Introduction to Semantics (EGG Wroclaw 05)

Transcription

1 Introduction to Semantics (EGG Wroclaw 05) 0. Preliminaries 0.1 Semantics vs. pragmatics Semantics only concerned with literal meaning as opposed to non-literal, or situational meaning, most of which is covered by pragmatics. (Division of labour) Examples: irony (= meaning the opposite of what is literally said), can only be accounted for on the basis of literal meaning. 0.2 Ambiguity What is interpreted is not the (superficial) form but the expression. Sometimes the same form may correspond to two expressions. Homonymy: book as a verb and as a noun (morpho-syntactic structure); bank (pure disambiguation, no structure: bank 1, bank 2,) Structural ambiguity: (0) John hit the donkey with the stick 2 constituent structures => expressions (0 ) Every man loves a woman. 2 LFs => 2 expressions Relevant level of structure (Logical Form) may be semantically motivated. 0.3 Lexical vs. logical semantics Lexical semantics asks: What is the meaning of a given simple expression? Logical semantics asks: What is the meaning of a complex expression, given its structure and the meanings of the simple expressions it contains? Answer given in terms of Compositionality: The meaning of a complex expression is determined by its structure (LF) and the meanings of its immediate parts. 1. Sentence meaning 1.1 Basic ideas Sentence meanings as starting points, then take meanings of other expressions as contributions to sentence meanings (Frege s strategy). Descriptive aspect of sentence meaning: sentences describe/characterize/classify situations (1) Laura is knocking at the door. 1.2 Descriptions Descriptions make a distinction between objects of a given domain: to describe something as a computer = to put it into the same category with other objects (= computers) and distingushing it from still others (= non-computers).

2 Mathematical model: domains as sets satisfying two principles: Extensionality Sets A and B are identical as soon as they have the same members. + Comprehension For every condition there is a set containing precisely those objects as members that meet the condition. Notation: {x x} (= the set of objects x such that x). distinctions as charateristic functions A function from set A to set B is a set of ordered pairs (x,y) [ arrows x > y] where x A and y B and such that, for any x A there is precisely (= at least and at most) one y B such that (x,y) ƒ. Notation: ƒ: A > B; ƒ is of type (AB) NB: Ordererd pairs individuated by members and order: (x,y) = (x',y') justincase x = x' and y = y'! A characteristic function on a set U (= the domain) is a function from U to t, the set of truth values ({0,1}). Simplification: Replace characteristic function by characterized set: {x ƒ(x) = 1} 1.3 Situations maximally specific: A situation talked about (say, this situation) has many unknown aspects that are nonetheless settled. temporally located/limited: (2) The German chancellor is a woman. false now, probably true in the future; i.e. false of this situation, probably true of (some) future situation spatially unlimited can talk about the president of the US, wherever he is, etc. Hence: We may as well identify a situation with the world (at large) at some particular time (interval). BUT NOT WITH THE TIME ITSELF because situations are:

3 not necessarily actual (3) The Pope is a woman. (4) The Roman emperor is a woman. There is no situation which (3) describes correctly; likewise for (4). Hence (3) and (4) would characterize the same set of situation unless SOME SITUATIONS ARE NON-ACTUAL (or MERELY POSSIBLE) WORLDS at particular times. Logical Space (s) contains all possibilities, i.e. all possible worlds at particular times (as ordered pairs (w,t)). [Metaphysical simplification: cross-world identity of time] Terminology: Index for point in s 1.4 Main definitions The intension of a sentence is a function of from s to t. Hence it is of type (st). Notation: S The content of a sentence is the set characterized by its intension. Notation: S The extension of a sentence (relative to some index (w,t)) is the truth value its intension determines at (w,t). Notation: S w,t Terminology: Among semanticists, proposition denotes both intensions and contents of sentences.

4 2. Predication 2.1 Content as Contribution (1) Olaf is coughing. (2) Olaf is coughing = {(w,t) Olaf is coughing in w at t } = Olaf "+" is coughing Olaf is coughing =? 1 =? 2 (3a) (b) (c) Olaf is coughing = {(w,t) Olaf is coughing in w at t } Tim is coughing = {(w,t) Tim is coughing in w at t } Tom is coughing = {(w,t) Tom is coughing in w at t } Kripke s Hypothesis Olaf = Olaf, Tim = Tim, Tom = Tom, More generally: NN = the bearer of NN (4) Olaf is coughing = {(w,t) Olaf is coughing in w at t } = Olaf "+" is coughing Olaf = Olaf is coughing =? 2 Contents as contributions (5) is coughing = Olaf is coughing Olaf = {(w,t) Olaf is coughing in w at t } Olaf = {(w,t) is coughing in w at t } Contributions as functions The content of the predicate must contain sufficient information to determine the proposition expressed by the sentence once the content of the subject is provided: Filling subject content Olaf Tim Tom Table 1: The content of is coughing into the predicate content yields {(w,t) Olaf is coughing in w at t } {(w,t) Tim is coughing in w at t } {(w,t) Tom is coughing in w at t }

5 The table can be thought of as (representing) a function. This function is taken to be the content of the predicate. More generally: Frege s strategy G. Frege: Die Grundlagen der Arithmetik. Breslau [sic] 1884 Unless independently identifiable (by the semanticist), the meaning of an expression E may be construed as the contribution E makes to the meaning of (larger) expressions in which E occurs, i.e. as a function that assigns the meaning of the whole to the meanings of alternative complementary part(s): from: Rest * E Rest E = r =? where * is the relevant syntactic combination and A is expression A s meaning (semanctic value, content, extension, intension,) to: Rest 1 * E = f (r 1 ) Rest 1 E = r 1 = f Rest 2 * E = f (r 2 ) Rest 2 E = r 2 = f, where ƒ is the function assigning to any Rest the value Rest * E. NB: Only one of the consituents (immediate parts) may receive its meaning by Frege s strategy. Semantic composition If one of the constituent s meaning is obtained by Frege s principle, then the meaning of the whole is obtained by functional application: r + ƒ = ƒ( r ) [= the value ƒ assigns to r ] Conclusion The content of the predicate is coughing and of predicates in general is a function from individuals to sets of indices. 2.2 Lambdas changed my life (B. Partee) x {(w,t) x is coughing in w at t } Table 1a: Typical line of (the table representing) the content of is coughing The typical line contains enough information to completely determine the whole table (and thus the function is coughing ); it may therefore be used as a name of the function. the

6 Notational Convention If a is a set (type), then: [λx a. x] denotes the function that assigns to every x in a whatever object x denotes. Definition e is the set of all (possible) individuals (persons, tables, cities, numbers,). With these notational conventions is coughing = [λx e.{(w,t) x is coughing in w at t }] Three logical laws concerning λ-notation Law of α-conversion general law of variable binding The x is schematic and can be replaced by any variable y. In particular, [λx a. x] and [λy a. y] denote the same function (provided that variable confusion is avoided): (α) [λx a. x] = [λy a. y] Example: [λx e.{(w,t) x is coughing in w at t }] = [λy e.{(w,t) y is coughing in w at t }] Law of β-conversion important in applications [ β-reduction ] The value obtained by applying a function [λx a. x] to some object A of type a can be described by substituting A for x in the right hand side: (β) [λx a. x] (A) = A Example: [λx e.{(w,t) x is coughing in w at t }] (Tom) = {(w,t) Tom is coughing in w at t } Law of η-conversion less important If ƒ is the name of a function of some type (ab), then ƒ assigns to any x in a the value ƒ(x) and can thus be described by the lambda-term [λx a. f (x) ] : (η) [λx a. f (x) ] = ƒ Example: [λy e.[λx e.{(w,t) x is coughing in w at t }] (y)] = [λx e.{(w,t) x is coughing in w at t }] 2.3 Generalizing Frege s strategy TWO STEPS Transfer the notion of extension from sentences to names. The truth value of a sentence S can be thought of as (an indicator of) whatever the sentence refers to at a given index i (viz. i itself if S is true, and nothing otherwise). By analogy, the extension of a name is its bearer. Apply Frege s strategy to extensions (in lieu of meanings) As a consequence, the extension of the predicate is coughing and of predicates in general is a function from individuals to truth values, i.e. of type (et), e.g.:

7 Individual (Type e) Olaf 1 Tim 0 Tom 0 truth value (t) Table 2: Extension of is coughing in a situation (w*,t*) in which only Olaf is coughing Using (and extending) λ-notation: (*) is coughing w*,t* = [λx e. [whether] x is coughing in w*att*] (This must be understood as a function assigning 1 if the condition in the whetherclause is met, and 0 otherwise. [whether-convention] In the future, we will sometimes omit the whether.) Again we obtain functional application as the mode of (extensional) composition: Olaf is coughing w*,t* = is coughing w*,t* ( Olaf w*,t* ) functional application = [λx e. x is coughing in w*att*](olaf) by (*) = 1 by Table 2 + the whether-convention NB. Extensions of predicates correspond to sets of individuals, viz. the sets they characterize; it will turn out to be convenient to think of them as sets. Intensions in general are functions assigning extensions to indices. If A is any expression: A = λi s A i Intensions of proper names assign their bearer to any index ; hence they are of type (se) Alice = λi s Alice Alice? Who the Intensions of predicates assign (charateristic functions of) sets of individuals to indices; hence they are of type (s(et)). is coughing = [λi s is coughing i ] = [λi s [λx e. x is coughing in the world of i at the time of i]] = [λ(w,t,) [λx e. x is coughing in w at t]] nested lambdas notational simplification

8 2.4 Iterating Frege s strategy Content (6) [ John [ loves Mary ] ] from (a) loves Mary =[λx. {(w,t) s x loves Mary in w at t}] loves Sue =[λx. {(w,t) s x loves Sue in w at t}] loves =? Mary = Mary loves =? Sue = Sue to: (b) loves Mary = loves ( Mary ) =[λx e.{(w,t) s x loves Mary in w at t}] etc. loves =[λy e.[λx e.{(w,t) s x loves y in w at t}]] Mary = Mary (7) [ John [ [ shows Mary ] Wroclaw ] ] from (a) shows Mary =[λy.[λx.{(w,t) s x shows y to Mary in w at t}]] shows Sue =[λy.[λx.{(w,t) s x shows y to Sue in w at t}]] shows =? Mary = Mary shows =? Sue = Sue to: (b) shows Mary = shows ( Mary ) =[λy.[λx.{(w,t) s x shows y to Mary in w at t}]] etc. shows =[λz e.[λy e.[λx e.{(w,t) s x shows y to z in w at t}]] Mary = Mary Extension loves wt =[λy e.[λx e. [whether] x loves y in w at t]] (= that function f of type (e(et)) such that f (y)(x) = 1 if[ and only if x loves y in w at t) shows wt =[λz e.[λy e.[λx e. [whether] x shows y to z in w at t]] (= that function f of type (e(e(et)) such that f (z)(y)(x) = 1 if[ and only if x loves y in w at t)

9 Intension loves =[λ(w,t).[λy e.[λx e. [whether] x loves y in w at t]]] shows =[λ(w,t).[λz e.[λy e.[λx e. [whether] x shows y to z in w at t]]] Type correspondence Category content extension intension Sentence p t st Name e e se intransitive ep et s(et) transitive e(ep) e(et) s(e(et)) ditransitive e(e(ep)) e(e(et)) s(e(e(et))) Table 5: Types of semantic values The (contribution to the) content is determined starting from propositions (as the contents of sententences) and individuals (as the contents of names) and applying Frege s strategy. [Russellian semantics; cf. Kaplan (1975)] The extension is determined starting from truth values (as the extensions sententences) and individuals (as the extensions of names) and applying Frege s strategy. [extensional semantics] The intension is the function that assigns the extension to each index. [Frege- Carnap semantics] 3. Quantification 3.1 Quantifiers in subject position (8) Nobody is coughing. (a) Nobody is coughing ={(w,t) s nobody is coughing in w at t } nobody =? is coughing =[λx e {(w,t) s x is coughing in w at t }] (b) Nobody is coughing w,t = [whether] nobody is coughing in w at t nobody w,t =? is coughing w,t = λx e [whether] x is coughing in w at t

10 Filling predicate content is coughing is laughing loves Mary X ep Table 4: The content of nobody into the subject content yields {(w,t) nobody is coughing in w at t } {(w,t) nobody is laughing in w at t } {(w,t) nobody loves Mary in w at t } {(w,t) nobody X in w at t } Filling predicate extension into the subject extension yields is coughing 1 iff nobody is couging in w* at t* is laughing 1 iff nobody is laughing in w* at t* loves Mary 1 iff nobody loves Mary in w* at t* P et 1 iff nobody P in w* at t* Table 5: The extension of nobody at an arbitrary index (w*,t*) PRIZE QUESTION: What are the typical lines of Tables 4 and 5? How do they depend on the (arbitrary) arguments X and P, respectively? PARTIAL ANSWER (pertaining to Table 5 only): nobody is couging in w* at t* iff no person is coughing in w* at t* iff nothing that is a person in w* at t* is coughing in w* at t* iff for no x that is a person in w* at t* do we have: x is coughing in w* at t* iff for no x that is a person in w* at t* do we have: is coughing w,t (x) =1 [iff {x x is a person in w*att*} is coughing w,t = ] P et 1 iff P(x) = 0, for all persons x in w* at t* Table 6: Typical line in the extension of nobody at an arbitrary index (w*,t*) Using λ-notation: nobody w,t =[λp et. P(x) = 0, for all persons x in w at t]

11 Similarly: no horse w,t =[λp et. P(x) = 0, for all horses x in w at t] every horse w,t =[λp et. P(x) = 1, for all horses x in w at t] someone w,t =[λp et. P(x) = 1, for at least one person x in w at t] a donkey w,t =[λp et. P(x) = 1, for at least one donkey x in w at t] REST OF ANSWER to prize question: Homework exercise (or look it up in my 2002 class notes) 3.2 Extensions of Determiners (9) every donkey w,t =[λp et. P(x) = 1, for all donkeys x in w at t] every w,t =? donkey w,t =? (10) = = = every donkey w,t [λp et. P(x) = 1, for all donkeys x in w at t] [λp et. P(x) = 1, for all x such that x is a donkey in w at t] [λp et. P(x) = 1, for all x such is a donkey w,t =1] Intelligent guess: donkey w,t = is a donkey w,t =[λx e. [whether] x is a donkey in w at t] and similarly for other (count) nouns (11) every donkey w,t =[λp et. P(x) = 1, for all donkeys x in w at t] every w,t =? donkey w,t =[λx e. [whether] x is a donkey in w at t] Applying Frege s method: every w,t =[λq et.[λp et. P(x) = 1, for any x such that Q(x) = 1]] (The set characterized by Q is a subset of the set characterized by P.) Similarly:: no w,t =[λq et.[λp et. P(x) = 0, for any x such that Q(x) = 1]] (The set characterized by Q is disjoint from the set characterized by P.) some w,t =[λq et.[λp et. P(x) = 1, for at least one x such that Q(x) = 1]] (The set characterized by Q overlaps the set characterized by P.)

12 References (mostly implicit, or made in class) Carnap, R.: Meaning and Necessity. Chicago/London Cresswell, M. J.: Logics and Languages. London Frege, G.: Die Grundlagen der Arithmetik. Breslau [translated into English] : Über Sinn und Bedeutung. Zeitschrift für Philosophie und philosophische Kritik 100 (1892), [various English translations available] Kaplan, D.: How to Russell a Frege-Church. Journal of Philosophy 72 (1975), Lewis, D. K.: On the Plurality of Worlds. Oxford Montague, R.: English as a Formal Language. In: B. Visentini (ed.), Linguaggi nella società e nella tecnica, Mailand : Universal Grammar. Theoria 36 (1970), Wittgenstein, L.: Tractatus logico-philosophicus [translated into English] Zimmermann, T. E.: Grundzüge der Semantik. Class notes, Uni Frankfurt, summer term (accessible from my home page)