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”): 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 p T F p q pq T F p q pq T 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 q pq T F p q peq T F
Material implication The logic of if…then statements is covered by the connective . If it rains, you’ll get wet. (pq, where p=it rains, q=you’ll get wet) p q pq T F 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. p q pq T F 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. p q pq (pq)p ((pq)p)q T F 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. p q pq (pq)p ((pq)p)q T F 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. p q p q pq (pq)q ((pq)q)p T F 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. p q p q pq (pq)q ((pq)q)p T F 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? p q p q pq (pq)q ((pq)q)p T F 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! p q p q pq (pq)q ((pq)q)p T F 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.