Lecture 10: Semantics of Predicate
Logic
1
Outline
✤
I. Definition of Truth-in-a-Model
✤
II. Some Simple Examples
✤
III. Philosophical Dimensions of the Definition
2
Outline
✤
I. Definition of Truth-in-a-Model
✤
II. Some Simple Examples
✤
III. Philosophical Dimensions of the Definition
3
Recall: Valuations
✤
So in the first two weeks, we spent
a lot of time on truth-tables.
✤
In lecture 6, we noted that
valuations were essentially a
description of a row of a truth-table:
a description of what truth-values
the basic propositional letters (p,q,
etc.) got in addition to a description
of the rules for ∧, ∨, →, ↔, ¬.
Let Wff be the set of formulas. Let {T,F} be
the set of truth-values true and false.
✤
Then a valuation is a function v: Wff → {T,F}
such that for all formulas ϕ, ψ:
✤
i. v(ϕ∧ψ)=T iff v(ϕ)=T and v(ψ)=T,
ii. v(ϕ∨ψ)=T iff v(ϕ)=T or v(ψ)=T,
iii. v(¬ϕ)=T iff v(ϕ)=F,
iv. v(ϕ→ψ)=F iff v(ϕ)=T and v(ψ)=F
v. v(ϕ↔ψ) =T iff v(ϕ)=v(ψ)=T
or v(ϕ)=v(ψ)=F
4
How to Read the Definition
Let Wff be the set of formulas. Let {T,F} be
the set of truth-values true and false.
✤
Then a valuation is a function v: Wff → {T,F}
such that for all formulas ϕ, ψ:
✤
✤
Recall that “iff” is an abbreviation for
“if and only if.” So if you say p iff q,
this is just to say that p holds precisely
when q holds.
✤
So the way to the read the definition is
as defining the left-hand side in terms
of the right-hand side.
✤
This makes sense because the things on
the right-hand side are simpler and
have already been defined by clauses
already appearing earlier on in the
definition.
i. v(ϕ∧ψ)=T iff v(ϕ)=T and v(ψ)=T,
ii. v(ϕ∨ψ)=T iff v(ϕ)=T or v(ψ)=T,
iii. v(¬ϕ)=T iff v(ϕ)=F,
iv. v(ϕ→ψ)=F iff v(ϕ)=T and v(ψ)=F
v. v(ϕ↔ψ) =T iff v(ϕ)=v(ψ)=T
or v(ϕ)=v(ψ)=F
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In a picture . . . . .
✤
✤
We calculated the below truthtable in lecture 4 (example 3).
We can think about each row of
the truth-table as a valuation,
which we enumerate as V1, V2, V3, V4
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1
2
3
4
5
p
q
¬p
¬q
(¬p ∧ ¬q)
V1
T
T
F
F
F
V2
T
F
F
T
F
V3
F
T
T
F
F
V4
F
F
T
T
T
The Idea In What Follows
✤
The idea in what follows is that:
✤
✤
in propositional logic, we
started with truth-tables and
then noted that valuations
were a description of a row
of a truth-table.
in predicate logic, we start
with valuations.
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✤
The valuations are defined in
terms of models, which we
introduced to last time.
✤
On the next slide, we recall
the definition of model.
✤
Then we present the
fundamental definition of
truth-in-a-model, which is the
same thing as valuations in
the setting of predicate logic.
Recall: Definition of a Model
✤
Recall that a language L of predicate logic is just a collection of
individual constants a,b,c,d, . . . plus predicate letters F,G,R,S,. . .,
where it is understood from context that some are properties Fx and
others are relations Rxy (cf. Gamut vol 1 pp. 74-75). Relative to a
language L, we define the notion of a model (cf. Gamut vol 1 p. 91).
✤
A model in language L is a set M, called the domain, along with an
interpretation function I such that (i) if a is an individual constant, then I(a) is a member of M.
(ii) if F is a unary predicate, then I(F) is a subset of M.
(iii) if R is a binary predicate, then I(R) is a subset of MxM.
(iv) if S is an ternary predicate, then I(S) is a subset of MxMxM.
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Truth in a Model, Gamut p. 91
✤
Suppose that M is a model in a language L, and that for all a in M there is individual constant c from
L with I(c)=a. Then we define a valuation function VM: Sentences ➝ {T,F} as follows:
✤
Clauses for Atomic Formulas:
If R is an n-ary relation symbol and c1, . . . , cn are individual constants, then VM(Rc1⋅⋅⋅cn)=T iff 〈I(c1), . . . , I(cn)〉 ∈ I(R).
✤
Clauses for Propositional Connectives:
VM(ϕ∧ψ)=T iff VM(ϕ)=T and VM(ψ)=T,
VM(ϕ∨ψ)=T iff VM(ϕ)=T or VM(ψ)=T,
VM(¬ϕ)=T iff VM(ϕ)=F,
VM(ϕ➝ψ)=F iff VM(ϕ)=T and VM(ψ)=F,
VM(ϕ ↔ ψ)=T iff VM(ϕ)=VM(ψ)=T or VM(ϕ)=VM(ψ)=F
✤
Clauses for Quantifiers:
VM(∃x ϕ(x))=T iff there’s at least one constant d from L with VM(ϕ(d))=T
VM(∀x ϕ(x))=T iff for all constants d from L one has that VM(ϕ(d))=T
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Memorize the previous slide; how
to do that.
✤
The previous slide is the most
important thing in this lecture.
✤
Print off the previous slide and
memorize it.
✤
Again, recall that “iff” is an
abbreviation for “if and only
if.”
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✤
So the way to the read the
definition of the previous page
is as defining the left-hand side
in terms of the right-hand side.
✤
This makes sense because the
things on the right-hand side
are simpler and have already
been defined by clauses already
appearing earlier on in the
definition.
Flagging An Assumption
✤
In our definition of truth-in-amodel, we assumed that
everything in our models was
named by some constant letter.
✤
It turns out that one can define
a notion of valuation or truth in
a model which does not require
this assumption.
✤
This assumption greatly
simplifies greatly the definition
of truth-in-a-model. But this
assumption seems like a deep
idealization. For, there are
probably a lot of things out
there without names.
✤
This is the difference between
Gamut section 3.6.2 and 3.6.3.
This difference really won’t
matter for what we’re doing
here. So primarily for this
reason, we’ll just keep with our
idealizing assumption.
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Outline: where we’re at
✤
I. Definition of Truth-in-a-Model
✤
II. Some Simple Examples
✤
III. Philosophical Dimensions of the Definition
12
Simple Example: Karate Club
✤
✤
So the members of Karate club are
Alice, Bob, and Claire. Alice is
treasurer (t), Bob is secretary (s), and
Claire is president (p). The members
are also ranked as to who is the
superior Karate person. We write Rxy
for “x is superior to y in Karate.”
Formally, the model K of Karate club
is given by:
K = {Alice, Bob, Claire}
I(p)=Claire, I(t)=Alice, I(s)=Bob,
I(R) = {〈Alice, Bob〉, 〈Bob, Claire〉, 〈Alice, Claire〉}
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✤
Example: VK(Rtp)=T
✤
For, by the atomic clauses, we have
that this is true precisely when
(*)
〈I(t), I(p)〉 ∈ I(R)
But given that I(t) = Alice, I(p)=Claire,
this is the case precisely when
(**)
〈Alice, Claire〉 ∈ I(R)
And we can see that this holds by
looking at the specification of I(R)
over to the left.
Simple Example: Chess Club
✤
✤
So the members of Chess club are
Alice, Bob, and Claire. Alice is
president (p), Bob is treasurer (t), and
Claire is secretary (s). Some members
can vote on financial matters before
the club but others cannot. We write
Fx as an abbreviation for “x can vote
on financial matters.”
Formally, the model of C chess club is
given by:
C = {Alice, Bob, Claire}
I(p)=Alice, I(t)=Bob, I(s)=Claire,
I(F) = {Alice, Bob}.
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✤
Example: VC(Fp ∧ Ft ∧ ¬Fs)=T
✤
For, by the clauses for conjunction and
negation, it suffices to show that:
1.
VC(Fp)=T
2.
VC(Ft) =T
3.
VC(Fs)=F
✤
But by clauses for atomics, this
happens when
1.’
I(p)∈ I(F)
2.’
I(t)∈ I(F)
3.’
I(s)∉ I(F)
But by inspection of the sets to the
right, we see that this is the case.
Simple Example: Drama Club
✤
✤
So the members of Drama club are
Alice, Bob, and Claire. Alice is
treasurer (t), Bob is president (p), and
Claire is secretary (s). Some are better
at Shakespeare than others. We write
Rxy for “x is superior at Shakespeare
to y.”
Formally, the model D of drama club
is given by:
D = {Alice, Bob, Claire}
I(p)=Bob, I(t)=Alice, I(s)=Claire,
I(R) = {〈Alice, Bob〉, 〈Claire, Bob〉, 〈Alice, Claire〉}
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✤
Example: VD(∃x Rxp)=T
✤
Note that I(t)=Alice, I(p)=Bob, and
〈Alice, Bob〉 ∈ I(R).
✤
Then 〈I(t), I(p)〉 ∈ I(R).
✤
Then by clause for atomics,
(*)
VD(Rtp)=T
✤
Then by clause for existentials,
(**)
VD(∃x Rxp)=T
Outline: where we’re at
✤
I. Definition of Truth-in-a-Model
✤
II. Some Simple Examples
✤
III. Philosophical Dimensions of the Definition
16
Models and Truth-Tables
✤
A natural question to ask at this
point is, what is the relation
between models and truth
tables?
✤
The basic idea is that individual
models correspond to
individual rows in a truth-table.
✤
An individual row of the truthtable tells us the truth-values of
the basic propositional letters.
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✤
An individual model tells us
the truth value of atomic
statements such as Fa and Fb.
✤
In propositional logic, there are
only finitely many rows. There
are infinitely many models, so
we can’t enumerate them.
✤
Similarly, there is an algorithm
for determining truth in a truthtable. There’s no algorithm for
predicate logic.
Aim of Truth-in-a-Model
✤
A natural question to ask at this
point is: what are we trying to
do with truth-in-a-model?
✤
The basic idea is that we’re
pursuing our basic goal of
demonstrating how the truth of
the whole sentence depends on
the truth of its parts.
✤
Except now we’ve added a new
innovation: the interpretation
function.
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✤
The interpretation function tells
us which things in the model
correspond to which things in
our language. Given a constant
symbol c of our language, the
interpretation function tells us
that it picks out the object I(c) in
the model.
✤
The truth of the whole sentence
depends on the truth of its parts
and the interpretation of its
component parts.
Aim of Truth-in-a-Model
✤
This idea of ours is intuitively
plausible: truth of atomic sentences
should depend on how our words
relate to the world and how the world
is.
✤
E.g. truth of Rsp in Karate Club
depends on the fact that (i) s picks out
Bob in Karate club, (ii) p picks out
Claire in Karate club, and (iii) R picks
out being superior at Karate, and (iv)
Bob is superior to Claire at Karate.
Drama Club
Karate Club
Alice
Bob
Claire
Superior at
Karate
Language L:
p
t
s
R
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Alice
Bob
Claire
Superior at
Shakespeare
Why care if truth-in-a-model is
truth?
✤
So truth can initially seem to be a
mysterious concept. What we know
about it seems unlike what we know
about red and mass and inflation.
✤
But Tarski pointed out that you can
read the left-hand sides of the
definition of truth-in-a-model (which
mention truth) as being defined in
terms of the right-hand sides, which
ultimately boil down to which
words pick out which things.
✤
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From Coffa’s history of the Vienna
Circle: “Carnap used to tell his
students a story about the first time
Tarski explained to him his ideas on
truth. They were at a coffeehouse,
and Carnap challenged Tarski to
explain how truth was defined for
an empirical sentence such as ‘This
table is black.’ Tarski answered that
‘This table is black’ is true iff this
table is black; and then, Carnap
explained, ‘the scales fell from my
eyes’ “ (Coffa, The Semantic Tradition
from Kant to Carnap: To the Vienna
Station p. 304).
Is Truth-in-a-model Truth?
✤
Well, what is the alternative?
✤
Another important idea is the
deflationary conception of truth.
According to this view, all there is
to truth is the following:
✤
✤
ϕ is true iff ϕ
This is the so-called
disquotational scheme.
✤
On this view, you don’t need to
say anything about how our
language corresponds to items in
the world.
✤
One traditional problem case for
this view has been so-called selfreferential sentences. Suppose you
walked into the room and there
was only one red sentence on the
screen and it said:
✤
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the red sentence on the screen
is false.
Ω
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