CAS LX 502

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CAS LX 502

• 3b. Truth and logic
• 4.1-4.4

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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

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Intuitions about logic

• If its Thursday, ER will be on at 10.Its
  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.

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Truth out there in the world

• A statement like Its 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. Lets
  just say we have a statementwell call it p. The
  statement (proposition) p can be either true or
  false. And another one, well call it q.

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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.

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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 T F
• It is not Thursday ?p. F T

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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
  ThursdayER is not on at 10.

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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.

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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.

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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. Thats
  true whenever q is false. ? is the negation
  operator (not).

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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.

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Truth tables

• We can show the effect of logical operators and
  connectives in truth tables.

p ?p
T F
F T
p q p?q
T T T
T F F
F T F
F F F
p q p?q
T T T
T F T
F T T
F F F
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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 ?.
• Were going to South Carolina or Oklahoma.
• Seems odd to say this if were 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.
• Doesnt seem to exclude people who both need
  special assistance and have small children.

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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 eitherorbut not both.
• Inclusive or (disjunction ?) is eitheroror
  both.

p q p?q
T T T
T F T
F T T
F F F
p q p?eq
T T F
T F T
F T T
F F F
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Material implication

• The logic of ifthen statements is covered by the
  connective ?.
• If it rains, youll get wet.(p?q, where pit
  rains, qyoull get wet)

p q p?q
T T T
T F F
F T T
F F T

• What is the truth value of If it rains, youll
  get wet?
• Well, its true if it rains and you get wet, its
  false if it rains and you dont get wet. But what
  if it doesnt rain?

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Material implication

• If John is at the party, Mary is. (p?q, where
  pJohn is at the party, qMary is at the party)
• Suppose thats true, and that John is at the
  party.
• We can conclude that Mary is at the party.

p q p?q
T T T
T F F
F T T
F F T

• That isp?q.p.q.

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Material implication

• If John is at the party, Mary is. (p?q, where
  pJohn is at the party, qMary is at the party)
• Suppose thats true, and that John is at the
  party.
• We can conclude that Mary is at the party.

p q p?q (p?q)?p ((p?q)?p)?q
T T T
T F F
F T T
F F T

• That isp?q.p.q.

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Material implication

• If John is at the party, Mary is. (p?q, where
  pJohn is at the party, qMary is at the party)
• Suppose thats true, and that John is at the
  party.
• We can conclude that Mary is at the party.

p q p?q (p?q)?p ((p?q)?p)?q
T T T T T
T F F F T
F T T F T
F F T F T

• That isp?q.p.q.

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Material implication

• If John is at the party, Mary is. (p?q, where
  pJohn is at the party, qMary is at the party)
• Suppose thats 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
T F F T F
F T T F T
F F T T T

• That isp?q.?q.?p.

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Material implication

• If John is at the party, Mary is. (p?q, where
  pJohn is at the party, qMary is at the party)
• Suppose thats 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
T F F T F F T
F T T F T F T
F F T T T T T

• That isp?q.?q.?p.

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Material implication

• If John is at the party, Mary is. (p?q, where
  pJohn is at the party, qMary is at the party)
• Suppose thats 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 ((p?q)?q)?p
T T F F T T
T F F T F F
F T T F T T
F F T T T F

• That isp?q.q.p.

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Material implication

• If John is at the party, Mary is. (p?q, where
  pJohn is at the party, qMary is at the party)
• Suppose thats 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 T
T F F T F F T
F T T F T T F
F F T T T F T

• That isp?q.q.p.

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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.

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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
  didnt.
• John Wilkes Booth acted alone and he didnt.
• The first is necessarily true, a tautology, the
  second is necessarily false, a contradiction.

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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 cant 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).

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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.

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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.

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Predicate logic

• Thus
• Man(x) ? Mortal(x) A(x) ? B(x)Man(Socrates)
  A(S)Mortal(Socrates) B(S)
• Note This is not exactly in the right form yet,
  but its close. The right form of the first
  premise is actually ?xMan(x)?Mortal(x). More on
  that later.

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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.

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Entailment

• This is the same relationship as p?q from before.
  If we know p is true, we know q is trueand 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 doesnt tell
  us one way or the other about whether p is
  trueand knowing p is false doesnt 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.).

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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.

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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, well 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, whats important is that we know its
  truth conditionswe know what the world must be
  like if it is true.

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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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