Download Quantification - Rutgers Philosophy

Quantification
by Sidney Felder
The truth-functional connectives constitute the expressive and deductive “engine” of propositional
logic. These combine and relate complete atomic sentences whose internal structures are given no
distinctively logical role whatsoever. From one formal point of view, quantificational logic, in all of
its infinite varieties, is marked by its penetration to a sentence’s sub-atomic level, a level from which
sub-sentential elements that are common to multiplicities of sentences can be discerned and
exploited. It has been found, specifically, that just by isolating those aspects of propositions that can
be interpreted as making assertions about all or some individuals of one domain or another, a
tremendously more powerful system of logic is created: In augmenting the propositional logic by the
universal and existential quantifiers “for all x” (symbolized (x) or (∀x)) and “there exists an x”
(symbolized (∃x)), and their associated apparatus, we both 1) enormously expand the sphere of
abstract ideas and structures that can be given precise and distinctive expression in formal terms and
2) vastly increase the range of propositions that can be brought into non-trivial expressive and deductive relationship with each other.
We are now going to define the vocabulary and concepts of what is variously called the First Order
Predicate Calculus, Predicate Logic, the Lower Predicate Calculus (LPC), and First Order Logic
(FOL). (Another (now seldom used) older name is the First Order Functional Calculus). First
Order Logic is simultaneously the most elementary and the most standard of the quantificational logics. (There is, for example, Second Order Logic, Third Order Logic, Quantified Modal Logic, Epistemic Logic, etc.).
The First Order system whose axioms include only logical axioms (only axioms that are logically
valid) singles out the class of logically valid formulae. However, there are many other concepts and
structures that can be expressed in a First Order language. Each augmentation of the logical axioms
(which belong to all First Order systems) by a logically consistent set of non-valid axioms (proper
axioms) defines a class of more definite structures. While many different choices of axioms will
single out the same class of structures, and while many conceptually natural structures are undefinable in First Order systems, the expressive power of First Order languages completely dwarfs that of
sentential calculi.
These notes are designed to provide a synoptic orientation to some of the simplest concepts, relations, and structures that can be expressed in a First Order Language. In addition to the sentences
and connectives of propositional logic, the basic elements of a First Order System include: 1) a nonempty domain or universe of individuals U; 2) a set of individual constants a, b, c,... each denoting
some definite individual belonging to U; 3) a set of predicate constants A, B, C,... designating properties and relations among the individuals of U (a monadic predicate corresponds to a subset of U
and an n-adic predicate for n greater than 1 corresponds to a set of n-tuples of elements of U); 4) a
set of variables x, y, z,... ranging over all the objects of U; 5) the quantifiers (∀x) and (∃x), representing the expressions ‘for all x’ and ‘there exists an x’ respectively; and 6) the logical constant ‘≈’,
invariably representing the relation of identity among individuals. While the designations of the
predicate constants (predicates) and the individual constants (constants) vary from interpretation to
interpretation, the quantifiers and the the identity relation are (like the truth-functional connectives)
Course Notes
Page 1
Quantification
treated as logical constants, and their interpretations are fixed. (Although an infinite number of distinct formulae can be composed from these linguistic elements, each well-formed formula of the language is assumed to be finite in length (i.e., to be composed of a finite number of symbolic
tokens)).
Now suppose that U is the set of physical objects and the predicate G is interpreted as the property
green. If the constant a is interpreted as ‘the largest leaf on Earth’ and the constant b is interpreted
as the sun, the sentence G(a) is interpreted as the sentence “The largest leaf on Earth is green”, and
the sentence G(b) is interpreted as the statement “The sun is green”. It is a consequence of the
interpretation chosen that G(a) is true and G(b) is false. Or, suppose that U is the set of natural
numbers {0,1,2,3,...}, that the symbol > represents, as usual, the relation ‘numerically greater than’,
and that the constants 2 and 3 represent, as usual, the numbers 2 and 3. Under this interpretation,
the sentence >(0,5) (more commonly written 0>5) , which represents the statement “0 is greater than
5” is false, but the sentence 5>0, which represents the statement “5 is greater than 0”, is true. On
the other hand, the simple predicate formula G(x) is neither true nor false categorically: G(x) is true
for some assignments of objects to x and false for others. Adopting a slightly different point of
view, we say that the formula G(x) is satisfied by some assignments of values to x and is not satisfied by others. Thus if we interpret the predicate G as the property green (or, equivalently, the set
of all green objects), G(x) will be satisfied by all green objects and will fail to be satisfied by any
non-green object. Analogously, x>y is true for some pairs of natural numbers and is false for others. Specifically x>y is satisfied by all ordered pairs of natural numbers whose first terms are
numerically greater than their second terms, and fails to be satisfied by any ordered pair whose first
term is not greater than its second term.
We now describe how the quantified formulae are to be understood. Given the non-empty domain
of discourse U of all physical objects, and supposing for the present that the predicate letter G is
interpreted as the color green, the formula (∀x)G(x) states that all objects of U possess the property
green, and the formula (∃x)G(x) states that there exists at least one object in U that possesses the
property green. In both of these formulae, all occurrences of x are bound, and we say that x
appears in this formula only as a bound or apparent variable. The variables embedded in the quantifiers themselves, (∀x) and (∃x), are always classified as bound, and the other occurrences of the
variable x in the above formulae are bound because (as indicated by the pattern of parentheses)
these occurrences fall within the scope of the quantifiers. Because we normally desire the proposition (∀x)G(x) to imply the proposition (∃x)G(x) (i.e., because we want (∃x)G(x) to be true whenever
(∀x)G(x) is true), it is standard to arrange things so that the formula (∀x)G(x) has existential
import, meaning that the proposition that all objects are green is interpreted in such a way as to
imply that there exists at least one green object. This is accomplished not by making any particular
assumption about the meaning of the universal quantifier (∀x), but rather by imposing the constraint
that the universe of discourse U can never be empty, that is, by imposing the demand that all admissible domains contain at least one object. Under the assumption that U has at least one object, the
truth of the statement that “All objects are green” presupposes that at least one object in U is green,
meaning that (∀x)G(x) is true in no domain in which (∃x)G(x) is false. If we admitted the possibility of a domain U with no objects, (∀x)G(x) would be vacuously true, and we would be admitting a
situation in which (∀x)G(x) was true and (∃x)G(x) was false. (The reader should be aware that
there are variations of the standard formalism, called free logics, in which empty domains are admitted).
What about a sentence of the form G(c), which states that the specific object designated by the
Course Notes
Page 2
Quantification
constant c is green? Such a statement is intermediate in logical strength between (∀x)G(x) and
(∃x)G(x) (i.e., is logically implied by (∀x)G(x), logically implies (∃x)G(x), and is logically equivalent to neither). For example, the statement that all objects are green is logically implies the statement that Joe’s coat is green, and the statement that Joe’s coat is green implies the statement that
there exists at least one green object. (In the peculiar but logically irreproachable universe whose
only object is Joe’s coat, the following statements are all simultaneously true: All objects are green;
Joe’s coat is green; Some object is green; Nothing other than Joe’s coat is green). Now, consider
any definite domain U. If it is not the case that all objects in U are green, there must exist some
object that is not green. Symbolically, the sentence ¬ (∀x)G(x) is equivalent to the statement
(∃x)(¬ G(x) (¬ G is of course the predicate ‘is not green’). Analogously, it is clear that if there does
not exist even a single green object in U, all objects in U (and there has to be at least one object in
U) must be non-green. In symbols, ¬ (∃x)G(x) is equivalent to (∀x)(¬ G(x). This means that it is
possible to define universal quantification in terms of existential quantification, and vice-versa. If we
take existential quantification as primitive, the universally quantified formula (∀x)G(x) corresponds
to the formula ¬ (∃x)¬ G(x). In words (interpreting the predicate constant G as green), the assertion
that all objects are green is logically equivalent to the assertion that there does not exist even a single non-green object. Conversely, if we take universal quantification as primitive, (∃x)G(x) is the
proposition ¬ (∀x)¬ G(x). In words, the statement that there exists at least one green object is logically equivalent to the statement that not all objects are non-green. (Recall the the manner in which
it is possible to define conjunction in terms of negation and disjunction, and disjunction in terms of
negation and conjunction).
We now consider cases of universal generalization, cases possessing the form of the proposition “All
ravens are black”. The first thing that needs to be said about this proposition is that its analysis
should not proceed under the assumption that this proposition is an instance of the ascription of a
predicate to a subject: Whatever the superficial linguistic form of this sentence, ‘All ravens’ is not its
subject. This proposition really concerns arbitrary individuals and the relationship, in a particular
world, of the properties ‘raven’ and ‘black’. Letting R designate the set of ravens and B designate
the class of black objects, this proposition is represented by the formula (∀x)(R(x)→B(x)), which is
read “for all x, if x is a raven, then x is black”. Here, the coordinated occurrences of x serve the
same function as do pronouns: This proposition states that for any particular object we choose, if
that object is a raven, then that object is black—where in each possible case, “the particular object”
and the two occurrences of “that” represent the same object. From a slightly different point of view,
this sentence states that R is a subset of B, i.e., that the set of ravens is a subset of the set of black
objects. When we reduce the subset relation to the relation of membership from which the subset
relation was originally defined, this means that each member of R is a member of B. Because we
conceive the variable x as ranging over the whole of U, what we really mean is: Given any arbitrary
element x of U, if x belongs to the set of ravens, then x belongs to the set of black objects. Given
that a universal generalization such as the one we are considering is in some sense a conjunction of
conditionals (in asserting it, we are asserting the statement “If a is a raven, then a is black and if b
is a raven, then b is black and if c is a raven, then c is black,...”), it is a generalization that does
not possess existential import. (Although the domain over which the variable x ranges is necessarily
non-empty, any particular subset of this domain, such as the set R of ravens, may be empty). This
is to be contrasted with the standard formal rendering of the proposition “Some ravens are black”,
which is “There exists at least one object that is both a raven and black”, symbolized by the formula (∃x)(R(x)∧B(x)). This is a formula that obviously (and by design) does carry existential
import.
Course Notes
Page 3
Quantification
Those who find it odd that the universal generalization (∀x)(R(x)→B(x)) is not logically stronger
than the existential generalization (∃x)(R(x)∧B(x)) will perhaps be mollified by the observation that
the true universal correlate of (∃x)(R(x)∧B(x)) is not (∀x)(R(x)→B(x)) but rather the sentence
(∀x)(R(x)∧B(x)), which says that all objects are black ravens. Confining ourselves to non-empty
domains (as is our invariable practice here), it is indeed the case that (∀x)(R(x)∧B(x)) is never true
when (∃x)(R(x)∧B(x)) is false. (The closest thing to the universal generalization (∀x)(R(x)→B(x))
that implies (∃x)(R(x)→B(x)) —that is, the logically weakest statement that implies both
(∀x)(R(x)→B(x)) and (∃x)(R(x)→B(x))—is the formula (∀x)(R(x)→B(x))∧(∃x)(R(x)). The logically
weaker existential analog of (∀x)(R(x)→B(x)) is the very weak proposition (∃x)(R(x)→B(x)), which
states that there exists at least one object x such that if x is a raven, then x is black, meaning
“Among the class of ravens, if any actually exist (which we don’t guarantee), at least one is black”.
This is a statement whose truth does not imply that any ravens exist. When considered together,
these formulae strongly suggest that the sentence (∀x)(R(x)→B(x)) is not about ravens, but rather
about the relations among the elements and subsets of the domain U. This is one reason that we
are able to take the logical equivalence between (∀x)(R(x)→B(x)) (“All ravens are black”) and its
contrapositive (∀x)(¬ B(x)→¬ R(x) (“All non-black things are non-ravens”) in stride.
The proposition (∀x)(R(x)→B(x)) must not be confused with the proposition (∀x)R(x)→(∀x)B(x)).
In the former sentence, the whole conditional expression (R(x)→B(x)) falls within the scope of the
quantifier (∀x), and hence all the occurrences of x in the expression are bound by the same quantifier. In the latter, the occurrence of the x in R(x) falls within the scope of the first quantifier, and
the occurrence of the x in B(x) falls within the scope of the second quantifier but not the first.
Consequently, the substitutions for x on the left of the disjunction sign are not coordinated with the
substitutions for x on the right of the disjunction sign, and hence we can without any alteration of
meaning use different variables on the two sides of the disjunction. So, for example, the formula
(∀x)R(x)→(∀y)B(y)) means the same thing as the formula (∀x)R(x)→(∀x)B(x)). We translate these
equivalent sentences as “If all objects are ravens, then all objects are black”. Although this is not a
statement that anyone would be inclined to utter, it is certainly true in any world reasonably similar
to ours. This is because the antecedent is manifestly false in this world or in any world very much
like it, which implies (given the definition of the material conditional →) that the whole conditional
is (vacuously) true whether or not all ravens are black.
An analogous contrast exists in the case of disjunction. The proposition (∀x)(A(x)∨ ¬ Ax)), which
states that for each object x, x has either the property A or the property ¬ A, is logically valid. The
quite distinct expression (∀x)A(x)∨(∀x)¬ A(x), on the other hand, states “All objects have the property A or all objects have the property ¬ A”. This is a contingent statement, and a rather severe
one. It implies 1) that if even a single object possesses property A, then all objects possess property
A and 2) that even if a single object does not possess property A, then none do. Looking now at a
particular instance, suppose U is the set of natural numbers N {0,1,2,3,...}, O the property ‘odd’, and
E the property ‘even’. Consider the propositions (∀x)(O(x)∨E(x)) and (∀x)O(x)∨(∀x)E(x). The
first says “For any natural number x, x is either odd or even”, a statement that is obviously and necessarily true. The second proposition makes the plainly absurd (and necessarily false) statement that
all natural numbers are odd or all natural numbers are even. There are of course situations in which
this kind of disjunction is appropriately asserted. If I believe that there is a good chance that a
large commercial aircraft has crashed into the seas near the Arctic Circle and a good chance that it
landed safely at its scheduled destination, it is extremely likely to be true that either all its passengers are alive or all its passengers are dead.
Course Notes
Page 4
Quantification
The expressive capabilities of quantificational language can be seen yet more clearly in formulae in
which n-place relations and a multiplicity of distinct variables appear. Phenomenae associated with
alternations of universal and existential quantifiers are particularly revealing. We must here restrict
ourselves to a consideration of the distinction between the meanings of the expressions
(∀x)(∃y)(y>x) and (∃x)(∀y)(x>y). Suppose U is again the set of natural numbers N, and that the
dyadic relation > is the irreflexive numerical relation ‘greater than’. Under this interpretation,
(∀x)(∃y)(y>x) states that for any natural number x, there exists a natural number y such that y is
greater than x. This states, in effect, that there is no largest natural number, a statement that is
obviously true. The sentence (∃x)(∀y)(x>y), on the other hand, states that there exists some natural
number x, such that given any natural number y, x is greater than y. This says, in other words, that
there is some particular natural number that is greater than any natural number—an assertion that is
more colloquially rendered by the statement that there is a single natural number that is simultaneously greater than all natural numbers, a statement that is patently false.1
The distinction between these two statements will perhaps be illuminated by a consideration of the
following two “games” between two opponents A and B: In the game corresponding to the sentence
(∀x)(∃y)(y>x), A has the first move, and chooses any natural number x he likes. If B cannot name
a number that is greater than x, A wins the whole game; if B can name a number greater than x, B
wins this round. Because A must select a definite natural number (though any number he pleases)
on each round, and because A must commit himself first on each round, B clearly has a winning
strategy because whatever A selects on any round, there is always available to B, for example, the
number that is one greater than the number A selects on that round. Although A’s choice of number
on round n+1 may always be a larger number than any of B’s choices on any of the rounds 1 to n,
the winner of each round n is determined by the relative magnitudes of the choices on round n, and
it is obvious that B has a winning strategy that determines the outcome of every round. Consider
now the game corresponding to (∃x)(∀y)(x>y). A goes first, and selects a natural number. However
large this number is, it is nevertheless a finite number. B wins this round by simply selecting its
immediate successor. There is in fact only one round in this game, which reflects the fact that
(∃x)(∀y)(x>y) asserts (falsely) that there is a single finite number that is greater than all finite numbers simultaneously, a single choice by A that exceeds all possible choices by B. In the case of
(∀x)(∃y)(y>x), B merely has to demonstrate that each number has one successor or another, not
necessarily the same for any two choices of A.
Sentences prefixed by the triple of alternating quantifiers (∀x)(∃y)(∀z) are critical in all mathematics
involving the concept of limit. Thus consider a sequence such as 1/2,1/3,1/4,1/5,..., a sequence
whose limit (in this case its greatest lower bound) is not itself contained as a term in the sequence.
The numerical differences between successive terms of this sequence obviously become always
smaller (i.e., smaller without exception) as we consider successively further terms in the sequence.
The terms also obviously get closer to the number 0 as we consider successively further terms.
Does the satisfaction of the two conditions of the last two sentences justify the assertion that a
sequence converges to 0 (i.e., that the limit of the sequence is 0) ? Consider the sequence obtained
from the sequence 1/2,1/3,1/4,1/5... by adding 1 to each term. Although the numerical differences
1
What about the formulae (∀x)(∃y)(x>y) and (∃x)(∀y)(y≥x) (≥ means ‘greater than or equal to’)? The first formula says that for all natural numbers x, there exists a natural number y such that y is less than x. The second formula says that there exists a natural number x such that for all natural numbers y, x is less than or equal to y. The
first formula states that there is no least natural number, a statement that is false. The second formula states that
there is a least natural number, a statement that is true.
Course Notes
Page 5
Quantification
between successive terms of the sequence 1+1/2,1+1/3,1+1/4,1+1/5... become irreversibly smaller as
we consider further terms in the sequence, and although the numerical values of these terms move
successively closer to 0, the limit of this sequence is 1 and not zero. This shows that these two
properties are not sufficient conditions for a sequence to have 0 as limit. Interestingly, neither of
these two properties is a necessary condition for a sequence to have 0 as limit. Thus consider the
sequence 1/2,1/3,1/4,2/2,2/3,2/4,1/5/1/6,1/7,2/5,2/6,2/7,1/8/1/9,1/10,..., which we will call s*. It is
neither the case that the numerical differences of successive terms always diminish as we move further along the series (the difference between 2/4 and 1/5 is greater than the difference between 1/3
and 1/4) nor the case that successive terms always move closer to 0 (2/5 is further from 0 than is
1/5) . However, the limit of this sequence is 0. In the intuitively salient sense, it is plain that there
is a progressive trend towards 0 as we consider ever more remote terms of the sequence, and it is
plain (and can be demonstrated) that eventually the sequence comes arbitrarily near 0 and to no
other number.
There is a precise expression of these intuitions in terms of alternating universal and existential
quantifiers. Let δ represent a numerical separation greater than 0, N a particular ordinal position in
a sequence, n a variable ranging over the positions of a sequence, and s(n) the numerical term occupying the position n in the sequence. (Remember, a sequence is a function s from the natural numbers to an arbitrary set. Thus the sequence 1/2.1/3,1/4,1/5,... is formally represented as the set of
ordered pairs (0,1/2) , (1,1/3) , (2,1/4) , (3,1/5) ,...). We say that the limit of a sequence is L iff for
any arbitrarily chosen non-zero numerical separation δ, there exists an ordinal position N in the
sequence s such that for all positions n of the sequence beyond N (i.e., for all n>N), the numerical
difference between the numerical value of the term in the nth position (n>N) and L is less than δ.
In symbolism, a magnitude L is the limit of a sequence s iff (∀δ>0) (∃N)(∀n>N)(s(n)-L<δ). The
sequence s* above whose upward and downward oscillations progressively diminish in magnitude
satisfies this condition, and hence the sequence s* converges to 0 as limit.
Among the infinitely many other logical and mathematical notions that quantificational language permits us to define, I will only mention two.
The first is the general notion of an equivalence relation. Recall that an equivalence relation is a
relation that possesses the properties of reflexivity, symmetry, and transitivity. (Numerical equivalence, logical equivalence, and the relation among natural numbers ‘leaves the same remainder
when divided by n’ are notable equivalence relations). In symbolic terms, these properties are represented as follows:
(∀x)(xRx) states that for all x, x is equivalent to itself—Reflexivity.
(∀x)(∀y)((xRy)→(yRx)) states that for any pair of elements x and y, if xRy then yRx—Symmetry.
(∀x)(∀y)(∀z)((xRy∧yRz)→(xRz)) states that for any three elements x, y, and z, if xRy and yRz,
then xRz—Transitivity.
With this same apparatus (the language of quantificational logic + Identity), we are also able to
specify the precise cardinality of any finite domain U. (∀x)(∀y)(x=y) states that for any assignment
of objects from U to the variables x and y, the object assigned to x is identical to the object
assigned to y. This sentence means that there exists at most one object (in U). When conjoined
with the constraint that all domains are non-empty, the truth of this sentence implies that there exists
exactly one object (i.e, that exactly one object belongs to U).
Course Notes
Page 6
Quantification
(∃x)(∃y)(x≠y)∧((∀z)((z=x)∨(z=y))) states that the domain in question contains an object that we can
assign to x and an object that we can assign to y such that 1) the objects assigned to x and y are
distinct and 2) for any assignment of an object in U to z, the object assigned to z is identical either
to the object assigned to x or to the object assigned to y. This sentence means that there exist
exactly two objects (i.e, that exactly two objects belong to U). This sentence may be regarded as
the conjunction of the sentence (∃x)(∃y)(x≠y), which states that there are at least two objects, and
the sentence ((∀z)((z=x)∨(z=y))), which states that there are at most two objects.
(∃x)(∃y)(∃z)((x≠y)∧(y≠z)∧(x≠z))∧(∀v)((v=x)∨(v=y)∨(v=z))) states that U contains an object that we
can assign to x and an object that we can assign to y and an object that we can assign to z such
that 1) no two of the objects assigned to x, y, and z are identical and 2) for any assignment of an
object in U to v, the object assigned to v is identical either to the object assigned to x, the object
assigned to y, or the object assigned to z. This sentence means that there exist exactly three objects
(i.e, that exactly three objects belong to U). This proposition may also be regarded as the conjunction of two sentences, the sentence (∃x)(∃y)(∃z)((x≠y)∧(y≠z)∧(x≠z)), which states that there are at
least three objects, and the sentence (∀v)((v=x)∨(v=y)∨(v=z)), which states that there are at most
three objects.
Clearly, analogous constructions exist for any finite cardinality.
We now turn to more formal account of the semantics of First Order systems, in particular of the
notions of logical validity, truth in an interpretation, and satisfiability. (This whole analysis is due
to the great logician Alfred Tarski (1901-1983) , and is expounded in his ground-breaking paper
“The Concept of Truth in Formalized Languages (1935) ). We begin by defining the general notion
of interpretation of a first order language.
First, we specify the language of First Order Logic With Identity (FOL≈). For purposes of exposition, we make the special assumption that the classes of variables, constants, predicates, and functions of the language each contain a denumerable set of elements, that is, a number of elements
whose cardinality is equal to the infinite set of natural numbers N.
To the atomic sentences A, B, C,... AA, BB, CC,... AAA, BBB, CCC, ..., ... and truth-functional
logical constants of propositional logic ∧, ∨, ¬ , ->, →
←, the language of FOL≈ adds the following:
1) an infinite set of variables x, y, z, x1 , y1 , z1 ,... x2 , y2 , z2 ,..., ...;
2) an infinite set of individual constants a, b, c,... a1 , b1 , c1 ,... a2 , b2 , c2 ,..., ...;
3) an infinite set of predicate constants divided into a)monadic predicate constants P1 , Q1 , R1 ,... (P,
Q, R,... for short), P11 , Q11 , R11 ,... P12 , Q12 , R12 ,..., P13 , Q13 , R13 ,...; b)dyadic predicate constants P2 , Q2 ,
R2 ,..., P21 , Q21 , R21 ,... P22 , Q22 , R22 ,..., P23 , Q23 , R23 ,...; c)triadic predicate constants P3 , Q3 , R3 ,..., P31 , Q31 ,
R31 ,... P32 , Q32 , R32 ,..., P33 , Q33 , R33 ,...; d)four-place predicate constants; e)five-place predicate constants,
etc.;
4) an infinite set of function constants, divided into a)monadic functional constants f1 , g1 , h1 ,... (f, g,
h,... for short), f 11 , g11 , h11 ,... f 12 , g12 , h12 ,..., f 13 , g13 , h13 ,...; b)dyadic functional constants f 2 , g2 , h2 ,..., f 21 ,
g21 , h21 ,... f 22 , g22 , h22 ,..., f 23 , g23 , g23 ,..., c)triadic functional constants, etc.;
5) the universal quantifier ∀ (frequently symbolized ( )) and the existential quantifier ∃;
6) the logical constant =.
Course Notes
Page 7
Quantification
An interpretation of the formulae of FOL= is a functional mapping I from linguistic entities to
(most typically) non-linguistic entities that is defined by the following:
1) the assignment of a definite truth value to each propositional letter A, B, C,... and the assignment
to the symbols ∧, ∨, ¬ , ⊃, ←
→ of the previously defined truth functions conjunction, disjunction,
negation, material conditional, and material biconditional respectively.
2) the specification of a non-empty universe of discourse U, which can possess any cardinality and
whose elements can possess any character, over which the variables x, y, z,... range. The elements
of U are some number of individual objects oi .
3) the assignment to the symbolic complex (∀x) (sometimes simply (x)) of the meaning ‘for all x in
U’ and to the symbolic complex (∃x) of the meaning ‘there exists at least one x in U’.
4) the assignment to each individual constant a, b, c,... of a definite object belonging to the domain
U (‘Socrates’ and ‘Joe’ are constants representing the definite individuals Socrates and Joe).
Although the interpretation I may assign the same object in U to a multiplicity of constants, each
constant can name only one individual in U.
5) the assignment to each monadic predicate constant P of a subset of U; the assignment to each
dyadic predicate constant P2 of a subset of the Cartesian Product U×U; the assignment to each triadic predicate constant P3 of a subset of U×U×U, etc. Thus P is a set of elements of U, P2 is a set
of ordered pairs of elements of U, P3 is a set of ordered triples of elements of U, etc. A subset of
U corresponds to a property (e.g., x is green); a set of ordered pairs corresponds to a two-place relation (e.g., x is to the left of y); a set of ordered triples corresponds to a three-place relation (e.g., y
is between x and z); etc.
6) the assignment to each monadic functional constant f of a one-variable function from objects of U
to objects of U; the assignment to each dyadic function constant f 2 of a two-variable function from
ordered pairs of objects of U to objects of U; the assignment to each triadic function constant f 3 of
a three-variable function from ordered triples of objects of U to objects of U; etc. Within the
domain of natural numbers, successor is a one-variable function (no two distinct numbers have the
same successor) and addition is a two-variable function (the operation of addition associates a single
number (their sum) to a pair of numbers). In the domain of human beings, mother of is a multivariable function from finite sets of human beings to human beings (more than one child can have
the same mother).
7) the assignment to the logical constant = of the relation is identical to. (Note that FOL is often
defined without a ‘built-in’ standard identity relation). The denotations of the logical constants ∧, ∨,
¬ , ⊃, ←
→, =, ∀, ∃ are the same in all interpretations. The denotations of propositional letters, individual constants, predicates, and functions in general vary from interpretation to interpretation; nevertheless, propositional letters always denote atomic propositions, individual constants always denote
individuals of U, predicates invariably denote sets (monadic predicates invariably denote subsets of
U; dyadic predicates invariably denote subsets of U×U; triadic predicates invariably denote subsets
of U×U×U; function symbols f nm invariably denote functions from n-tuples of elements of U to elements of U. Unless there were some invariants in the interpretation of the elements of a formal system, the propositions corresponding to these expressions would have no form, and hence the basic
idea that logical inferences and truths are valid in virtue of their form would collapse into nothingness.
We now discuss some of the most illuminating of the elementary aspects of the notions satisfiability,
logical validity, and truth in an interpretation. We first define the truth of formulae in which no
Course Notes
Page 8
Quantification
variables are present, and then go on to define the truth of formulae that contain variables.
The formula P(c) is true in interpretation I if and only if the object that the interpretation I assigns
to the constant c is an element of the subset of U that I assigns to the monadic predicate P, and is
false if the object designated by c does not belong to the set corresponding to P. (In this latter case,
c belongs to the set corresponding to the complement of I(P) in U). Thus assume that we’ve chosen
an interpretation in which I(Mortal) (the meaning the interpretation I assigns to the predicate constant Mortal) is the set of mortals, and that I(Socrates) (the meaning the interpretation I assigns to
the individual constant Socrates) is the man Socrates. Under this interpretation, the sentence
‘Socrates is mortal’ is true because the object that the interpretation I assigns to the constant
Socrates is an element of the subset of U that I assigns to the monadic predicate Mortal. The sentence ‘Jack is located directly to the left of Joan’ is true if the objects I assigns to the constants
‘Jack’ and ‘Joan’ form an ordered pair that belongs to the set of ordered pairs that I assigns to the
dyadic predicate ‘left of’. If Jack is indeed directly to the left of Jill, the ordered pair (Jack,Jill)
belongs to the set of ordered pairs representing the relation ‘directly to the left of’, and hence the
sentence ‘Jack is directly to the left of Jill’ is true. On the other hand, because (assuming we are
operating in the context of a space that is at least locally Euclidean) Jill cannot be directly to the
left of Jack if Jack is directly to the left of Jill (the relation ‘directly to the left of’ is plainly an
asymmetrical relation), the ordered pair (Jill,Jack) does not belong to the set of ordered pairs representing the relation ‘directly to the left of’. And, assuming that we have chosen an interpretation in
which the constants 0, 1, 2, 3,... are the natural numbers 0, 1, 2, 3,... and are considering the relation ‘x, y, and z are mutually distinct’, the statement that 2, 4, and 3 are mutually distinct is true
because the ordered triple (2,4,3) will belong to the set of all triples of numbers whose components
are distinct, and the statement that 2, 3, and 3 are mutually distinct will be false because the ordered
triple (2,3,3) does not belong to the set of all triples whose components are distinct.
Note that while an interpretation I directly assigns a truth value to the elementary propositional sentences, I assigns an object of U to each individual constant and a subset of U to each predicate constant. However, just as the truth value of all propositions compounded from the elementary sentences by use of the truth-functional connectives is uniquely determined by the truth values of the
elementary sentences and the fixed denotations of the connectives, the truth value of every formula
that associates n individual constants to an n-place predicate constant is a consequence—it should be
clear—of the interpretations of the individual and predicate constants. Because a particular object
either belongs to a particular set or doesn’t, formulae of the form P(a), P2 (a,b), P3 (a,b,c) are either
true or false in a given interpretation, and hence they can enter into truth-functional combinations
with each other as well as with straight propositional formulae.
We now go on to the case of formulae that contain variables. The truth and falsity of these formulae are not in general determined by a specification of an interpretation. In order to define the truth
of formulae with variables, we need to augment the given interpretation by a further semantical
operation called an assignment or valuation, which associates objects of the domain to variables.
We begin with the simple formula G(x). We assume that we are working within an interpretation I
that assigns the property green to the predicate constant G. G(x) can be taken as meaning ‘x is
green’. As we have emphasized before, such a sentence, like the arithmetical statement ‘X is a
prime number’, is neither true nor false. There are some numbers whose substitution for X will
yield a true statement, and some numbers whose substitution for X will yield a false statement.
From a somewhat different point of view, in which we keep our focus on the variable expression ‘X
is a prime number’, we say that certain numbers satisfy the condition or property ‘is a prime
Course Notes
Page 9
Quantification
number’ and certain numbers do not. In the case ‘x is green’, the variable x is conceived to range
over the entirety of the domain U. The property green corresponds to a particular subset of U (the
set of green objects): The individual objects that belong to the subset of green objects, such as
blades of grass and leaves, satisfy the formula G(x); the objects that fall outside the subset of green
objects, such as fire engines, roses, and ripe tomatoes, do not satisfy this formula. In relation to a
given interpretation, G(x) is a propositional function, a function whose domain is U and whose codomain is the set of definite propositions obtained when a definite object in U is substituted for
(assigned to) the individual variable x.
Consider the dyadic predicate ‘less than’, assumed to be defined on the set Z of integers. The
dyadic predicate ‘less than’, <, corresponds to the set containing all ordered pairs of integers whose
first terms are less than their second. Thus this set contains an infinite number of elements (ordered
pairs) such as (2,3) , (-10,-3) , (0,27) , etc., and it excludes an infinite set of ordered pairs such as
(5,5) , (3,2) , (-3,-10) , (27,0) . The formula x<y is satisfied by all ordered pairs of numbers that
belong to the first set (all ordered pairs of integers whose first terms are less than their second), and
is not satisfied by any ordered pair in the second set (whose second components are smaller than or
equal to their first). Sticking with the same underlying Universe (the set of integers), consider now
the triadic predicate ‘x is between y and z’. This predicate corresponds to the set of all ordered
triples (x,y,z) the magnitude of whose first component falls strictly between the magnitudes of the
secondq and third component. Among the infinite number of ordered triples that belong to the set
representing this triadic predicate are (2,1,3) , (2,3,1) , (10,-17,30) , and (-2,120,-19) . Finally, we
consider the four-place predicate ‘numbers x, y, and z add up to a’, which corresponds to the set of
all ordered quadruples whose first three terms add up to the fourth. The formula x+y+z=a is satisfied by quadruples such as (0,0,0,0) , (1,2,3,6) , (2,3,1,6) , and (17,-10,-20,-13) , but is not satisfied by
quadruples such as (0,0,0,1) , (-1,-2,-3,8) , and (5,-5,5,11) .
Given the above interpretation of the monadic predicate constant G as the color green, the formula
(∃x)G(x), obtained by attaching as prefix the quantifier (∃x) to G(x), is interpreted as the statement
‘There exists an object x, such that x is green’, or, more colloquially, ‘There exists at least one
green object’. Given the known character of our world, this statement (given the domain specified
in our interpretation) is obviously true. Our present task is to relate the truth of this quantified
statement to the satisfiability of the unquantified formula G(x), a formula that is not prefixed by the
quantifier (∃x) or any other quantifier. Most simply stated, the quantified sentence (∃x)G(x) is true
if and only if there exists some object that satisfies the open formula G(x). Note the manner in
which the ascription of truth to one expression depends upon the satisfiability of another. Some
objects satisfy the formula G(x) and some do not, but because at least one object answers to the
description Green, and hence satisfies the formula G(x), a formally quite distinct but formally related
expression (∃x)G(x) is categorically (i.e., unqualifiedly, unconditionally) true. Analogously, because
not all objects are green, not all objects satisfy the formula G(x), and hence (∀x)G(x), the statement
that all objects satisfy the formula G(x), is simply, unqualifiedly false.
This pattern reflects the essential distinction between an open formula on the one hand, and a closed
formula or sentence on the other. When there is no expression of either the form (∃x) or (∀x) that
is attached as prefix to the formula G(x), we say that the occurrence of x in G(x) is free or that x
occurs in G(x) as a free variable. When G(x) is prefixed by either (∃x) or (∀x), the variable x in
G(x) is said to fall within the scope of the quantifier (either (∃x) or (∀x), as the case may be), and
we say that the occurrence of x in G(x) is bound or that x occurs in G(x) as a bound variable. (An
occurrence of a variable in either the quantifier (∃x) or (∀x) is always classified as bound).
Course Notes
Page 10
Quantification
Whether G(x) is satisfied depends upon whether or not the object in U that is associated with the
variable x by a given assignment belongs to the set corresponding to G. In other words, some
assignments of objects to x produce true statements and some assignments of objects to x produce
false statements. Indeed, G(x) may be conceived as a function from U to the set containing the two
values true and false. On the other hand, because there exists at least one green object, (∃x)G(x) is
simply true, and because there exists at least one object that is not green, (∀x)G(x) is simply false.
So although the variable x appears in the expression (∃x)G(x), it does not behave like a true variable
because the truth of these quantified statements does not depend upon what object is assigned to x.
Because the attachment of the universal or existential quantifiers as prefix to G(x) has the effect of
removing all variation from the quantified expression no less completely than would the substitution
of a constant for the variable x, it has the effect of making the truth of these quantified statements
something completely unconditional. For this reason, a variable x in a formula is sometimes
referred to as a real variable when it does not fall within the scope of a quantifier, and is referred to
as an apparent or dummy variable when all its occurrences fall within the scope of a quantifier.
Thus x functions as a real variable in G(x), and functions as an apparent variable in (∀x)G(x) and
(∃x)G(x).
There is a powerful and elegant device, invented by Tarski, that permits a unified treatment of all
formulae possessing any number of free and bound variables. It works as follows: We fix an
arrangement of all individual variables of the language into a denumerably infinite sequence
(x,y,z...). An assignment or valuation v of set of variables {x,y,z,...} associates with each variable in
the set of variables {x,y,z,...} exactly one individual of the domain U. Although no variable is
assigned more than one element of U (v is a function), any number of variables of the sequence can
be assigned the same individual of U, meaning that among the class of possible valuations, there are
an infinite number of valuations that assign the same individual to all variables. Each possible
denumerable sequence s=(o0 ,o1 ,o2 ,...) of individuals from U represents a particular assignment v, a
comprehensive assignment of objects of U to variables, associating o0 with x, o1 with y, o3 with z,
etc. (The subscripts attached to the letter ‘o’ are present merely to distinguish among objects of U,
and are not meant to indicate any ordering among these objects. Thus o1 and o4 may be the same
object). There are a super-denumerably infinite number of possible assignments, as many as there
are distinct ways to associate a fixed denumerable set of objects {o0 ,o1 ,o2 ,...} to the fixed infinite
sequence of variables (x,y,z,...). (The cardinality of the set of possible assignments V is equal to the
cardinality of R, the set of real numbers). In other words, there is a one-to-one correspondence
between the class of possible valuations V and the class of infinite sequences S that can be formed
from elements of U. Thus, for example, suppose that our underlying domain U is N, the set of natural numbers {0,1,2,3,...}. The sequence of numbers (1,1,3,1,1,1,...) defines an assignment of 1 to x,
1 to y, 3 to z, and 1 to every other variable; the distinct sequence (1,3,1,1,1...) defines an alternative
assignment, one that associates 1 to x, 3 to y, and the number 1 to every other variable.
We say that the sequence s=(o0 ,o1 ,o2 ,...) satisfies a formula F if and only if the objects that are
assigned to the free variables in F by the comprehensive assignment of objects to all variables (represented by the sequence (o0 ,o1 ,o2 ,...)) satisfy the conditions defined by F. For example (a number
of examples are indispensable here), suppose as given an interpretation I that assigns to the monadic
predicate G the property green, and consider the sequence (the world’s largest blade of
grass,Mars,the orange I just ate,the fire engine that just went by,the sky at night,the sky at night,...).
The formula G(x) will be satisfied by this particular sequence, as well as by any assignment v that
assigns a green object (say a particular blade of grass) to the variable x, whatever objects v assigns
to the other variables y, z, etc. In other words, the formula G(x) is satisfied by all sequences, such
Course Notes
Page 11
Quantification
as the one above whose first six terms we listed, whose first element is a green object. What about
the formula G(y)? Because y is the second variable in the canonical sequence of variables, G(y) is
satisfied by all sequences whose second place is occupied by a green object, whatever color the
objects are that occupy the first place, the third place, or any place beyond the third.
Consider now an interpretation J that possesses as domain the set of natural numbers {0,1,2,...} and
that assigns to the two-place predicate G the relation ‘greater than’. The formula G(y,z) (y>z) is
satisfied by all sequences whose second terms are greater than their third terms. Thus, for example,
the sequences (5,7,6,1,6,1,6,1,6,...) and (9,8,4,9,9,9,...) satisfy the formula G(y,z), and the sequences
(5,6,7,5,6,5,6,5,6,...) and (9,5,5,4,9,9,9,...) do not. Finally, consider the formula A(x,y,z), where A is
interpreted as the predicate ‘x+y=z’. This formula is satisfied by the sequence (5,6,11,2,3,4,...) as
well as by all other sequences possessing the property that the sum of the magnitudes of the first
and second components is equal to the magnitude of the third.
Note the following general characteristic shared by all these cases: In determining whether a
sequence s satisfies a formula F, we only need to examine the terms of s that correspond to the
positions (in the fixed ordering of variables) of the free variables of F. Thus supposing that G(x) is
the simple formula above whose only free variable is x, we only need to look at the first element of
a sequence in order to tell whether that sequence satisfies G(x): If the first term of the sequence is a
green object, the sequence satisfies G(x) whatever objects occupy the other places of the sequence,
and hence every sequence whose first term is a green object satisfies G(x). In other words, the possession by a sequence of a green object in its first position is a necessary and sufficient condition
for its satisfaction of the formula G(x). Likewise, we need only look at he second element of a
sequence to determine whether that sequence satisfies G(y). All sequences whose second position is
occupied by a green object satisfy G(y), whatever the color of the objects that occupy either the first
position or any position beyond the second. This pattern generalizes in the natural way to formulae
containing more than one free variable. Determining whether a sequence of numbers satisfies the
formula A(x,y,z) requires an examination of only three positions in the sequence—given the ordering
of variables we have fixed above, the three salient positions in the sequence are the first, the second,
and the third. Every sequence whose first and second term adds up to its third term satisfies
A(x,y,z), the other positions of the sequence playing no role whatsoever. We suppose (as is standard) that every particular formula of our language possesses a finite number of symbols, and hence
a finite number of free variables. Why then do we bother with infinite sequences in these definitions? The reason that we employ infinite sequences is that there is no set bound either on the
finite number of free variables a formula contains or on the position in the ordering of variables a
free variable in a formula may be selected from.
Truth functional combinations of formulae with free variables behave as would naturally be
expected. If G denotes the set of green objects, ¬ G denotes the complement of G in U, the set of
“non-green” objects, i.e., the set of all objects that are not green. A sequence s satisfies ¬ G(x) if
and only if s does not satisfy G(x). In other words, any sequence that does not satisfy G(x) satisfies
¬ G(y), and vice-versa. Extending and generalizing to possibly polyadic (n-place) formulae F, G, H,
etc. with any number of free or bound formulae, we have the following: 1) A sequence s satisfies the
formula ¬ F if and only if s does not satisfy F. 2) A sequence s satisfies the formula F∧G, the conjunction of formulae F and G, if and only if s satisfies F and s satisfies G. 3) A sequence s satisfies
the formula F∨G, the disjunction of formulae F and G, if and only if s satisfies F or s satisfies G
(i.e., if and only if s satisfies at least one of the formulae F and G). 4) A sequence s satisfies the
formula F->G, the conditional with formula F as antecedent and G as consequence, if and only if s
Course Notes
Page 12
Quantification
satisfies ¬ F or s satisfies G. 5) A sequence s satisfies the formula F←
→G, the biconditional whose
first formula is F and whose second formula is G, if and only if s satisfies both F and G or s satisfies neither F nor G.
The non-obvious cases involve the relationships between the satisfaction of formulae with free variables x, y, z, etc. and the satisfaction of formulae in which these variables x, y ,z, etc. are bound by
quantifiers. Because by this point the extensions to formulae with a multiplicity of free variables
should be reasonably clear, we restrict ourselves to the simplest case of a formula with a single free
variable, G(x). Consider again the interpretation I that assigns the property green to the predicate G.
The basic idea is that the satisfaction of (∃x)G(x) by an arbitrarily chosen sequence (in a given
interpretation I) is contingent upon the existence of a sequence whose first term is a green object.
In other words, an arbitrarily chosen sequence s satisfies the closed formula (∃x)G(x) with the variable x bound (quantified) because there exists a sequence that satisfies the formula obtained from
(∃x)G(x) by removing the prefix that binds (quantifies) the variable x.
More precisely (now hold on to your hats!), we say that a particular sequence s satisfies the closed
formula (∃x)G(x) if and only if there exists a sequence differing from s in at most the first place
that satisfies the open formula G(x). (Any sequence that differs from s in at most a single place w
is called a w-variant of s). In other words, s satisfies (∃x)G(x) in the case that s itself satisfies
G(x), as well as in the case that a distinct sequence—one that differs from s in its first position—
satisfies G(x). Thus a sequence s whose first term is the fire engine that just went by satisfies
(∃x)G(x) because there exists a sequence—in this case a sequence distinct from s—that satisfies
G(x), for example, any chosen sequence whose first term is the largest leaf on Earth. In this case,
the sequence s whose first term represents the fire engine that just went by satisfies (∃x)G(x)
because there exists a sequence differing from s solely in its first place that satisfies G(x). In the
case of a sequence s* whose first term is the world’s largest green leaf, s* satisfies (∃x)G(x) because
s* is a sequence that doesn’t differ in any place at all from a sequence (s* itself) that satisfies G(x).
It is because we want to accommodate both cases in a single clause that we employ the superficially
puzzling phrase ‘at most’. Given what was said before about the relevance of only those positions
in a sequence corresponding to the free variables of a formula, the definition of satisfaction for an
existentially quantified formula implies that if one sequence satisfies G(x), then all sequences satisfy
(∃x)G(x).
The case of universally quantified formulae is exactly analogous. A sequence s satisfies (∀x)G(x) if
and only if all sequences that differ from s in at most the first place satisfies G(x). Because not all
objects are green, there exist sequences whose first positions are not occupied by green objects,
which implies that there exist sequences that do not satisfy the formula G(x). Thus suppose that s
is a sequence whose first term is a green object such as the world’s largest leaf. Because there exist
sequences differing from s in at most the first place that do not satisfy G(x), s itself does not satisfy
the formula (∀x)G(x). Consequently, if there exists even a single sequence that does not satisfy
G(x), then no sequence satisfies (∀x)G(x). More generally, any formula that contains no free variables is satisfied either by all sequences or by none.
We can now define the notion of truth for an interpretation. A formula F is true for an interpretation I if and only if—in relation to the interpretation I—all sequences satisfy F. A formula F is
false for an interpretation I if and only if—in relation to the interpretation I—no sequence satisfies
F. According to this definition, every formula F without any free variables—whether it is the case
that F contains no variables at all or it is the case that all its variables are bound by quantifiers—is
Course Notes
Page 13
Quantification
either true or false for an arbitrary interpretation I. (Of course, F will be true for some interpretations and false for others). A formula F is true for an interpretation I if and only if ¬ F is false for
I, and F is false for an interpretation I if and only if ¬ F is true for I. An interpretation relative to
which each member of a set of formulae Γ is true is called a model of Γ. (According to our definitions, any truth-functional tautology is true for all interpretations).
A formula is logically valid if and only if it is true in all possible interpretations, and is logically
false if and only if it is false in all interpretations. A formula F, whether it is open or closed, is
satisfiable if and only if there exists a sequence that satisfies F in at least one interpretation. Symmetrically, F is unsatisfiable or contradictory if no sequence satisfies F in any interpretation. A formula F is logically valid if and only if ¬ F is unsatisfiable, and F is satisfiable if and only if ¬ F is
not logically valid.
A set of formulae Γ is simultaneously satisfiable if and only if there exists at least one sequence s
that satisfies all formulae in Γ. A set of formula Γ logically implies a formula F, or (equivalently)
F is a logical consequence of Γ, if and only if there is no sequence that simultaneously satisfies
each formula of Γ but fails to satisfy F. This means that every model of Γ is a model of F. Two
formulae F and G are logically equivalent if and only if F logically implies G and G logically
implies F, which means that F and G are true in the same models.
Now it might seem that 1) if the monadic predicate G can be assigned the property ‘green’ in one
interpretation and the property ‘red’ or even ‘not green’ in another, and 2) if the the dyadic predicate
G can be assigned the relation ‘greater than’ in one interpretation, ‘less than’ in another interpretation, and ‘is equal to’ in a third, and 3) if the domain U may contain any number of elements of any
character whatsoever, it will be impossible to find any formula that will be true in all interpretations
or false in all interpretations. This impression is wrong. As we emphasized earlier, U is always a
non-empty set, G is always a subset of U, and all variables and constants are assigned elements of
U. Thus, for example, whatever non-empty set the domain U corresponds to, and whatever subset
of U the predicate G stands for, it will always be true that any element of U will either belong to G
or to its complement in U, and hence the formula (∀x)(G(x) ∨ ¬ G(x)) is logically valid. At the
same time, because no element of U can both belong to a subset of U and not belong to that same
subset of U, the formula (∀x)(G(x)∧¬ G(x)) is logically contradictory. The formula
(∀x)(G(x)∧H(x)→G(x)) is logically valid because G(x)∧H(x) is a subset of G(x) whatever the identities of G and H, which means that any object that belongs to G and H necessarily belongs to G.
As a final example, we mention the formula ((∃x)(∀y)(xRy))->((∀y)(∃x)(xRy)), which means that if
at least one particular object in a domain U possesses the relation R to all objects in U simultaneously, then for each object y in U, there exists at least one object in U that has the relation R to y.
(For example, the existence of even one truly misanthropic person—the existence of a person who
shuns everyone—is sufficient to ensure the truth of the statement that everyone is shunned by someone or other. (On the other hand, the fact that each person is shunned by someone does not mean
that there exists a person who shuns everyone)).
A formula that is valid for all interpretations in a particular domain is said to be valid in that
domain. For FOL=, it is easy to demonstrate that any formula that is valid in a domain with a definite finite number n of objects is also valid in any domain possessing n objects, and that any formula that is satisfiable in a domain possessing a definite finite number n of objects is satisfiable in
any domain possessing n objects. Even more striking is that for FOL without Identity, it can be
demonstrated 1) that any formula that is valid in a domain with a definite finite number n of objects
Course Notes
Page 14
Quantification
is also valid in every domain possessing n or fewer objects, and 2) that any formula that is satisfiable in a domain possessing a definite finite number n of objects is satisfiable in every domain—
finite or infinite—possessing n or more objects. (Note that because any formula that specifies the
cardinality of a domain is false in all domains that possess a different cardinality, no such formula
can be logically valid.)
Finally, something should be said about the relationship between provability and logical consequence. A well-formed formula (wff) F is provable or deducible from a set of formulae Γ of FOL
if F can be derived from Γ according to certain intuitively truth-preserving but formally well-defined
rules (transformations) given with the definition of the system. A formula F is a logical consequence of a set of formulae Γ if it is impossible for all of the formulae in Γ to be true while F is
false. A logically valid formula F of FOL is a formula that no conditions can make false, and an
unsatisfiable (or self-contradictory) formula F of FOL is a formula that no conditions can make true.
(A central objective of the present exposition is to make the notions of logical consequence, logical
validity, and unsatisfiability precise). In 1930, Kurt Gödel proved that all logically valid formulae of
FOL are formally derivable from a small set of transparently logically valid formulae of FOL, the
(valid) logical axioms of FOL, a circumstance that could not be assumed to hold in advance. The
proof that all logical consequences of the logically valid axioms of FOL are deducible from the
axioms of FOL (i.e., are theorems of FOL), together with the far simpler proof that all formulae
derivable from the axioms of FOL are logically valid, implies the critical metalogical result that the
set of theorems of FOL and the set of logically valid formulae of FOL are co-extensive.2
2
This result must not be confused with Gödel’s far more well-known and surprising Incompleteness Theorem
(usually denoted “Gödel’s Proof”), published in 1931. Very roughly stated, this latter remarkable metamathematical result implies that any consistent finitary formal system rich enough to express all arithmetical propositions necessarily permits the construction of a formula that is demonstrably both true and unprovable. In other terms (still
very rough), consider the class of systems yielded by the augmentation of the generic logical axioms by further
axioms that 1) permit the definition of the three operations ‘successor of’, ‘addition’, and ‘multiplication’ and
2) permit us to formally distinguish in finite terms between the well-formed formulae that are axioms and the wellformed formulae that are not. Gödel demonstrated that for any such system, there exists a definite and unambiguous proposition (closed formula) G—not necessarily the same for each system—such that neither G nor its negation
can be derived from the axioms of that system.
Course Notes
Page 15