now consider some examples of this sort.
When all the quantifiers are the same, there is usually no problem, but when both universal
and existential quantifiers occur, the potential for misunderstanding and misformulating the
sentence increases.
z EXAMPLE 2.1 - 7
Translate the PL-sentence ∀x∀y(x · y = 0 ↔ x = 0 ∨ y = 0) into good mathematical
English. Assume the usual meanings of the symbols, and take the universe of discourse to
be the set of real numbers R.
Solution
This is a universal sentence with two universal quantifiers; what is said about x and y
is meant to be completely general.
Put into words, this sentence asserts that “the product of two real numbers is zero iff
at least one of them is zero.”
z EXAMPLE 2.1 - 8
Translate the PL-sentence ∀n(On ↔ ∃k(n = 2k + 1)) into good mathematical English,
using the following interpretation key:
KEY :
U = N = {0, 1, 2, . . .}
On: n is odd
Solution
The universe of discourse here is the set of natural numbers; On indicates the property
of being odd. The other symbols are meant in their customary sense.
The sentence given is a universally quantified biconditional with On as one of the
conditions. We thus have a sentence that could be taken as a definition of being odd. A
literal translation says that “for all natural numbers n, n is odd iff there is some natural
number k such that n is equal to 2k + 1.” (Note that the range of each quantifier is the set
of natural numbers; k must be a natural number as well as n.)
In more idiomatic mathematical English, we would say that “a natural number is odd
iff it is one more than an even number.”
The two different quantifiers in the last example were not that difficult to interpret correctly.
In some sentences, however, this will not be nearly so easy. The difficulty arises when the
quantifiers appear together and quantify the same inner sentence. The order in which they
appear in the formulation makes all the difference in the meaning of the sentence. We’ll now
look at an example of this sort.
z EXAMPLE 2.1 - 9
Translate the PL-sentence ∀x∃x̄(x + x̄ = 0 = x̄ + x) into good mathematical English, using
the following interpretation key:
KEY :
U = R, the set of real numbers
2.1 -9
Solution
The only thing identified by our key is the universe of discourse. Since no other symbols
are assigned meaning, we may assume that they are to be taken in their usual sense relative
to the real number system: + is ordinary addition, = is ordinary equality. Both x and x̄
are variables ranging over the set of real numbers. The double equation given is a standard
way to abbreviate the conjunction of two equations: (x + x̄ = 0) ∧ (0 = x̄ + x).
Read literally, this sentence says, ‘For every real number x there is a real number x̄
such that x + x̄ = 0 and 0 = x̄ + x’.
How should this be put into good mathematical English? The key is properly placing
the quantifiers. To abbreviate the inner sentence, let’s call x̄ an additive inverse of x, since
their sum in either order yields the additive identity 0 (this is standard terminology).
Do we translate the above sentence, then, as “Every real number has an additive inverse’, or as ‘There is an additive inverse for every real number”?
Since ordinary English is somewhat ambiguous, especially the way some people use it,
these two options may seem to say the same thing to you. But there is a very important
difference in them. The first sentence says that given any real number x there is an associated x̄ that is its additive inverse. Strictly interpreted, however, the second sentence says
there is a real number x̄ that is the additive inverse for every real number x. In the first
case, x̄ may change, depending on the value of x; but in the second case x̄ is the same for
all x. The second sentence corresponds to a ∃-∀ sentence, not to a ∀-∃ sentence, as given.
The correct formulation of the given PL sentence, therefore, is the following: “Every
real number x has an additive inverse x̄.” This captures the meaning of the original sentence: it says that every real number has an inverse, not that there is a single number x̄
which is the inverse for every real number.
Our final examples illustrate the reverse process of translating from informal mathematical
English into the formal language of PL. As mentioned above, this can be difficult to do, particularly with multiple quantifiers. As a check on your work, you should carefully translate your
answer back into an informal statement and see if it is equivalent to what you started with.
z EXAMPLE 2.1 - 10
Formulate the following definition of the number-theoretic relation “divides” in the formal
language of PL: d divides a, denoted by d | a, iff a is a multiple of d. Take the set of positive
natural numbers as the universe of discourse: U = N+ .
Solution
We are given a definition holding for all positive natural numbers, not just for two
special numbers d and a, so a universal sentence is obviously understood. Our sentence will
thus begin ∀d∀a, followed by the defining sentence involving a and d.
The definition says that d | a iff a is a multiple of d. Rather than use a special symbol
to denote the relationship of “being a multiple of,” we will proceed more mathematically
and spell out exactly what this means: a is a multiple of d iff a = m · d for some m.
Proceeding further in our analysis, the use of the phrase “for some m” indicates the
occurrence of an existential quantifier. We know from arithmetic that this value of m must
be unique, but that is not what is being asserted here, so we will use the ordinary existential
quantifier, not the unique existence quantifier. The position and scope of ∃m is not too
difficult to determine: it relates to the equation a = m · d.
Putting all these things together, we obtain the following.
KEY :
U = N+
FORMULATION :
∀d∀a(d | a ↔ ∃m(a = m · d))
2.1 -10
z EXAMPLE 2.1 - 11
Translate the following characterization of the greatest common divisor into the formal
language of PL: A positive integer is the greatest common divisor of two positive natural
numbers iff it is both a common divisor and a multiple of any common divisor .
Solution
This is a proposition from elementary number theory, so once again the most appropriate
universe of discourse is probably U = N+ .
The binary relation of “divides” is implicitly present below the surface of this statement;
we will use it to define “common divisor.” That c is a common divisor of two natural
numbers a and b can be formulated by c | a ∧ c | b, using the notation for “divides”
introduced in the last example.
We will use the symbolism ‘GCD’ to stand for the greatest common divisor function.
It is a function of two variables and yields a positive integer as its result. We thus have the
following symbols to use in our formulation.
KEY :
U = N+
x | y: x divides y
GCD(x, y): the greatest common divisor of x and y
To obtain our formulation, we will use the variables a and b to stand for any two
positive natural numbers, d to stand for their greatest common divisor, and c to stand for
any common divisor of a and b. It is clear that we should use universal quantifiers for a
and b, since the definition applies to any positive natural numbers. The universal quantifier
should also be used with c, since the definition mentions “any common divisor.” What do
we do about d? Here, too, a universal quantifier is understood, not an existential quantifier.
The definition can be thought of as saying that a (i.e., any) positive integer d will be the
greatest common divisor iff such and such a property is the case. It does not claim the
existence or unique existence of such an entity, even though there are such. (That would
have to be stated separately and proved as a theorem of number theory.) Our formulation,
then, is the following one.
FORMULATION :
∀a∀b∀d(d = GCD(a, b) ↔ (d | a ∧ d | b) ∧ ∀c(c | a ∧ c | b → c | d)))
Our final example is the famous parallel postulate of Euclidean geometry, in the version
made popular by John Playfair about 1800. This example also introduces the notion of variables
of different sorts, which mathematicians use all the time to further abbreviate their symbolic
formulations.
z EXAMPLE 2.1 - 12
Formulate Playfair’s parallel postulate in the formal language of PL: Given a line and a
point not on the line, there is a unique line passing through the given point parallel to the
given line.
Solution
We have here two sorts of objects: points and lines. We will use lower case letters,
such as l and m, to stand for lines, and upper case letters, like P , to denote points. This
will simplify our formulation considerably. Quantified variables will be tacitly restricted
by their case instead of by using either a restricted quantifier or a predicate qualifier. Our
universe of discourse, therefore, will need to contain two sub-universes or sorts of objects:
points and lines. Since Playfair’s postulate is a sentence of plane geometry, we will specify
the universe of discourse U as consisting of all points and lines in a single, unspecified plane.
Among the lines of U there is the binary relation “is parallel to,” which we will symbolize by means of the suggestive mathematical notation k . Between lines and points there is
2.1 -11
the binary relation of “incidence,” of a point lying on a line. Thinking of a line as being
a set of points, it is easiest to use the set-theoretic notation P ∈ l to indicate that point
P lies on line l, rather than choosing some two-place predicate letter like LP l to stand for
the relation “lies on.” “Not lying on a line” is then symbolized by P ∈
/ l.
Besides the relations of “being parallel to” and “lying on” nothing else needs special
symbols.
What quantifiers and connectives need to be used? Let’s first look at the quantifiers.
The original sentence talks about “a line” and “a point,” but the intent is to state something in general about any line and any point, so we will be using the universal quantifier
∀ twice. It also claims the existence of a unique line, so we will also be using the unique
existence quantifier ∃! .
How about sentential connectives? There is an explicit negation in the relation “not
on.” An “and” appears in the original sentence, but this merely indicates that the two
universal quantifiers “for all lines l” and “for all points P ” are being successively asserted;
it is not a genuine sentential connective.* This is written simply by putting one quantifier
after the other: ∀l∀P . It would be inappropriate to use ∧ here because we are conjoining
quantifiers, not sentences.
On the other hand, the sentence does contain a conjunction, though the word “and” is
not present to indicate it. We are given two properties that the unique line must simultaneously satisfy: it must pass through the given point and it must be parallel to the given
line.
There is one more connective that could appear in our formal version; we still need to
restrict our point P to those that lie off the given line l. This is done for universal quantifiers, as noted above, by means of a conditional sentence with the restriction occurring in
its antecedent. Alternatively, we can use a restricted quantifier to shorten the formulation.
We are now ready to make our formal translation. Note the type and order of the
quantifiers that appear; convince yourself that this is just what is needed when you are
done reading the translation.
KEY :
U = {x : x is a point or x is a line in some common plane}
P ∈ l: point P lies on line l
m k l: line m is parallel to line l
CONDENSED FORMULATION :
∀l(∀P ∈
/ l)∃!m(P ∈ m ∧ m k l)
NON-RESTRICTED FORMULATION :
∀l∀P (P ∈
/ l → ∃!m(P ∈ m ∧ m k l))
The examples in this section begin to exhibit the degree of logical complexity mathematical
statements can have. The PL versions probably seem much more complex to you than the
corresponding formulations in mathematical English, and they are certainly more artificial
looking. This is partly because English has been your native (or second) language for some
time now, but it is also due to the logical complexity involved, which informal sentences cover
up. The logical structure of a mathematical sentence does not become completely explicit until
the sentence is analyzed and symbolized as we have done, using the tools of PL. In order to
familiarize yourself with the deductive practices legitimized by the inference system of PL, such
as negating quantified sentences, you will have to learn, at least temporarily, to live with the
additional linguistic complexity that arises out of making the logical structure explicit.
* “And” will indicate a sentential connective, however, if the properties of “being a point” and “being a line” are symbolized
by means of an interior qualifying predicate letter. Then the assertion “l is a line” must be conjoined with “P is a point.”
See Exercise 59.
2.1 -12
EXERCISE SET 2.1
Problems 1 - 8: Reading and Evaluating Singly-Quantified Sentences
Translate the following sentences of PL into idiomatic sentences of informal mathematical English. Then
tell whether the sentences are true or false under the given interpretations.
Keys:
N+ is the set of positive integers, and N is the set of natural numbers (the positive integers and 0);
Z is the set of integers, Q is the set of rational numbers, and R is the set of real numbers.
The symbol ∈ denotes ordinary set membership: x ∈ S means x belongs to the set S.
1. ∀x(x + 0 = x)
Key: U = Z
2. ∀x(x · 0 = 0)
Key: U = N
*3. ∀x(x < x2 )
Key: U = R
4. ∀x∀y∃z(x + z = y)
Key: U = Z
*5. ∀x(x 6= 0 → ∃y(x · y = 1))
Key: U = Q
*6. ∀x∀y(x ∈
/ Q∧y ∈
/ Q→x·y ∈
/ Q)
Key: U = R
7. ∀x∀y∀z(x · (y + z) = x · y + x · z)
Key: U = R
8. ∀x∀y∀z(x < y → x · z < y · z)
Key: U = R
Problems 9 - 19: Reading and Evaluating Multiply-Quantified Sentences
Translate the following sentences of PL into idiomatic sentences of informal mathematical English. Then
tell whether the sentences are true or false under the given interpretations.
Keys: N+ , N, Z, Q, and R are as in Exercises 1 - 8.
*9. ∀x∃y(y ≤ x)
Key: U = N
*10. ∃y∀x(y ≤ x)
Key: U = N
11. ∀x∃y(y > x)
Key: U = N
12. ∃y∀x(y > x)
Key: U = N
x+1
y +1
13. ∀x∀y x 6= 1 ∧ y 6= 1 → y =
↔x=
x−1
y −1
Key: U = R
14. ∀a∀b∀p(P p ∧ p | ab → p | a ∨ p | b)
Key: U = N; P n: n is prime; p | a: p divides a
15. ∀p(P p ↔ ∀m∀n(p = m · n → m = 1 ∨ n = 1))
Key: U = N; P n: n is prime
16. ∀a∀b(a 6= 0 ∧ b 6= 0 → ∃!x∃!y(x · a = b ∧ a · y = b))
Key: U = R
17. ∀P ∀Q(P 6= Q → ∃!l(P ∈ l ∧ Q ∈ l))
Key: see Example 12
2.1 -13
*18. ∀l∀m(l k m ↔ ¬∃P (P ∈ l ∧ P ∈ m))
Key: see Example 12
19. L = lim f(x) ↔ (∀ > 0)(∃δ > 0)∀x(0 < |x − a| < δ → |f(x) − L| < )
x→a
Key: U = R
Problems 20 - 34: Writing Singly-Quantified Sentences
Determine whether each of the following sentences are true or false. Then translate it from mathematical
English into the symbolism of PL. For each sentence, set up a key that gives the universe of discourse
and that states the interpretation of any constant, function, operation, property, or relation symbols. Use
standard mathematical notation wherever you can, but explain any non-standard symbols you use.
*20. All equilateral triangles are equiangular.
*21. No scalene triangle is isosceles.
*22. Some rectangles are squares.
*23. Some rectangles are not squares.
*24. Not all triangles are congruent.
25. No odd number is composite.
26. Some prime numbers are even.
27. All isosceles right triangles are equilateral.
28. No real number satisfies the equation x2 + 1 = 0.
29. Some real numbers satisfy the inequality a2 < a.
30. Some real numbers do not satisfy the inequality a2 < a.
31. Every natural number is either even or odd.
32. All natural numbers are both integers and rational numbers.
33. The square root of 2 is not rational.
34. There is exactly one even prime number.
Problems 35 - 38: True or False
Are the following statements true or false? Explain your answer.
35. Any consequence that logically follows from a set of premises can be proved from them using Sentential
Logic’s inference rules.
36. Predicate Logic enables one to analyze the complete logical structure of a mathematical statement.
37. Predicate Logic is an extension of Sentential Logic.
*38. Universal sentences usually contain an inner conjunction when formulated with the machinery of Predicate Logic.
Problems 39 - 59: Writing Multiply-Quantified Mathematical Sentences
Determine whether each of the following sentences are true or false. Then translate it from mathematical
English into the symbolism of PL. For each sentence, set up a key that gives the universe of discourse
and that states the interpretation of any constant, function, operation, property, or relation symbols. Use
standard mathematical notation wherever you can, but explain any non-standard symbols you use.
*39. Two real numbers can be multiplied together in either order to give the same result.
*40. Given any two positive real numbers a and b, there is a natural number n such that na > b.
a+b
41. The arithmetic mean,
, of any two positive real numbers a and b is always less than or equal to
2
√
ab, their geometric mean.
42. The sum of two rational numbers is rational.
43. If a and b are rational numbers, then so is ab .
2.1 -14
44. All positive integers can be factored into a product of two primes.
45. Every natural number is smaller than some other natural number.
46. There is an integer that is less than all other integers.
47. All positive integers greater than one have a prime factor.
*48. Between any two real numbers there is a rational number.
49. For integers a, b, and c, if a 6= 0, then ab = ac iff b = c.
50. Every non-empty subset of natural numbers has a least element.
51. A set S of real numbers is bounded above iff there is some real number M that is greater than or equal
to any x in S.
52. A function f is monotone increasing on the set of real numbers iff whenever x1 < x2 , f(x1 ) < f(x2 ).
53. A function f is one-to-one iff different inputs x1 and x2 yield different outputs f(x1 ) and f(x2 ).
∞
∞
X
X
54. If 0 ≤ an ≤ bn for all n, then the series
an converges if the series
bn does.
n=1
n=1
55. If f is a continuous function on the closed interval [a, b] and if f(a) · f(b) < 0, then there is a number c
between a and b such that f(c) = 0.
56. If f is a differentiable function over a closed interval [a, b] that has f(a) = f(b), then there is a number
c between a and b such that f 0 (c) = 0.
EC
57. Two points determine a line.
58. Exactly one circle can be drawn through three non-collinear points.
59. Formulate Playfair’s parallel postulate (see Example 12) using predicate symbols to stand for the properties of being a point and being a line and qualifying the variables within the sentence. Compare your
result with the formulation given earlier, which used variables of different sorts.
Problems 60 - 66: Writing Multiply-Quantified Ordinary Language Sentences
Translate the following English sentences into the language of PL. Choose an appropriate universe of discourse and provide a key for interpreting the various non-logical symbols you use. Discuss any ambiguities
in the original sentences and note how you resolved them in your formulation.
60. If all test grades are high, then no test grades are low.
61. If everyone is the same, then nobody is different.
62. Everybody loves somebody.
63. Everyone is loved by somebody.
EC
64. Nobody loves everybody.
65. Everything is affected by everything.
66. You can fool some of the people all of the time, and you can fool all of the people some of the time, but
you can’t fool all of the people all of the time.
67. Equivalents for Uniqueness Assertions
a. Formulate the sentence ∃!xP x in an equivalent way, using only the quantifier ∃, along with whatever
sentential connectives and variables are required.
b. Formulate the sentence ∃!xP x in an equivalent way, using both ∃ and ∀, along with whatever sentential
connectives and variables are required.
2.1 -15
HINTS TO STARRED EXERCISES 2.1
3. See Example 3.
5. See Example 8.
6. See Example 7.
9. See Example 9.
10. Note that this is not the same sentence as Exercise 9. The order of quantifiers affects the meaning of a
sentence.
18. The lowercase letters in this sentence represent lines, while the uppercase letters represent points.
20. Let U be the set of all triangles and recall what quantifier and connective usually occur with universal
statements.
21. This is a universal statement involving a negation.
22. Take U to be the set of all rectangles and note that this asserts the existence of squares.
23. This is similar to Exercise 22.
24. Pay attention to how negation enters into this sentence.
38. [No hint.]
39. Begin the sentence with two universal quantifiers.
40. You can make use of a restricted existential quantifier to shorten this sentence.
48. Begin your sentence with two universal quantifiers.
2.2 Predicate Logic Syntax and Semantics
In Section 2.1 we started using the formal approach of Predicate Logic to transcribe mathematical sentences. This process begins with an interpretation key: choose an appropriate
universe of discourse and assign various symbols to denote the constants, functions, operations, properties, and relations that are mentioned. Then, analyzing the logical structure of
the sentence, we combine symbols using the appropriate quantifiers and connectives so that
the final result says what it should.
For illustration purposes, we focused on a variety of individual sentences, changing the
key as we went, but mathematicians normally work within the context of given mathematical
theories, such as number theory or geometry. Associated with each theory there is a specialized PL language appropriate for expressing its particular results. Common to this family of
mathematical PL languages will be the logical symbols, the way in which we formulate words
and sentences (syntax), and the way in which we interpret them (semantics). We will focus on
these matters here to systematize and extend what we did less formally in the last section.
The Syntax of PL: Terms, Formulas, and Sentences
The process of translating sentences into a PL language presupposes the ability to combine
the various symbols to construct well-formed formulas. Just as for a natural language, not any
old string of letters and words from a PL language will make a meaningful phrase or sentence.
Syntactically correct sentences of any language are constructed from properly formed words,
put together according to its rules of grammar or syntax, using appropriate punctuation.
The conventions behind mathematical sentence formation are not completely new to you.
You have been writing some sentences (equations, inequalities) for years. The language of PL
merely builds on mathematical modes of expression, making sentences fully symbolic. Given
that the main creators of PL were mathematicians, this is hardly surprising.
Thinking about the logico-lingual structure exhibited by the examples and exercises of
Section 2.1, we can note the following summary features.
Mathematical sentences affirm or deny that one or more properties or relations hold of
or among some or all objects of certain types.
The rules for the syntax of a PL language, therefore, must do two main things:
(1) they must specify how to denote objects (names), and
(2) they must stipulate the ways in which properties and relations can be asserted of these
objects (formulas or sentences).
The first task is covered by rules governing term formation, the second by rules for the
formation of well-formed formulas. PL formulas can be classified as either open or closed
sentences, depending upon how quantification enters into the formulation. We will discuss
each of these points in turn.
(1) Primitive and Derived Terms
The terms of PL consist of strings of symbols that can represent an individual object. These
consist first of all of primitive terms, and secondly, of derived terms.
The primitive terms of a PL language are its constants and variables. These symbols name
definite or unspecified objects of the universe of discourse respectively. Derived terms are built
up from existing terms. They are constructed either by applying function symbols to given
terms, getting F(t1 , t2 , . . . , tn ), where F is an n-place function symbol and all ti are terms, or
by placing an operation symbol in front of/between/behind terms, using prefix/infix/postfix
notation, such as ∗ t1 , t1 ∗ t2 , or t1 ∗, where ∗ represents an operation performed upon the
objects named by t1 and t2 . The resulting expression is a term standing for some object in the
universe of discourse.
2.2 -1
√
For example, ‘ 2 ’ is a derived
from the primitive term ‘2’ by prefixing
√ term constructed
the square root function symbol
. The term a−1 is similarly formed from the constant term
‘a’ using the multiplicative inverse operation. The derived term ‘3x2 + x − 1’ is constructed in
several steps from the primitive terms ‘3’, ‘x’, ‘2’, and ‘1’ by means of the operation symbols
for multiplication (simple juxtaposition), exponentiation, addition, and subtraction. Standard
algebraic terminology calls this a trinomial (three-term) expression, but it is also a single
compound term standing for an (unspecified) object in the universe of discourse.
(2) Well-Formed Formulas
The well-formed formulas (wffs) or simply formulas of a PL language are strings of symbols
that make meaningful statements about the objects named by the terms. These consist of
atomic formulas and compound formulas.
The atomic formulas are those wffs that contain no sub-statements. They are complete
statements that cannot be split into smaller statements. The atomic formulas of PL are all
of the form Rt1 t2 · · · tn , where R is a predicate symbol denoting an n-place relation in the
universe of discourse and each of the ti are terms.
As with functional terms, predicates may be denoted using prefix, infix, or postfix notation.
For specific binary relations, such as “divides” or “equals”, we normally use infix notation:
3 | 12 ; 3x2 + x − 1 = 0. But if B stands for the ternary (three-place) betweenness relation
in geometry and P , Q, and R denote points, we might write BP QR to indicate that Q lies
between P and R. We symbolize a one-place relation or property similarly. For example, if P
stands for the property of being prime, we can assert that 7 is prime by writing P 7.*
Mathematical practice may vary from standard logical notation on this point. Mathematicians sometimes take an extensional viewpoint, letting the property symbol P stand for the
set of all objects having that property. Then one can assert that x has property P by writing
the set-theoretic statement x ∈ P (read: x is in P). For example, if P stands for the set of all
primes, 7 ∈ P asserts that 7 is prime. This differs from the logical notation P 7; we will use
both ways to express property assertions in this text.
Compound formulas of PL are derived from atomic formulas by operating upon them with
logical connectives and logical quantifiers. Connectives are applied in the normal way to wellformed formulas P and Q, yielding ¬P, P ∧ Q, P ∨ Q, P → Q, and P ↔ Q. Also, if Px is
a well-formed formula containing an unquantified variable x, then the formulas ∀xPx, ∃xPx,
and ∃! xPx are compound well-formed formulas.
For example, the formula P n ∧ n 6= 2 → On of Example 2.1-3 is a compound formula
obtained by joining atomic formulas via sentential connectives. The formula ∃k(n = 2k + 1)
of Example 2.1-8 is the result of applying a quantifier to an atomic formula. The compound
formulas ∀d∀a(d | a ↔ ∃m(a = m · d)) and ∀l∀P (P ∈
/ l → ∃!m(m k l ∧ P ∈ m)) of Examples
2.1-10 and 2.1-12 are obtained by applying both quantifiers and connectives in appropriate
ways to subformulas.
People sometimes wrongly confuse terms and formulas because they don’t keep straight
the different purposes of language and symbolism. This seems to happen on all levels of
mathematics, particularly when students begin studying a highly symbolic field, such as algebra
or logic. This is probably because at first the subject seems too abstract or meaningless to
them. Algebraic expressions containing variables, which are terms, are sometimes taken to
be equations, which are formulas. Students occasionally treat the expression x2 + 3x + 2, for
example, as an equation to be solved, x2 + 3x + 2 = 0. Similarly, in translating sentences from
mathematical English into the language of PL, logic students sometimes put down a complex
term as their formulation when what is needed is a statement. Keeping in mind the distinction
between terms and formulas may help you avoid this sort of confusion. As we noted before,
* Some logic texts use parentheses and commas to denote properties and relations, like P (7) and B(P, Q, R); our notation
is a bit more Spartan and avoids making them look like function values.
2.2 -2
failure to recognize the difference between a compound term (the sentence P → Q, formed from
P and Q by means of the connective →) and an assertion (the claim P = Q) is likewise at the
base of the confusion that interprets the logical operator → as the logical relation “implies.”
(3) Open and Closed Sentences
We will call each well-formed formula of PL a sentence, since each of them makes a complete
statement. Sentences are classified as either open or closed, depending on the quantification
status of the variables they contain.
An occurrence of a variable in a formula is free or unbound iff it does not lie within the scope
of a quantifier, as indicated by parentheses; otherwise it is bound . In the sentence ∃x(x+x = x),
all occurrences of the variable x are bound. In the formula ∀a(a + ā = 0), the variable a is
bound, but ā is free. As written, the formula ∀nP n ∧ n 6= 2 → On has one bound occurrence
of n and two free occurrences; this is because the scope of a quantifier in a formula is taken
by convention to be as small as possible. If you want more of a sentence to be quantified,
you have to use parentheses: ∀n(P n ∧ n 6= 2 → On) is the universally quantified sentence of
Example 2.1-3.
A closed sentence is one that has no free variables. It thus makes a definite statement that
is either true or false when the various constant, predicate, function, and operation symbols
are interpreted according to an interpretation key. An open sentence is one that is not closed;
i.e., one that contains one or more free variables. Such a sentence generally has no definite
truth value, not even after being interpreted. It remains an indefinite statement due to the
unquantified variables it contains and so is neither true or false.
In using the term ‘sentence’ to cover both open and closed well-formed formulas, we are
consciously departing from our original use of the term. According to the definition given in
Section 1.2, only closed sentences ought to count as sentences, since only they are either true or
false. However, since we wish to use the inference rules of SL later on for open sentences as well
as closed sentences (in keeping with ordinary mathematical practice), we are now stretching the
term ‘sentence’ to cover this new case for PL. Open sentences become closed as soon as all the
variables are given definite values or are quantified. You can think of them as being potential
sentences in the earlier sense, about which you can argue much as with closed sentences.
z EXAMPLE 2.2 - 1
Give examples of closed and open sentences and discuss their truth values.
Solution
The sentence ∀x∃x̄(x + x̄ = 0 = x̄ + x) of Example 2.1-9 is a closed formula. It is true
when R is taken as the universe of discourse and the symbols +, 0, and = are interpreted
in the usual way. Every real number x does have an additive inverse x̄ that is also a real
number. However, it is false if the universe of discourse is the set of natural numbers N, for
the number 1 (or any non-zero number) has no additive inverse that is a natural number.
The subformulas ∃x̄(x + x̄ = 0 = x̄ + x) and x + x̄ = 0 = x̄ + x are both open sentences.
The first formula contains free occurrences of the variable x, the second one of both x and
x̄. Normally, neither one can be considered true or false,* regardless of the universe of
discourse chosen or the interpretation given to the symbols.
This example nicely illustrates the fact that truth values for most closed sentences of PL are
interpretation-dependent. A closed sentence asserts a definite result that must be either true or
false, but what truth value it takes on depends upon the universe of discourse and the meaning
given to the symbols. Open sentences, on the other hand, are usually neither true nor false:
their meaning is still open or variable. The free variables still need to be given replacement
values or quantified out before such a sentence will be converted into a definite statement.
* Actually, we must hedge a bit with open sentences, here and elsewhere, because if our interpretation is highly specialized,
such as U = {0} under ordinary addition, these will both be true. This will also be the case for some other situations.
2.2 -3
The Semantics of PL: Interpretation and Truth Values
By means of a key stipulating the meaning of its symbols, a closed PL sentence becomes
an unambiguous statement about the universe of discourse. However, such a sentence might
be given many different interpretations: the universe of discourse might be changed, or the
meanings of the non-logical symbols drastically altered. PL sentences thus have a certain
generic or abstract meaning that becomes fully specified or concrete only when a specific
interpretation key is provided. While this may seem like a liability when you are symbolizing
definite mathematical sentences, it turns out to be an asset, both from the point of view of logic
and mathematics. The possibility of multiple interpretations for PL sentences is a necessary
feature of the semantics of PL, but it is also a characteristic feature of contemporary branches
of abstract mathematics, such as algebra, geometry, topology, and advanced analysis.
The semantics of any system of logic addresses the issues of how to interpret formal sentences
and how to assign truth values to them. In Section 1.1 we learned that both of these are
important for explicating the notion of logical implication, which is central to the study of logic.
However, when we dealt with the notion of logical implication relative to SL (Section 1.3), we
were able to ignore the matter of interpretation and focus solely on truth-value assignments.
This approach also worked for treating the notions of logical equivalence, logical truth and
logical falsehood, as well as consistency, independence, and theory completeness.
We could ignore interpretations in all those cases because Sentential Logic remained up on
the macroscopic level of whole sentences. In order to determine, for example, whether a set
of premises logically implied a certain conclusion, we simply had to know the truth-functional
meaning of the connectives as stipulated by their truth tables. To consider all possibilities, we
assigned all conceivable truth values to the atomic sentences involved and then calculated the
truth values of the compound sentences. Considering all possible truth-value assignments to
the atomic sentences was our way of considering all possible interpretations for SL.
Now that we are probing the logical infra-structure of sentences, however, this must change.
We cannot casually and independently assign truth values to sentences without considering
what the various atomic sentences assert relative to one another. There may be logical properties of these sentences or logical relations between them that depend upon their internal logical
structure. Certain sentences being true or false, other ones may already have their truth values
determined. And so not all truth-value assignments will be legitimate; some may be impossible.
For example, if the sentence ∀x(0 ≤ x) is true, then 0 ≤ a must also be true for any constant a
on account of the logical form of the two sentences, regardless of the intended meaning of ≤
or the value of a. Similarly, if ∃x(x 6= 0) is true, then ∀x(x = 0) must be false: these two
sentences contradict one another and so have opposite truth values.
To assign truth values to related sentences of PL in an intelligent way, then, we must know
in an abstract way what the atomic sentences assert about a potential universe of discourse.
Thus we will have to proceed deeper into sentence semantics than we have done so far.
Interpreting Closed Sentences of PL
Since we need to allow multiple meanings for our sentences in order to define properties
such as logical truth and relations such as logical implication, we will consider all well-formed
formulas at the outset as uninterpreted formulas, as sentences abstracted from all particular
meaning or non-logical content.
There are several phases in making an interpretation for sentences of a given PL language.
We must first of all identify a suitable referential structure for the language involved. This
consists of a non-empty set of objects or universe of discourse that contains potential referents
for the terms of the language. The various constant, function, operation, and predicate symbols
of the language must then be assigned meanings or interpreted in the structure in a way that
respects their intended category types. Thus, we must interpret constants as distinguished
2.2 -4
elements of the structure, property symbols as properties of the structure’s members, binary
predicate symbols as binary relations on the structure, one-place function symbols as functions
of a single variable, and so on.
While we are relatively free within this category constraint to give any meaning whatsoever
to the non-logical terms, the logical symbols will be treated differently. By mutual agreement,
we will always interpret them in a fixed way according to the standard conventions of logic.
The logical connective symbols ∧, ∨, ¬, →, and ↔ will denote the standard truth-functional
connectives “and,” “or,” “not,” “if-then,” and “iff”; the identity symbol ‘=’ will always stands
for the binary relation “is identical to”; and the quantifiers ∀x, ∃x, and ∃! x will be taken to
mean “for all x in the universe of discourse,” “there is an x in the universe of discourse,” and
“there is a unique x in the universe of discourse” respectively.
The above approach is sufficient for dealing with closed sentences of PL. Any closed sentence of the language so interpreted makes a definite assertion that is either true or false of the
structure under consideration. In keeping with the correspondence theory of truth for propositions that was adopted already in Section 1.2, such a sentence is considered true if what it
asserts is actually the case for that structure and is false otherwise.
z EXAMPLE 2.2 - 2
Give two interpretations for the following sentences and discuss their resulting truth values.
0 < 1;
1 < (1 + 1);
1 · 1 = 1;
∃x(x · x < x);
∀x(x 6= 0 → 0 < x).
Solution
These would appear to be simple statements taken from ordinary arithmetic. To obtain a
definite interpretation for these sentences, we will specify a universe of discourse and then
interpret the various non-logical symbols 0, 1, +, · , and < .
– The intended meaning of these sentences is specified by the following interpretation.
KEY :
U = N = {0, 1, 2, . . .}
0: the number zero
1: the number one
+: ordinary operation of addition of natural numbers
· : ordinary operation of multiplication of natural numbers
< : ordinary relation of ‘less than’ for natural numbers
Our symbol assignment could have been done more compactly, simply by saying “Let the
various symbols have their usual meaning.” Writing everything out, however, highlights the
fact that symbols do not come with inherent meaning. Symbols are freely chosen by people
to represent various things; they must be interpreted in order to communicate meaning.
That some symbols have come to be associated with a fixed meaning due to long and
familiar usage should not obscure this fact.
Under this usual interpretation, the fourth sentence is false while the others are true.
– We can give many different interpretations for the above sentences. As a second one, take
the universe of discourse to be R, the set of all real numbers, in place of N, leaving the
various symbols interpreted as before, except relative to R.
Under this new interpretation, the first four sentences become true, while the last one is
false. The first three are obviously true; the fourth sentence is also true because we can
find a real number x so that x · x < x: let x = 1/2. To establish the falsehood of the last
sentence, we need a counterexample to show that x 6= 0 → 0 < x is not universally true.
The real number −1 (or any negative real number) will serve this purpose: it is not zero,
yet it is also not greater than 0. Thus (via Neg Cndnl ), the fifth sentence is false.
We can also give an interpretation for Example 2 where all the sentences become true
(see Exercise 15). Still other interpretations might make other subsets of these sentences true.
We will give one more example to illustrate the method of finite universe interpretation. In
2.2 -5
cases like this, one concocts a finite universe and then specifies which abstract properties and
relations are to be considered true. The final result may look (and is) contrived, but such an
artificial product can reveal important logical possibilities and impossibilities holding for and
among sentences.
z EXAMPLE 2.2 - 3
Give a finite universe interpretation for the sentences of Example 2 (repeated below) that
makes the second one false and the others true.
0 < 1;
1 < (1 + 1);
1 · 1 = 1;
∃x(x · x < x);
∀x(x 6= 0 → 0 < x).
Solution
Since the only constants here are 0 and 1, we will try to construct an interpretation whose
universe of discourse contains only those elements. (We need at least two distinct elements,
else the second sentence will necessarily be true if the first one is. Why?)
We will now specify what things we want to be true of 0 and 1 regarding the order relation <
and the operations + and · so that the above sentences have the desired truth values. Recall
that we are free to interpret these symbols in unusual ways, subject to category restrictions:
we must interpret < as an order relation on U and + and · as binary operations on U .
– The first and third sentences give definite results that 0 and 1 must satisfy; specifying 0 < 1
will also satisfy the last sentence if all we have to consider are 0 and 1.
– To make the fourth sentence true, we have to take 0 · 0 < 0 or 1 < 1. Suppose we do the
latter. Then to falsify the second sentence, we must choose 1 + 1 = 0 and take 1 6< 0.
Putting all this together, we get the following.
KEY :
U = {0, 1}
< : Let 0 < 1, 1 < 1 completely define the order-relation < . Thus, 1 6< 0.
[Think of < as specifying that each element is below the element 1 of U ].
+: Let 0 + 0 = 0, 0 + 1 = 0, 1 + 0 = 0, and 1 + 1 = 0 define addition on U .
[Think of + as annihilation, or as always choosing the smallest element 0 of U ].
· : Let 0 · 0 = 0, 0 · 1 = 1, 1 · 0 = 1, and 1 · 1 = 1 define multiplication on U .
[Think of · as always choosing the larger of the two elements of U ].
Choosing precisely these truths about U , we can now recheck that the second sentence is
false, since 1 6< 0, while all the others are true.
Interpreting Open Sentences of PL
It is possible to become very technical in setting up the semantics for PL, treating the
key notion of satisfaction (needed here) in a highly set-theoretic manner.* Our object here,
however, is not to study logic for its own sake nor even for the sake of foundations or meta-logic,
but to get a good foundation for learning the art of mathematical proof construction. Since
that can be done without delving deeply into the set-theoretic semantics of PL, our treatment
will continue to be fairly informal and intuitive, though it will still involve some technicalities.
Thus far we have only seen how to assign definite truth values to closed sentences relative
to an interpretation. We cannot say of open sentences that they are either true or false under a
given interpretation, since no fixed meaning is assigned to free variables in such formulas. This
is not a limitation of how interpretations are set up: variables are supposed to be indeterminate.
They should be able to take on any values inside the structure.
This places us in a predicament: truth values need to be assigned to open sentences if they
are to participate in the further development of PL, and yet they have no definite truth value.
* Such an approach originates with Tarski, who showed in the mid-1930’s how to make the notion of truth precise for all
well-formed formulas of Predicate Logic. This approach can be found in any rigorous logic textbook.
2.2 -6
To get around this obstacle, we will introduce the notion of an “evaluation” and use it to define
truth values for open sentences.
An evaluation of the free variables of a sentence is a particular assignment of meaning
to these variables as specific objects in the universe of discourse. Evaluating these variables
converts an open sentence into a definite, unequivocal statement about the universe of discourse.
An evaluated interpretation of a formula of PL is thus an interpretation of the logical and nonlogical symbols together with an evaluation of its free variables. Generally speaking, some of
these evaluated interpretations will convert open sentences into true statements about things
in the structure, while others will make them false; but in any case, each sentence obtains a
definite truth value in this way.
An evaluated interpretation which makes an open sentence true is said to satisfy the formula. Using evaluated interpretations, we can now discuss the semantics for PL. Before doing
so, however, we will illustrate this concept with an example.
z EXAMPLE 2.2 - 4
Discuss the truth values of the following sentence in the theory of natural number arithmetic
under different evaluations.
0 < 1;
w < (w + w);
1 · x = 1;
y · y < y;
z 6= 0 → 0 < z.
Solution
Here we are taking the intended interpretation of Example 2, with U = N = {0, 1, 2, . . .}.
Given this interpretation, we look to see what effect different evaluations have on these
sentences.
1) The first sentence, which is closed, is true under any evaluation, since there are no free
variables to be evaluated. 2) The truth value of the second sentence will vary; it is false if w
is evaluated as 0, but is true otherwise since every positive natural number is less than its
double. 3) The third one also has an indefinite truth value; it will be true iff x is evaluated
as the number one. 4) The fourth sentence is always false; no natural number value can be
assigned to y that will turn it into a true statement. 5) The fifth statement, however, is
true no matter what value is assigned to z; 0 is smaller than all other natural numbers.
Truth and Falsehood for Sentences of PL; Models
The central notions of “true” and “false” for PL sentences in general can now be explicated
using the notion of evaluated interpretations. This definition builds upon our earlier definition
that a statement is true iff what it says is the way things actually are.
DEFINITION 2.2 - 1: Truth Values for PL Sentences
a) A sentence is true under a given interpretation iff it is true under every possible evaluation for that interpretation.
b) A sentence is false under a given interpretation iff it is false under every possible evaluation for that interpretation.
c) A sentence is indeterminate under a given interpretation iff it is true under some evaluations and false under other evaluations for that interpretation.
A closed sentence, according to these definitions, is already either true or false once an
interpretation is specified. Evaluations add absolutely nothing to an interpretation of a closed
sentence and so do not affect its truth value. The sentences 0 < 1 and ∃x(x · x < x), for
instance, are respectively true and false under the usual arithmetic interpretation, regardless
of how variables are evaluated. They do not contain any free variables and so are unaffected
by an evaluation. We will mostly work with closed sentences in what follows, in which case the
above definitions aren’t really needed; but in order to work with open sentences, these are the
definitions we require.
2.2 -7
An open sentence first needs to be evaluated before we can say whether it is then true or
false. It may then be always true, always false, or sometimes true and sometimes false. The
last three sentences of Example 4 illustrate these possibilities. Under the given interpretation,
the sentence 1 · x = 1 is indeterminate; its truth value depends on how x is evaluated. The
sentences y · y < y and z 6= 0 → 0 < z, however, do have definite truth values under the usual
interpretation. The first one is false and the second one true under all possible evaluations of
the variables x and y.
If a set of sentences is true under a given interpretation, the mathematical structure involved
is called a model * for those sentences under that interpretation. The same formal sentence of PL
may, of course, be true under one interpretation and false under another, even if the underlying
universe of discourse U remains fixed. The sentence ∀x(x 6= 0 → 0 < x), for instance, is true of
the natural numbers under the ordinary interpretation; that is, N is a model for this sentence.
However, this same sentence is false when < is (perversely) taken to mean “is greater than”:
N is not a model under this interpretation of the sentence.
As we saw in Example 2, changing the universe of discourse but leaving the meaning of
the symbols the same (relative to the new structure) may lead to different truth values for
sentences, but it may also keep them the same. The following two sentences illustrate this:
(1) ∃x(x · x < x), and
(2) ∀x(x 6= 0 → 0 < x).
Considered as statements about N under the usual interpretation of the symbols, the first
sentence is false while the second one is true. Treating them as statements about the integers Z,
the first sentence retains its truth value of F , but the second one changes from T to F . Treating
them as statements about the real numbers R, the first sentence becomes T and the second one
F . Finally, treating them as statements about non-negative rational numbers, both are true.
Logical truths and falsehoods can also be defined in terms of evaluated interpretations.
Intuitively, a sentence is logically true/false iff it is true/false simply on account of its logical
form or structure and not on account of the meaning of the non-logical terms involved. Thus,
we should be able to reinterpret and reevaluate all the non-logical content symbols (adhering
to category restrictions) and end up with a sentence that is still true/false under that evaluated
interpretation. This gives us the following definition, which builds upon the last one.
DEFINITION 2.2 - 2: Logically True, False, and Indeterminate PL Sentences
a) A sentence is logically true iff it is true under every possible interpretation.
b) A sentence is logically false iff it is false under every possible interpretation.
c) A sentence is logically indeterminate iff it is true under one interpretation and false
under another one.
Thus, a sentence is logically true iff every interpretation yields a model for it. Such sentences
are sometimes said to be true of all possible worlds, where such “worlds” or universes of
discourse range over all possible non-logical content or meaning. In order to show that a
sentence is logically true, however, we do not actually have to examine every possible structure,
interpretation, and evaluation. We only need to demonstrate that the sentence must be true of
every structure under any evaluated interpretation, regardless of what these might be.
Tautologies of SL can be used to generate logically true formulas for PL. For example,
Px ∨ ¬Px and Px → Px, as well as their universal closures ∀x(Px ∨ ¬Px) and ∀x(Px → Px),
are logical truths for any one-place predicate symbol P and any variable x. For, regardless of
how you interpret P and evaluate x, any element x in any universe U either has property P
or it doesn’t; and if it has it, then it has it.
Additional logical truths are due to the logical structure contributed by the quantifiers and
the identity relation. Here is an example that illustrates this.
* Unfortunately, logical usage of the term ‘model’ is in direct conflict with the way scientists and applied mathematicians
use this term. In logic a model is the mathematical structure satisfying the theory, while in applied mathematics, it is
the theory that is the model for the concrete situation.
2.2 -8
z EXAMPLE 2.2 - 5
Show that the following are logical truths.
a) Law of Identity: ∀x(x = x)
b) ∀xPx → ∃xPx
Solution
a) Regardless of how the universe of discourse U is chosen, since ‘=’ always means identity
and since every object in the structure is identical to itself, this sentence must be true
under all possible interpretations. It is therefore a logical truth. In fact, all unquantified
formulas t = t, where t is any term, are logically true: any evaluated interpretation
will be true regardless of how any variables are interpreted.
b) This sentence says that if everything has a given property, then something has to have
it. The truth of this statement depends upon the fact, unstressed until now, that only
non-empty sets qualify (by convention) as bona fide universes of discourse. Regardless
of the universe of discourse U chosen, then, or the interpretation given to the property
symbol P, if ∀xPx is true, then all members of U have property P. Hence some member
of U has property P; ∃xPx holds. Thus the given conditional sentence is true under
every interpretation. The truth of this result cannot be shown via a truth table, as in
SL, but our knowledge of the truth table for → nevertheless helps us see that it is a
logical truth.
A sentence is a logical falsehood iff its negation is a logical truth (which gives us one way
to generate them): if it is never true, then its negation is always true; and conversely. Thus,
a sentence is a logical falsehood iff it has absolutely no models. Contradictory sentence-forms
from SL, such as P ∧ ¬P, can be used to generate similar contradictions in PL: Px ∧ ¬Px
or ∃x(Px ∧ ¬Px). Other logical falsehoods, such as ∀xPx ∧ ¬∃xPx, depend upon the logical
structure of PL sentences contributed by quantifiers or the identity relation.
A closed sentence is logically indeterminate iff both it and its negation have models. Open
sentences are often good candidates for being indeterminate, but closed sentences can also be
logically indeterminate. The sentence ∀x(x 6= 0 → 0 < x) from Example 2, for instance, is
logically indeterminate: it is true of N, but not of R.
z EXAMPLE 2.2 - 6
Determine the logical status of the following closed sentences.
a) ∀x(P x ∨ Qx).
b) ∃x(x = a ∧ (P a ∧ ¬P x)). Assume here that a is constant.
c) ∃x∀yP xy.
Solution
a) This sentence is logically indeterminate. We can exhibit one interpretation under which
it is true and another under which it is false. Since the sentence is closed, we do not
need to consider evaluations, only interpretations. We will take U = N in both cases.
First let P x mean x is prime and Qx mean x is odd. Here the statement becomes false:
4 is neither prime nor odd.
But if we take P x to mean x is even and Qx to mean x is odd, then the statement is
true: every natural number is either even or odd.
b) This sentence is a logical falsehood. Given any universe of discourse U and any element
a belonging to the set, if x denotes the same object as a, which is what x = a demands,
then x must have the same properties as a. It is therefore impossible for a to have
property P while x does not have property P ; the sentence is always false.
c) This sentence is logically indeterminate, for we can give it two interpretations, one
making it true and the other making it false. We take U = {2, 3, 4, . . .} for both cases.
2.2 -9
If P xy means that x is a prime factor of y, then our sentence is false, for it says that
some number is a prime factor of every positive integer greater than one. Every number
does have a prime factor, but it isn’t the same one for all of them; i.e., while ∀y∃xP xy is
true under this interpretation, ∃x∀yP xy is not. We will look more closely at the logical
connection between these two sentence forms shortly.
The given sentence is true if we take P xy to mean x ≤ y, for there is an element of U
that is less than or equal to all members of U , namely, 2.
Logical Implication in PL
Just as logical truth and falsehood depend solely on the logical structure of sentences, so
the validity of arguments depends only on logical form, not on non-logical content. Regardless
of the particular truth values that a set of premises and conclusion may in fact have under a
given meaning of the symbols, we must know what truth-value assignments may be associated
with arguments of that form under any interpretation. Multiple interpretations and evaluations
of the free variables are thus essential for demonstrating that a relation of logical implication
holds among sentences of PL.
We will begin exploring the topic of logical implication by restricting our attention to closed
sentences, where evaluations may be ignored.
DEFINITION 2.2 - 3: Logical Implication for Closed PL Sentences
A set of closed sentences P logically implies a closed sentence Q iff whenever the premises
are true under an interpretation, the conclusion is also true under that interpretation.
As before, we do not have to consider all possible interpretations one by one. We can use a
generic argument to show that regardless of the interpretation, such and such will be the case,
and thus logical implication holds. The following examples illustrate this procedure. The forms
of reasoning we employ here depend on the fixed meaning of the logical operators involved and
correspond to informal versions of what we will adopt later as Inference Rules for PL.
z EXAMPLE 2.2 - 7
Justify the following implication claims.
a) ∀x(P x → Qx), ∃xP x = ∃xQx
b) ∀y∃x(x < y) 6 = ∀x(x 6= 0 → 0 < x)
Solution
a) Suppose ∀x(P x → Qx) and ∃xP x are true for some universe of discourse U and some
properties P and Q.
Then some element a in U has property P (premise 2): P a is true.
Furthermore, since any element of U having property P also has property Q (premise 1),
this is the case for element a: P a → Qa is true.
Hence (by Modus Ponens) a must have property Q: Qa is true.
And so the conclusion ∃xQx must be true whenever the premises are true.
b) To show that this is an invalid argument, we must construct a counter-argument; that
is, we must find an interpretation for the sentences that makes the premise true and
the conclusion false.
– Let U = Z and take the usual meaning for the various mathematical symbols.
Then the first sentence is true: given any integer y, we can find another integer x that
is less than it (take x = y − 1). But the second sentence is false because −1 is different
from 0 but not greater than 0.
– An alternative counter-argument to this claim is given by the following finite universe
interpretation: take U = {0, 1}, and let 0 < 0 and 1 < 1 be the only order relations
holding. Then the premise is true while the conclusion is false.
2.2 -10
In order to mirror mathematical arguments in our formal deductions, we will want to make
valid arguments that involve open as well as closed sentences, for these occur frequently in
mathematical proofs. We must therefore extend the notion of logical implication to hold for
them.
DEFINITION 2.2 - 4: Logical Implication for PL Sentences
A set of sentences P logically implies a sentence Q, written P = Q, iff any evaluated
interpretation that makes the premises true also makes the conclusion true.
Thus, P = Q iff whenever an evaluated interpretation satisfies the premises, it also satisfies the conclusion. This definition reduces to the one given above in the case when only
closed sentences are involved: then no evaluations are needed. But whenever an open formula
appears in an argument, evaluated interpretations must be used to determine whether or not
the argument is valid.
z EXAMPLE 2.2 - 8
Show that the argument “x = 1 ∨ x = −1, x 6= 1; therefore, x = −1” is valid.
Solution
Suppose the two premises are true under some evaluated interpretation; say, when x is
interpreted as some constant value a in U .
Then, since the first premise is true, either a names the same object as 1 or a is the same
object as −1.
Since the second premise is also true, a is not 1.
Thus, by process of elimination (i.e., Disjunctive Syllogism), we conclude that a is −1.
The sentence x = −1 is thus true under any evaluated interpretation that makes both
premises true. Therefore the above argument is valid.
The last two examples illustrate the fact that there may be two essential parts to demonstrating the validity of an argument: interpreting Predicate Logic symbolism (quantifiers, identity, etc.) and evaluating variables relative to a given universe of discourse, and then arguing
via SL reasoning with the statements instantiated to elements of U . Thus we are building here
on what we already know from SL.
Logical Equivalence in PL; Quantifier Order in Sentences
The semantic notion of logical equivalence in PL is likewise dependent upon interpretations
and evaluations. It can be defined in terms of logical implication.
DEFINITION 2.2 - 5: Logical Equivalence for PL Sentences
Two sentences P and Q of PL are logically equivalent, written P = Q, iff each sentence
logically implies the other.
Given our PL notion of logical implication, this means P and Q are logically equivalent
iff every evaluated interpretation that makes one formula true also makes the other one true,
and conversely. Hence, equivalent sentences are alike true or false of every structure under any
evaluated interpretation.
We will give two examples here to illustrate the notion of logical equivalence. The first one
is trivial, but it illustrates a general principle worth noting; namely, the particular variables
used in writing quantified sentences of PL are irrelevant. Such variables are only apparent;
they’re “dummy variables.”
2.2 -11
z EXAMPLE 2.2 - 9
Determine the truth of the following equivalence claims.
a) ∀xPx = ∀yPy?
b) ∃y∀xPxy = ∀x∃yPxy?
Solution
a) Since both sentences here assert of every object in the universe of discourse that it has
property P, they are obviously alike both true or false. Thus, the two sentences here
are logically equivalent.
b) These sentences are not logically equivalent:
∃y∀xPxy = ∀x∃yPxy, but ∀x∃yPxy 6 = ∃y∀xPxy.
Implication holds between the first and second sentence because the existence of a single
element y in U having some relationship P to every possible x in U implies that every
x is in relationship P to some y.
However, the reverse implication fails to hold. Just because every x is in relationship P
with some y (a local instance) does not guarantee that the same y (a global instance)
is in that relationship to every x. The value of y may well depend upon the choice of
x (local instances need not be global).
As a counterexample, take N to be the universe of discourse, and let Pxy denote the
relationship ‘x is a prime number greater than y’. Every natural number is less than
some prime number, since there are infinitely many of them, but certainly not all natural
numbers are less than the same prime number.
The last example points out the need to pay close attention to the order in which quantifiers
appear in a sentence. Quantifier order may seem a rather subtle matter, but it is nevertheless
important. Implicit failure to distinguish these two sentence structures led mathematicians
in certain historical instances to confuse concepts that needed to be separated before further
progress could be made. This happened in the theory of calculus. Nineteenth-century mathematicians finally realized in mid-century that they had to distinguish uniform convergence (a
global property) from pointwise convergence (a local property) and uniform continuity from
pointwise continuity. Without going into detail, the pointwise concepts require that a certain
number exist for each member of a given set (must only satisfy a ∀-∃ sentence), while the
uniform concepts require a single number to work throughout the entire set (must now satisfy
a ∃-∀ sentence). The sole difference in their formulations, but one that makes all the difference,
is the relative order of the existential and universal quantifiers. The global, uniform concept is
logically stronger than the local, pointwise concept.
Multiple Interpretations and Meta-Mathematical Concerns
The semantics of Predicate Logic has forced us to adopt an abstract point of view on the
meaning of sentences: formulas must be able to have multiple interpretations in order to deal
with their truth and consequences. The value of this approach goes beyond the confines of
logic, however. It has also become important in mathematical research and education. In the
remainder of this section, we will briefly point out how abstractness has come to characterize
much of contemporary mathematics.
What advantage might there be to treating mathematical sentences as abstract, uninterpreted formulas? Twentieth-century developments in mathematical logic provide one answer:
multiple interpretations clarify the logical structure of mathematical theories. This is true even
of areas like arithmetic and geometry, where a standard interpretation is intended. In order
to investigate the possible logical relationships holding among the various axioms of a theory,
we have to be able to vary the interpretations. Taking an abstract viewpoint allows us to
determine whether a sentence is or is not a logical consequence of a set of premises.
2.2 -12
We will elaborate on this by discussing some basic foundational notions for axiomatic
theories formulated in the language of PL. This will repeat and extend what we said earlier in
connection with SL (see Section 1.3). To keep matters simple, we will restrict our attention to
closed sentences. Since the axioms of a deductive theory are always closed sentences anyway,
this is no real limitation.
DEFINITION 2.2 - 6: Logical Consistency
A set of sentences A is logically consistent iff an interpretation exists under which they are
all true; i.e., iff a model for A exists. A set of sentences A is logically inconsistent iff no
model for A exists.
To illustrate our ideas, we will return to the sentences introduced in Example 2. We repeat
them here for easy reference:
0 < 1;
1 < (1 + 1);
1 · 1 = 1;
∃x(x · x < x);
∀x(x 6= 0 → 0 < x).
In Example 2, we looked at two different interpretations of these sentences. The first one
was the intended number-theoretic interpretation of the above sentences whose universe was N,
providing a model for all but the fourth sentence. The second interpretation took R as the
universe of discourse under the standard interpretation of the symbols; it was a model for all
of the sentences but the last. And Example 3 presented a structure that satisfied all but the
second sentence. The five formulas taken together do form a logically consistent set, though:
it is possible to find an interpretation that makes them all true (see Exercise 15).
Logically true sentences are individually or jointly consistent, since all structures of the
language are models for them. On the other hand, logical falsehoods are inconsistent; no
structure can be a model for a contradiction.
Mathematicians are often concerned not with the consistency of an entire set of sentences,
but with whether a single sentence is consistent with certain other sentences whose truth or
consistency is already known or assumed. This can be defined in terms of the other concept.
DEFINITION 2.2 - 7: Joint Consistency
A sentence Q is consistent with a set of consistent sentences A iff the entire set is logically
consistent; i.e., iff there is a model of A that is also a model for Q. Q is inconsistent with
A iff it is not consistent with them; i.e., no model for all the sentences exists.
Example 2 allows us to conclude, for instance, both that the fifth sentence above is consistent
with the first three and also that the fourth sentence is consistent with them.
Logical consistency is a weaker notion than logical implication. If a consistent set of sentences A logically implies a conclusion Q, then certainly Q is consistent with A: every model
of A is a model of Q, which is more than is required for consistency. The converse, however,
does not hold. As in SL, a sentence Q can be consistent with A without being implied by
them. Example 2 demonstrates, for instance, that the fourth sentence is not implied by the
first three since we gave an interpretation making the first three true and the fourth false.
DEFINITION 2.2 - 8: Logical Independence
A sentence Q is logically independent of a set of sentences A iff neither Q nor ¬Q are
logically implied by A.
This agrees with our intuitive notion of being independent, but it is phrased negatively. Q
is logically independent of A iff there are two models for A, one that is not a model for Q and
another that is not a model for ¬Q. Finding one model of A that is a model for ¬Q (hence,
not a model for Q) and another one that is a model for Q (and so not for ¬Q) is sufficient
for showing that Q is logically independent of A. It therefore suffices to show that Q and ¬Q
are each consistent with A; this establishes the independence of Q. Since we are dealing with
closed sentences, this sufficient condition is also a necessary condition. Any model that is not
a model of Q must necessarily be one for ¬Q, and conversely.
2.2 -13
So we have the following, more positive formulations for logical independence:
Q is logically independent of A iff there are two models for A, one that is a model for
Q and another that is a model for ¬Q.
Equivalently, Q is logically independent of A iff both Q and ¬Q are consistent with A.
The interpretations of Example 2 allow us to conclude, for instance, not only that the final
sentence is logically consistent with the first three, but that it is independent of them.
Finally, we can define theory completeness for any set of sentences.
DEFINITION 2.2 - 9: Theory Completeness
A set of sentences A is theory complete iff no sentence in the language of the theory is
independent of A.
Thus, given any sentence Q, either A = Q or A = ¬Q. This is a very strong requirement
for a set of premises or axioms A. It should not be confused with deductive completeness,
however, which says that if a conclusion Q follows from A, then it can be proved from them;
that is an entirely different matter. This concept relates more to the content of A than to
the logic, since it claims for a complete set of sentences that all propositions formulated in the
appropriate language are logically determined by them: either such a sentence follows from A
or else its opposite does. If there are any independent sentences in the theory, then A is not
theory complete, and conversely.
The semantic notions of consistency, independence, and theory completeness have become
very important topics of research during the past century and are studied in detail for various
axiomatic theories in advanced courses in mathematical logic. They form central concerns of
the technical foundations of mathematics. Important progress has been made in this area for
a wide range of theories, something you can learn more about from any advanced textbook on
mathematical logic.
Abstract Formalism in Mathematics
While we have shown that logic and the foundations of mathematics benefit from taking
an abstract viewpoint on the meaning of sentences, it is not clear yet that mathematics itself
has much to gain from such a perspective. Don’t mathematicians make definite assertions with
their sentences? And if they do, why would they want to ignore that meaning in their work?
These questions can be addressed on several levels. For some, viewing mathematical sentences as meaningless or uninterpreted strings of symbols might be attributed to a sort of
philosophical agnosticism, to an outlook known as formalism. Mathematicians, according to
this point of view, merely play games with formal strings of symbols according to certain rules,
much like they might play a game of chess. Any meaning that might lie behind the game is irrelevant. Meaning is something scientists and others worry about when they use mathematics,
not something mathematicians need to be concerned with. It is the task of those who use mathematics to interpret mathematical formulas for their field of investigation. Mathematicians,
this school of philosophy says, merely generate formal, deductive systems of results.
This approach isn’t seriously entertained by many practicing mathematicians, though the
ideas of Hilbert and others in the first half of the twentieth century tended to encourage such
an outlook. Mathematicians have intuitions about their areas of research that guide them in
their work just as any scientist’s do. They do not merely push symbols around on a piece of
paper according to the syntactic and inferential rules of logic. They may be working abstractly
and axiomatically, but they are directed in their research and demonstrations by their general
mathematical intuitions as well as by their knowledge of specific models.
2.2 -14
Nevertheless, even rejecting a formalist philosophy of mathematics, there are still good
reasons for allowing formulas to have multiple interpretations and for treating them as uninterpreted sentences. Comparing similar results that hold in a number of very different settings,
mathematicians can abstract from the particular details of any one structure and develop a
general theory to cover the central common features of all such structures. This tendency is
particularly characteristic of much of twentieth-century mathematics. This makes possible a
real economy of thought. Proving an abstract result, it necessarily holds for all the different
interpretations or models for the theory. You do not have to prove that result separately for
each situation.
By taking an abstract viewpoint, mathematical rigor is also promoted. Ignoring the specific content associated with a particular interpretation, you are better able to guard against
illegally smuggling unwarranted or implicit assumptions into a proof, something you might be
tempted to do by being overly familiar with what the terms are supposed to mean (the intended
interpretation). Concern for deductive rigor is precisely how the abstract viewpoint entered
mathematics in the late nineteenth century with the work of Pasch in geometry. After 2000
years, properties of the relation of “betweenness” were finally recognized by Pasch, Hilbert, and
others and were explicitly incorporated into an axiomatic system for modern geometry. Their
proposal to treat propositions of mathematics as uninterpreted sentences arose in precisely this
context.
If you proceed further into parts of contemporary mathematics, you will meet up with
this abstract approach time and again. Up to this point, though, you may not be acquainted
with theories designed in this way. The theories of ordinary arithmetic, elementary algebra,
geometry, and calculus are intended as theories of a single mathematical structure, having a
fixed, standard interpretation. But not all theories have this character. The theory of vector
spaces (linear algebra), for instance, applies to structures that are very different with respect
to the objects they contain and the operations they support. The objects may be matrices or
functions or numbers or n-tuples of numbers; the distinguished objects (those represented by
the constant symbols of the theory) may not be at all similar; and the operations involved may
be very different. They may also satisfy different properties: for some multiplication may be
commutative, for others not. Yet all of them are vector spaces, sharing a common algebraic
structure. The theory of vector spaces, appropriately interpreted, applies to each one of them.
A similar thing happens in other parts of algebra. We will see this in Chapter 7 when we
look at partially ordered sets, lattices, and Boolean algebras: many distinct models will be
comprehended by a given theory because of the common structure they share.
The abstract, axiomatic approach to mathematics takes to the limit a tendency that has
long been present in algebra. All symbols except logical ones are viewed as “variable” in
the sense that they can be given multiple interpretations. The subject matter or possible
interpretations are permitted to be as varied as possible, subject only to category restrictions.
This approach now characterizes much of modern mathematics, not just algebra. This mode
of thinking is present in geometry, real analysis, topology, graph theory, and other areas. Our
discussion of Predicate Logic in this section introduces you to this trend.
2.2 -15
EXERCISE SET 2.2
Problems 1 - 14: PL Syntax
In the problems below, interpret all standard constant, operation, and relation symbols in the usual way, the
letters from the middle or the end of the alphabet as variables, and the letters from the front of the alphabet
as constants. Then do the following:
a) Tell whether the given symbol strings are terms, formulas, or neither.
b) If the symbol strings are formulas, identify them as either open or closed sentences. For those sentences that are open, point out the free variable occurrences.
c) Identify all primitive terms and all derived terms occurring in the given symbol-strings.
*1. ∃x(x2 < 0)
*2. ∀n(m + n − 2)
3. ∠1 ∼
= ∠2
*4. y 6= 0 → ∃x(x < y)
5. | − 7|
*6. |x| = x ∨ −x
*7. (x − 1)(x + 1)
8. a3 + 1 = 0 → a = −1
9. ∀x(x + 2 = 5 ↔ x = 3)
10. 2 | 6
11. x ÷ y
12. ∀x∀y(x | y → x | yz)
√
13. sin(π/2) = 2
14. x − tan x ∧ ∃x(x = 0)
*15. Give an interpretation for the sentences of Example 2, repeated here, that will make them all true.
0 < 1;
1 < (1 + 1);
1 · 1 = 1;
∃x(x · x < x);
∀x(x 6= 0 → 0 < x).
16. Determine the truth value of ∃x(x2 − 2 = 0) for the following interpretations. The symbols N, Z, Q,
and R stand for sets of the natural numbers, the integers, the rational numbers, and the real numbers
respectively.
a. U = N
b. U = Z
c. U = Q
d. U = R
Problems 17 - 23: Interpretations, Models, and Truth Values for Arithmetic
Determine the truth value of the following three sentences under each of the given interpretations.
The symbol 0 should be taken as a constant with its usual meaning, while s is a one-place function symbol
(s is the first letter of successor ; Problem 17 gives its standard interpretation). The symbols N and Z stand
for the set of natural numbers and the set of integers respectively.
a. ∀x(s(x) 6= 0)
b. ∀x∀y(s(x) = s(y) → x = y)
c. ∀y(y 6= 0 → ∃x(y = s(x)))
*17. U = N
s(x): s(x) = x + 1
*18. U = N
s(x): s(x) = x + 2
*19. U = Z
s(x): s(x) = x + 1
20. U = N
s(x): s(x) = x2
*21. U = Z
s(x): s(x) = x2 + 1
2.2 -16
EC
22. U = N
x + 1 : x even
x+1
: x odd
2
23. U = Z12 = {0, 1, . . ., 10, 11}
s(x): s(x) = x +12 1 (mod-twelve addition; e.g., 8 +12 5 = 1)
s(x): s(x) =
(
Problems 24 - 32: Interpretations, Models, and Truth Values for Geometry
Determine the truth value of the following five sentences under each of the given interpretations. The terms
‘point’ and ‘line’ refer to objects of different sorts in the universe of discourse U . Assume in each case that
U contains all possible (interpreted) ‘points’ and ‘lines’ of a single plane, and that ‘passes through’ has the
ordinary meaning of ‘contains’.
a. For each pair of distinct points there is at least one line passing through them.
b. For each pair of distinct points there is at most one line passing through them.
c. Each line passes through at least two distinct points.
d. Each line misses (does not pass through) at least one point.
e. There exists at least one line.
*24. Point: point in the plane
Line: line in the plane
25. Point: point on the x-axis
Line: x-axis
*26. Point: point in the plane with integer coordinates (integral lattice point)
Line: line in the plane
27. Point: point in the plane with integer coordinates (integral lattice point)
Line: line in the plane passing through the origin
*28. Point: point in the plane with integer coordinates (integral lattice point)
Line: line in the plane passing through the origin whose slope is either undefined (y-axis) or is a rational
number
29. Point: point in the plane with integer coordinates whose values are strictly between −10 and 10
Line: line in the plane passing through the origin whose slope is either undefined (y-axis) or is a rational
number
*30. Point: point in the plane with integer coordinates (integral lattice point)
Line: rectangle in the plane with sides parallel to the axes and vertices at integral lattice points
31. Point: point in the plane with integer coordinates (integral lattice point)
Line: rectangle in the plane with sides parallel to the axes and vertices at integral lattice points, one of
which is the origin
32. Point: point in the plane with integer coordinates whose values are either 0 or 1
Line: rectangle in the plane with sides parallel to the axes and vertices at integral lattice points, one of
which is the origin
Problems 33 - 40: Interpretations, Models, and Truth Values for Algebra
Determine the truth value of the following four sentences under each of the given interpretations. The
symbol e denotes a constant, while ∗ denotes a binary operation. The symbols N and Q stand for the set of
natural numbers and the set of rational numbers respectively.
a. ∀x∀y∀z((x ∗ y) ∗ z = x ∗ (y ∗ z))
b. ∀x(e ∗ x = x = x ∗ e)
c. ∀x∃y(x ∗ y = e = y ∗ x)
d. ∀x∀y(x ∗ y = y ∗ x)
33. U = N
e: 0
∗: ordinary addition
2.2 -17
34. U = N, the set of natural numbers
e: 0
∗: a ∗ b = min{a, b}
35. U = Q, the set of rational numbers
e: 1
∗: ordinary multiplication
36. U = Q, the set of rational numbers
e: 0
a+b
∗: a ∗ b =
2
37. U = N
e: 0
∗: a ∗ b = |a − b|
38. U = the set of all real numbers except −1
e: 0
∗: a ∗ b = a + a · b + b
39. U = Z12 = {0, 1, 2, . . ., 10, 11}
e: 0
∗: a ∗ b = a +12 b (Twelve-clock addition; e.g., 8 +12 5 = 1)
40. U = {mx + b : m ∈ Q, m 6= 0; b ∈ Q}, the set of linear functions having rational coefficients
e: x
∗: (m1 x + b1 ) ∗ (m2 x + b2 ) = m1 (m2 x + b2 ) + b1 = m1 m2 x + m1 b2 + b1 (substitution or composition of
functions)
Problems 41 - 47: Truth Values for PL Sentences
Determine whether the following sentences are logically true, logically false, or logically indeterminate sentences. Explain your answers, using the notions of interpretations and models (finite or infinite).
41. ∃xP x
*42. ∀x(P x ∧ ¬P x)
43. ∀x((P x ∧ ¬Qx) ↔ ¬(P x → Qx))
44. ∀x(P x ∧ Qx) → ∃xP x ∧ ∃xQx
45. ∀x(P x → ¬P x)
46. ∀x∀y(P xy ∨ P yx)
47. ∀x∀y(P xy ↔ P yx)
Problems 48 - 62: Truth Values for PL Sentences
Determine whether the following mathematical sentences are logically true, logically false, or logically indeterminate sentences. Explain your answers, using the notions of interpretations and models (finite or
infinite). Note: you should not assume that the familiar symbols/words have their familiar meanings!
*48. 1 < 2
*49. x = 1 → x2 = 1
*50. x = 0 ∧ x = 1 → 0 = 1
51. ∀x(1 · x = x)
52. ∀x∀y(x · y = y · x) → 2 · 3 = 3 · 2
53. ∀x(x2 > 0)
54. ∀x(x = 0 ∨ x 6= 0)
55. ∃x(x 6= 0)
56. ∃x(x = 0)
2.2 -18
57.
√
2 is rational → 0 = 1
58. All isosceles triangles are equilateral triangles.
59. Some triangles are right triangles.
60. All right triangles are triangles.
61. All squares are rectangles.
62. Not all rectangles are squares.
Problems 63 - 69: Logical Implication for PL Sentences
Show that the following implications hold. Argue their validity in terms of arbitrary interpretations and
what you know about SL.
63. ∃x(P x ∧ Qx) = ∃x(P x ∨ Qx)
64. ∀x(P x → Qx) = ∀x(P x ∧ Rx → Qx)
*65. ∀x(P x ∨ Qx), ∃x(¬P x) = ∃xQx
66. ∀x(P x ↔ Qx), ∃x(¬P x) = ∃x(¬Qx)
67. ∃x(¬(P x ∧ Qx)), ∀xP x = ∃x(¬Qx)
68. ∀x(P x ∨ Qx) = ∀xP x ∨ ∃xQx
69. ∀x(P x → Qx) = ∃xP x → ∃xQx
Problems 70 - 73: Aristotelian Logic and Logical Implication
Show that the following implications, which represent certain syllogistic argument forms from Aristotelian
logic, are valid. Argue their validity in terms of arbitrary interpretations and what you know about SL.
*70. ∃x(Px ∧ Qx), ∀x(Qx → Rx) = ∃x(Px ∧ Rx)
71. ∀x(Px → Qx), ∀x(Rx → ¬Qx) = ∀x(Px → ¬Rx)
72. ∃x(Px ∧ Qx), ∀x(Rx → ¬Qx) = ∃x(Px ∧ ¬Rx)
73. ∀x(Qx → ¬Px), ∀x(Rx → Qx) = ∀x(Px → ¬Rx)
Problems 74 - 79: Logical Equivalence of PL Sentences
Show that the following equivalences hold. Argue their validity in terms of arbitrary interpretations and
what you know about SL.
74. ∃x(Px ∨ Qx) = ∃x(Qx ∨ Px)
75. ∀x(Px → Qx) = ∀x(¬Qx → ¬Px)
76. ∀x(Px → Qx) = ∀x(¬Px ∨ Qx)
77. ∃x(¬(Px ∧ Qx)) = ∃x(¬Px ∨ ¬Qx)
78. ∀x(Px → Qx ∧ Rx) = ∀x((Px → Qx) ∧ (Px → Rx))
79. ∀x(Px ↔ Qx) = ∀x((Px ∧ Qx) ∨ (¬Px ∧ ¬Qx))
Problems 80 - 82: Order of Quantifiers
Work the following problems, which illustrate the importance of quantifier order.
80. Determine the truth value of ∀x∃y(y ≤ x) for the following interpretations. Interpret ≤ as usual.
a. U = N
b. U = Z
c. U = Q
d. U = R
81. Determine the truth value of ∃y∀x(y ≤ x) for the following interpretations. Interpret ≤ as usual.
a. U = N
b. U = Z
c. U = Q
d. U = R
82. How do your answers to Exercises 80 and 81 illustrate Example 9b? Be specific. Which formula is
logically stronger, ∃y∀xPxy or ∀x∃yPxy? Why?
2.2 -19
EC
EC
Problems 83 - 92: Logical Implication and Equivalence of PL Sentences
Work the following problems to explore the distribution-like relations of implication and equivalence holding
between the given statements. Support positive claims of implication/equivalence with an abstract argument
and negative claims by giving a counterargument, using finite or infinite interpretation domains.
83. a. Does ∀x(Px ∨ Qx) = ∀xPx ∨ ∀xQx?
b. Does ∀xPx ∨ ∀xQx = ∀x(Px ∨ Qx)?
84. a. Does ∃x(Px ∨ Qx) = ∃xPx ∨ ∃xQx?
b. Does ∃xPx ∨ ∃xQx = ∃x(Px ∨ Qx)?
85. a. Does ∀x(Px ∧ Qx) = ∀xPx ∧ ∀xQx?
b. Does ∀xPx ∧ ∀xQx = ∀x(Px ∧ Qx)?
86. a. Does ∃x(Px ∧ Qx) = ∃xPx ∧ ∃xQx?
b. Does ∃xPx ∧ ∃xQx = ∃x(Px ∧ Qx)?
87. a. Does ∀x(¬Px) = ¬∀xPx?
b. Does ¬∀xPx = ∀x(¬Px)?
88. a. Does ∃x(¬Px) = ¬∃xPx?
b. Does ¬∃xPx = ∃x(¬Px)?
89. a. Does ∀x(Px → Qx) = ∀xPx → ∀xQx?
b. Does ∀xPx → ∀xQx = ∀x(Px → Qx)?
90. a. Does ∃x(Px → Qx) = ∃xPx → ∃xQx?
b. Does ∃xPx → ∃xQx = ∃x(Px → Qx)?
91. a. Does ∀x(Px ↔ Qx) = ∀xPx ↔ ∀xQx?
b. Does ∀xPx ↔ ∀xQx = ∀x(Px ↔ Qx)?
92. a. Does ∃x(Px ↔ Qx) = ∃xPx ↔ ∃xQx?
b. Does ∃xPx ↔ ∃xQx = ∃x(Px ↔ Qx)?
Problems 93 - 96: Consistency and Independence
Work the following problems dealing with metalogical properties of PL sentences.
93. Show that the sentences given in Example 2 are independent of one another. You may cite any results
that are relevant from the examples worked in the text.
94. Show that the sentences given for working Exercises 17 - 23 are independent of one another. Cite any
results that are relevant from your work on those problems.
95. Show that the sentences given for working Exercises 24 - 32 are independent of one another. Cite any
results that are relevant from your work on those problems.
96. Show that the sentences given for working Exercises 33 - 40 are independent of one another. Cite any
results that are relevant from your work on those problems.
Problems 97 - 98: Exploring the Ideas of Abstraction and Formalism
Answer the following essay questions.
97. What areas of mathematics are you familiar with that use an abstract , multiple-interpretations approach?
How do you respond to such an approach? Based upon your experience, if any, and upon what has been
said about abstraction in the text, state in your own words what you think to be the advantages and
disadvantages of taking an abstract, deductive approach in mathematics.
98. A formal approach to mathematics is one that insists on formulating every proposition of a mathematical
theory in the language of some system of logic, such as PL, and then using its rules of inference to justify
conclusions drawn from the axioms of that theory. How is this formal approach related to an abstract
approach? Can an approach be abstract without being formal? Can it be formal without being abstract?
Explain your answers.
2.2 -20
HINTS TO STARRED EXERCISES 2.2
1. [No hint.]
2. [No hint.]
4. [No hint.]
6. [No hint.]
7. [No hint.]
15. A familiar universe of discourse exists so that when all the symbols in the sentences are interpreted in
the usual way, the sentences are true.
17. [No hint.]
18. [No hint.]
19. [No hint.]
21. [No hint.]
24. [No hint.]
26. Use the given interpretation of point, not the standard one.
28. Use the given interpretations of point and line, not the standard ones.
30. Use the given interpretations of point and line, not the standard ones.
42. What logical law from SL does this generalize?
48. Consider all possible interpretations of 1, 2, and < , not just the standard one.
49. Consider all possible interpretations of 1 and the squaring function, not just the standard one.
50. Consider all possible interpretations of 0 and 1, not just the standard one.
65. The first premise says everything has property P or property Q, and the second states that something
lacks property P . What SL inference rule applies here?
70. The first premise says something has property P and property Q. The second premise says anything
with property Q has property R as well. What SL inference rules apply here?
2.3 Rules for Identity & Universal Quantifiers
The Natural Deduction System we will devise for PL will extend and modify that of SL in
the following ways. In the first place, all the Inference Rules for SL will remain in force, except
now they will be taken to apply to any sentences, whether open or closed . Some logic texts
restrict inference rules to closed sentences, but this makes formal arguments more involved and
fails to mirror the way people normally reason. So we will also permit deductions that involve
open sentences.
For example, mathematicians typically argue in the following way (see also Exercise 1):
x = 1 ∨ x = −1,
x 6< 0,
−1 < 0;
therefore, x = 1.
We will set up our Natural Deduction System to account directly for the validity of such
reasoning by taking SL’s Inference Rules to apply to well-formed formulas in general. The
symbols P, Q, and R appearing in the formulation of SL’s rules will now be taken to stand for
complete sentences of either type, open or closed.
Secondly, in addition to SL’s Deduction System, modified as noted, we will accept a number
of other rules that are attuned to the internal logical structure of the formulas involved. The
inference rules peculiar to PL are ones that dictate how we may deduce sentences based on the
binary identity relation “=” and the logical quantifiers ∀, ∃, and ∃ ! .
As before, the primary rules of inference that govern the use of identity and the quantifiers
are of two types: they allow us to conclude some formula from an equation or a quantified
sentence already gotten (Elimination Rules), or they allow us to conclude an equation or a
quantified formula from one or more preceding lines (Introduction Rules). In addition, there
will be two Replacement Rules for negating quantified sentences and one for replacing unique
existence statements.
The rules for identity are pretty obvious; they codify moves we normally take for granted
in reasoning about equations. Two of the Int-Elim Rules for quantifiers and the negation
Replacement Rules are also quite natural. The Introduction Rule for the universal quantifier ∀,
the Replacement Rule for the unique existence quantifier ∃ ! , and the Elimination Rule for
the existential quantifier ∃ are more complex, but they underlie some fairly important proof
techniques used in mathematics, so they must also be included. We will study these rules
rather informally and leave most of the technical subtleties in the background. Our main goal
here is to learn how to construct mathematical proofs involving quantifiers and equations. We
will show how to do formal proofs as well, but our stress will be increasingly upon the value of
the rules for developing mathematical proof strategies.
Since Introduction and Elimination Rules are often both needed within a single deduction,
we will discuss them simultaneously for each logical operator. In this section we will treat
the inference rules for identity and the universal quantifier. The remaining rules relating to
existential sentences and the Replacement Rules will be covered in Section 2.4.
Rules of Inference For Identity: Substitution
A logic for mathematical argumentation should certainly include axioms or rules for identity, for mathematical theories are full of sentences (equations) that assert the identity of two
differently denoted objects. In elementary settings we mostly use the equal sign ‘=’ to connect
2.3 -1
two expressions standing for exactly the same quantity (arithmetic and algebra),* but “equals”
occurs in other contexts, too, such as set theory.
In geometry we sometimes say two line segments or two angles or two triangles are equal;
this was Euclid’s practice, for instance, though it is frowned upon in today’s geometry. Here,
we will take “=” to mean “is completely identical with” or “is the same in all respects with.”
We can use other terms to treat equality in the broader sense: two line segments or triangles
can be said to be congruent if they can be made to coincide or if they have identical measures.
Since careful mathematics uses ‘=’ in the sense of identity, and since many non-mathematical theories also make use of identity, it seems appropriate to think of ‘=’ as a logical
symbol. We will therefore treat the theory of identity as a part of Predicate Logic and will
always interpret ‘=’ as “is the same as.” We will not permit multiple interpretations of ‘=’ as
we do with mathematical relations.
Intuitively, two objects are identical iff they are indiscernible from all points of view. This
characterization is Leibniz’s Law of Indiscernibility. We can phrase this more precisely, using
the symbolism of PL. If t1 and t2 are two terms in the language, then t1 = t2 iff given any
formula P(· x ·), P(· t2·) holds whenever P(· t1·) does, and conversely. Here we use P(−) to
represent any statement in the language (and not merely one asserting that some property
P holds) and · x · to indicate a particular occurrence of x in the statement P about x; this
occurrence is replaced by t1 and t2 in the formulas P(· t1·) and P(· t2·) .
DEFINITION 2.3 - 1: Leibniz's Law of Indiscernibility for Identity
t1 = t2 iff P(· t1·) = P(· t2·) for all statements P.
The objects denoted by terms t1 and t2 thus have exactly the same properties and relations.
They cannot be distinguished on the basis of any statement about them and thus should be
considered the same object (see also Exercise 19).
The above definition explicates the intuitive meaning of equality of terms and so enables
us in theory to determine when two terms denote the same object. However, we will rarely use
it in practice to conclude that two differently denoted objects are identical, since that would
involve the impossible task of checking all possible properties or statements with respect to
the objects denoted by t1 and t2 . Instead we will use other criteria for equality. For the most
part, these will be Introduction Rules for identity. In a few cases, however, (for example, in
set theory or formal arithmetic), equality will be asserted by means of some axiom or theorem
that gives an independent mathematical criterion for when two objects are identical.
Considering Leibniz’s Law of Indiscernibility more or less in the forwards direction, as a
statement of what can be asserted about equal terms, we obtain a rule for drawing conclusions
from premises that involve equations. If you know P(· t1·) and t1 = t2 , you may legitimately
conclude P(· t2·). In fact, you ought to be able to substitute t2 for t1 as many times as you
wish. Multiple substitution is formulated by the following rule of inference, which we will call
Substitution of Equals (Sub).**
Sub
P(·t1·)
t1 = t2
P(·t2·)
* Mathematicians sometimes distinguish between what they call an equation and an identity. The former is what we are
calling an identity. The latter (a mathematical identity) is an equation meant to hold for all values of a certain domain;
i.e., it is the universal closure of an equation. While this distinction is important, we will continue to use the terms
‘equation’ and ‘identity’ interchangeably, since an equation asserts the identity of the objects named.
** In a more detailed development of logic, this substitution rule would need to be further qualified. See Michael Resnik’s
Elementary Logic, for example.
2.3 -2
Rules of Inference for Identity: Equivalence Properties
Substitution of Equals is the main Elimination Rule governing the use of identity. There is
also an Introduction Rule for identity. In addition, there are two more Int-Elim Rules, both of
which can be derived using Sub. Since it is handy to have all three of these rules immediately
available for making deductions, though, we will take them as primitive rules of inference
for identity. These three rules do not characterize identity, however, in the same way that
Leibniz’s Law of Indiscernibility does. Besides identity, many other binary relations, known
as equivalence relations (congruence is one such relation), satisfy the same three properties.
Thus, in addition to these three new rules, we will continue to make use of Sub, which captures
one direction of the defining Law of Indiscernibility for identity.
The Introduction Rule for “=” asserts the logical truth known as the Law of Identity.
Since the identity relation is reflexive, each object being identical with itself, you may write
this down at any line in a deduction: t = t. No premises are required in order to draw this
conclusion. You can refer to this rule either by writing Iden or by writing Reflex . Its soundness
may be demonstrated by appealing to the Law of Indiscernibility as capturing the meaning of
“identity” (see Exercise 16a). Schematically, this rule of inference is as follows:
Iden / Ref lex
t=t
You will sometimes see the closed sentence ∀x(x = x) given as the Law of Identity. This
differs only slightly from our formulation. From the above rule, taking x as our term, we can
conclude x = x. Since this is a logical truth, its universal closure ∀x(x = x) also holds (see
Exercise 29). In our formulation of the law, we have broadened the scope of the conclusion to
allow any term t to be used, not just a variable x.
The Law of Identity is required to obtain certain results in PL, such as x = x (see also
Exercise 17), but it doesn’t get applied very often. It would seem to be even less useful in
mathematics; when would you ever need to claim that something is equal to itself? Actually,
though, there are a few occasions where we will find it indispensable. It is sometimes used in the
anchor step of certain proofs by mathematical induction (see Section 3.1) and in certain other
types of arguments (see Exercise 48). But its most important use in elementary mathematics
is probably as the opening move in a rigorous argument establishing an identity.* In order to
prove something like tan2 x + 1 = sec2 x, we ordinarily begin with one side of the identity and
try to generate a sequence of algebraically equivalent expressions until we arrive at the other
side (see below). Arguments are composed of sentences, however, not just a list of identical
terms. The standard approach can be made completely rigorous by beginning with an identity
via Iden and then using Sub to change one side of the equation into the next expression.
A result analogous to the Law of Identity is sometimes used in certain geometry proofs,
only there it is really congruence, not identity, that is involved. For instance, the Greek mathematician Pappus, who lived several centuries after Euclid, presented a rather elegant proof of
the fact that an isosceles triangle has congruent base angles. Taking the congruency criterion
SAS for triangles and considering a given isosceles triangle from two different perspectives, he
showed that the triangles are congruent by using the reflexive property of congruence; i.e., by
using the fact that any angle is congruent to itself. The details of this argument will be left for
you to work out (see Exercise 22).
Besides being reflexive, congruency shares two further properties with identity and all other
equivalence relations: symmetry, and transitivity (cf. Exercise 1.3 - 10). The rule of inference
* Here the more specialized mathematical meaning of the term identity is relevant. See the first note above.
2.3 -3
known as Transitivity of Identity (Trans) concludes t1 = t3 from t1 = t2 and t2 = t3 . This
can be considered to be a special case of Sub (see Exercise 17b).
Schematically, we have the following:
T rans
t1 = t2
t2 = t3
t1 = t3
The soundness of this rule is easily argued: if a single object is denoted by both t1 and t2
and another one by both t2 and t3 , then these objects must be identical since t2 names them
both. Thus t1 and t3 also name the same object and so are equal. This rule can be derived
via Leibniz’s Law of Indiscernibility (see Exercise 16c).
The rule of inference known as Symmetry of Identity (Sym) concludes t2 = t1 from t1 = t2 .
Like Trans, it is both an Introduction and Elimination Rule for = .
Schematically, we have:
t1 = t2
Sym
t2 = t1
The soundness of this rule of inference is evident from the fixed meaning of “=”. This rule can
also be demonstrated by means of the Law of Indiscernibility (Exercise 16b) or using the rule
Sub (Exercise 17a).
While the symmetry of “=” may be fairly clear to you, young students often fail to realize the soundness of this rule. The equals sign is read in certain settings as if it meant
“yields” or “produces,” which effectively converts the bidirectional relation of equality into a
one-directional relation. The distributive law, for example, which says a(b + c) = ab + ac,
is often taken to mean something quite different from its reversed form, ab + ac = a(b + c).
Moving from left to right, the first is a rule for expanding a compound product, while the
latter is seen as a rule for factoring. It is important to realize that a given equation such as this
one legitimizes going in both directions. Sometimes you want to expand an expression, while
other times you want to factor it. The ability to reverse a procedure seems to be involved in
recognizing the symmetry of equality in cases such as these. Symmetry is not automatically
acknowledged by everyone simply because the equals sign has been used.
Sym can be used to generalize the Substitution rule, so you can substitute a term for its
equal no matter how the equality is asserted in the first place. Trans can also be generalized
using the symmetric property of identity, so that if t1 and t2 appear in either order on line
i while t2 and t3 appear in either order on line j, you may conclude t1 = t3 on line k (see
Exercise 18). In other words, things equal to the same thing are equal to each other. As you
argue in these generalized ways in your deductions, you may simply cite Sub or Trans as your
reason, though strictly speaking it might require some fussing with the premises using Sym
before applying Sub or Trans to obtain the conclusion.
Using Equations in Mathematical Proofs
The inference rules for identity dictate what you can do with equations in mathematical
proofs. You can substitute one term for another in a given formula using Sub, you can write
down that something is equal to itself with Iden, you can rearrange a given equation by means
of Sym, and you can equate two things that are equal to a third via Trans. And that’s all.
2.3 -4
Well, almost all. In a less formal mathematical setting, you can and should do something
a bit more than apply these primitive rules to string a series of linked equations together. For
instance, in showing carefully how to multiply binomials using the distributive law, mentioned
above, you should present your work as in the following example.
z EXAMPLE 2.3 - 1
Expand the polynomial (x + 2)(x + 3) in a step by step fashion.
Solution
We multiply these binomials as follows:
(x + 2)(x + 3) = (x + 2)x + (x + 2)3
= x2 + 2x + 3x + 6
= x2 + 5x + 6.
The equations in this example could also have been written all together on one line as
(x + 2)(x + 3) = (x + 2)x + (x + 2)3 = x2 + 2x + 3x + 6 = x2 + 5x + 6,
but this fails to highlight as well the fact that (x + 2)(x + 3) = x2 + 5x + 6, which is the answer
sought. Stringing these equations together vertically aligned on ‘=’ as in Example 1 indicates
the same series of equalities but better indicates the conclusion we want: the top left expression
equals the bottom right one. A formal argument would use Trans repeatedly to justify the
conclusion (see Exercise 18b).
This proof notation is well-designed, not only because of its brevity, but also because it
suggests the proper way to string equations together in composing a proof. To prove an
equation, you begin with the expression on one side and you work with it to obtain something
equal to it. Then you find another result equal to that one, and so on, continuing forward from
the term just gotten until you arrive at the expression of the other side.
Unfortunately, a bad proof habit that violates this sage advice is often cultivated in elementary algebra and trigonometry. Students sometimes take the result that is to be proved, such
as tan2 x + 1 = sec2 x, and work with both sides of the identity at once, transforming the result
into a series of new equations until an equation is finally gotten that is known to be correct.
Such a (bad) argument might look like the following sequence of formulas:
tan2 x + 1 = sec2 x
sin2 x/ cos2 x + 1 = 1/ cos2 x
sin2 x + cos2 x = 1.
This practice is misleading even when all the separate equations are correct, for it is no
longer clear what is given, what has been proved, and what still needs to be argued. Moreover,
this procedure is unsound in general (it starts out by asserting the very thing that needs
proving!); it can easily lead to wrong results. The sequence of equations generated must be
reversible for it to be valid. Following exactly the same procedure, we can show that 1 = 2:
1=2
0=0
[multiplying by 0]
While we’ve started with the patently false statement 1 = 2, after multiplying both sides by
0 (a perfectly legitimate but irreversible operation) we get 0 = 0, which is certainly true. But
getting this conclusion hardly qualifies as a valid demonstration of the original statement.
2.3 -5
A modified version of this approach is acceptable, however. As long as we keep the forward
and backward directions separate, we may work with both sides of an equation simultaneously,
though not by creating a series of transformed equations. We can start with one side of the
equation in the top left corner and with the other side in the bottom right corner. Then finding
values equal to the first side, we list them downwards from the top right; we list values equal
to the other side upwards from the bottom right. When we reach a common value, we have a
proof of our result. Taking the above example, we would have:
tan2 x + 1 = (sin2 x/ cos2 x) + 1
= (sin2 x + cos2 x)/ cos2 x
= 1/ cos2 x
..
.
[adding fractions]
[because sin2 x + cos2 x = 1]
= 1/ cos2 x
= sec2 x
This dual approach is completely legitimate. In fact, since you generally do not know at
the outset which side of an equation will be easier to work with, it makes good sense to use
both a forward and backward approach in generating a proof. But keep straight what it is
you are doing. To repeat the earlier warning: do not organize your argument as a sequence of
transformed equations.
Eliminating Universal Quantifiers: Universal Instantiation
To understand the elimination rule for the universal quantifier, called Universal Instantiation (UI ), we only need to recall the intended meaning of the quantifier ‘for all’. If the sentence
being asserted is a universally quantified one, such as ∀xP(x), then once an interpretation is
provided, the sentence P(x) holds for every object x in the universe of discourse. Thus, if t
is any constant term (a name denoting some definite object in the universe of discourse), the
closed sentence P(t|x), obtained from P(x) by substituting t for x everywhere in the formula
(uniformly evaluating x at t) must be true if the original sentence was.
Schematically, we have the following rule:
∀xP(x)
UI
P(t|x)
Such a rule is obviously sound: if a formula P(x) is true under an interpretation for all x in
the universe, it certainly holds for a particular thing t. We can thus legitimately instantiate
the universal statement ∀xP(x) to t, obtaining the particular sentence P(t|x).
It is helpful in this connection (and others) to think of the universal statement ∀xP(x) as
asserting a grand conjunction, containing one conjunct P(t|x) for every t in the universe of
discourse. According to the Sentential Logic rule of inference Simp, appropriately generalized to
handle more than two conjuncts, you can validly conclude any conjunct P(t|x), no matter what
t denotes. UI thus acts as a PL extension of Simp, designed to handle any size conjunction.
To use UI , just as with any Int-Elim rule, our sentence must be of the right type overall.
We can only apply this rule to universal sentences. UI cannot be applied to only part of a
sentence; the whole sentence must be quantified by ∀. For instance, it is invalid to conclude
¬P a from ¬∀xP x; even if not every object has property P , we have no reason to conclude that
a particular object a fails to have it. Not every positive integer is prime, for example, but 7
certainly is.
2.3 -6
The Role of UI in Mathematical Proofs
UI is used over and over again in mathematics, though this is not usually made explicit.
In the process of proving that some result holds for a given object, we usually apply previous
results, axioms, or definitions, known to hold universally for a certain class of objects, to the
particular object we are interested in, because it is one of those objects. This is valid because
UI is sound.
Applications of this rule can be found in virtually every mathematical proof. For example,
in order to prove that 4ABC is congruent to 4A0 B 0 C 0 , you take a universal congruency
criterion, such as SAS, and apply it to the instance under consideration. Similarly, if n is an
odd number, we can take the definition of being odd — a number is odd iff it can be written as
2k + 1 for some number k — and specify it for n, thereby concluding that there is some number
k such that n = 2k + 1. In all such cases, the operation of UI is presumed: since a result is
true of all objects of a given type, it must be true of the object under consideration.
We will give two arguments from mathematics, both of them quite simple, in order to
illustrate the use of UI and some of the earlier rules for identity in formal proof diagrams. The
first one comes from the theory of natural number arithmetic; the second one is a simple result
from abstract algebra.
z EXAMPLE 2.3 - 2
Show that ∀x(s(x) > 0), 1 = s(0) − 1 > 0. [Here s stands for the successor function,
which relates each natural number to the next one greater than it; 0 denotes the number
zero; and 1 is defined by the second equation as s(0).]
Solution
The following proof diagram establishes the claim. Note that step 4 could be omitted, using
a generalized version of Sub in 5, but we have included it to show how to apply the rules
in the exact format in which they were given.
1
2
∀x(s(x) > 0)
1 = s(0)
Prem
Prem
3
4
5
s(0) > 0
s(0) = 1
1>0
UI 1
Sym 2
Sub 3, 4
z EXAMPLE 2.3 - 3
Show that ∀x(x ∗ e = x), ∀x(x−1 ∗ x = e) − e−1 = e, where e is a constant, ∗ is a binary
operation, and ( )−1 denotes a function that assigns an inverse a−1 to each element a in
the universe of discourse.
Solution
The following proof diagram establishes the claim. We will give an abridged demonstration
by using generalized versions of our inference rules for “=.” Constructing a longer proof
will be left as an exercise (see Exercise 23).
1
2
∀x(x ∗ e = x)
∀x(x−1 ∗ x = e)
Prem
Prem
3
4
5
e−1 ∗ e = e
e−1 ∗ e = e−1
e−1 = e
UI 2
UI 1
Trans 3, 4
2.3 -7
Deducing Universal Sentences: Universal Generalization
One of the primary goals of mathematics, as of any theoretical science, is to investigate
and systematically exhibit the patterns of uniformity and regularity (laws) that exist in its
field of study. Thus a great many mathematical propositions are universal sentences, law-like
statements asserted to hold for an entire class of objects. Proving them requires our being able
to deduce universal sentences.
In order to prove that some assertion P(x) holds for all elements x of a universe of discourse U, we must somehow show that membership in such a class entails P(x). Now if U
is a small enough finite domain, it might be possible to take each member in turn and show
that it has the desired property. But in most instances such a tack will be hopeless or even
impossible. Most mathematical theories have infinite models as their intended or potential
interpretations. In this context it is well to heed the time-worn adage, “you can never prove
anything by examples.” You can verify that a universal sentence holds in certain cases by looking at specific examples, but this does not prove it in general. Nevertheless, this fact should not
keep you from trying out several well chosen or even random examples, for working through
such examples may help you see just what it is that needs demonstrating and how it might be
related more generally to what is already known. This is particularly good advice when you’re
not sure whether a universal statement is even true or, assuming it is, how to go about proving
it. If nothing else, it gives you something to do while you’re waiting for inspiration to strike,
and it builds up some confidence in the truth of the result.
The main problem with arguing for the truth of a proposition by means of specific cases
lies in the fact that each such argument may turn upon the particular nature of the objects
chosen, and so not be valid for other members of the universe. On the other hand, if it were
possible to find an object that was completely representative, a generic element of the universe
of discourse, as it were, then an argument framed in terms of such an object ought to work
for each member of the universe and so should be considered to yield a proof of the universal
sentence.
Now, in reality there are no such things as generic or “brand x” objects in a universe of
discourse. Every object has its own peculiar properties that distinguish it from all other ones.
However, if we were to choose an unassigned letter a to stand for an arbitrary member of a
model for the theory being developed and were then careful to assert only those things about
it that followed solely from its belonging to such a domain, we would then have achieved a sort
of proof by generic example. It is precisely this proof strategy that is embedded in the rule of
inference known as Universal Generalization (UG). In fact, we have already used this form of
reasoning in our meta-arguments several times above (see Example 2.2-11, for instance).
We will describe this proof method more precisely before symbolizing it, because we need
to make a decision about how UG should be schematized and applied. If a sentence P(a|x)
can be proved to hold for an arbitrary representative individual denoted by a, where nothing
particular is assumed or known regarding the nature of this object except for its denoting
a member of the universe of discourse, then according to UG you may assert the universal
statement ∀xP(x).
A proof using UG would thus seem to require the supposition “let a be an arbitrary element
in the universe of discourse.” We could symbolize this as a supposition a ∈ U ,* but strictly
speaking, this notation isn’t available to us inside PL; it’s part of the informal meta-language
outside PL instead. As a matter of fact, using a as a constant symbol automatically assumes,
according to the standard convention, that a denotes some member of U , so we don’t need
to state this. And even if we did, it wouldn’t function in any genuine deductive way in the
subargument. Nevertheless, in informal arguments, the fact that a denotes an arbitrary, indefinite object of the universe of discourse is often explicitly stated as a supposition. It serves as
* This is the approach taken by Michael Resnik in Elementary Logic.
2.3 -8
an indicator that the instance is supposed to be perfectly general. Since our purpose in this
text is to provide a basis for informal mathematical argument, we might follow this practice
by writing a ∈ U as a supposition beginning a subproof. Doing this, however, will make our
formal proofs much more involved, so we will not do so. We will instead just be careful that
the letter used as an arbitrary constant in our argument hasn’t had anything asserted about it
that isn’t true of every element of our universe of discourse. We will then go on to prove that
P holds for a; that is, that P(a|x) is the case. Based upon the fact that P(a|x) is the case for
an arbitrarily chosen element a, we will then conclude that P(x) holds for all x.
Schematically, then, we have the following form for UG:
P(a|x)
UG
∀xP(x)
In order to use UG in a formal deduction, therefore, we must be certain that nothing has
been asserted about a prior to the line in which P(a) occurs that would limit its range of
reference. This requirement is fairly obvious; otherwise a might stand for a specific member
that is not typical of the members of the set in all respects. In addition, a must not already
occur in P(x), the statement we want to generalize. Else here, too, a might represent some
definite, atypical object of the universe of discourse. Unless both of these restrictions are
followed, inferring a universal sentence from an instance may be invalid.
There is nothing special about the letter a, of course; in practice you will probably use
whatever letter or symbol helps you remember what sort of object the letter stands for. In
fact, mathematical proofs often use the variable x in a dual role, both as the variable to be
generalized upon and as the arbitrary individual to be argued about. In formal proofs, however,
we will distinguish these two roles.
z EXAMPLE 2.3 - 4
Show that ∀x(P(x) → Q(x)), ∀xP(x) − ∀xQ(x).
Solution
This is established by the following proof diagram.
1
2
∀x(P(x) → Q(x))
∀xP(x)
Prem
Prem
3
4
5
6
P(a) → Q(a)
P(a)
Q(a)
∀xQ(x)
UI 1
UI 2
MP 3, 4
UG 5
The next example gives a slightly abbreviated deduction by instantiating more than one
universally quantified variable at a time, something that strictly requires several applications of
our rules to legitimize. This maneuver occurs quite often in informal mathematical arguments.
Similarly, it is also possible to universally generalize more than one variable at a time, which
will shorten other deductions. Neither of these moves is necessarily sound, however, if the
two quantifiers involved aren’t both universal quantifiers, so you shouldn’t draw any further
conclusions from what we are doing here.
z EXAMPLE 2.3 - 5
Show that ∀x∀y(x + y = y + x), ∀x(x + 0 = x) − ∀x(0 + x = x).
2.3 -9
Solution
This claim is established by the following proof diagram.
1
2
∀x∀y(x + y = y + x)
∀x(x + 0 = x)
Prem
Prem
3
4
5
6
a+0 = a
a+0 = 0+a
0+a=a
∀x(0 + x = x)
UI 2
UI 1 (twice)
Sub 3, 4
UG 5
Although UG is indeed proof by generic example, it should probably be stressed once again
that you cannot just take a particular member of the universe of discourse at random and show
that the result holds for it. A particular member, however chosen, is not a generic element;
it is always a specific concrete element. What you can do with particular examples, however,
is to determine how you might next give a perfectly general argument in terms of a constant
symbol a taken to stand for any individual member of the domain whatsoever.
The soundness of UG is intuitively clear.* If we can show that a typical element a of the
universe of discourse satisfies statement P, then that will work no matter which particular
element we choose. And so every element will satisfy statement P. Here, again, it is helpful to
think of the universal sentence as asserting a conjunction, with one conjunct P(a|x) present for
each element a of the structure. Having shown that each conjunct holds by arguing in terms of
a generic element, we ought to be able to conjoin them all to obtain the universal statement.
UG can thus be thought of as a PL extension of the SL Introduction Rule Conj .
The Role of UG in Mathematical Proofs
Like many other rules of inference we have studied, UG is often tacitly assumed in mathematical proofs. In fact, many times the theorem to be proved will already be formulated in
terms of an arbitrary instance.
For example, you will sometimes see statements like “If a real-valued function f is differentiable, then it is continuous” or “If triangle 4 ABC is isosceles, then its base angles
are congruent.” This isn’t the recommended way to write mathematical propositions, but it
happens. Such informal writing partly masks the fact that these sentences are intended as
universally quantified sentences; hence UG seems superfluous. For such sentences you merely
begin arguing in terms of the function f or in terms of the triangle 4 ABC. However, if
these sentences were formulated to make the hidden quantifier apparent, as “All differentiable
real-valued functions are continuous” and “All isosceles triangles have congruent base angles,”
then you would begin the proofs by saying “Let f be any differentiable, real-valued function”
and “Let 4 ABC be an arbitrary isosceles triangle” (here we are using f and 4 ABC rather
than a to stand for the arbitrary instance being considered). But regardless of how the original
statement is formulated, you want to know that such results hold universally, because they will
inevitably be used later on to apply (via UI ) to a variety of different situations. The rule that
permits you to conclude that they hold universally after an argument has been made in terms
of a generic case is UG.
While this rule of inference is often left implicit in a mathematical argument, this does not
mean it is unimportant for proof-making. In fact, quite the opposite seems to be true. Many
people run stuck before they get started. Using a proof strategy based on UG, however, you
can always get going on a proof of a universal statement. Knowing what sort of object you
are being asked to prove something about, you can begin by supposing that a is just such an
* Proving this for a rigorous formulation of UG , however, involves technical complications due to the restrictions on a.
2.3 -10
object. Draw a diagram for it if that will help you visualize the situation. In doing these things,
you will have overcome the initial psychological barrier associated with constructing a proof.
Strange as it sounds, this may be just what is needed. Naturally, UG won’t tell you exactly
how to construct the proof, but it will get you rolling. The Backward-Forward Method of proof
analysis will then help keep you moving. Analyze how you would prove the desired result about
such an object, given what you know about such objects in general, and take stock of what you
know and what you still want to show. If you know the domain can be split up in a natural
way into several types of objects, you may want to try to use proof by Cases to generate the
result needed for each type of object and then generalize (see Exercise 43, for example).
Mathematical proofs of universal statements are not always of the sort indicated by our
formal discussion of UG. Often the set of elements involved in the generalization is the universe
of discourse, as assumed above, but there are many other times when it may be some subset of
the universe of discourse instead. Such a proof will proceed, however, much as it would using
ordinary UG, as the following example shows.
z EXAMPLE 2.3 - 6
Analyze the beginning and overall proof strategy to use in proving the proposition “All
positive real numbers have real square roots.”
Solution
If this result is part of a development of real number arithmetic, the universe of discourse
will be the set of real numbers R, but this sentence only mentions positive real numbers,
so we will have to add that restriction in our sentence formulation. Using our knowledge
of what is meant by “square root,” we can reformulate the assertion that such a number
has a square root by saying that it is the square of some number.
Two slightly different proof strategies are available to us, depending on whether restricted
quantifiers are used to write our sentence’s formulation.
1) First suppose we use an unrestricted universal quantifier. Then the proposition to prove
is as follows: ∀x(x > 0 → ∃y(y 2 = x)).
To get started on a proof here, we would suppose r is any real number, and we would
try to prove that r > 0 → ∃y(y 2 = r), using r as the arbitrary term to generalize on.
Since this is a conditional sentence, we would next get ready to apply CP : supposing
r > 0, we would try to prove ∃y(y 2 = r).
2) A second approach applies if we use a restricted quantifier; this immediately combines
what was done in two separate steps in the last procedure.
The sentence to be proved in this case is the following: (∀x > 0)(∃y)(y 2 = x).
To prove this, we would suppose that r is an arbitrary positive real number, and we
would then try to deduce ∃y(y 2 = r).
In either case, once we have proved the inside sentence, we would generalize. The overall
proof strategy here is nothing more than UG.
An informal proof would typically follow the second strategy: we would compress the two
suppositions, and without using a new letter for the instance, start by saying “Suppose x
is a positive real number.” Once having shown that x has a square root y, the proof would
be considered complete, tacitly assuming UG.
2.3 -11
EXERCISE SET 2.3
*1. Give a formal deduction for the following argument from the second paragraph of this section, citing
any SL rules of inference used: (x = 1) ∨ (x = −1), x 6< 0, −1 < 0 − x = 1.
Problems 2 - 5: Completing Deductions
Fill in the reasons for the following deductions.
*2. a = b, b = c, d = c, d = e − a = e
1 a=b
2 b=c
3 d=c
4 d=e
5
6
7
a=c
c=e
a=e
3. a = c, b = d − P ab ↔ P cd
1 a=c
2 b=d
3
P ab
4
5
6
7
a=c
P cb
b=d
P cd
8
P cd
9
10
11
12
a=c
P ad
b=d
P ab
13
P ab ↔ P cd
4. ∀xP x − ∀yP y
1 ∀xP x
2
3
Pa
∀yP y
*5. ∀x(P x ∨ Qx), ∀x(¬Qx ∨ Rx) − ∀x(P x ∨ Rx)
1 ∀x(P x ∨ Qx)
2 ∀x(¬Qx ∨ Rx)
3
¬P a
4
5
6
7
8
9
∀x(P x ∨ Qx)
P a ∨ Qa
Qa
∀x(¬Qx ∨ Rx)
¬Qa ∨ Ra
Ra
10
11
P a ∨ Ra
∀x(P x ∨ Rx)
2.3 -12
Problems 6 - 9: Logical Implication and Conclusive Deductions
Determine whether the following claims of logical implication are true or false, and explain why. Then
determine whether the deductions that are given are conclusive or inconclusive. Carefully point out each
place where a rule of inference is being used incorrectly.
6. x2 + 3x − 2, x = 1 ∨ x = 2 = x2 + 3x − 2 = 2 ∨ 8
1 x2 + 3x − 2
Prem
2 x = 1∨x = 2
Prem
3
x=1
2
Spsn 1 for Cases
4
5
x + 3x − 2
x2 + 3x − 2 = 2
Reit 1
Sub 3, 4
6
x=2
Spsn 2 for Cases
7
8
x2 + 3x − 2
x2 + 3x − 2 = 8
Reit 4
Sub 6, 7
9
x2 + 3x − 2 = 2 ∨ 8
*7. ∀x(P x → Qb), ∀xP x = ∀xQx
1 ∀x(P x → Qb)
2 ∀xP x
3
4
Qb
∀xQx
Cases 5, 8
Prem
Prem
MP 1, 2
UG 3
*8. ∀x∀y(x 6= y → P xy), ∀x∀y(x = y → Qxy) = ∀x∀y(¬P xy → Qxy)
Prem
1 ∀x∀y(x 6= y → P xy)
2 ∀x∀y(x = y → Qxy)
Prem
3
¬P xy
Spsn for CP
4
5
6
7
8
∀x∀y(x 6= y → P xy)
¬(x 6= y)
x=y
∀x∀y(x = y → Qxy)
Qxy
Reit 1
MT 4, 3
DN 5
Reit 2
MP 7, 6
9
10
¬P xy → Qxy
∀x∀y(¬P xy → Qxy)
9. ∀xP x → ∀yQy, ¬Qb = ¬∀xP x
1 ∀xP x → ∀yQy
2 ¬Qb
3
4
5
P a → Qb
¬P a
¬∀xP x
CP 3-8
UG 9
Prem
Prem
UI 1
MT 3, 2
UG 4
*10. Criticize the following informal proof involving equations, which purports to show that 0 = 1. Here the
problem lies with the mathematics involved.
Let x = 1.
Squaring gives x2 = 1, so x2 = x by substitution.
Subtracting 1 from each side, we get x2 − 1 = x − 1.
Factoring yields (x − 1)(x + 1) = x − 1.
Dividing both sides by x − 1, we get x + 1 = 1.
Subtracting 1 from both sides, x = 0.
But since x = 1, too, we conclude that 0 = 1.
2.3 -13
Problems 11 - 14: True or False
Are the following statements true or false? Explain your answer.
*11. In PL, the symbol ‘=’ is open to a variety of interpretations, since PL works with abstract sentences.
12. In order to prove an identity, one manipulates both sides of the equation until a true identity results.
13. The inference rule UI is used to deduce a universal sentence.
*14. To prove a universal statement, one can show that the result holds for an arbitrary element of the
universe of discourse.
15. Explain Leibniz’s Law of Indiscernibility in your own words.
Problems 16 - 22: Exploring Rules for Identities
Work the following problems, which explore the connections among and implications of the various rules
we’ve adopted for identities.
16. Identity is an Equivalence Relation
Using the original definition of identity given by Leibniz’s Law of Indiscernibility, show that identity is
an equivalence relation on terms. That is, show the following:
a. Reflexive Law : − t = t
b. Symmetric Law : t1 = t2 − t2 = t1
c. Transitive Law : t1 = t2 , t2 = t3 − t1 = t3
17. Prove the following properties of identity by means of Substitution of Equals and the Law of Identity.
a. Symmetry: t1 = t2 − t2 = t1
b. Transitivity: t1 = t2 , t2 = t3 − t1 = t3
18. Generalized Transitivity of Identity
a. How many different forms of the generalized rule Trans are there? Pick one other than the one given
and derive it, using Trans (in its original formulation) along with Sym.
b. Use Trans to show the following: x1 = x2 , x2 = x3 , x3 = x4 − x1 = x4 .
*19. Distinguishing Objects
*a. Using a formal proof diagram, show that two objects must be distinct if a property P exists that
distinguishes them; that is, show P a, ¬P b − a 6= b.
*b. Conversely, using Leibniz’s Law of Indiscernibility, determine what follows when a 6= b.
20. Give a deduction to show that if t(a) is a compound term containing the term a, and if a = b, then
t(a) = t(b).
*21. Prove that P x, x = a ∨ x = b − P a ∨ P b.
22. Show that the base angles of an isosceles triangle are congruent by means of Pappus’ proof (see the text,
p. 2.3-3); that is, by using SAS and the fact that an angle is congruent to itself.
23. Construct a longer version of the proof given in Example 3, using the rules of inference precisely as they
were originally given instead of in their generalized forms. Thus the abbreviated version is valid.
Problems 24 - 28: Proving Mathematical Identities
Prove the following identities in algebra and trigonometry, combining equations in the proper way, as indicated in the text. You may assume the rules of arithmetic, definitions of trigonometric functions, and the
basic identity sin2 x + cos2 x = 1.
24. (a − b)(a + b) = a2 − b2
25. (x + y)3 = x3 + 3x2 y + 3xy2 + y3
*26. tan x/ sin x = sec x
27. cot2 x = csc2 x − 1
2.3 -14
EC
28. Formulate symbolically and then prove: The difference between two consecutive squares (of integers) is
the sum of these integers.
Problems 29 - 33: Deducing Logical Truths
Using the rules of inference for identity and the universal quantifier demonstrate the following logical truths.
29. − ∀x(x = x)
30. − ∀x∀y(x = y → y = x)
31. − ∀x∀y(x = y ↔ y = x)
32. − ∀x∀y∀z(x = y ∧ y = z → x = z)
33. − ∀x∀y∀z(x 6= z → x 6= y ∨ y 6= z)
Problems 34 - 35: Aristotelian Logic and Derivations
Deduce the following arguments, which represent certain syllogistic forms from Aristotelian Logic, using
PL’s inference rules for universal sentences.
34. ∀x(Px → Qx), ∀x(Qx → Rx) − ∀x(Px → Rx)
*35. ∀x(Px → Qx), ∀x(Rx → ¬Qx) − ∀x(Px → ¬Rx)
Problems 36 - 43: Deductions
Deduce the following, using PL’s inference rules for universal sentences.
36. ∀x(P x → Qx), P a − Qa
37. ∀x(P x → Qx) − ∀x(P x ∧ Rx → Qx)
38. ∀x(P x ∨ Qx → Rx) − ∀x(P x → Rx)
39. ∀x(¬(P x ∧ Qx)), ∀xP x − ∀x(¬Qx)
40. ∀x(P x ∨ Qx), ∀x(P x → ¬Rx) − ∀x(Rx → Qx)
*41. ∀xP x ∨ ∀xQx − ∀x(P x ∨ Qx)
42. ∀x(P x → Qx) − ∀xP x → ∀xQx
43. ∀x(P x ↔ Qx) − ∀xP x ↔ ∀xQx
Problems 44 - 46: Interderivability
Show the following interderivability results, using PL’s inference rules for universal sentences.
44. ∀x(P x ∧ Qx) − ∀xP x ∧ ∀xQx
45. ∀x(P x → Qx ∧ Rx) − ∀x((P x → Qx) ∧ (P x → Rx))
46. ∀x(P x ↔ Qx) − ∀x((P x ∧ Qx) ∨ (¬P x ∧ ¬Qx))
Problems 47 - 55: Additional Deductions
Deduce the following, using PL’s inference rules for identity and universal sentences.
47. Using the argument given in the text on page 2.3-6 as your guide, give a formal proof of the universal
identity ∀x(tan2 x + 1 = sec2 x). Determine and include as premises the universal sentences relating
tan x and sec x to sin x and cos x as well as the key identity relating sin x and cos x.
48. Show that if ∗ denotes a binary operation, then the following general results hold. Thus, in particular,
if ∗ stands for addition, we obtain the rule “Equals added to equals are equal.”
a. a = b − a ∗ c = b ∗ c
b. a = b ∧ c = d − a ∗ c = b ∗ d
c. Give an informal proof that a < b → a + c < b + c for any real numbers a, b, and c. You may use the
above results as well as the definition of less than (x < y iff x +z = y for some z > 0), the associativity
of addition ((x + y) + z = x + (y + z)), and the commutativity of addition (x + y = y + x).
49. Show that ∀x∀y∀z((x ∗ y) ∗ z = x ∗ (y ∗ z)), ∀x∀y(x ∗ y = y ∗ x) − ∀x∀y∀z((x ∗ y) ∗ z = (z ∗ y) ∗ x),
where ∗ is a binary operation.
2.3 -15
EC
50. Show ∀x∀y(N x ∧ P y ↔ Bx0y), ∀x(N x ∨ P x) − ∀x∀y(¬Bx0y → P x ∨ N y). [One interpretation:
think of N x as saying x ≤ 0, P x as saying x ≥ 0, and Bx0y as saying that x ≤ 0 ≤ y.]
51. Show that ∀x∀y(x ≤ y ∨ y ≤ x), ∀x∀y∀z(x ≤ y ∧ y ≤ z → x ≤ z), a ≤ b, a 6≤ c − c ≤ b.
52. Show ∀a∀b∀c∀d((a, b) (c, d) = ac + bd), ∀x∀y(xy = yx) − ∀a∀b∀c∀d((a, b) (c, d) = (c, d) (a, b)).
[This asserts according to one interpretation that the dot product for vectors is commutative.]
53. Show ∀x∀y∀z((x + y) + z = x + (y + z)), ∀x∀y(x + y = y + x), ∀x∀y(x · y = y · x),
∀x∀y(x ∗ y = (x + (x · y)) + y) − ∀x∀y(x ∗ y = y ∗ x), where +, ·, and ∗ are all binary operations.
54. Show ∀x∀y∀z(x ⊗ (y ⊕ z) = (x ⊗ y) ⊕ (x ⊗ z)), ∀x(x ⊕ 0 = x), ∀x(x ⊕ x0 = 1), ∀x(x ⊗ 1 = x), ∀x(x ⊗ x0 =
0) − ∀x(x ⊗ x = x), where 0 and 1 are constants, ⊕ and ⊗ are binary operations, and ( )0 is a unary
operation or function.
55. Show ∀x(min x ↔ ∀y(x ≤ y)), ∀x∀y(x ≤ y ∧ y ≤ x → x = y) − min a ∧ min b → a = b. In words, if a
set has a minimum for a relation ≤ that is antisymmetric (i.e., satisfies the second premise), it must be
unique.
2.3 -16
HINTS TO STARRED EXERCISES 2.3
1. Use either Cases or NI as your main proof strategy.
2. [No hint.]
5. [No hint.]
7. Not much is good in this deduction.
8. Recall that Sentential Int-Elim Rules need to be applied to full sentences.
10. What mathematical operations have restrictions on them?
11. [No hint.]
14. [No hint.]
19. a. Use Proof by Contradiction here.
b. Use NBE on Leibniz’s Law of Indiscernibility.
21. Use either Cases or EO for your main proof strategy.
26. Use the mathematical identity tan x = sin x/ cos x.
35. Start by instantiating the premises to an arbitrary element a; finish by generalizing on what you prove
about a.
41. Use Cases as your main proof strategy.
Inference Rules for Predicate Logic
SENTENTIAL RULES OF INFERENCE
See Inference Rules For Sentential Logic
Take all sentential letters now as standing for formulas (open or closed sentences) of PL
= INTRODUCTION AND ELIMINATION RULES
Substitution of Equals (Sub)
P(·t1 ·)
t1 = t2
Transitivity of = (Trans)
t1 = t2
t2 = t3
P(·t2 ·)
t1 = t3
Reflexivity of = (Reflex/Iden)
Symmetry of = (Sym)
t1 = t2
t=t
t2 = t1
∀ ELIMINATION AND INTRODUCTION RULES
Universal Instantiation (UI )
∀xP(x)
P(t|x)
Universal Generalization (UG )
P(a|x)
a arb new constant
t any term
∀xP(x)
∃ INTRODUCTION AND ELIMINATION RULES
Existential Generalization (EG )
P(t|x)
t any term
∃xP(x)
Existential Instantiation (EI )
∃xP(x)
P(a|x)
a arb new constant
Q
Q
a not in Q
RULES OF REPLACEMENT FOR NEGATIONS
Universal Negation (UN )
¬∀xP(x) = ∃x(¬P(x))
Existential Negation (EN )
¬∃xP(x) = ∀x(¬P(x))
2.4 Rules for Existential Quantifiers
In the last section we explored the Int-Elim Rules for identity and universal quantifiers.
In this section we will consider the Int-Elim Rules for existential quantifiers. In connection
with the existential Introduction Rule, called Existential Generalization, we will highlight the
Method of Analysis, a method of some historical importance for mathematics. We will also
treat the Replacement Rules for negating both universal and existential sentences. Finally, we
will look at how to prove unique-existence statements.
With this section, our Natural Deduction System for PL will be complete, and we will
be ready to move beyond logic to take up various topics in discrete mathematics. There we
will make use of a more informal and less stylized method of deduction, but the various proof
strategies we’ve learned from our study of SL and PL will continue to give us insight into what
we are doing and how we should proceed.
Deducing Existential Sentences: Existential Generalization
It is often said by mathematics instructors that you can never prove anything with examples.
As you may recall from Section 2.3, this was more or less the warning we gave regarding proving
universal statements. Specific cases prove specific results; they do not demonstrate general ones.
Nevertheless, as we also noted there, the way in which universal sentences are proved via UG is
in a sense proof by generic example; so we have already seen that this advice must be tempered,
though we must take care to deal with typical or generic examples.
For the case at hand, we will go a step further and refute the advice altogether. In order
to prove an existential sentence ∃xP(x), our main proof strategy will be precisely to look for
an example that will demonstrate it. If we can find an instance t for which P(t|x) holds,* we
are then entitled to conclude ∃xP(x).
Existential Generalization (EG) is the name given to the rule of inference that captures
this mode of inference. It is schematized as follows:
P(t|x)
EG
∃xP(x)
To apply EG, therefore, we must do two things. We must locate a potential candidate t, and
we must show that it is, in fact, an instance having the requisite property: that P(t|x) is true.
Having done this, we are certainly entitled to conclude ∃xP(x), for we actually know more
than we are claiming in the existential sentence: we know a definite instance that satisfies
statement P(x). EG is thus a sound rule of inference.
Just as it was helpful to think of a universal sentence as being a conjunction of sorts, so
here it helps to think of an existential sentence ∃xP(x) as being a generalized disjunction.
Saying that there is some object satisfying P(x) is analogous to asserting a grand disjunction
containing one disjunct P(t|x) for each t in the universe of discourse: “either this object
satisfies statement P, or this other one does, or . . . .” Taking this view of existential sentences,
EG is a PL rule that parallels and extends Add : knowing one of the disjuncts for certain, we
may infer the whole disjunction, which is weaker.
We’ll illustrate this rule of inference first of all with some simple mathematical examples. To
show in geometry that given a line l and a point P lying off l there is a line m parallel to l that
passes through P , you merely demonstrate how to construct such an m by means of congruent,
alternate interior angles. To demonstrate that continuity does not entail differentiability in
* Recall that P(t|x) indicates the formula gotten by substituting t everywhere for x in the formula P(x).
2.4 -1
elementary calculus, that there are continuous functions that are not differentiable, you hunt
up a (counter)example. The most common example is the function f (x) = |x|, which is
continuous at x = 0 but not differentiable there. In both of these cases, EG is used to establish
the existential result, though in ordinary mathematical arguments the role of EG remains below
the surface. A mathematical proof usually quits as soon as an instance is found that satisfies
the formula or requirement.
Existential Generalization generates existential sentences; the main logical operator is the
existential quantifier. EG cannot be directly applied to parts of sentences. However, it can be
and often is used in the intermediate steps of a deduction to prove an existential component
sub-sentence. For example, consider the proposition that an odd number m times an odd
number n is odd. This sentence is the universal closure of the conditional sentence, “If m is
odd and n is odd, then m · n is odd.” A proof of this result proceeds according to CP , assuming
that both m and n are odd, and showing on that basis that m · n is odd. This latter conclusion
requires EG; here the exemplar generalized on is non-specific because m and n are. For, since
m and n are odd, according to the definition there are some particular numbers j and k such
that m = 2j + 1 and n = 2k + 1. Thus m · n = (2j + 1) · (2k + 1) = 2(2jk + j + k) + 1, using the
rules for identity and the laws of arithmetic* to justify our calculations. Since 2jk + j + k is
an integer, it yields an instance that shows that m · n is odd. Thus, by EG, we conclude that
there is an integer i such that m · n = 2i + 1, and so m · n is odd according to the definition of
being odd.
The following two examples begin to illustrate the use of EG in a formal setting. The first
one depends on the fact that every structure of the language is built upon a non-empty domain,
so there must be elements that have the desired property (see also Example 2.2-7).
z EXAMPLE 2.4 - 1
Show that ∀xP x − ∃xP x.
Solution
The following proof diagram establishes the claim.
1
∀xP x
Prem
2
3
Pa
∃xP x
UI 1
EG 2
The next example illustrates a subtle but important point about EG that is easily misunderstood. In passing from P(t|x) to ∃xP(x), we need not replace every occurrence of t in
the formula P(t|x) by x. In fact, doing this may not yield P(x), for the existential formula
we want to prove may well itself contain an occurrence of the instantiating constant t. A few
moments’ thought should convince you that EG, as formulated above, validates generalizing
from a sentence containing a constant t to a sentence obtained from it by replacing any number
of occurrences of t with x and prefixing the quantifier ∃x (see Exercise 70). If we consciously
use the Backward Method of Proof Analysis and keep fixed in our mind what we are proving,
we will be able to identify the appropriate sentence P(x) and see how to apply EG.
z EXAMPLE 2.4 - 2
Show that ∀x(x < s(x)) − ∃x(x < s(0)). Here 0 is a constant, < is a binary relation,
and s is a function, possibly the natural number successor function.
* The laws of arithmetic involved are various universal sentences, such as the commutative and distributive laws. These
require to be instantiated via UI in order to be used formally in an argument such as the one given.
2.4 -2
Solution
The following proof diagram establishes our argument. Note that since we want to end up
proving ∃x(x < s(0)) via EG, our formula P (x) will be x < s(0), not 0 < s(x) or x < s(x).
Thus in moving to step 3 we only replace the first occurrence of 0 by x.
[To elaborate: from step 2 we can also conclude (∃x)(0 < s(x)) or (∃x)(x < s(x)) using the
other options mentioned for P (x). We don’t, though, because that’s not what is wanted.
How a sentence is existentially generalized depends upon the desired conclusion.]
1
∀x(x < s(x))
Prem
2
3
0 < s(0)
∃x(x < s(0))
UI 1
EG 2
EG and the Method of Analysis
As you work mathematical proofs that require Existential Generalization, you will soon
learn that applying the rule may require some mathematical ingenuity. Sometimes locating
the instance t will pose no problem; it will be relatively close at hand. But other times it
may be extremely difficult to locate the instance, requiring a good deal of familiarity with
the field. The existence of everywhere-continuous, nowhere-differentiable functions is a good
example of this; before mathematicians generated such functions in the mid-nineteenth century,
people thought continuous functions had to be differentiable nearly everywhere. Generating a
(counter)example took some ingenuity. Often, then, most of the work going into such a proof
happens behind the scenes and consists of figuring out which object might be a candidate for
the instance, even though, strictly speaking, that isn’t part of the deduction. Nevertheless,
while locating the instance may lack deductive status, a presentation that fails to show where
the particular instance comes from is highly unsatisfying.
Consider the following illustration from elementary algebra. Suppose you are asked to show
that 2x2 +x−6 = 0 has a positive solution. You might just exhibit a positive number satisfying
the equation. Showing 2(3/2)2 + 3/2 − 6 = 0 gives the solution x = 3/2 to the problem.
However, this demonstration fails to show where the number 3/2 materialized from. Maybe
you made a lucky guess or got the answer from someone else, or maybe you used a bad solution
procedure but still got the right answer. Such a minimal approach to problem solving, therefore,
may be greeted with more suspicion than enthusiasm by a mathematics instructor, who usually
wants to see all your work. In solving an equation you ought to provide some supporting
documentation to indicate how you arrived at the solution.
To present a solution to this equation, therefore, you should probably write down something
like the following sequence of steps, bringing in the constraint that the solution must be positive
just before the conclusion.
2x2 + x − 6 = 0
(2x − 3)(x + 2) = 0
2x − 3 = 0 ∨ x + 2 = 0
x = 3/2
∨
x = −2
But x > 0, so x = 3/2.
And, to satisfy a picky prof, you should also include a check for your answer:
2(3/2)2 + 3/2 − 6 = 0 and 3/2 > 0.
Let’s analyze the proof-character of this argument. If all we want to do is establish the
existence of a positive solution to 2x2 + x − 6 = 0, then the solution method can be considered
2.4 -3
part of the proof only if we take the notion of proof in its broadest sense, as discourse that
includes whatever is needed to convince someone of the argument’s validity. Properly speaking,
however, the proof, as we’ve been using the term in this book, consists primarily of the step
that is often left off and is somewhat denigrated by being called “the check” when it is present.
Since 3/2 is positive and satisfies the equation, it is the instance that establishes the existence
of a positive solution via EG.
On the other hand, it can be argued that solving an equation consists in finding all those
values that satisfy it, so we are actually looking for a necessary and sufficient condition on
the variables involved. Viewed in this way, the steps preceding the check provides a necessary
condition and so comprises half the deduction. In terms of solving the equation, they exhibit
the discovery process that unearthed the various candidates to be checked out as solutions of
the equation. But they also demonstrate that if 2x2 + x − 6 = 0 and x > 0, then x = 3/2. To
finish, we still need a conditional going in the other direction: if x = 3/2, then 2x2 + x − 6 = 0
and x > 0. This sufficient condition holds if the solution checks out.
The discovery process in algebra (solving an equation) and elsewhere is extremely interesting
from both a logical and an epistemological point of view. Essentially, in making such an
argument, we assume that we have or know that there is a solution (denoted by x), and we
then argue in a logically forward direction to determine what its value must be, thus subjecting
the algebraic expressions involving x to the laws of arithmetic. We are thus assuming the truth
of the statement to be proved , and seeing what must follow from the existence of such an x. If
there is such an x, then it must be 3/2. We then turn around to see whether 3/2 actually is an
instance. For this, we don’t retrace our steps; we just substitute it in to check that it satisfies
the equation.
This procedure — assuming that the proposition to be proved is true and arguing forward to
determine what follows from it — is known as the Method of Analysis. Stated baldly, it seems
either illegal or wrong-headed: “Why is it assumed that the proposition is true, when we still
need to show that it is? Isn’t that a blatant case of circular reasoning?” But when you keep
the logic in mind, remembering that the final conclusion only gives you a necessary condition
for the result, you realize that the Method of Analysis is not the end of the matter, but only
a fruitful beginning. Analysis is a powerful tool of mathematical discovery, but it is not yet a
(full) proof. The Method of Analysis must be followed up by the Method of Synthesis; that is,
by a deductive proof of the result. Synthesis begins where analysis leaves off. You must take
the information gotten by analysis and use it to construct a proof of the proposition. This may
mean trying to reverse the steps gotten, showing that the conditions found to be necessary are
also sufficient (or can be modified in certain ways to become sufficient ones), or it may mean
using the conclusion, such as in the above example, to show that it satisfies the given formula.
The Method of Analysis has been used throughout the history of mathematics as a tool of
discovery. Its value was first recognized in ancient Greek geometry. There certain auxiliary
constructions based upon assuming the correctness of the result to be proved were used in order
to get some idea of how a reversed proof might proceed and what it might involve. The Method
of Analysis was later reinterpreted and institutionalized in elementary algebra by Viète (1591),
Descartes (1637), and Fermat in their works on the theory of equations. Here the Method
of Analysis drives the underlying process of solving equations, though the solution is usually
checked directly rather than by reversing the argument.
Eliminating Existential Quantifiers: Existential Instantiation
The elimination rule for existentially quantified sentences, Existential Instantiation (EI ), is
the most complex rule of all the quantifier Int-Elim Rules, though UG rivals it in difficulty in a
formal setting. Once again, we will keep our discussion fairly informal to avoid technicalities as
much as possible, concentrating on becoming familiar with the mathematical proof technique
based upon EI .
2.4 -4
EI proceeds as follows. If an existential sentence, such as ∃xP(x), is true under an interpretation, then we know that some object in the associated universe of discourse satisfies
statement P(x). Thus, even though we may not know precisely which object it is, we can
temporarily use an arbitrary unassigned constant symbol a to name this object. Our argument
may now be continued in terms of a: assuming P(a|x), we use it as a supposition for proving
some result Q. That result follows from the given existential formula, because we do have such
an x; all we don’t know is which particular one it is.
Formally speaking, EI proceeds by taking P(a|x) as a supposition for a subproof that is
preceded by the existential sentence ∃xP(x). The conclusion Q may then be inferred from
∃xP(x) on the basis of its having been proved from P(a|x), with the proviso that a denotes
an arbitrary element of the structure. Schematically, we have the following:
∃xP(x)
P(a|x)
EI
Q
Q
The requirement that a is completely arbitrary means we must be sure a does not denote
some specific object in the universe of discourse about which we have already made some
assertion. Thus a should not occur in any line of the outer proof preceding the supposition
P(a|x), including that of ∃xP(x) itself. In this way no unwarranted assumptions about the
instance will be smuggled into the argument. We must argue solely in terms of a being a
completely generic element of the domain satisfying statement P(x). Also, a should not occur
in our conclusion Q, because a is only a temporary name for the instance; we are not asserting
that a actually is the instance and so should not treat it as such in the main part of the proof.
Any result Q that is deduced from P(a|x) and that does not itself mention a can be exported
from the subproof back into the main proof as a consequence of the original existential sentence.
EI is a complex suppositional form of argument, but we can gain additional intuition on it
by considering existential sentences once again as disjunctions. ∃xP(x) then says that either
this object of the universe of discourse satisfies statement P(x), or that one does, and so on.
Not knowing which one really does, we can consider each case in turn and try to prove the
result from assuming that it might be the one. If each element cooperates, we can then use a
generalization of Cases to conclude the desired result from the grand disjunction. Naturally,
we won’t be able to consider each element individually if the universe of discourse is large,
because then the disjunction is unmanageable. So what we do instead is take a typical, generic
element a, and, supposing only that it satisfies P(x) (and anything else that holds for an
arbitrary element in the universe of discourse), show that it does indeed give the result wanted.
This proves Q on the basis of assuming any case P(a|x) whatsoever. This is exactly what EI
permits. Just as in applying Cases, once we have proved Q from each alternative, we may
conclude Q as our conclusion in the main part of the proof.
EI is a sound rule of inference. If the original existential sentence ∃xP(x) is true, Q will be
as well. To show that this is always the case is quite involved; we will accept it as intuitively
sound based on the explanations we have given of its meaning and on the analogy between it
and Cases, which is sound. Note, however, that to apply EI correctly, the sentence preceding
the subproof must be an existential sentence overall and the premise of the subproof must be
an instance of that sentence; otherwise invalid or inconclusive arguments can result.
To illustrate the use of EI , suppose we want to prove some result about an odd number n.
We would start with the definition, as before: n is odd iff there is some integer k such that
n = 2k+1. To use this existential defining condition, we then let some constant symbol k0 stand
for the integer asserted to exist and argue in terms of this value: n = 2k0 + 1. Any conclusion
2.4 -5
that follows from this equation will then also follow from the existentially quantified sentence
by EI . In informal mathematical arguments, we use the same symbol k both as a variable and
as a constant representing the instance; the existential quantifier is merely dropped off and the
argument continues. A strictly formal approach uses different symbols for the variable and the
arbitrary constant denoting the instance, as we just indicated.
We will now use EI in two formal examples. These will show you how to cite EI and how
to label the supposition it generates in a proof diagram. The first one formulates a so-called
immediate inference from Aristotelian Logic (see Example 2.2-11); the second one deduces the
valid interchange between existential and universal quantifiers (see Example 2.2-15). Note that
when your premise set includes both existential and universal sentences (as in Example 3),
you should invariably first instantiate the existential sentence so that you know what value to
instantiate the universal sentence to.
z EXAMPLE 2.4 - 3
Show that ∀x(P x → Qx), ∃xP x − ∃xQx.
Solution
The following proof diagram establishes this claim.
1
2
∀x(P x → Qx)
∃xP x
Prem
Prem
3
Pa
Spsn for EI
4
5
6
7
∀x(P x → Qx)
P a → Qa
Qa
∃xQx
Reit 1
UI 4
MP 5, 3
EG 6
8
∃xQx
EI 2, 3-7
z EXAMPLE 2.4 - 4
Show that ∃x∀yP xy − ∀y∃xP xy.
Solution
Let’s first use the Backward-Forward Method of Proof Analysis to see how to proceed. To
obtain our conclusion, it suffices to get ∃xP xb for an arbitrary b. This can be gotten from
P ab, if we can find an instance a that satisfies the formula. We therefore try for P ab from
the given premise. Since the premise is an existential sentence, we apply EI to begin. The
resulting proof diagram establishes the claim.
1
∃x∀yP xy
2
∀yP ay
Spsn for EI
3
4
5
P ab
∃xP xb
∀y∃xP xy
UI 2
EG 3
UG 4
6
∀y∃xP xy
Prem
EI 1, 2-5
This example gives the only way in which existential and universal quantifiers can be
interchanged; the other direction is invalid, as we saw earlier. Nevertheless, it may not be
immediately apparent why a deduction similar to the one just given won’t prove this wrong
direction. We will leave it as an exercise for you to puzzle out why such a deduction is invalid
(see Exercise 6).
2.4 -6
Simplifying Negated Quantified Sentences
UG and EG are the main ways to prove universal and existential sentences. Both of them
are direct proof strategies. However, as always, there is also an indirect proof strategy available,
namely, Proof by Contradiction. To show that a quantified sentence holds, we can use NE :
assuming the sentence’s opposite for the sake of argument and deducing from it a pair of
contradictory sentences. Since this is absurd, we then conclude the original sentence instead.
In order to make use of negated quantified sentences as premises, we need to know what can
be validly concluded from them. We will give two Replacement Rules that cover this, beginning
with one for negated universal statements.
Suppose, then, that ¬∀xP(x) is true. This means not all elements of U satisfy statement
P(x). Thus, at least one element of U must not satisfy statement P(x); that is, ∃x(¬P(x))
is true. This sentence turns out to be equivalent to the original negation. For now suppose
∃x(¬P(x)) is true. Then some object of U does not satisfy statement P(x). Hence not all
objects satisfy P(x); and so ¬∀xP(x) is true. These two sentences are therefore logically equivalent. This gives us the first of the following Replacement Rules, known as Universal Negation
(UN ). The second one shows how to negate an existential sentence; an argument similar to
what we just gave shows that Existential Negation (EN ) is also sound (see Exercise 11).
UN
EN
¬∀xP(x) : : ∃x(¬P(x))
¬∃xP(x) : : ∀x(¬P(x))
The soundness of these Replacement Rules can be shown more formally as well by deducing
each of the equivalents from its mate using only the Int-Elim Rules for quantified sentences
and Proof by Contradiction (see Exercises 12-13). This means that we do not really have
to incorporate them separately into our Natural Deduction System for PL. We will do so
regardless, however, for the equivalences they give make it possible to negate sentences readily.
Another way we could argue in favor of these rules is by appealing to the intuitive meaning
of universal and existential sentences as generalized conjunctions and disjunctions. From this
point of view, UN is simply a generalization of one of the sound DeM Replacement Rules
(negated conjunctions are disjoined negations) while EN is the other DeM rule (negated disjunctions are conjoined negations). These connections, as well as others pointed out along the
way, demonstrate the fruitfulness of the intuitive extended-SL viewpoint we have adopted in
thinking about universal and existential sentences.
Using the above Replacement Rules in the forward direction, we are now able to deduce
consequences from a negated quantified sentence, whether it appears as the supposition of a
Proof by Contradiction or as a sentence anywhere in a proof. These rules can also be used in
the backward direction to deduce a negated quantified sentence from its equivalent, passing
from the right hand side of the rule to the left hand side. Both forward and backward maneuvers are quite useful in constructing deductions involving negated universal and existential
sentences. Moreover, as is the case with all Replacement Rules, we are permitted to replace
either equivalent by the other one anywhere within a sentence. Thus, the scope of these rules is
considerably broader than merely replacing a whole sentence by its equivalent. However, using
these rules as bidirectional inference rules on entire sentences is probably their most useful
function.
Using these two rules, along with substitution of equivalents, which does not change the
truth values of the sentences, we can negate and simplify more complex sentences. The symbolic pattern common to both negations is the following: to simplify a negated quantified
sentence, move the negation sign past the quantifier and change the given quantifier to the
other one. Applying the quantifier to the inside formula and using equivalents associated with
the Replacement Rules for SL, we may be able to simplify the sentence further.
2.4 -7
z EXAMPLE 2.4 - 5
Negate the sentence ∀x(P x → Qx) and simplify the result.
Solution
By UN, ¬∀x(P x → Qx) = ∃x(¬(P x → Qx)).
This gives us an equivalent sentence, but we can simplify it further, making it an existentially quantified conjunction via Neg Cndnl .
This gives ∃x(P x ∧ ¬Qx) as our final sentence.
As noted in Section 2.1, many mathematical sentences are most simply formulated by means
of restricted quantifiers. When such sentences are negated as above, their results agree with
what is done when the sentences are expressed in expanded form. This is shown in the next
example (see also Exercise 57).
z EXAMPLE 2.4 - 6
Negate the sentence (∀x ∈ P )Qx and compare the result to that of the last example.
Solution
We first note that (∀x ∈ P )Qx is an alternative way to formulate ∀x(P x → Qx), using a
restricted quantifier.
Negating restricted quantifiers as we do ordinary quantifiers yields (∃x ∈ P )(¬Qx), which
can’t be further simplified. In fact, this procedure is valid, for if it is not the case that all
x in P have property Q, then some x in P must fail to have property P .
Note from this that we do not negate the quantifier restriction on x; this only limits the
universe of discourse. You must instead negate what is being said about such x, the additional properties it has.
As you can verify, this final result is an abbreviated form for Example 5’s conclusion:
∃x(P x ∧ ¬Qx).
z EXAMPLE 2.4 - 7
Negate and simplify the following version of Playfair’s Euclidean Parallel Postulate: given
any line l and any point P not on l there is a line m through P that is parallel to l.
Solution
The negated sentence is symbolized in the following way (compare Example 2.1-12):
¬∀l∀P (P ∈
/ l → ∃m(P ∈ m ∧ m k l)).
We can now pass to the following logical equivalents:
∃l∃P (¬(P ∈
/ l → ∃m(P ∈ m ∧ m k l)))
via UN
∃l∃P (P ∈
/ l ∧ ¬∃m(P ∈ m ∧ m k l))
via Neg Cndnl
Translating this sentence back into decent mathematical English, we have the following:
there is a line l and a point P not on l so that there is no line m passing through P that
is parallel to l.
We have worked the sentence to a certain point, but we might go still further, obtaining
the following:
∃l∃P (P ∈
/ l ∧ ∀m(¬(P ∈ m ∧ m k l)))
via EN
∃l∃P (P ∈
/ l ∧ ∀m(P ∈
/ m ∨ m 6 k l))
via DeM
∃l∃P (P ∈
/ l ∧ ∀m(P ∈ m → m 6 k l))
via Cndnl
Translating this final sentence back into good mathematical English, we have the following:
there is a line l and there is a point P not on l so that all lines m passing through P are
not parallel to l.
This example raises the issue of which sentence should be taken as the best negation of the
original. Generally, if the negation contains a quantified clause that is an existential sentence,
this clause should be a quantified conjunction; while if it contains a universal clause, it should
2.4 -8
be a quantified conditional (see Section 2.1). That is why we chose to use Cndnl on the second
to last sentence in giving our final simplification.
In mathematical practice you will usually negate a sentence informally to get another
sentence without ever passing through a series of formal sentences like the ones above. When
that is done, the problem of which negation to choose among several is less likely to arise.
Instead, the problems are a bit different. Negating a mathematical sentences is generally done
by getting a good understanding of what it says and then trying to formulate a sentence that
says the exact opposite. But if you are uncertain as to what the negation of a given sentence
really is, you may want to formalize the sentence and negate it as we just did, translating back
to good mathematical English at some point and asking whether such a sentence sounds like
the logical opposite of what was originally given. If not, a check on the translation procedure
or the negation process is in order. If everything has been done correctly (a crucial issue
when using the formal apparatus of PL), then the sentence obtained will be a negation of the
original. Using typed variables and restricted quantifiers where possible will usually help keep
the formulation and negation process manageable.
Deductions Using Replacement Rules
We will now look at two examples that use UN and EN in formal deductions. These will
also involve the Int-Elim Rules for existential sentences.
z EXAMPLE 2.4 - 8
Show that ∀x(P x ∧ Qx), ¬∀x(P x ∧ Rx) − ¬∀x(P x → Rx).
Solution
To conclude ¬∀x(P x → Rx), we will prove its equivalent, ∃x(¬(P x → Rx)). Due to Neg
Cndnl , it suffices to prove ∃x(P x ∧ ¬Rx). We will do this by finding an instance a such
that (P a ∧ ¬Ra) holds. The first premise easily yields P a; the second one must be needed
to obtain ¬Ra. UN applied to this premise gets us started. The following proof diagram
implements this plan.
1
2
∀x(P x ∧ Qx)
¬∀x(P x ∧ Rx)
Prem
Prem
3
∃x(¬(P x ∧ Rx)
UN 2
4
5
6
7
8
9
10
11
12
13
14
¬(P a ∧ Ra)
Spsn for EI
¬P a ∨ ¬Ra
∀x(P x ∧ Qx)
P a ∧ Qa
Pa
¬Ra
P a ∧ ¬Ra
∃x(P x ∧ ¬Rx)
∃x(¬(P x → Rx))
¬∀x(P x → Rx)
DeM 4
Reit 1
UI 6
Simp 7
DS 5, 8
Conj 8, 9
EG 10
Neg Cndnl 11
UN 12
¬∀x(P x → Rx)
EI 3, 4-13
The Replacement Rules we’ve adopted provide us with two procedures for deducing negated
quantified sentences. In order to refute ∀xP(x), we must show ∃x(¬P(x)); that is, we must
show that there is an object — a counterexample — that fails to satisfy the sentence. We can
2.4 -9
do this directly via EG, producing the counterexample, or indirectly, by showing that its
negation is absurd. In the last example, we generated a counterexample directly. An alternative
approach would be to use indirect proof (NI ). The next example proves a negated universal
sentence of arithmetic by showing directly that 0 is a counterexample.
z EXAMPLE 2.4 - 9
Show that ∀x(0 6= s(x)) − ¬∀y∃x(y = s(x)), where 0 is a constant and s is a function.
Solution
In order to prove the conclusion, we will deduce its equivalent, ∃y∀x(y 6= s(x)). Since this
is an existential sentence, we will try to locate an instance a that makes ∀x(a 6= s(x)) true.
But this is immediate, given the premise: we take a = 0. Armed with this Backward Proof
Analysis, it is now very easy to construct the formal proof diagram.
1
∀x(0 6= s(x))
Prem
2
3
4
∃y∀x(y 6= s(x))
∃y(¬∃x(y = s(x)))
¬∀y∃x(y = s(x))
EG 1
UN 2
EN 3
Uniqueness Assertions in Deductions
Our system of PL has three quantifiers: ∀, ∃, and ∃! . The first two are the primary ones; the
third can be defined using the other two, though we have treated it as primitive (see Section 2.1).
Given the fixed interpretations of these quantifiers, it is clear that the sentence ∃! xP(x) means
the same as/is logically equivalent to the conjunction ∃xP(x) ∧ ∀x∀y(P(x) ∧ P(y) → x = y)
(see Exercise 2.1-67). We will thus take as our final Replacement Rule the following one, which
we will call Unique Existence (Uniq Exis).*
U niq Exis
∃!xP(x) : : ∃xP(x) ∧ ∀x∀y(P(x) ∧ P(y) → x = y)
This last sentence can be readily separated into an existential sentence and a universal sentence
by means of Simp, or recombined from these two components by means of Conj . This makes
it possible to handle unique-existence sentences in deductions by means of the other rules of
inference for ∀ and ∃; no additional Int-Elim Rules need to be adopted.
The elimination procedure for the quantifier ∃! is much as it was for the quantifier ∃. Given
the sentence ∃!xP(x), you can first conclude that ∃xP(x) by means of Uniq Exis and Simp.
Then, using EI , you can suppose that a is the instance guaranteed and proceed to argue further
in terms of it. Whatever is proved by using P(a|x) will then follow from the original sentence.
In informal proofs, of course, you will not go through such an elaborate process. You
will simply go from the unique existence statement directly to supposing that a is the unique
instance satisfying P(x), bypassing Uniq Exis and Simp as being obvious and not worth noting.
In order to prove a uniqueness assertion ∃!xP(x), you must first deduce the two conjuncts
∃xP(x) and ∀x∀y(P(x) ∧ P(y) → x = y). You must thus show that there is such an object x,
and you must show that if x and y denote any objects satisfying statement P(−), then they
denote the same object.
Naturally, either of these conjuncts can be demonstrated first. The approach that seems
most intuitive is to first show that there is such an object, and then to show that it is unique.
Why try to show something is unique if you don’t even know that it exists?
* In a more formal treatment of PL, this equivalence might be taken as the definition of the uniqueness quantifier. Our
approach is to adopt it as a (minor) Replacement Rule. See also Exercises 32-33.
2.4 -10
Nevertheless, there are many times when this common-sense approach turns out to be the
“wrong” course of action to take. Often, by supposing that x and y denote objects satisfying
statement P(−), you not only find out that there is at most one object, but also what that
object must be. In fact, you may only need to suppose that x is an object satisfying P(−)
(the Method of Analysis again) in order to determine what the object must be. A necessary
condition of x satisfying P(−) may be that it be such and such, say, some value t. If this
happens, then for y to satisfy P(−), it too must be equal to t. From x = t and y = t, you can
conclude that x = y, which shows the uniqueness result you want (the second conjunct).
The Method of Analysis, then, can sometimes give you the uniqueness conjunct in addition
to an instance to try out for the existence clause. Applying EG, you may then be able to
show that this object t actually satisfies P(−), which gives you what you need to complete the
uniqueness proof.
An alternative approach to proving the universally quantified uniqueness conjunct is to
prove it by Contradiction. This usually goes as follows. Supposing that there are two distinct
objects satisfying P(−) (the negation of the second conjunct, via UN and Neg Cndnl ), you
arrive at a contradiction. You then conclude that there is at most one object.
While this second approach is legitimate, often such proofs lack elegance and can be immediately converted into direct proofs of the first type mentioned above, especially if what is
contradicted is that these objects are distinct. In other words, supposing that both x and y
satisfy P(−), you arrive at the conclusion that the objects are identical. But this is just what
you need to draw the original conclusion via CP and UG. If this happens and the original
supposition that the objects were distinct does not enter into the subproof except to contradict
the final conclusion that they are not distinct, the argument is better reformulated in the direct
way mentioned above.
To illustrate how uniqueness assertions are proved, we will show that the additive identity
for real-number arithmetic is unique. We will do this informally, leaving a longer formal version
for an exercise (Exercise 81).
z EXAMPLE 2.4 - 10
Show that 0 is the unique additive identity for ordinary arithmetic:
i.e., show that ∃!z∀x(x + z = x ∧ z + x = x).
Solution
To begin our proof, we first note that 0 is such an instance: given any real number x,
x + 0 = x and x = 0 + x.
To show uniqueness, we next suppose that z denotes an object that is a “zero” or additive
identity. Then z + 0 = z (since 0 is an identity) and z + 0 = 0 (because z is an identity).
Equating equals, z = 0. Thus the identity is unique.
An analogue of the argument just given is used repeatedly in abstract algebra to show the
uniqueness of identities for groups, rings, fields, vector spaces, etc. If a binary operation has
an identity, then it must be unique according to such an argument. Proofs for the uniqueness
of inverses proceed similarly (see Exercise 71).
Comparing Formal and Informal Proofs: Why Study Logic?
We have presented the deductive systems of Sentential Logic and Predicate Logic in an
abstract partly-formal fashion, using letters for sentences, for constants, predicates, and so on,
both in formulating the rules themselves, and in working examples and exercises. This was
done for two reasons. We’ve already mentioned one of these in Section 2.2: the science of logic,
which studies patterns of valid inference and deductive argument, is by nature an abstract
2.4 -11
discipline, applying to sentences regardless of the meaning or subject matter and requiring the
possibility of multiple interpretations in order to be developed.
There is another reason for our formal approach as well: most mathematical arguments
are tremendously complex from a logical point of view. They are thus difficult to analyze and
construct in detail until after you have become familiar with all the various inference rules.
We have occasionally used mathematical arguments to illustrate or practice different proof
strategies, but this was more the exception than the rule. They were included to remind you at
regular intervals that the ultimate goal of our study is to learn to read and write mathematical
proofs.
Now that we have the entire arsenal of PL’s deduction system at our disposal, we are finally
at the point where we can begin to tackle genuine mathematical arguments. However, as you
will notice from the final example given below, even the simplest mathematical argument may
become unbearably long and complicated when all its logical detail is exposed; informal proofs
generally take many things for granted that are not explicitly stated.
Now, the reason for our study of logic is definitely not to encourage you to swamp your
mathematical proofs with intricate logical detail. Rather, logic is being studied to accomplish
two other things with respect to mathematical proofs.
The first thing our study of logic is meant to do is convince you that full logical detail
can be provided if it is wanted , that it is possible in principle to make deductions completely
rigorous from a logical standpoint. It is thus incorrect to say, as some mathematicians do, that
total rigor is unattainable in mathematics, citing as their evidence the historical fact that each
era of mathematicians seems to go beyond the preceding one in what it counts as axiomatic and
as a valid argument. On the contrary, over the past century and a half, logic has progressed
dramatically, to the point where the logical structure of mathematical sentences can now be
fully analyzed, the various rules of inference used in valid deductive arguments recognized and
cataloged, and the nature of mathematical argument itself clarified and codified by means of
a natural deduction system. Mathematical logicians now believe, with pretty good evidence,
that a system of Predicate Logic like the one we have presented is well equipped to handle any
sort of mathematical proof. This is an achievement of great significance; its value has yet to
be fully appreciated by mathematicians.
But while rigor is now attainable in mathematical proofs, this certainly does not mean
that it should be aspired to. For accomplishing certain foundational tasks or for thoroughly
mechanizing the process of deduction as is done for automatic theorem-proving in computer
science, this degree of rigor may be important. However, it is certainly not desirable for practicing mathematics and computer science students to incorporate every logical detail into their
arguments. After a while, a point of diminishing returns is reached in ordinary argumentation,
where there seems to be a trade-off between logical rigor and clarity. Our method of constructing natural deduction proofs helps us gain an overview of a deduction since the argument is
structured by a systematic use of component subproofs and since tautologies are rarely pulled
in from outside the proof, but given the limits of human comprehension, the accumulation of
logical details in a long proof eventually begins to obscure just what it is that the proof is
doing.
This being the case and given the overwhelming logical complexity of most mathematical
arguments, why have we spent so much time learning the rules of logical inference for SL and
PL? Well, because these rules underlie the various proof techniques that are used day in and
day out for deductive argumentation. Learning them in a simpler formal context, students
can begin to internalize the reasoning they codify and learn to distinguish between valid and
invalid arguments.
The second and more important thing that your study of logic is meant to do, therefore,
is teach you proof strategies that will help you construct your own mathematical proofs. You
should be familiar with the overall strategy of the Backward-Forward Method of Proof Analysis
and with using subproofs to obtain certain key results as intermediate steps in your proofs;
2.4 -12
and you should know just what specific proof strategies to try, based on your knowledge of the
logical form of the premises (or sentences already proved) and particularly of the conclusion.
The intention is that the logic you’ve learned will function quietly and unobtrusively, behind
the scenes, on reserve and ready for active duty when needed. To become familiar with each of
the basic proof techniques, we studied them in turn, separately and in some detail. As you go
on from here to become more proficient in creating mathematical proofs, such things should
become second nature to you. You will eventually forget what the strange terms Modus Tollens
and Disjunctive Syllogism refer to, and you will no longer be able to create formal proofs on
the spot. That’s just fine, so long as you have absorbed the general thrust of your study of
logic and are better able as a result to follow the various maneuvers in a proof constructed by
others and to map out a strategy for making your own proofs.
We will close this lesson with an example that illustrates the striking difference between an
informal proof and a formal one. The difference, you will see, is due in part to logical structure,
but it is also due to additional premises that are more or less taken for granted in an informal
presentation but cannot be left implicit in a formal deduction. We will first give the informal
version, then the formal one. Enjoy them both for what they are.
z EXAMPLE 2.4 - 11
Prove the transitive law for the “less-than” relation of ordinary arithmetic — if a < b and
b < c, then a < c — given the rules for calculating with real numbers and the definition
that x < y iff there is a positive real number z such that x + z = y.
Solution
Suppose a < b and b < c. Then there are positive real numbers p1 and p2 such that a+p1 = b
and b + p2 = c. Substituting, we get (a + p1 ) + p2 = c. This gives us a + (p1 + p2) = c.
Since p1 + p2 is itself positive, a < c.
z EXAMPLE 2.4 - 12
Rework the last example by means of a formal proof diagram.
Solution
We will first explain our approach and then append the formal proof.
We must begin by determining the premises; these are not automatically supplied for a
mathematical argument. We will also formulate the sentences involved using standard PL
symbolism.
Since the transitive law is meant as a general law, it must be a universal sentence. We
will formulate it using variables x, y, and z, saving a, b, and c as constants from which to
generalize: ∀x∀y∀z(x < y ∧ y < z → x < z).
The premises certainly include < ’s definition:
1) ∀x∀y(x < y ↔ (∃z)(z > 0 ∧ x + z = y)).
Analyzing the informal argument, it seems they must also include the facts that:
2) the sum of positive numbers is positive: ∀x∀y(x > 0 ∧ y > 0 → x + y > 0); and
3) addition is associative: ∀x∀y∀z((x + y) + z = x + (y + z)).
This turns out to be all that’s needed; the formal proof given below based on these premises
establishes the conclusion.
Although the deduction is long, it has been shortened a fair bit by combining multiple
universal instantiations (steps 7, 10, 18, 24, 29), the two existential instantiation subproofs
(steps 12-28), and the multiple universal generalizations (step 32). Without this, the proof
would have had quite a few more lines and even a fourth level subproof, since one existential
instantiation argument would need to be done inside the other one.
2.4 -13
Here, then, is the abbreviated formal deduction:
1 ∀x∀y(x < y ↔ ∃z(z > 0 ∧ x + z = y))
2 ∀x∀y(x > 0 ∧ y > 0 → x + y > 0)
3 ∀x∀y∀z((x + y) + z = x + (y + z))
Prem
Prem
Prem
4
a<b ∧ b<c
Spsn for CP
a<b
∀x∀y(x < y ↔ ∃z(z > 0 ∧ x + z = y))
a < b ↔ ∃z(z > 0 ∧ a + z = b)
∃z(z > 0 ∧ a + z = b)
b<c
b < c ↔ ∃z(z > 0 ∧ b + z = c)
∃z(z > 0 ∧ b + z = c)
Simp 4
Reit 1
UI 6 (2×)
BE 7, 5
Simp 4
UI 6 (2×)
BE 10, 9
5
6
7
8
9
10
11
12
13
z1 > 0 ∧ a + z1 = b
z2 > 0 ∧ b + z2 = c
Spsn for EI
Spsn for EI
14
15
16
17
18
19
20
21
22
23
24
25
26
27
z1 > 0
z2 > 0
z1 > 0 ∧ z2 > 0
∀x∀y(x > 0 ∧ y > 0 → x + y > 0)
z1 > 0 ∧ z2 > 0 → z1 + z2 > 0
z1 + z2 > 0
a + z1 = b
b + z2 = c
(a + z1 ) + z2 = c
∀x∀y∀z((x + y) + z = x + (y + z))
(a + z1 ) + z2 = a + (z1 + z2 )
a + (z1 + z2 ) = c
z1 + z2 > 0 ∧ a + (z1 + z2 ) = c
∃z(z > 0 ∧ a + z = c)
Simp 12
Simp 13
Conj 14, 15
Reit 2
UI 17 (2×)
MP 18, 16
Simp 12
Simp 13
Sub 20, 21
Reit 3
UI 23 (2×)
Sub 22, 24
Conj 19, 25
EG 26
28
29
30
∃z(z > 0 ∧ a + z = c)
a < c ↔ ∃z(z > 0 ∧ a + z = c)
a<c
31 a < b ∧ b < c → a < c
32 ∀x∀y∀z(x < y ∧ y < z → x < z)
2.4 -14
EI 8, 11, 12-27
UI 6 (2×)
BE 29, 28
CP 4-30
UG 31 (3×)
EXERCISE SET 2.4
Problems 1 - 4: Completing Deductions
Fill in the reasons for the following deductions.
*1. ∃xP x, ∀x(Qx → ¬P x) − ¬∀xQx
1 ∃xP x
2 ∀x(Qx → ¬P x)
3
Pa
4
5
6
7
8
∀x(Qx → ¬P x)
Qa → ¬P a
¬Qa
∃x(¬Qx)
¬∀xQx
9
¬∀xQx
*2. ¬∃x(P x ∧ Qx), ∃x(Rx ∧ Qx) − ¬∀x(Rx → P x)
1 ¬∃x(P x ∧ Qx)
2 ∃x(Rx ∧ Qx)
3
4
5
6
7
8
9
10
11
12
13
14
15
Ra ∧ Qa
¬∃x(P x ∧ Qx)
∀x(¬(P x ∧ Qx))
¬(P a ∧ Qa)
¬P a ∨ ¬Qa
Qa
¬P a
Ra
Ra ∧ ¬P a
∃x(Rx ∧ ¬P x)
∃x(Rx ∧ ¬P x)
∃x(¬(Rx → P x))
¬∀x(Rx → P x)
*3. ∃x(P xx ∨ Qx), ∀x∀y(P xy → Qy) − ∃xQx
1 ∃x(P xx ∨ Qx)
2 ∀x∀y(P xy → Qy)
3
P (a, a) ∨ Qa
4
5
6
7
∀x∀y(P xy → Qy)
∀y(P ay → Qy)
P aa → Qa
P aa
8
9
P aa → Qa
Qa
10
Qa
11
Qa
12
13
Qa
∃xQx
14 ∃xQx
2.4 -15
4. ∃x(P x ∧ ¬Qx) − ¬∀x(P x → Qx)
1 ∃x(P x ∧ ¬Qx)
2
P a ∧ ¬Qa
3
∀x(P x → Qx)
4
5
6
7
8
P a ∧ ¬Qa
P a → Qa
Pa
Qa
¬Qa
9
¬∀x(P x → Qx)
10 ¬∀x(P x → Qx)
Problems 5 - 8: Logical Implication and Conclusive Deductions
Determine whether the following claims of logical implication are true or false. Then determine whether
the deductions that are given are conclusive or inconclusive. Carefully point out each place where a rule of
inference is being used incorrectly.
*5. ∀xP x → ∃yQy, ¬∀yQy = ¬∃xP x
1 ∀xP x → ∃yQy
Prem
2 ¬∀yQy
Prem
3
4
5
6
7
8
EC
∀y(¬Qy)
¬∃yQy → ¬(∀x)P x
∀y(¬Qy) → ¬(∀x)P x
¬∀xP x
∃x(¬P x)
¬∃xP x
6. ∀x∃yP xy = ∃y∀xP xy
1 ∀x∃yP xy
2
3
4
5
6
∃yP ay
P ab
∀xP xb
∃y∀xP xy
∃y∀xP xy
UN 2
Conpsn 1
EN 4
MP 5, 3
UN 6
EN 7
Prem
UI 1
Spsn for EI
UG 3
EG 4
EI 2, 3-5
7. ∀xP x → ∃yQy, ¬∀yQy = ∀x(¬P x)
1
2
∀xP x → ∃yQy
¬∀yQy
Prem
Prem
3
4
P a → ∃yQy
P a → Qb
UI 1
Spsn for EI
5
6
7
8
¬∀yQy
¬Qb
¬P a
¬∃xP x
9
10
¬∃xP x
∀x(¬P x)
Reit 2
UI 5
MT 4, 6
EG 7
EI 3, 4–8
EN 9
2.4 -16
8. ∃xP x, ∀y(P y ∨ Qy) = ¬∀yQy
1 ∃xP x
2 ∀y(P y ∨ Qy)
3
4
5
6
7
Pa
P a ∨ Qa
¬Qa
∃y(¬Qy)
¬∀yQy
Prem
Prem
EI 1
UI 2
DS 4, 3
EG 5
UN 6
Problems 9 - 10: True or False
Are the following statements true or false? Explain your answer.
*9. Predicate Logic provides mathematicians with the tools for making their arguments perfectly rigorous.
10. The Method of Analysis is a procedure for proving that the solutions found in solving an equation satisfy
the equation.
Problems 11 - 13: Soundness of UN and EN
Work the following problems related to the Replacement Rules UN and EN.
11. Show the soundness of EN by explaining why the two sentence forms involved are logically equivalent.
12. Show without using UN or EN that ¬∀xP x − ∃x(¬P x).
13. Show without using UN or EN that ¬∃xP x − ∀x(¬P x).
Problems 14 - 25: Deductions
Deduce the following, using PL’s inference rules.
14. ∃x(P x ∧ Qx) − ∃x(P x ∨ Qx)
*15. ∀x(P x ∨ Qx), ∃x(¬P x) − ∃xQx
16. ∀x(P x ↔ Qx), ∃x(¬P x) − ∃xQx
17. ∃x(¬(P x ∧ Qx)), ∀xP x − ∃x(¬Qx)
18. ∃x(P x ∧ Qx) − ∃xP x ∧ ∃xQx
19. ∃xP x → ∃xQx − ∃x(P x → Qx)
20. ∀x(P x ∨ Qx) − ∀xP x ∨ ∃xQx
21. ∀xP x → ∀xQx − ∃x(P x → Qx)
22. ∀x(P x → Qx) − ∃xP x → ∃xQx
23. ∀x(P x ↔ Qx) − ∃xP x ↔ ∃xQx
*24. − ∀x(P x → Qx) ∨ ∃x(P x ∧ ¬Qx)
25. ∃xP x, ∀x(P x → x = b ∨ x = c) − P b ∨ P c
Problems 26 - 27: Aristotelian Logic and Derivations
Deduce the following arguments, which represent certain syllogistic forms from Aristotelian Logic, using any
of PL’s inference rules.
*26. ∃x(P x ∧ Qx), ∀x(Qx → Rx) − ∃x(P x ∧ Rx)
27. ∃x(P x ∧ Qx), ∀x(Rx → ¬Qx) − ∃x(P x ∧ ¬Rx)
Problems 28 - 31: Interderivability
Deduce the following interderivability results, using any of PL’s inference rules.
28. ∃x(P x ∨ Qx) − ∃xP x ∨ ∃xQx
29. ∃x(P x → Qx) − ∀xP x → ∃xQx
30. ∃x(P x ∧ ∀yQxy) − ∃x∀y(P x ∧ Qxy)
2.4 -17
31. ∀x(P x → P b ∨ P c) − ∃xP x → P b ∨ P c
Problems 32 - 33: Logical Equivalences
Give arguments for the validity of the following equivalence claims. Thus each of the quantifiers could be
defined in terms of the other one and negation if so desired.
*32. ∃xP(x) = ¬∀x(¬P(x))
33. ∀xP(x) = ¬∃x(¬P(x))
Problems 34 - 43: Logical Equivalences and Negations
Determine logical equivalents for the following negations. Simplify the results using negation Replacement
Rules from SL to put them in their most natural form.
34. ¬∃x(P x ∧ Qx)
35. ¬∃x(P x ∨ Qx)
36. ¬∃x(P x → Qx)
37. ¬∃x(P x ↔ Qx)
38. ¬∀x(P x ∧ Qx)
39. ¬∀x(P x ∨ Qx)
40. ¬∀x(P x → Qx)
41. ¬∀x(P x ↔ Qx)
42. ¬∀x∀y∃z(x + z = y)
*43. ¬∀x∀y∃m(N (m) ∧ mx > y)
Problems 44 - 50: Negating Definitions
On the supposition that in a given situation the antecedent of each biconditional does not hold, explain what
can be concluded (via NBE ); that is, negate the defining condition (the clause following the double arrow).
Put your answer in simplest form.
44. E(a) ↔ ∃k(a = 2k).
45. O(a) ↔ ∃k(a = 2k + 1)
46. a | b ↔ ∃m(a = mb)
*47. x < y ↔ ∃z(z > 0 ∧ x + z = y)
*48. S ⊆ T ↔ ∀x(x ∈ S → x ∈ T )
49. I(z) ↔ ∀x(x + z = x = z + x)
50. P = Q ↔ ∀v(v(P ) = T → v(Q) = T )
Problems 51 - 56: Definitions, Restricted Quantifiers, and Negated Sentences
Formulate the following definitions in the language of PL, identifying your universe of discourse and any
non–standard symbolism. You may use restricted quantifiers to help simplify your formulations. Once you
have your definition, negate it according to the instructions given above.
*51. A positive integer greater than 1 is prime iff it has no positive integral factors except itself and 1.
52. An ordered field F is Archimedean iff given any two members of F there is a positive integral multiple
of one of them that is greater than the other.
53. A number l is a least upper bound for a set S, written l = lub(S), iff l is greater than every element of
S and l is less than or equal to every number u that is greater than every element of S.
54. A function f is continuous at a iff for every positive real number there is a positive real number δ such
that whenever the absolute difference |x − a| is positive and less than δ, then the absolute difference
|f(x) − f(a)| is less than .
2.4 -18
EC
55. A number L is the limit of a sequence {xn }, written L = lim xn , iff for every positive real number n→∞
there is a natural number N such that the absolute difference |xn − L| is less than for all n greater
than N .
56. {xn } is a Cauchy Sequence iff for every positive real number there is a natural number N such that
the absolute difference |xm − xn | is less than for all m and n greater than N .
57. Negating Restricted and Unrestricted Existentials
a. Formulate the sentence (∃x ∈ P)Q(x) without using a restricted quantifier.
b. Negate and simplify both sentences given in part a. Then expand the sentence with the restricted
quantifier to show that the negation there is the same as the one containing the unrestricted quantifier.
Problems 58 - 69: Implication, Equivalence and Derivability
Determine whether or not the following pairs of sentences are equivalent. If they are, show it by giving a
deduction of each sentence from the other, using any of PL’s rules of inference. If they are not, tell whether
either sentence logically implies the other. If one follows from the other, show it by giving a deduction. If
neither one implies the other, give counterarguments to show why not.
[P should be taken to be a sentence that does not contain x as a free variable.]
58. ∀x(P ∧ Q(x)) = P ∧ ∀xQ(x)
59. ∃x(P ∧ Q(x)) = P ∧ ∃xQ(x)
60. ∀x(P ∨ Q(x)) = P ∨ ∀xQ(x)
61. ∃x(P ∨ Q(x)) = P ∨ ∃xQ(x)
62. ∀x(P → Q(x)) = P → ∀xQ(x)
63. ∃x(P → Q(x)) = P → ∃xQ(x)
64. ∀x(Q(x) → P) = ∀xQ(x) → P
65. ∃x(Q(x) → P) = ∃xQ(x) → P
66. ∀x(Q(x) → P) = ∃xQ(x) → P
67. ∃x(Q(x) → P) = ∀xQ(x) → P
68. ∀x(P ↔ Q(x)) = P ↔ ∀xQ(x)
69. ∃x(P ↔ Q(x)) = P ↔ ∃xQ(x)
70. Explain why EG validates generalizing from a sentence containing a constant a to a sentence obtained
from it by replacing any number of occurrences of a with x and prefixing the quantifier ∃x. Hint:
choose P(x) judiciously.
Problems 71 - 76: Informal Deductions Involving Quantifiers
Work the following mathematical problems using informal paragraph proofs instead of formal deductions.
71. Show by means of an informal argument similar to that of Example 10 that the additive inverse of any
real number is unique; i.e., show that for any given real number a there is exactly one real number a
such that a + a = 0 = a + a.
72. Using the premises identified in Example 12 along with the commutative law for addition, construct an
informal proof of the following proposition: if a < b and c < d, then a + c < b + d.
73. Pigeonhole Principle
a. A wedding party is given for selected friends and family of the bride and groom. Show that if 100
people attend in all, including the bride and groom, at least two of them must have the same number
of acquaintances among all those present. [Modified from an example of Paul R. Halmos in The
Thrills of Abstraction, Two Year College Mathematics Journal (Sept 1982), p. 248.]
b. Show that if a is an odd integer, then there is a positive integer b such that a | 2b −1. Hint: Consider
differences. [Taken from the Problem Corner in the January 1990 issue of Quantum, pp. 33, 53.]
2.4 -19
74. Prove the following special case of the general result that says that polynomials of different degrees must
represent different functions: x2 6= mx + b for any choice of m or b. That is, prove the proposition
¬∃m∃b∀x(x2 = mx + b). Give an informal proof, but point out where you are using rules of inference
from PL. Hint: prove this result by contradiction, using values of x to specify what m and b must be.
75. Give an informal argument to show that there is exactly one prime number that occurs as a member of
two different twin primes. (Twin primes are primes that are nearly identical, differing by two.) Hint:
to prove this, first look at some such triplets, whether primes or not, and note that a particular prime
factor always crops up for one of the numbers.
76. Assuming that every real number r has a cube root (which is proved using the Intermediate Value
Theorem for continuous functions), give an informal argument to show that it is unique. Point out the
main rules of inference that your argument depends upon. Hint: use what you know about factoring
the difference of two cubes, x3 − r 3 , and the quadratic formula.
Problems 77 - 82: Formal and Informal Mathematical Proofs
Construct deductions for the following mathematical results, as instructed. Take Examples 11 and 12 as
models for how much logical detail to supply in your deductions.
*77. Prove the following argument in two ways, first by giving an informal argument, and secondly by giving
a formal argument: All rational numbers are algebraic. All transcendental numbers are not algebraic.
Some real numbers are transcendental. Therefore, some real numbers are irrational.
78. Give a formal proof of the proposition ∀x∀y(E(x) → E(xy)), which states that multiples of even integers
are even. You may assume as premises the definition for being even, ∀x(E(x) ↔ ∃k(x = 2k)), and the
associativity law for multiplication, ∀x∀y∀z((xy)z = x(yz)).
79. Using the definition for divisibility of integers, ∀x∀y(x | y ↔ ∃m(y = mx)), formally prove that if a | b
and b | c, then a | c. You may assume any of the following laws of arithmetic as premises: the associative
laws ∀x∀y∀z((x + y) + z = x + (y + z)) and ∀x∀y∀z((xy)z = x(yz)); the commutative laws ∀x∀y(x + y =
y + x) and ∀x∀y(xy = yx); and the left and right distributive laws ∀x∀y∀z(x(y + z) = xy + xz) and
∀x∀y∀z((x + y)z = xz + yz).
EC
80. Using any laws of algebra (see Exercise 79, for example) plus the definition for being odd, ∀x(Ox ↔
∃y(x = 2y + 1)), construct a formal proof of the fact that the product of two odd numbers is odd. The
argument is sketched just prior to Example 1.
81. Give a formal proof to show that ∀x(x + 0 = x), ∀x(0 + x = x) − ∃!z∀x(x + z = x ∧ z + x = x) (see
Example 9).
EC
82. Give a formal proof to show that ∀x∃y(y = exp x), ∀x∀y(y = exp x ↔ x = ln y), ∀x∀y(y = ex ↔ x =
ln y) − ∀x(exp x = ex ) .
2.4 -20
HINTS TO STARRED EXERCISES 2.4
1. [No hint.]
2. [No hint.]
3. [No hint.]
5. Review the rules for negating quantified sentences.
9. [No hint.]
15. Remember to work with specific results before using general ones.
24. Note that you’re deducing an “or” statement, and use an appropriate proof strategy from SL.
26. Start off by using EI , using the first premise.
32. Apply the appropriate Negation Replacement Rules.
43. Your inner sentence should be a universal statement, so put it into its most natural ∀(→) format.
47. Use SL Replacement Rules to put your final answer in its simplest, most natural format.
48. Use SL Replacement Rules to put your final answer in its simplest, most natural format.
51. To formulate the definition symbolically, put the defining clause into an if-then format. Then negate
this final clause to show what follows if a positive integer is not prime.
77. Take U = R, and let Qx, Ax, and T x stand for x is rational, algebraic, and transcendental respectively.
Argue as in Examples 11 and 12 for the informal and formal deductions.