Semantics Truth and Meaning

Truth and meaning The basis of formal semantics: knowing the meaning of a sentence is knowing under what conditions it is true. ! Semantics ! We wish to describe meaning (truth conditions) precisely and in such a way as to predict our intuitions about meanings—we will do this by using a logical language as a metalanguage. ! Infinite use, finite means ! ! A fundamental property of language is its recursive nature—we can create unboundedly many new sentences, and understand what they mean. Infinite use, finite means ! “Infinite use of finite means,” one of the main reasons to suppose that our knowledge of language is systematic, that language is not a collection of habits and analogy, but must be described by a grammar. Syntax ! The generally accepted view of syntax breaks sentences down into hierarchical parts. There are nouns, there are verbs, there are units made of verbs and nouns. New sentences can be created by mixing and matching these components together. ! ! [S Pat [AuxP will [VP eat [NP the sandwich]]]] [S The students [AuxP have [VP risen [PP in protest]]]] Formal semantics, a.k.a. truth conditional semantics, a.k.a. model-theoretic semantics, related to Montague Grammar. In the domain of syntax, the task is primarily to describe/explain why some arrangements of words count as sentences of English, others don’t, and more broadly, how this system relates to those underlying other languages, and how this system can arise. Semantics ! ! ! We’re not here to study syntax, we’re here to study semantics, but we’re going to delve a bit into both. The syntactic system that defines what are “good sentences of English” provides hierarchical structures, but we know not only what sequences of words might be classified as “English” but we know what those sequences of words mean. Just as there must be a grammar that defines what sequences of words are English, there must also be a grammar that tells us how the meanings of the parts contribute to the meaning of the whole. 1 F1 ! F1 To that end, we are going to create a “mini-grammar of English”, a fragment. This grammar will provide both the syntactic structure of a small number of English sentences and the rules by which we can understand their meaning. By doing this, we can start to understand what is involved in the grammar of semantics more generally. ! Using the syntax of F1 ! We start with S (we are building a sentence). ! ! Using the syntax of F1 ! ! We start with S (we are building a sentence). Several different rules can apply. We can either rewrite S as N VP, or as S conj S, or as neg S. Let’s pick N VP. Now, N can be rewritten as Pavarotti, Loren, or Bond. ! S VP ! Bond N ! Pavarotti, Loren, Bond S ! S conj S Vi ! is boring, is hungry, is cute S ! neg S Vt ! likes VP ! Vt N Conj ! and, or VP ! Vi Neg ! it is not the case that We start with S (we are building a sentence). Several different rules can apply. We can either rewrite S as N VP, or as S conj S, or as neg S. Let’s pick N VP. S VP N Using the syntax of F1 ! N S ! N VP Using the syntax of F1 S ! Rewrite rules (the syntax): ! We start with S (we are building a sentence). Several different rules can apply. We can either rewrite S as N VP, or as S conj S, or as neg S. Let’s pick N VP. Now, N can be rewritten as Pavarotti, Loren, or Bond. And VP can be rewritten either as Vt N or Vi. S N Bond VP Vi 2 Using the syntax of F1 Using the syntax of F1 ! ! ! ! ! We start with S (we are building a sentence). Several different rules can apply. We can either rewrite S as N VP, or as S conj S, or as neg S. Let’s pick N VP. Now, N can be rewritten as Pavarotti, Loren, or Bond. And VP can be rewritten either as Vt N or Vi. And Vi can be rewritten as is boring, is hungry, or is cute. ! S With this little grammar, we can already create an unbounded number of sentences. ! VP N Bond Vi is hungry Using the syntax of F1 ! It is not the case that Bond is boring or Loren is hungry. Using the syntax of F1 It is not the case that Bond is boring or Loren is hungry. ! It is not the case that Bond is boring or Loren is hungry. S S Neg S It is not the case that S S Conj N VP Bond Vi or S VP Loren Vi It is not the case that is hungry Use the rules and only the rules. ! ! ! You may or may not have had experience with syntax before. And it may or may not have involved trees like the ones we’ve just seen. Probably it involved more complicated trees (and for good reasons, which are explored in the syntax class). But here, it’s fine to just approximate the syntax by using the PS rules just given. When drawing trees, don’t try to remember what you learned about them in LX250 or LX522. Just rewrite the way the rules allow you to. No IP, no CP. Just what the rules allow. or N VP Bond Vi S N VP Loren Vi is hungry is boring A word of warning ! S Neg N is boring Conj Compositionality ! ! ! A fundamental assumption about how it is that we can know what novel sentences mean is that meaning is compositional. The meaning of the whole is derived from the meaning of the parts and how the parts are arranged. The syntax gives us the parts and how they are arranged, now we must approach the question of how the meaning is assigned to the parts and from there to the whole. 3 The meaning of names ! ! ! We talked about a meaning for names (like Bond, say) as being something like “pointing to an individual that exists in the world.” We need a way to formalize this kind of intuitive idea: a model. A model contains two relevant things: A set of the individuals in the universe, and a “pointing function” that associates names with those individuals. Models and pointing ! ! We’ll call the set of individuals in the universe “U” (for “Universe”), and the pointing function “F” (for “function”—or maybe “finger”). Both of those together constitute a model, which we will often call “M” (for “model”). ! M = . So, to evaluate the meaning of a name, we see which individual in the world the name points to. ! F(Pavarotti), then, is “the individual named (in this model) by Pavarotti”. Evaluating the meaning of bits of tree ! ! Our goal in F1 is to create simple sentence structures with the syntax, and a assign a meaning (compositionally) to the whole sentence that matches our intuitions. ! So, we need to evaluate the meaning of individual nodes in the tree as well. The simplest case would be evaluating the meaning of the a node like Pavarotti at the bottom of the tree. Evaluating the node: [Pavarotti] M The meaning of names: [Pavarotti]M = F(Pavarotti) = Pavarotti ! The ultimate interpretation assigned to this node is the individual Pavarotti. U Evaluating meaning: [ ]M Translating from a node in a syntactic structure to a semantic meaning is accomplished by what we call an evaluation function. Given a syntactic node, its result is the semantic interpretation of that node. ! Simplest case ! … is hungry … Bond Loren Pavarotti … F The interpretation depends on the model, so we also need to specify with respect to what model a node is being evaluated. Predicates/properties ! ! ! ! ! ! So we have a meaning assigned for one node in S the tree. How about the “verb” is VP N hungry? What is [is hungry]M? A way we can think of Bond Vi properties is as something that divides the universe of individuals into two [Bond]M = is hungry groups, those that have F(Bond) = the property and those [is hungry]M Bond that do not. 4 Predicates/properties ! ! ! Predicates/properties One simple and intuitive way to implement this is to S say that []M of a property is a set containing those VP individuals that have the N property. Like we did for names of Bond Vi individuals, we can suppose that the name of a property “points” to the M is hungry set of individuals that has [Bond] = the property. F(Bond) = [is hungry]M That is, this can be part of Bond the job that F does. ! ! ! Suppose Bond and Pavarotti are the hungry ones in the universe of individuals in this model. F(is hungry) = {Bond, Pavarotti} ! ! VP Vi ! N Bond ! ! VP Vi is hungry [N]M = F(Bond) [VP]M = [Vi]M = F(is hungry) { x: x is hungry in M} [S]M = true iff [N]M " [VP]M = true iff F(Bond) " { x: x is hungry in M} ! U ! [S]M1 = F1(Bond) " F1(is hungry) = Bond " {Bond, Loren} U1 In the specific situation M1 . [ ]M We now need to assign interpretations to the rest of the nodes of the tree. There are no new meaningful elements, so the meanings will all be formed on the basis of Bond or is hungry or both. … is hungry … Bond Loren Pavarotti … F1 S [ ]M ! Vi Bond is hungry is hungry ! Bond [Bond]M = is hungry F(Bond) = [is hungry]M = Bond F(is hungry) = {Bond, Pavarotti} … is hungry … Bond Loren Pavarotti … F S Bond VP N Great, 2 down, 4 to go. Bond is hungry N S S N Bond ! VP Vi [Bond]M = is hungry Meaning is compositional. F(Bond) = [is hungry]M = Bond F(is hungry) = So, what’s [N]M? {Bond, Pavarotti} ! Based on the principle of compositionality, we can assume/deduce the that nodes above share the same denotation as the nodes below, in cases where there is no combination happening. [N]M = Bond S N Bond VP Vi [Bond]M = is hungry F(Bond) = [is hungry]M = Bond F(is hungry) = {Bond, Pavarotti} 5 [ ]M ! ! ! ! ! Now, to determine the meaning of the S as a whole… What do we want? Well, this should be true only when Bond is hungry. And that’s true if Bond is in the F(is hungry) set. [ ]M S ! VP N Bond Vi [Bond]M = is hungry F(Bond) = [is hungry]M = Bond F(is hungry) = {Bond, Pavarotti} That is, [S]M = true just in case [N]M is in the set [VP]M. ! [S N VP]M = true iff [N]M " [VP]M, otherwise false. ! ! Thus, we end up with an interpretation of this sentence that goes like this: ! [S]M = true iff F(Bond) " F(is hungry), otherwise false. Given this particular model, that boils down to ! [S]M = true iff Bond " {Bond, Pavarotti}, otherwise false. (True in this situation) N Bond VP Vi [Bond]M = is hungry F(Bond) = [is hungry]M = Bond F(is hungry) = {Bond, Pavarotti} A semantic rule for every structural rule [ ]M ! S We can define a semantic rule for interpretation that says just that: ! S N Bond VP Our goal is to design a semantics for F1 that can provide an interpretation (truth conditions) for any structure that the syntax can provide. Vi [Bond]M = is hungry F(Bond) = [is hungry]M = Bond F(is hungry) = {Bond, Pavarotti} ! Neg S Neg ! For Neg S, we want it to be false whenever S is true, and true whenever S is false. ! ! [Neg S]M = ! ! However, this is not quite enough—we want to have an interpretation for every node in the tree. This gives us an interpretation of [S Neg S], but what is the interpretation of Neg? false if [S]M = true, true if [S]M = false. So, we also need rules for structures like S conj S, neg S, Vt N. ! What Neg does is takes the truth value of the S it is next to and reverses it. It is a function—it takes the truth value of the S it is next to as an argument, and returns a truth value (the opposite one). [it is not the case that] M = true ! false false ! true 6 Neg S ! ! ! [S [Neg It is not the case that] [S´ Pavarotti is boring]]. [Neg] M = [It is not the case that] M = true ! false false ! true [S´] M = true iff [N]M " [VP]M, otherwise false = true iff [Pavarotti]M " [Vi] M, otherwise false = true iff [Pavarotti]M " [is boring]M, otherwise false = F(Pavarotti) " F(is boring), otherwise false Neg S ! [S [Neg It is not the case that] [S´ Pavarotti is boring]]. ! And, so [S Neg S´] M = [Neg]M ( [S´]M ). ! Resulting in: ! [S]M = false if F(Pavarotti) " F(is boring), otherwise true. And ! For dealing with and and or, we also want to define a function. We want S 1 and S2 to be true when S1 is true and S 2 is true, and false under any other circumstance. [S S1 Conj S2] M = [Conj] M ( < [S1]M, [S2]M > ) ! [and] M = ! < true, true > ! true < true, false > ! false < false, true > ! false < false, false > ! false Or ! For dealing with and and or, we also want to define a function. We want S 1 or S2 to be false when S 1 is false and S 2 is false , and true under any other circumstance. [S S1 Conj S2] M = [Conj] M ( < [S1]M, [S2]M > ) ! [or] M = ! Transitive verbs ! The one piece of the model that we have not addressed yet are transitive verbs, like likes. ! ! ! ! S ! N VP VP ! Vt N Vt ! likes We want to be able to evaluate [S N VP] M the same way whether VP is built from a transitive verb or an intransitive verb. That is, we want [VP]M to be a predicate, a set of individuals. < true, true > ! true < true, false > ! true < false, true > ! true < false, false > ! false Transitive verbs ! ! Essentially, we want [likes Bond]M to be a set of those individuals that like Bond in M. However, we need a definition for [likes] M (we already have one for [Bond]M). It should be something that creates a set of individuals that depends on the individual next to it in the structure. A function again. 7 Transitive verbs Transitive verbs ! And then, we define a rule that will interpret the VP in a sentence with a transitive verb: ! Like and, likes relates two things, although likes relates two individuals, and and relates two sentences. ! So, we build a two-place predicate, in the same way: ! [VP Vt ! [likes]M = { : x likes y in M } ! ! For example, if P likes L, L likes B and that’s all the liking in this situation, then [likes] M = { , } S ! N VP [S N VP]M = true iff [N]M " [VP]M, otherwise false S ! S Conj S S ! Neg S [S S1 Conj S2 ]M = [Conj]M ( < [S1]M, [S 2 ]M > ) ! We have created a little fragment describing a (very small) subset of English, generating structural descriptions of syntactically valid sentences and providing the means to determine the truth conditions of these sentences. ! We did this by formulating a set of syntactic rewrite rules, each accompanied by a semantic rule of interpretation, such that every syntactic step can be interpreted compositionally. VP ! Vi N ! Pavarotti, … [Pavarotti]M = F(Pavarotti) Vi ! is boring, … [is boring]M = {x: x is boring in M} [likes]M = { : x likes y in M } Vt ! likes Conj ! and, … So if [N]M = Bond, then [VP Vt N] M is the set containing those individuals who like Bond in M. What we have [S Neg S´]M = [Neg]M ( [S´]M ). [VP Vt N]M = {x : < x, [N]M > " [Vt]M } VP ! Vt N N]M = {x : < x, [N] M > " [Vt]M } [and]M = {<,true>, ,false>, …} Neg ! it is not the [iintct]M = {, } case that One step more general ! ! Looking over the rules that we have, we can actually go a step further in generalizing our semantic rules (helpful as we expand our fragment’s coverage). There are basically two kinds of rules we have: Those that combine meanings of adjacent (sister) nodes in the syntactic structure, and those that define intrinsic (non-compositional) meanings. 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 Bond are of type , and sentences like Bond is hungry are of type . 8 Characteristic functions ! For predicates like is hungry, we have considered these to be sets of individuals (e.g., those that are hungry in the model). ! We can look at those same individuals in a slightly different way, using the characteristic function of the set. ! A characteristic function is a function that, given an argument, will return true iff the argument was a member of the set, and false otherwise. The same information content as the set. Predicates as functions ! ! Semantic type ! Predicates like is hungry can then be said to have semantic type . That is, a function from individuals to truth values. ! Similarly, it is not the case that can be taken to be of type , a function from truth values to truth values. Transitive verbs That is, we can define [likes]M as something like this: ! [likes]M = x ! f where f is a function from individuals to truth values and f(y) = true iff y likes x in M, otherwise false. ! That is, [likes]M is a function from individuals to functions (from individuals to truth values): semantic type >. For transitive verbs, what we want is a relation between two individuals, resulting in a truth value. The way we have it set up now, a verb like likes will combine with the object to form a simpler predicate like likes Bond, at which point it acts just like is boring. So, here, we want likes to take an argument of type and return a predicate of type . So, we define it as a function of type >. Why we’re doing this ! ! [is hungry] M = x ! true if x is hungry in M x ! false otherwise Transitive verbs ! ! So, without losing information, we can view predicates from the perspective of their characteristic functions and define is hungry to instead be a function that, given an individual, will return true if the individual is hungry in the model. ! ! Once we have defined things in terms of semantic type, and in terms of functions and arguments, we can collapse a number of our semantic interpretation rules into more general rules. Functional application: [# $]M = [#]M ([$]M ) or [$]M ([#]M), whichever is defined. Pass up: [$ #]M = [#]M 9