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Home , Extension (semantics), Logical connective, Possible world, Syllogism

as conditions

CAS LX 502 ! 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 2a. , Compositionality, ! The meaning of p: A specification of possible worlds. 2.1-2.3

Recall the trick we can do How do we arrive at truth conditions?

! Homer stands. ! Homer stands. Marge stands. ! True iff Homer stands. ! True iff Homer stands. ! True iff Marge stands.

Two parts of Homer stands The Homer part and the stands part

! Homer stands. Marge stands. ! Homer stands. ! We use the Homer to ! Homer stands. Bart’s father stands. refer to that guy. ! Homer stands is true when that guy has the 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.

1 Unsaturated Unsaturated propositions

! A with a “hole” in it is ! A proposition with a “hole” in it is called an unsaturated proposition. called an unsaturated proposition. ! It’s something that, once we fill in the ! Thus: hole, will be true or (of a given ). ! true ! false ! Portner draws these like so: ! true ! false

Unsaturated propositions Meaning is compositional

! Although perhaps we could come up ! It seems that there is something common with a better picture, the idea is that: across all the propositions we might express using Homer. ! Homer stands: the possible worlds in which ! And something in common across all the Homer stands propositions we might express using stands. ! Homer stands. Homer snores. Marge stands. ! Stands: (unsaturated) Given a referent x, the possible worlds in which x stands. ! Given that each word seems to have a consistent contribution to the meaning, (to ! true some extent) regardless of the sentence in ! Homer: ! false which it appears…

Meaning is compositional Meaning is compositional

! We hypothesize this: ! 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 ! Meaning is compositional haven’t heard the sentence before—and The meaning of a sentence is formed have to compute the meaning. from the meanings of its parts, and the way in which they are arranged. ! So the project here is really: ! Understand the pieces of meaning ! Understand how they combine to form larger units of ! Homer strangles Bart. Bart strangles Homer. meaning

2 Where are we so far? Unsaturated propositions

! In the of things that we’ve been ! We’ve added the idea of an considering as part of meaning: unsaturated proposition, which would be a proposition, but for the lack of an ! Possible world: A state of affairs. individual. ! One special possible world is the actual world, ! Given an individual, it would be a set w . 0 of possible worlds. ! Individuals: Referents, like Homer. ! It’s “waiting for an individual.” ! Propositions: Sets of possible worlds ! It, in a sense, turns individuals into sets ! In which the proposition is “true.” of possible worlds.

Limiting our attention to wk Semantic type

! For simplicity in presentation, let’s stop ! The entire semantics that we are creating thinking about sets of possible worlds briefly, here depends on two types of things, and limit out to specific possible individuals and truth values. worlds. ! We can label individuals as being of type “e” (traditional, think “entity”), and truth ! One good candidate would be w , but it 0 values as being of type “t”. doesn’t have to be that one necessarily. ! If we do this, we can consider a proposition ! In these terms, like Homer are of type to be either true or false. , and sentences like Homer stands are of ! Though in the back of our minds, we know that this is type . in a particular possible world.

A Functions of semantic types ! is a basic type. ! A transforms one thing into another. ! is a basic type. ! We can define the squaring function as a function that takes a number and gives ! If " and # are types, <",#> is a type. back that number multiplied with itself ! <",#> is a function that takes something of ! Square(n) = n $ n type " and returns something of type #. ! This is a function from numbers to numbers. It takes a number, it gives back a number. ! is a type. <,>> is a type. ! is not, nor is <>.

3 Functions functions

! A function doesn’t need to give back the ! An intransitive like stands can be same kind of thing it gets. Usually, the thing it viewed as a function from individuals to truth gives back depends on the thing it gets, but values. Given an individual x, it will return it doesn’t need to be of the same type. true if x is boring, or false if x is not boring.

! Change-machine($n-bill) = 4 $ n quarters. ! Stands(x) = true if x stands; false otherwise.

! This is a function from bills to quarters. ! This is a function from individuals (type ) to truth values (type ). That is, it has type .

Enter the % % [ return value ]

! There is a way to write functions that we will get some ! Change-machine($n-bill) = 4 $ n quarters. experience with as the semester progresses, using ! Change-machine = % $n-bill [ 4 $ n quarters ] lambda notation. Here’s a first introduction ! The structure of a function written in lambda notation is: % argument [ return value ] ! Square = % n [ n $ n ]

! So, for the meaning of stands, we might write this: ! Not very complicated, just a short way to write “that function f such that, given

! % x [ x stands (in wk) ] argument, returns return value.” ! Type

value % argument [ return value ] value % argument [ return value ]

! Square = % n [ n $ n ] ! Strictly speaking, there’s an intermediate step, ! Square(3) = 3 $ 3 = 9 which is written like so: ! Square(4) = 4 $ 4 = 16. ! Square = % n [ n $ n ] ! Square(3) = 3 % n [ n $ n ] = 3 $ 3 = 9 ! To evaluate a function, we take the value ! Square(4) = 4 % n [ n $ n ] = 4 $ 4 = 16. and substitute it in for the argument within the return value. If we give it a 3, and the ! What value % argument [ return value ] means is: argument is n, then we replace all of the ns Replace every instance of argument within return with 3s and evaluate the return value. value with value, then evaluate return value. ! This operation goes by the name lambda conversion.

4 value % argument [ return value ] Desiderata for a theory of meaning

! One last piece of terminology: Instances of ! A is synonymous with B argument within return value are said to be ! A has the same meaning as B variables that bound by the lambda operator. ! A entails B ! If A holds then B automatically holds ! Triple = % n [ 3 $ n ] ! 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 ! A is automatically true, regardless of the Lambda operator ! A is a Bound variable ! A is automatically false, regardless of the facts

Intuitions about logic Truth out there in the world

! If it’s Thursday, ER will be on at 10. ! A like It’s Thursday is either true It’s Thursday. (corresponding to the facts of the world) or it is false ER will be on at 10. (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 ! Logic is essentially the study of valid argumentation argument, no matter what statements we use. Let’s and . just say we have a statement—we’ll call it p. The ! If the are true, the conclusion will be true. statement (proposition) p can be either true or false. And another one, we’ll call it q.

Modus ponens An invalid argument

! So, whatever p and q are: ! Incidentally, some things are not valid . Modus ponens and modus ! If p then q. tollens are. This is not: p. q. ! If it is Thursday, then ER is on at 10. It is not Thursday ! Granting the premises If p then q and *ER is not on at 10. p, we can conclude q.

5 Other forms of valid argument Other forms of valid argument

! If it is Thursday, then ER is on. ! Pat is watching TV or Pat is asleep. If ER is on, Pat will watch TV. Pat is not asleep. If it is Thursday, the Pat will watch TV. Pat is watching TV. Hypothetical

! If p then q. ! p or q. If q then r. ¬q. If p then r. p.

Logical Logical connectives

! A proposition, say p, has a . In light of the ! We can combine propositions with facts of the world, it is either true or false. The conditions under which p is true is are called its truth connectives like and, or. In logical conditions. notation, “p and q” is written with the & (“and”): p & q; “p ! We can also create complex expressions by or q” is written with ' (“or”): p ' q. combining propositions. For example, ¬q. That’s true whenever q is false. ¬ is the operator ! p & q is true whenever p is true and q is (“not”). true. Whenever either p or q is false, p & q is false.

Truth tables Or v. ' v. 'e

! We can show the effect of logical operators ! The meaning we give to or in English (or any other natural language) is not quite the same as the and connectives in truth tables. 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 p ¬p p q p&q p q p'q Oklahoma. ! You will pay the fine or you will go to jail. T F T T T T T T ! Seems a bit unfair if you get put in jail even after paying the fine. F T ! We will preboard anyone who has small children or needs special T F F T F T assistance. ! Doesn’t seem to exclude people who both need special assistance F T F F T T and have small children. F F F F F F

6 Or v. ' v. 'e Material implication

! There are two interpretations of or, differing in their ! The logic of if…then statements is with respect to what happens if both covered by the connective !. connected propositions are true. ! ! ('e) is “either…or…but not both.” If it rains, you’ll get wet. ! Inclusive or (disjunction; ') is “either…or…or both.” (p!q, where p=it rains, q=you’ll get wet)

p q p q p q p!q p q ' p q 'e ! What is the truth value of T T T T T F T T T If it rains, you’ll get wet? T F T T F T T F F ! Well, it’s true if it rains and you get wet, it’s false if it F T T F T T F T T rains and you don’t get wet. F F F F F F F F T But what if it doesn’t rain?

Truth and the world Limits of propositional logic

! There are some kinds of logical intuitions that are ! In most cases, the truth or falsity of a statement has to do not captured by propositional logic. For example: with the facts of the world. We cannot know without checking. It is contingent on the facts of the world ! All men are mortal. (synthetic). Socrates is a man. Socrates is mortal. ! John Wilkes Booth acted alone. ! Try as we might, we can’t prove this logically with ! Sometimes, though, the very form of the statement guarantees that it is true no matter what the world is like only p, q, and r to work with, but it nevertheless (analytic). seems to have the same deductive quality as other ! Either John Wilkes Booth acted alone or he didn’t. (like modus ponens). ! John Wilkes Booth acted alone and he didn’t. ! The first is necessarily true, a tautology, the second is necessarily false, a contradiction.

Predicate logic logic

! Propositional logic is about predicting the truth and ! Predicate logic is an of falsity of propositions when combined with one another and subjected to operators like negation. propositional logic that allows us to do ! What we need for the All men are mortal case is this. something like: ! Mortal(Socrates) ! For any individual x, if x is a man, then x is mortal. True if the predicate Mortal holds of the ! That is, we need to be able to look inside the individual Socrates. sentence, to refer to predicates (properties) not just to truth and falsities of entire propositions. ! Individuals have properties, and just like we labeled our propositions p, q, r, we can label properties abstractly like A, B, C.

7 Predicate logic

! Thus: ! From the standpoint of linguistic knowledge of ! Man(x) ! Mortal(x) A(x) ! B(x) meaning (intuition), there are sentences that stand Man(Socrates) A(S) in a implicational relation, where the truth of the first Mortal(Socrates) B(S) guarantees the truth of the second. ! The anarchist assassinated the emperor. ! Note: This is not exactly in the right form yet, but it’s close. The ! The emperor died. right form of the first is actually (x[Man(x)! Mortal(x)]. More on that later. ! It is part of the meaning of assassinate that the unlucky recipient dies. So, the first sentence entails the second.

Entailment Synonymy

! For a paraphrase to be a good one, and accurate ! This is the same relationship as p!q from before. If we rendering of the meaning, the sentence should know p is true, we know q is true—and if we know q is entail its paraphrase and the paraphrase should false, we know p is false. entail the sentence. ! The anarchist assassinated the emperor. ! The dog ate my homework. ! The emperor died. ! My homework was eaten by the dog. ! At the same time, knowing q is true doesn’t tell us one ! This kind of mutual entailment (like from earlier) is way or the other about whether p is true—and knowing ) p is false doesn’t tell us one way or the other about a requirement for synonymy. whether q is false. ! We take entailment to be those that specifically arise from linguistic structure (synonymy, hyponymy, etc.).

Truth and meaning Truth and meaning

! A young boy named Rickie burned down the library at ! If we know what a sentence means Alexandria in 639 AD by accidentally failing to extinguish his cigarette properly. we know (at least) the conditions under which it is true. ! True? Well, we’ll pretty much never know (though perhaps we can rate its likelihood). ! On that assumption, we proceed in But knowing whether it is true or not is not a our quest to understand meaning in prerequisite for knowing its meaning. terms of truth conditions. ! Rather, what’s important is that we know its Understanding how the words and truth conditions—we know what the world structures combine to predict the truth must be like if it is true. conditions of sentences.

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