CAS LX 502
3b. Truth and logic
4.1-4.4
Desiderata for a theory of
meaning
• A is synonymous with B
• A has the same meaning as B
• A entails B
• If A holds then B automatically holds
• A contradicts B
• A is inconsistent with B
• A presupposes B
• B is part of the assumed background against which A is said.
• A is a tautology
• A is automatically true, regardless of the facts
• A is a contradiction
• A is automatically false, regardless of the facts
Intuitions about logic
• If it’s Thursday, ER will be on at 10.
It’s Thursday.
ER will be on at 10.
Modus Ponens
• Logic is essentially the study of valid
argumentation and inferences.
• If the premises are true, the conclusion will be true.
Truth out there in the world
• A statement like It’s Thursday is either true
(corresponding to the facts of the world) or it is
false (not corresponding to the facts of the world).
• Same for the statement ER is on at 10.
• It turns out that modus ponens is a valid form of
argument, no matter what statements we use. Let’s
just say we have a statement—we’ll call it p. The
statement (proposition) p can be either true or
false. And another one, we’ll call it q.
Modus ponens
• So, whatever p and q are:
• If p then q.
p.
q.
• Granting the premises If p then q and p, we
can conclude q.
Other forms of valid argument
• If it is Thursday, then ER is on at 10.
ER is not on at 10.
It is not Thursday.
Modus Tollens
• If p then q.
q.
p.
• It is Thursday = p
• It is not Thursday = p.
T
F
F
T
An invalid argument
• Incidentally, some things are not valid
arguments. Modus ponens and modus
tollens are. This is not:
• If it is Thursday, then ER is on at 10.
It is not Thursday
*ER is not on at 10.
Other forms of valid argument
• If it is Thursday, then ER is on.
If ER is on, Pat will watch TV.
If it is Thursday, the Pat will watch TV.
Hypothetical syllogism
• If p then q.
If q then r.
If p then r.
Other forms of valid argument
• Pat is watching TV or Pat is asleep.
Pat is not asleep.
Pat is watching TV.
Disjunctive syllogism
• p or q.
q.
p.
Logical syntax
• A proposition, say p, has a truth value. In light of
the facts of the world, it is either true or false. The
conditions under which p is true is are called its
truth conditions.
• We can also create complex expressions by
combining propositions. For example, q. That’s
true whenever q is false.  is the negation
operator (“not”).
Logical connectives
• We can combine propositions with
connectives like and, or. In logical notation,
“p and q” is written with the logical
connective  (“and”): p  q; “p or q” is
written with  (“or”): p  q.
• p  q is true whenever p is true and q is
true. Whenever either p or q is false, p  q
is false.
Truth tables
• We can show the effect of logical operators
and connectives in truth tables.
p
T
F
p
F
T
p
q
pq
p
q
pq
T
T
T
T
T
T
T
F
F
T
F
T
F
T
F
F
T
T
F
F
F
F
F
F
Or v.  v. e
• The meaning we give to or in English (or any
other natural language) is not quite the same as the
meaning that of the logical connective .
• We’re going to South Carolina or Oklahoma.
• Seems odd to say this if we’re going to both South Carolina
and Oklahoma.
• You will pay the fine or you will go to jail.
• Seems a bit unfair if you get put in jail even after paying the
fine.
• We will preboard anyone who has small children or
needs special assistance.
• Doesn’t seem to exclude people who both need special
assistance and have small children.
Or v.  v. e
• There are two interpretations of or, differing in their
interpretation with respect to what happens if both
connected propositions are true.
• Exclusive or (e) is “either…or…but not both.”
• Inclusive or (disjunction; ) is “either…or…or both.”
p
T
T
F
F
q
T
F
T
F
pq
T
T
T
F
p
T
T
F
F
q
T
F
T
F
peq
F
T
T
F
Material implication
• The logic of if…then statements is covered
by the connective .
• If it rains, you’ll get wet.
(pq, where p=it rains, q=you’ll get wet)
p
T
T
F
F
q
T
F
T
F
pq
T
F
T
T
• What is the truth value of If it
rains, you’ll get wet?
• Well, it’s true if it rains and you
get wet, it’s false if it rains and
you don’t get wet. But what if it
doesn’t rain?
Material implication
• If John is at the party, Mary is.
(pq, where p=John is at the party, q=Mary is at the party)
• Suppose that’s true, and that John is at the party.
• We can conclude that Mary is at the party.
p
T
T
F
F
q
T
F
T
F
pq
T
F
T
T
• That is:
pq.
p.
q.
Material implication
• If John is at the party, Mary is.
(pq, where p=John is at the party, q=Mary is at the party)
• Suppose that’s true, and that John is at the party.
• We can conclude that Mary is at the party.
p
T
T
F
F
q
T
F
T
F
pq (pq)p ((pq)p)q
T
F
T
T
• That is:
pq.
p.
q.
Material implication
• If John is at the party, Mary is.
(pq, where p=John is at the party, q=Mary is at the party)
• Suppose that’s true, and that John is at the party.
• We can conclude that Mary is at the party.
p
T
T
F
F
q
T
F
T
F
pq
T
F
T
T
(pq)p
T
F
F
F
((pq)p)q
T
T
T
T
• That is:
pq.
p.
q.
Material implication
• If John is at the party, Mary is.
(pq, where p=John is at the party, q=Mary is at the party)
• Suppose that’s true, and that Mary is not at the party.
• We can conclude that John is not at the party.
p q p q pq (pq)q ((pq)q)p
T
T
F
F
T
F
T
F
F
F
T
T
F
T
F
T
T
F
T
T
• That is:
pq.
q.
p.
Material implication
• If John is at the party, Mary is.
(pq, where p=John is at the party, q=Mary is at the party)
• Suppose that’s true, and that Mary is not at the party.
• We can conclude that John is not at the party.
p q p q pq (pq)q ((pq)q)p
T
T
F
F
T
F
T
F
F
F
T
T
F
T
F
T
T
F
T
T
F
F
F
T
T
T
T
T
• That is:
pq.
q.
p.
Material implication
• If John is at the party, Mary is.
(pq, where p=John is at the party, q=Mary is at the party)
• Suppose that’s true, and that Mary is at the party.
• Can we conclude that John is at the party?
p q p q pq (pq)q
T
T
F
F
T
F
T
F
F
F
T
T
F
T
F
T
T
F
T
T
T
F
T
F
((pq)q)p
• That is:
pq.
q.
p.
Material implication
• If John is at the party, Mary is.
(pq, where p=John is at the party, q=Mary is at the party)
• Suppose that’s true, and that Mary is at the party.
• Can we conclude that John is at the party? NOPE!
p q p q pq (pq)q
((pq)q)p
T
T
F
F
T
T
F
T
T
F
T
F
F
F
T
T
F
T
F
T
T
F
T
T
T
F
T
F
• That is:
pq.
q.
p.
Biconditional
• The last basic logical connective is the
biconditional  or  (“if and only if”).
• pq is the same as (pq)(qp).
• It says essentially that p and q have the
same truth value.
Truth and the world
• In most cases, the truth or falsity of a statement has to do
with the facts of the world. We cannot know without
checking. It is contingent on the facts of the world
(synthetic).
• John Wilkes Booth acted alone.
• Sometimes, though, the very form of the statement
guarantees that it is true no matter what the world is like
(analytic).
• Either John Wilkes Booth acted alone or he didn’t.
• John Wilkes Booth acted alone and he didn’t.
• The first is necessarily true, a tautology, the second is
necessarily false, a contradiction.
Limits of propositional logic
• There are some kinds of logical intuitions that are
not captured by propositional logic. For example:
• All men are mortal.
Socrates is a man.
Socrates is mortal.
• Try as we might, we can’t prove this logically with
only p, q, and r to work with, but it nevertheless
seems to have the same deductive quality as other
syllogisms (like modus ponens).
Predicate logic
• Propositional logic is about predicting the truth
and falsity of propositions when combined with
one another and subjected to operators like
negation.
• What we need for the All men are mortal case is
something like:
• For any individual x, if x is a man, then x is mortal.
• That is, we need to be able to look inside the
sentence, to refer to predicates (properties) not just
to truth and falsities of entire propositions.
Predicate logic
• Predicate logic is an extension of
propositional logic that allows us to do this.
• Mortal(Socrates)
True if the predicate Mortal holds of the
individual Socrates.
• Individuals have properties, and just like we
labeled our propositions p, q, r, we can label
properties abstractly like A, B, C.
Predicate logic
• Thus:
• Man(x)  Mortal(x)
Man(Socrates)
Mortal(Socrates)
A(x)  B(x)
A(S)
B(S)
• Note: This is not exactly in the right form yet, but it’s
close. The right form of the first premise is actually
x[Man(x)Mortal(x)]. More on that later.
Entailment
• From the standpoint of linguistic knowledge of
meaning (intuition), there are sentences that stand
in a implicational relation, where the truth of the
first guarantees the truth of the second.
• The anarchist assassinated the emperor.
• The emperor died.
• It is part of the meaning of assassinate that the
unlucky recipient dies. So, the first sentence
entails the second.
Entailment
• This is the same relationship as pq from before. If we
know p is true, we know q is true—and if we know q is
false, we know p is false.
• The anarchist assassinated the emperor.
• The emperor died.
• At the same time, knowing q is true doesn’t tell us one
way or the other about whether p is true—and knowing p
is false doesn’t tell us one way or the other about whether
q is false.
• We take entailment relations to be those that specifically arise
from linguistic structure (synonymy, hyponymy, etc.).
Synonymy
• For a paraphrase to be a good one, and accurate
rendering of the meaning, the sentence should
entail its paraphrase and the paraphrase should
entail the sentence.
• The dog ate my homework.
• My homework was eaten by the dog.
• This kind of mutual entailment (like  from
earlier) is a requirement for synonymy.
Truth and meaning
• A young boy named Rickie burned down the library at
Alexandria in 639 AD by accidentally failing to
extinguish his cigarette properly.
• True? Well, we’ll pretty much never know (though
perhaps we can rate its likelihood). But knowing
whether it is true or not is not a prerequisite for
knowing its meaning.
• Rather, what’s important is that we know its truth
conditions—we know what the world must be like
if it is true.
Truth and meaning
• If we know what a sentence means we
know (at least) the conditions under which
it is true.
• On that assumption, we proceed in our
quest to understand meaning in terms of
truth conditions. Understanding how the
words and structures combine to predict the
truth conditions of sentences.
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