CAS LX 502 - PowerPoint PPT Presentation

1 / 34
About This Presentation
Title:

CAS LX 502

Description:

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 ... – PowerPoint PPT presentation

Number of Views:77
Avg rating:3.0/5.0
Slides: 35
Provided by: PaulH248
Learn more at: https://www.bu.edu
Category:

less

Transcript and Presenter's Notes

Title: CAS LX 502


1
CAS LX 502
  • 3b. Truth and logic
  • 4.1-4.4

2
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

3
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.

4
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.

5
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.

6
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

7
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.

8
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.

9
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.

10
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).

11
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.

12
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
13
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.

14
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
15
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?

16
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.

17
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.

18
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.

19
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.

20
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.

21
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.

22
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.

23
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.

24
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.

25
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).

26
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.

27
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.

28
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.

29
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.

30
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.).

31
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.

32
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.

33
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.

34
?
  • ? ?
  • ?
  • ? ?
  • ? ?
  • ?
  • ?
Write a Comment
User Comments (0)
About PowerShow.com