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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
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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 T F • It is not Thursday = p. 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”): pq; “p or q” is written with (“or”): p q. • pq is true whenever p is true and q is true. Whenever either p or q is false, pqis false.
Truth tables • We can show the effect of logical operators and connectives in truth tables.
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.”
Material implication • The logic of if…thenstatements is covered by the connective . • If it rains, you’ll get wet.(pq, where p=it rains, q=you’ll get wet) • 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. (pq, 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. • That is:pq.p.q.
Material implication • If John is at the party, Mary is. (pq, 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. • That is:pq.p.q.
Material implication • If John is at the party, Mary is. (pq, 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. • That is:pq.p.q.
Material implication • If John is at the party, Mary is. (pq, 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. • That is:pq.q.p.
Material implication • If John is at the party, Mary is. (pq, 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. • That is:pq.q.p.
Material implication • If John is at the party, Mary is. (pq, 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? • That is:pq.q.p.
Material implication • If John is at the party, Mary is. (pq, 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! • That is:pq.q.p.
Biconditional • The last basic logical connective is the biconditional or (“if and only if”). • pq is the same as (pq)(qp). • 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) A(x) B(x)Man(Socrates) A(S)Mortal(Socrates) 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 pq 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.
