CAS LX 502 Semantics

CAS LX 502Semantics 2a. Reference, Compositionality, Logic 2.1-2.3

Meaning as truth conditions • We know the meaning of p if we know the conditions under which p is true. • conditions under which p is true = which are the possible worlds in which p holds • possible worlds = ways things might be • The meaning of p: A specification of possible worlds.

Recall the trick we can do • Homer stands. • True iff Homer stands.

How do we arrive at truth conditions? • Homer stands. Marge stands. • True iff Homer stands. • True iff Marge stands.

Two parts of Homer stands • Homer stands. Marge stands. • Homer stands. Bart’s father stands.

The Homer part and the stands part • Homer stands. • We use the name Homer to refer to that guy. • Homer stands is true when that guy has the property that he stands (being upright on his feet). • Other things/people can stand, and we feel that standing should be basically the same regardless of who we say holds that property.

Unsaturated propositions • A proposition with a “hole” in it is called an unsaturated proposition. • It’s something that, once we fill in the hole, will be true or false (of a given possible world). • Portner drawsthese like so:  true  false

Unsaturated propositions • A proposition with a “hole” in it is called an unsaturated proposition. • Thus:  true  false

Unsaturated propositions • Although perhaps we could come up with a better picture, the idea is that: • Homer stands: the possible worlds in which Homer stands • Stands: (unsaturated) Given a referent x, the possible worlds in which x stands. • Homer:  true  false

Meaning is compositional • It seems that there is something common across all the propositions we might express using Homer. • And something in common across all the propositions we might express using stands. • Homer stands. Homer snores. Marge stands. • Given that each word seems to have a consistent contribution to the meaning, (to some extent) regardless of the sentence in which it appears…

Meaning is compositional • We hypothesize this: • Meaning is compositionalThe meaning of a sentence is formed from the meanings of its parts, and the way in which they are arranged. • Homer strangles Bart. Bart strangles Homer.

Meaning is compositional • And it really has to be compositional. We after all know what the world has to look like in order for a sentence to be true, even if we haven’t heard the sentence before—and have to compute the meaning. • So the project here is really: • Understand the pieces of meaning • Understand how they combine to form larger units of meaning

Where are we so far? • In the set of things that we’ve been considering as part of meaning: • Possible world: A state of affairs. • One special possible world is the actual world, w0. • Individuals: Referents, like Homer. • Propositions: Sets of possible worlds • In which the proposition is “true.”

Unsaturated propositions • We’ve added the idea of an unsaturated proposition, which would be a proposition, but for the lack of an individual. • Given an individual, it would be a set of possible worlds. • It’s “waiting for an individual.” • It, in a sense, turns individuals into sets of possible worlds.

Limiting our attention to wk • For simplicity in presentation, let’s stop thinking about sets of possible worlds briefly, and limit out focus to specific possible worlds. • One good candidate would be w0, but it doesn’t have to be that one necessarily. • If we do this, we can consider a proposition to be either true or false. • Though in the back of our minds, we know that this is in a particular possible world.

Semantic type • The entire semantics that we are creating here depends on two types of things, individuals and truth values. • We can label individuals as being of type “e” (traditional, think “entity”), and truth values as being of type “t”. • In these terms, names like Homer are of type <e>, and sentences like Homer stands are of type <t>.

A formal systemof semantic types • <e> is a basic type. • <t> is a basic type. • If a and b are types, <a,b> is a type. • <a,b> is a function that takes something of type a and returns something of type b. • <e,t> is a type. <<e,t>,<e,<e,t>>> is a type. • <e,t,e> is not, nor is <<e,t>>.

Functions • A function transforms one thing into another. • We can define the squaring function as a function that takes a number and gives back that number multiplied with itself • Square(n) = nn • This is a function from numbers to numbers. It takes a number, it gives back a number.

Functions • A function doesn’t need to give back the same kind of thing it gets. Usually, the thing it gives back depends on the thing it gets, but it doesn’t need to be of the same type. • Change-machine($n-bill) = 4 n quarters. • This is a function from bills to quarters.

<e,t> functions • An intransitive verb like stands can be viewed as a function from individuals to truth values. Given an individual x, it will return trueif x is boring, or false if x is not boring. • Stands(x) = true if x stands; false otherwise. • This is a function from individuals (type <e>) to truth values (type <t>). That is, it has type <e,t>.

Enter the l • There is a way to write functions that we will get some experience with as the semester progresses, using lambda notation. Here’s a first introduction • The structure of a function written in lambda notation is:argument [ return value ] • So, for the meaning of stands, we might write this: • x [ x stands (in wk) ] • Type <e,t>

argument [ return value ] • Change-machine($n-bill) = 4 n quarters. • Change-machine =  $n-bill [ 4 n quarters ] • Square = n [ nn ] • Not very complicated, just a short way to write “that function f such that, given argument, returns return value.”

value argument [ return value ] • Square = n [ nn ] • Square(3) = 3  3 = 9 • Square(4) = 4  4 = 16. • To evaluate a function, we take the value and substitute it in for the argument within the return value. If we give it a 3, and the argument is n, then we replace all of the ns with 3s and evaluate the return value.

value argument [ return value ] • Strictly speaking, there’s an intermediate step, which is written like so: • Square = n [ nn ] • Square(3) = 3 n [ nn ] = 3  3 = 9 • Square(4) = 4 n [ nn ] = 4  4 = 16. • What value argument [ return value ] means is: Replace every instance of argument within return value with value, then evaluate return value. • This operation goes by the name lambda conversion.

value argument [ return value ] • One last piece of terminology: Instances of argument within return value are said to be variablesthat bound by the lambda operator. • Triple = n [ 3 n ] Lambda operator Bound variable

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.

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qis 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.(pq, 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?

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 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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CAS LX 502 Semantics