Download IM_FA16-03-PredicateLogic

INF385T: Information Modeling — Class 03
Unit 1: First Order Logic
Predicate Logic: Introduction
Slides for September 7
Karen M. Wickett
School of Information
University of Texas at Austin
Fall 2016
1
Unit 1 Roadmap
• Class 00: Basic Concepts
• Class 01: Sentence Logic: Syntax and Semantics
• Class 02: Sentence Logic: Inferencing
• Class 03: Predicate Logic: Introduction
• Class 04: Predicate Logic: Transcription and Validity
• Class 05: Predicate Logic: Inferencing: Truth Trees
• Class 06: Metatheory
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The Problem
How do you show, in SL, that the following argument is valid?
•
All men are mortal
All politicians are men
Therefore all politicians are mortal.
The SL formulation of the above argument is…
1)
P
2)
Q
3)
Therefore, R
The validity verdict for this formalization is: invalid
[There is an assignment of truth values to the atomic sentences that will make
the premises true and the conclusion false, namely P=True, Q=True, R=False]
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The Problem with Sentence Logic
• SL has let us down. It can’t represent intuitively valida arguments in way that
allows us to determine validity
• More particularly, it cannot represent the internal structure of atomic sentences.
• SL represents assertions atomically,
or as compounds of atomic sentences (etc.)
• It does not let us talk about things and properties of things.
The problem is not just in the inferencing system, but in the formal language itself:
SL is just not sufficiently expressive,
it cannot expose the logically relevant features.
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The PL Project - Goals
The objective…
A formal language that can remedy the shortcomings of SL, modeling more
arguments, supporting the derivation of inferences from modeled information.
• More specifically, a language that allows us to attribute properties to things
• Using predication to express logically relevant facts about individuals
—such as the properties those individuals have (or don’t have)
—and the relations they participate in
• In a way that will allow us to reason
— about those individuals
— about the domain generally
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The PL Project - Language
The language consists of a syntax and a semantics
• PL Syntax
• The vocabulary of PL and a recursive/inductive definition of how the
vocabulary items of PL can be combined
• More specifically:
what is, and what isn’t, a sentence of PL.
• PL Semantics
• A formal account of what it means for a sentence in PL to be true.
• And of closely related semantic notions
(consistency, logical truth, logical consequence, equivalence, etc)
• A precise definition of a relevant variety of (deductive) validity.
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The PL Project - Inference
For predicate logic to be useful we need some techniques for
determining valid inferences.
PL Inferencing Systems (aka “proof systems”)
—Natural Deduction
(see Teller if interested in the PL version)
—Truth Trees
(what we’ll learn)
—axiom-based proof systems
—Resolution
Will increased expressive power have an impact on inference systems?
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Sentences (w/o qunatifiers) in PL — Intuitively
Atomic sentences with no variables or quantifiers
F, G, H, …
are predicates,
acting like adjectives and verbs
a, b, c, …
are names
acting like proper names
Fa
Ha
Lab
Gafb
(pronounced “a is F”)
… Alvin is funny.
… Alvin is handsome.
… Alvin loves Belinda
… Alvin gave Fido to Belinda.
Compounds of those
• (Fa v Ha)
(Fa & Ha)
Lab  Lba
(Ga v A)
• (Fa v Ha) & Lab
The connectives have their familiar meanings.
• So (Fa & Ha) is true only when (Fa) is true and (Ha) is true.
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Quantified Sentences in PL - Intuitively
Vocabulary
x, y, z, …
are variables
similar to pronouns or variables in algebra
 and 
are quantifiers, universal and existential
similar to
“For all…” ()
“There is a…” ()
Quantified sentences and their “readings”
(top to bottom = logic-like to English-like)
(x)(Fx v Gx)
—“For all x, either x is funny or x is gross”
— “Every x is such that x is funny or x is gross.”
— “Everything is such that it is funny or it is gross”
— “Everything is either funny or gross.”
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Existentially Quantified Sentences - Intuitively
(x)(Fx)
“There exists an x such that x is funny”
“Some x is such that x is funny”
“Something is such that it is funny”
“Something is funny.”
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Negating a Universally Quantified Statement
If it isn’t that case that everything is funny
then there must be at least one thing that isn’t funny.
(and vice versa)
~(x)(Fx)
“It is not the case that for all x, x is funny.”
“For some x, it is not the case that x is funny.”
“Something is not funny.”
This is equivalent to “(x)~(Fx)”.
More on “quantifier interchange” later.
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Nested Quantifiers - intuitively
(x)(y)(Lxy v Ux)
For all x, there exists a y such that (x loves y or x is stingy)
alternatively: “for all” might be read as “each” or “every”
English: For everything there is something that is loved by that thing or that
thing (the one not doing any loving) is stingy.
• English has trouble unambiguously connecting multiple instances of its single
impersonal pronoun with the correct antecedents.
• English has only a few pronouns (or pronominal locutions like “that thing”),
so it is sometimes unclear what they refer to.
Bill saw a pie on the TV; he turned it off and went home to eat some.
Bill saw a pie on the TV; he decided it was what he wanted for lunch.
• Logic has lots of pronouns (it has as many variables as we want to use),
which can make things clearer (if less elegant).
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Informal Metalinguistic Interlude
• We use a metalanguage to talk about an object language.
• SL and PL are object languages for us;
we define them and talk about them in a metalanguage.
• Formation rules and valuation rules, are clearly in the metalanguage.
• And so, less obviously, are equivalence rules
• The actual well-formed expressions of SL and PL are in the object language.
• The metalanguage, more or less carefully written logic-like English, is not
formally defined, with the exception of a few special notions.
• Of course we could formally define the metalanguage using another
metalanguage,
• but then …
•
(well, you see the problem.)
[Could we have one language that is its own metalanguage? English is like that after all
(Answer: No, it is not a good idea. More later)]
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Metalinguist conventions
In a metalanguage we often need to refer to expressions in the object language.
We do that two ways:
• Forming the name of the expression with quotation marks
—e.g.: “x” is a variable.
• Using a metalinguistic variable
e.g.:
— If u is a variable then Fu is a sentence.
— If X is a sentence of SL then X & Y is a sentence of SL.
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Use and Mention
Why do we need to be so fussy?
So that we don’t make mistakes in the metalanguage
e.g.
1) Boston has two syllables
2) Boston has two million people
3) Therefore something has two syllables and two million people.
Now what exactly is that thing? A word? A city? Something that is both?
Really, there is no such weird thing which is both a word and a city.
Compare this argument with the one above.
1) “Boston” has two syllables
2) Boston has two million people
3) Therefore something has two syllables and two million people?
Anything closer to home? The colloquial use of the Dublin Core metadata system as well as
early RDF use has some use/mention disasters that are even now obstacles to the federation
of Dublin Core metadata and the consistent use of RDF and other semantic web languages.
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Metalinguistic (aka “Metasyntactic”) Variables
Ok in the metalanguage…
•
To refer to PL variables (x, y, z…) we use metalinguistic variables u, v.
•
To refer to the PL names (a, b, c…) we use metalinguistic variables s, t.
•
To refer in the metalanguage, to PL sentences we use metalinguistic variables X, Y.
[By refer, above, I mean refer as pronouns do.]
•
To directly name, in the metalanguage, any vocabulary item, we use quotation marks surrounding the
item referred to, unless the placement of the item in the sentence, on the page, or on the slide makes it
obvious we are mentioning not using the item: “a” “x” “F”, “u”, “s”, “X”.
An aside…
Two metalinguistic variables commonly used (mostly informally) in the computing community are
“foo” and “bar”. The classic discussion, not be be missed, is in the famous jargon file:
•
There is also an internet RFC (!) devoted to their etymology:
•
RFC 3092 Etymology of “Foo”, D. Eastlake, C. Manros, E. Raymond, 2001.
http://www.ietf.org/rfc/rfc3092.txt?number=3092
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Syntax for Predicate Logic
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Semiformal Syntax for PL: Vocabulary (Teller’s terminology)
Names = {a, b, c, …}
•
Used to refer to specific individual things; aka “individual constants”;
Variables = {x, y, z, …}
•
Used with quantifiers to support general reference to individual things in assertions;
aka “individual variables”.
(Names and variables may be collectively referred to as terms.)
Predicates = {F, G, H, … }
•
These attribute properties or relations to things
•
Always followed by one or more terms as arguments: “Fa”, “Gab”; “Fx”, “Gxy”.
•
How many “places” a predicate has (its “arity”) doesn’t vary for a given language.
•
Order counts, both syntactically and semantically. “Fab” is a different sentence than
“Fba”, and may well have a different truth value.
Connectives:
Quantifiers = {, }
(familiar truth-functional connectives) = {~, v, &, , }
Sentence letters = {A, B, C, … }
Parentheses = { {,(, ),} }
NB: metalinguistic variables aren’t part of the PL vocabulary and do not occur in PL expressions.
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Semiformal Syntax for PL: Grammar
(In what follows a term is a name or a variable.)
1. All atomic sentences of SL are atomic sentences of PL
2. Predicates followed by the appropriate number of terms (names or variables) are
atomic sentences of PL.
3. If X is a sentence of PL and u is a variable, then (u)X is a sentence and (u)X
is a sentence of PL.
4. If X and Y are sentences of PL then
( ~ X ), ( X v Y ), (X & Y), ( X  Y ), ( X  Y) are sentences of PL.
5. Nothing else is a sentence of PL.
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Quantification Terminology
•
A quantifier governs the shortest full sentence which follows it and binds the variables
in the sentence it governs …
•
[meaning] the quantifier applies to all of the occurrences of the same variable in
the shortest full following sentence.
(x)Fx  (x)Gx
The quantifier “(x)” governs “Fx”; “(x)” governs “Gx”.
•
The shortest full sentence following a quantifier is the quantifier’s scope.
•
(“scope” and “governed sentence” here are essentially synonymous).
•
A variable u is bound just in case it occurs in the scope of the quantifier (u) or (u).
•
A variable u is free just in case it is not bound; that is, does not occur in the scope of
any quantifier, (u) or (u). (Teller p30).
•
An open sentence is a sentence with one or more free variables.
•
A closed sentence has no free variables, all are bound.
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Semantics for PL: The basic ideas
In SL we used valuation rules to define the truth value of compound sentences in terms of
their components, and ultimately in terms of their atomic sentences.
First the basic ideas…
•
There is a world of things, and those things have properties
(including properties that are relationships between or amongst things).
•
The specific things that there are, and the specific properties they have, determine which
atomic PL sentences are true.
•
Which atomic PL sentences are true determines which non-atomic PL sentences are true.
•
SL comparison: which atomic SL sentence are true, determines which compound SL sentences
are true.
•
A PL argument is valid just when there is no possible way properties can be distributed
to things that would make the premises true and the conclusion false.
•
Which is to say, no set of atomic PL sentences, or negations of atomic PL sentences, that
would make the premises true and the conclusion false.
•
SL comparison: an SL argument is valid iff there is no possible assignment of truth values to
its atomic propositions that would make the premises true and the conclusion false.
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Semiformal Semantics for PL: The basic ideas
•
There is a world of things, and those things have properties
•
The specific things that there are, and the specific properties they have,
determine which atomic PL sentences are true.
• This information is given by an interpretation
•
Which atomic PL sentences are true
determines which non-atomic PL sentences are true.
•
This is determined by the meanings of the connectives
(including the quantifiers)
•
A PL argument is valid (PL-valid) just when there is no possible way that
properties can be distributed to things that would make the premises true and the
conclusion false.
•
A PL argument is valid if and only if there are no interpretations that make the
premises true and the conclusion false
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Semiformal Semantics for PL: Interpretation [Teller]
In SL we used truth tables to describe the semantics of the language, and to
keep track of what propositions were true or false.
But truth tables don’t suffice for PL, we need interpretations.
An interpretation is
a) A non-empty collection of objects (the domain)
b) An assignment of each object to (at least one) name.
c) A list of predicates.
d) An assignment to each n-place predicate of a set of sequences of n
objects of which the predicate is true, and the sequence of n objects
for which the predicate is false.*
(This will determine for all possible atomic sentences formed, which are
true and which false)
e) Assignments of truth values to sentence letters, if any.
*Strictly speaking, the clause in blue is unnecessary. The object sequences for which a
predicate is false can be determined by simply noting that there is no assignment of the
sequence to the predicate. (optional: cf. “the closed world hypothesis”)
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Semiformal Semantics for PL: Interpretation Teller
Teller defines interpretation as on the preceding slide.
He goes on to develop this way of specifying an interpretation:
a) A non-empty collection of objects (the domain), indicated by a set of
names.
b) A conjunction that, for a given set of predicates and the domain, includes
each possible atomic sentence or its negation. (nb: “or its negation”)
A Tellerian specification of an interpretation looks like this:
— D = {b, s, t}
— Gs & ~Gb & ~Gt
& Kb & Ks & Kt
& Lsb, & Lbs, & Ltt, & ~Lss, & ~Lst, & ~Lbb, & ~Lbt, & ~Lts, &
~Ltb.
Given that this is how he wants us to specify interpretations it is not surprising he
defined interpretation as his did. But there is more to say on this…
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Interpretation of a Sentence
•
An interpretation of a sentence
An interpretation specification that includes all the names,
predicates, and SL sentence letters that occur in that sentence.
If it meets that criterion it will contain enough information to determine
whether or not the sentence is true or false in the interpretation.
•
The following examples are primarily to assure ourselves of what an
interpretation of a sentence is.
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Universal interpretation example
•
(x)Fx
•
Interpretations of (x)Fx :
[D={a}; Fa ]
(x)Fx is true in that interpretation
[D={a,b}, ~Fa & Fb ]
(x)Fx is false in this interpretation
•
Not an interpretation of (x)Fx :
[ D={a, b}, Ga & Gb]
—
(must include either “Fa” or “~Fa” and either “Fb” or “~Fb”)
26
Interpretation of an existential sentence
• (x)(Fx & Rxe)
• Interpretations of (x)(Fx & Rxe)
[D={e}, Fe & Ree ]
Is the sentence true in this interpretation?
[D={a,e}, ~Fa & Fe & ~Ree & ~Rae & Rea & Raa]
• Not interpretations of (x)(Fx & Rxe)
—[D={a,e}, ~Fa & ~Fe]
Nothing about R
—[D={a,e}, ~Fa&~Ree&~Rae&Rea&Raa] Nothing about Fe
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Possible Confusions about “interpretations”
•
The meaning of the phrase "interpretation of a sentence" clashes completely and
radically with the natural English sense one might give to it.
•
Note that a sentence and its negation always have exactly the same set of possible
interpretations. That's right, even though they "say the opposite" they have the same
sets of interpretations!
•
For instance, these two sentences have exactly the same set of interpretations if you
use Teller's rule for interpretation-of-a-sentence construction:
•
"Fa v Fb" and "~(Fa v Fb)". Namely they each have these four interpretations...
— D={a, b} Fa&Fb
— D={a, b} ~Fa&Fb
— D={a, b} Fa&~Fb
— D={a, b} ~Fa&~Fb
•
In forming an "interpretation of a sentence" the only thing that matters is what names
and predicates were used in the sentence, not how they were used!
•
An interpretation of a sentence is a way the things talked about in a sentence could be.
•
And the set of all interpretations of a sentence is the set of all the ways the things
talked about could be.
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Minimal Interpretations: the recipe.
To spell it out in detail, here's how to form a (Tellerian) minimal interpretation of a
sentence.
Preparation, step by step. (involves three scratch lists)...
1. list the predicates used in the sentence (list A)
2. list the names used in the sentence (list B)
if there are no names, just variables, put the name "a" in list B anyway;
this must be done to satisfy the "no empty domain" rule.
3. from list A and B form every combinatorially possible atomic sentence (list C)
i.e. every atomic sentence that can be formed given those lists.
4. form the domain set D from the list of names in list B.
5. form the fact-conjunction this way:
•
for each atomic sentence in list C (the list of all possible atomic sentences
using the names and predicates that figure in the original PL sentence), include
as a conjunct in the fact-conjunction either that atomic sentence itself, or the
negation of that atomic sentence.
29
Notes on Minimal Interpretations
•
The recipe does not produce a unique minimal interpretation.
•
Minimal interpretations are not unique.
•
Every sentence has a number of minimal interpretations (even “Fa” has two)
depending on what atomic sentences are affirmed or denied in the factconjunctions.
•
You form the one you want to examine, or all of them if you need to.
•
Why do we call these minimal interpretations?
•
Because this is the simplest way to form interpretations that will always let us
determine T or F for the PL sentence in question.
•
Its simplicity lies in two things:
1.
2.
it ignores connectives and quantifiers and
given that it ignores connectives and quantifiers, there are no interpretations that
have fewer parts and can be assured to determine the truth value.
30
Substitution Instance Defined
•
For any universally quantified sentence (u)(…u…), the substitution instance of
the sentence, with the name s substituted for the variable u is (…s….),
•
the sentence formed by dropping the initial universal quantifier and writing s
for all free occurrences of u in (…u…).
•
For any existentially quantified sentence (u)(…u…) the substitution instance of
the sentence, with the name s substituted for the variable u is (…s….),
•
the sentence formed by dropping the initial existential quantifier and writing s
for all free occurrences of u in (…u…).
•
Yes, these are almost identical, so we could also say something like…
•
For any  quantified sentence (u)(…u…) the substitution instance of the
sentence, with the name s substituted for the variable u is (…s….),
— the sentence formed by dropping the initial  quantifier and writing s for all free
occurrences of u in (…u…).
31
Substitution Instance examples
•
(x)(Fx & Gx) has substitution instances such as …
Fa & Ga [a/x] and
•
Fb & Gb [b/x] etc.
(x)(Fx & Gx) has substitution instances such as …
Fa & Ga [a/x] and Fb & Gb [b/x] etc.
•
(x)( y)(Fx  Gy) has substitution instances such as…
(y)(Fa  Gy) [a/x] and (y)(Fb  Gy) [b/x] etc.
32
True in an interpretation: universally quantified sentences
•
A universally quantified closed sentence is true in an interpretation =df
all of the sentence’s substitution instances formed with names in the
interpretation are true in the interpretation.
•
Since (x)Fx is true in an interpretation just when all of its substitution
instances formed with names in the interpretation are true in that
interpretation,
•
it is true in an interpretation just when the conjunction of all of those
substitution instances is true in the interpretation.
• Which is to say that any universally quantified sentence is equivalent to
the conjunction all of its substitution instances.
33
example
(x)(Fx v Gx)
• Let I be
D= {a,b}
Fa & ~Fb & ~Ga & Gb
• Under I, (x)(Fx v Gx) has substitution instances
(Fa v Ga) and (Fb v Gb)
• Since the first disjunct, Fa, of the first substitution instance (Fa v Ga) is true
in I, the first substitution instance is true in I;
• since the second disjunct (Gb) of the second substitution instance is (Fb v Gb)
true in I, the second substitution instance is true in I
• Since both (all) substitution instances of (x)(Fx v Gx) are true in I,
(x)(Fx v Gx) is true in I
34
True in an interpretation: existentially quantified sentences
• An existentially quantified closed sentence is true in an interpretation =df
at least one of the sentence’s substitution instances formed with names in the
interpretation is true in the interpretation.
• Since an existentially quantified sentence is true in an interpretation just when
one of its substitution instances are true in the interpretation,
• an existentially quantified sentence is true in an interpretation just when
the disjunction of all of its substitution instances is true in the
interpretation.
• Which is to say that any existentially quantified sentence is equivalent to the
disjunction all of its substitution instances.
35
True in an interpretation: existentially quantified sentences example
(x)(Fx v Gx)
• Let I be
D= {a,b}
Fa & ~Fb & ~Ga & ~Gb
• Under I (x)(Fx v Gx) has substitution instances (Fa v Ga) and (Fb v Gb)
• Since the first disjunct, Fa, of the first substitution instance (Fa v Ga) is true in
I, the first substitution instance is true in I;
• since both disjuncts of the second substitution instance (Fb v Gb) are false in
I, the second substitution instance is false in I .
• Since at least one substitution instance of (x)(Fx v Gx) is true in I
(x)(Fx v Gx) is true in I
36
Quantifiers and their logically equivalent corresponding compounds
• The definitions just given imply that in an interpretation where D = {a, b, c … }
the following biconditionals are true for arbitrary quantified sentences…
(x)Fx  Fa & Fb & Fc …
• That is, since a universally quantified sentence is true if and only if every
one of its substitution instances is true, then it is true if and only if the
conjunction of its substitution instances is true.
(x)Fx  Fa v Fb v Fc …
• That is, since a existentially quantified sentence is true if and only if at least
one of its substitution instances is true, then it is true if and only if the
disjunction of its substitution instances is true.
• We can call these corresponding compounds [for a given interpretation].
37
Multiple Quantifiers
The situation with multiple quantifiers is interesting and not completely handled
in Teller.
There are two sorts of cases:
(i) where the sentence is a compound sentence containing quantifiers:
(x)(Fx) & (x)(Gx)
(ii) where the sentence is a quantified sentence with nested quantifiers:
(x)(y)(Fx  Gy)
Developing substitution instances and corresponding compounds for these are
applications of what we already know and the first case is in fact fairly
straightforward.
38
Multiple Quantifiers
We can apply the definitions of “substitution instance” to a sentence with nested
quantifiers and ultimately reach a substitution instance that contains no variables,
only closed atomic propositions and negations of closed atomic propositions.
We illustrate with an example where the main connective is a universal quantifier and
nested sentence (the immediate constituent) is an existential quantifier.
1.
A sentence with nested quantifiers; a domain with two objects.
(x)(y)(Fx  Gy) D={a, b}
2.
We rewrite the universally quantified sentence as an equivalent conjunction of its
substitution instances.
(y)(Fa  Gy) & (y)(Fb  Gy)
3.
We rewrite each existentially quantified sentence as an equivalent disjunction of its
substitution instances. And note that all quantifiers are gone (!).
((Fa  Gb) v (Fa  Ga) ) & ( (Fb  Ga) v (Fb  Gb))
39
Where this is all going, eventually…
• Once we get a quantified sentence expanded into a quantifier-free
equivalent expression (for a domain), we can compare it to the factconjunction for any interpretation of that sentence
• and using truth-functional methods (directly or indirectly) discover
whether or not the sentence is true in that interpretation.
• Why do truth functional methods suffice?
• Because for every atomic sentence in the quantifier-free compound
either that atomic sentence or its negation occurs in the fact
conjunction.
40
Quantifier Interchange: Intuitions
•
The preceding definitions of “true in an interpretation” can be demonstrated (by reasoning
using DeMorgan’s Law) to imply the following two equivalences (using Teller’s
notation)…
•
(u)(…u…)  ~u~(…u…)
•
(u)(…u…)  ~u~(…u…)
•
It may be easier to see the relationship by considering actual simple PL sentences a
examples:
•
(x)(Fx)  ~( x)~(Fx)
•
( x)(Fx)  ~(x)~(Fx)
•
Apart from the proof Teller gives and I reprise on the next slide these equivalences should
make intuitive sense to you, so before you go to the next slide be sure, again, that you see
that the following two observations are true,
•
Everything has a property F if and only if it is not the case that there is something that
lacks F.
[(x)(Fx)  ~( x)~(Fx)]
•
Something has a property F if and only if it is not the case that everything lacks F.
[(x)(Fx)  ~(x)~(Fx)]
41
Quantifier Interchange: Proof
We now reason about PL semantics to show that
•
(u)(…u…)  ~u~(…u…)
[From which it follows easily by DN that: (u)(…u…)  ~u~(…u…)]
•
•
(u)(…u…)  ~u~(…u…)
1.
(x)Fx
(Fa & Fb & Fc…)
2.
~(~Fa v ~Fb v ~Fc…)
3.
~(x)~Fx
Given / or Assumption
from 1) by defn of “true in an interpretation” for  sentences
from 2) by Double Negation + DeMorgan’s
from 3) by defn of “true in an interpretation” for  sentences
~u~(…u…)  (u)(…u…) .
1.
~[(x)~Fx]
2.
~[~Fa v ~Fb v ~Fc…)]
3.
~[~(Fa & Fb & Fc…)]
4.
(Fa & Fb & Fc…)
5.
(x)Fx
Given / or Assumption
from 1) by defn of “true in an interpretation” for  sentences
from 2) by DeMorgan’s
from 3) by Double negation
from 4) by defn of “true in an interpretation” for  sentences
•
So since (u)(…u…)  ~u~(…u…) and ~u~(…u…)  (u)(…u…)
then (u)(…u…)  ~u~(…u…)
•
NB: this is not a proof in PL, this is a proof about PL!
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Semiformal Semantics for PL: Interpretation [a revision of Teller]
[This is a slight variation of Teller’s original definition, closer to standard practice, and more
formally correct. This brings Teller closer to common practice, helps prevent some larger
misunderstandings, and provides us with some useful insights into how interpretations work.]
An interpretation is
a) A non-empty collection of objects (the domain)
b) An assignment of each object to (at least one) name.
c) A list of predicates.
d) An assignment of sets of sequences of n objects to each n-place predicate
(Indicating which objects the predicates are true of in the interpretation.
And thereby determining for all possible atomic sentences formed which are true
and which are false)
e) Assignments of truth values to sentence letters, if any.
Negation as absence of assertion
NB: This approach indicates false atomic propositions only indirectly.
[By not including in the list of assignments of sequences to predicates the
assignment of a particular object sequence to a particular predicate.]
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An intuitive example
An intuitive example interpretation,
a) Domain:
{Bill, Sue,Tom}
b) Name assignments:
c) Predicates:
Sue gets “s”, Bill gets “b”, Tom gets “t”
“G” ( _is-glib); “K” (_is-kind); “L” (“_loves_”);
d) Object sequences assigned to predicates:
“G” gets <s>;
“K” gets <b>, and <s> and <t>
“L” get <s,b>, <b,s>, <t,t>
This is a description of the situation where Sue is glib, everyone is kind, Sue
and Bill love each other; Tom loves himself.
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Haddock’s Eyes
Haddock’s Eyes. [Alice and the White Knight]
From Alice Through the Looking Glass
by the famous Oxford logician Charles Lutwidge Dodgson, also known as Lewis Caroll.
Alice was walking beside the White Knight in Looking Glass Land.
"You are sad." the Knight said in an anxious tone: "let me sing you a song to comfort you."
[...] The name of the song is called 'Haddocks' Eyes.'”
"Oh, that's the name of the song, is it?" Alice said, trying to feel interested.
"No, you don't understand," the Knight said, looking a little vexed. "That's what the name is called.
The name really is 'The Aged, Aged Man.'"
"Then I ought to have said 'That's what the song is called'?" Alice corrected herself.
"No you oughtn't: that's another thing. The song is called 'Ways and Means' but that's only what it's
called, you know!"
"Well, what is the song then?" said Alice, who was by this time completely bewildered.
"I was coming to that," the Knight said. "The song really is 'A-sitting On a Gate': and the tune's my
own invention."
So saying, he stopped his horse and let the reins fall on its neck: then slowly beating time with one
hand, and with a faint smile lighting up his gentle, foolish face, he began…
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