CAS LX 502 Semantics 2a. Reference, Comositionality, Logic 2.1-2.3 Meaning as truth conditions! We know the meaning of if we know the conditions under which is true.! conditions under which is true = which are the ossible worlds in which holds! ossible worlds = ways things might be! he meaning of : A secification of ossible worlds. Recall the trick we can do! Homer stands.! rue iff Homer stands. How do we arrive at truth conditions?! Homer stands. Marge stands.! rue iff Homer stands.! rue iff Marge stands. wo arts of Homer stands! Homer stands. Marge stands.! Homer stands. Bart s father stands. he Homer art and the stands art! Homer stands.! We use the name Homer to refer to that guy.! Homer stands is true when that guy has the roerty that he stands (being uright on his feet).! Other things/eole can stand, and we feel that standing should be basically the same regardless of who we say holds that roerty. 1
Unsaturated roositions! A roosition with a hole in it is called an unsaturated roosition.! It s something that, once we fill in the hole, will be true or false (of a given ossible world).! Portner draws these like so:! true! false Unsaturated roositions! A roosition with a hole in it is called an unsaturated roosition.! hus:! true! false Unsaturated roositions! Although erhas we could come u with a better icture, the idea is that:! Homer stands: the ossible worlds in which Homer stands! Stands: (unsaturated) Given a referent x, the ossible worlds in which x stands.! Homer:! true! false Meaning is comositional! It seems that there is something common across all the roositions we might exress using Homer.! And something in common across all the roositions we might exress 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 aears Meaning is comositional! We hyothesize this:! Meaning is comositional he meaning of a sentence is formed from the meanings of its arts, and the way in which they are arranged.! Homer strangles Bart. Bart strangles Homer. Meaning is comositional! And it really has to be comositional. 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 comute the meaning.! So the roject here is really:! Understand the ieces of meaning! Understand how they combine to form larger units of meaning 2
Where are we so far?! In the set of things that we ve been considering as art of meaning:! Possible world: A state of affairs.! One secial ossible world is the actual world, w 0.! Individuals: Referents, like Homer.! Proositions: Sets of ossible worlds! In which the roosition is true. Unsaturated roositions! We ve added the idea of an unsaturated roosition, which would be a roosition, but for the lack of an individual.! Given an individual, it would be a set of ossible worlds.! It s waiting for an individual.! It, in a sense, turns individuals into sets of ossible worlds. Limiting our attention to w k! or simlicity in resentation, let s sto thinking about sets of ossible worlds briefly, and limit out focus to secific ossible worlds.! One good candidate would be w 0, but it doesn t have to be that one necessarily.! If we do this, we can consider a roosition to be either true or false.! hough in the back of our minds, we know that this is in a articular ossible world. Semantic tye! he entire semantics that we are creating here deends on two tyes of things, individuals and truth values.! We can label individuals as being of tye e (traditional, think entity ), and truth values as being of tye t.! In these terms, names like Homer are of tye <e>, and sentences like Homer stands are of tye <t>. A formal system of semantic tyes! <e> is a basic tye.! <t> is a basic tye.! If " and # are tyes, <",#> is a tye.! <",#> is a function that takes something of tye " and returns something of tye #.! <e,t> is a tye. <<e,t>,<e,<e,t>>> is a tye.! <e,t,e> is not, nor is <<e,t>>. unctions! A function transforms one thing into another.! We can define the suaring function as a function that takes a number and gives back that number multilied with itself! Suare(n) = n $ n! his is a function from numbers to numbers. It takes a number, it gives back a number. 3
unctions! A function doesn t need to give back the same kind of thing it gets. Usually, the thing it gives back deends on the thing it gets, but it doesn t need to be of the same tye.! Change-machine($n-bill) = 4 $ n uarters.! his is a function from bills to uarters. <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 true if x is boring, or false if x is not boring.! Stands(x) = true if x stands; false otherwise.! his is a function from individuals (tye <e>) to truth values (tye <t>). hat is, it has tye <e,t>. Enter the % % argument [ return value ]! here is a way to write functions that we will get some exerience with as the semester rogresses, using lambda notation. Here s a first introduction! he structure of a function written in lambda notation is: % argument [ return value ]! Change-machine($n-bill) = 4 $ n uarters.! Change-machine = % $n-bill [ 4 $ n uarters ]! Suare = % n [ n $ n ]! So, for the meaning of stands, we might write this:! % x [ x stands (in w k ) ]! ye <e,t>! Not very comlicated, just a short way to write that function f such that, given argument, returns return value. value % argument [ return value ]! Suare = % n [ n $ n ]! Suare(3) = 3 $ 3 = 9! Suare(4) = 4 $ 4 = 16.! o 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 relace all of the ns with 3s and evaluate the return value. value % argument [ return value ]! Strictly seaking, there s an intermediate ste, which is written like so:! Suare = % n [ n $ n ]! Suare(3) = 3 % n [ n $ n ] = 3 $ 3 = 9! Suare(4) = 4 % n [ n $ n ] = 4 $ 4 = 16.! What value % argument [ return value ] means is: Relace every instance of argument within return value with value, then evaluate return value.! his oeration goes by the name lambda conversion. 4
value % argument [ return value ]! One last iece of terminology: Instances of argument within return value are said to be variables that bound by the lambda oerator.! rile = % n [ 3 $ n ] Lambda oerator 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 resuoses B! B is art 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 hursday, ER will be on at 10. It s hursday. ER will be on at 10. Modus Ponens! Logic is essentially the study of valid argumentation and inferences.! If the remises are true, the conclusion will be true. ruth out there in the world! A statement like It s hursday is either true (corresonding to the facts of the world) or it is false (not corresonding to the facts of the world).! Same for the statement ER is on at 10.! It turns out that modus onens is a valid form of argument, no matter what statements we use. Let s just say we have a statement we ll call it. he statement (roosition) can be either true or false. And another one, we ll call it. Modus onens! So, whatever and are:! If then...! Granting the remises If then and, we can conclude. An invalid argument! Incidentally, some things are not valid arguments. Modus onens and modus tollens are. his is not:! If it is hursday, then ER is on at 10. It is not hursday *ER is not on at 10. 5
Other forms of valid argument! If it is hursday, then ER is on. If ER is on, Pat will watch V. If it is hursday, the Pat will watch V. Hyothetical syllogism! If then. If then r. If then r. Other forms of valid argument! Pat is watching V or Pat is aslee. Pat is not aslee. Pat is watching V. Disjunctive syllogism! or... Logical syntax! A roosition, say, has a truth value. In light of the facts of the world, it is either true or false. he conditions under which is true is are called its truth conditions.! We can also create comlex exressions by combining roositions. or examle,. hat s true whenever is false. is the negation oerator ( not ). Logical connectives! We can combine roositions with connectives like and, or. In logical notation, and is written with the logical connective & ( and ): & ; or is written with ' ( or ): '.! & is true whenever is true and is true. Whenever either or is false, & is false. ruth tables! We can show the effect of logical oerators and connectives in truth tables. & ' Or v. ' v. ' e! he meaning we give to or in English (or any other natural language) is not uite 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 ay the fine or you will go to jail.! Seems a bit unfair if you get ut in jail even after aying the fine.! We will reboard anyone who has small children or needs secial assistance.! Doesn t seem to exclude eole who both need secial assistance and have small children. 6
Or v. ' v. ' e! here are two interretations of or, differing in their interretation with resect to what haens if both connected roositions are true.! Exclusive or (' e ) is either or but not both.! Inclusive or (disjunction; ') is either or or both. Material imlication! he logic of if then statements is covered by the connective!.! If it rains, you ll get wet. (!, where =it rains, =you ll get wet) ' ' e!! 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? ruth 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.! he first is necessarily true, a tautology, the second is necessarily false, a contradiction. Limits of roositional logic! here are some kinds of logical intuitions that are not catured by roositional logic. or examle:! All men are mortal. Socrates is a man. Socrates is mortal.! ry as we might, we can t rove this logically with only,, and r to work with, but it nevertheless seems to have the same deductive uality as other syllogisms (like modus onens). Predicate logic! Proositional logic is about redicting the truth and falsity of roositions when combined with one another and subjected to oerators like negation.! What we need for the All men are mortal case is something like:! or any individual x, if x is a man, then x is mortal.! hat is, we need to be able to look inside the sentence, to refer to redicates (roerties) not just to truth and falsities of entire roositions. Predicate logic! Predicate logic is an extension of roositional logic that allows us to do this.! Mortal(Socrates) rue if the redicate Mortal holds of the individual Socrates.! Individuals have roerties, and just like we labeled our roositions,, r, we can label roerties abstractly like A, B, C. 7
! hus: Predicate logic! Man(x)! Mortal(x) A(x)! B(x) Man(Socrates) A(S) Mortal(Socrates) B(S)! Note: his is not exactly in the right form yet, but it s close. he right form of the first remise is actually (x[man(x)! Mortal(x)]. More on that later. Entailment! rom the standoint of linguistic knowledge of meaning (intuition), there are sentences that stand in a imlicational relation, where the truth of the first guarantees the truth of the second.! he anarchist assassinated the emeror.! he emeror died.! It is art of the meaning of assassinate that the unlucky reciient dies. So, the first sentence entails the second. Entailment! his is the same relationshi as! from before. If we know is true, we know is true and if we know is false, we know is false.! he anarchist assassinated the emeror.! he emeror died.! At the same time, knowing is true doesn t tell us one way or the other about whether is true and knowing is false doesn t tell us one way or the other about whether is false.! We take entailment relations to be those that secifically arise from linguistic structure (synonymy, hyonymy, etc.). Synonymy! or a arahrase to be a good one, and accurate rendering of the meaning, the sentence should entail its arahrase and the arahrase should entail the sentence.! he dog ate my homework.! My homework was eaten by the dog.! his kind of mutual entailment (like ) from earlier) is a reuirement for synonymy. ruth and meaning! A young boy named Rickie burned down the library at Alexandria in 639 AD by accidentally failing to extinguish his cigarette roerly.! rue? Well, we ll retty much never know (though erhas we can rate its likelihood). But knowing whether it is true or not is not a rereuisite for knowing its meaning.! Rather, what s imortant is that we know its truth conditions we know what the world must be like if it is true. ruth and meaning! If we know what a sentence means we know (at least) the conditions under which it is true.! On that assumtion, we roceed in our uest to understand meaning in terms of truth conditions. Understanding how the words and structures combine to redict the truth conditions of sentences. 8
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