Meaning as truth conditions CAS LX 502 ! We know the meaning of p if we know the conditions under which p is true. Semantics ! conditions under which p is true = which are the possible worlds in which p holds ! possible worlds = ways things might be 2a. Reference, Compositionality, ! The meaning of p: A specification of Logic 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 name Homer to ! Homer stands. Bart’s father stands. 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. 1 Unsaturated propositions Unsaturated propositions ! A proposition 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 false (of a given possible world). ! 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 set 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 focus 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, names like Homer are of type to be either true or false. <e>, and sentences like Homer stands are of ! Though in the back of our minds, we know that this is type <t>. in a particular possible world. A formal system Functions of semantic types ! <e> is a basic type. ! A function transforms one thing into another. ! <t> 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. ! <e,t> is a type. <<e,t>,<e,<e,t>>> is a type. ! <e,t,e> is not, nor is <<e,t>>. 3 Functions <e,t> functions ! A function doesn’t need to give back the ! An intransitive verb 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 <e>) to truth values (type <t>). That is, it has type <e,t>. Enter the % % argument [ 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 <e,t> 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 tautology ! A is automatically true, regardless of the facts Lambda operator ! A is a contradiction 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 statement 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). Modus Ponens ! 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 inferences. just say we have a statement—we’ll call it p. The ! If the premises 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 arguments. 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 syllogism Disjunctive syllogism ! If p then q. ! p or q. If q then r. ¬q. If p then r. p. Logical syntax Logical connectives ! A proposition, say p, has a truth value. 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 logical connective & (“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.