Michael Franke Fritz Hamm. January 26, 2011

Transcription

1 Michael Franke Fritz Hamm Seminar für Sprachwissenschaft January 26, 2011

2 Three n would, p would, l would, n might p might l might

3 Basic intuition (1) If that match had been scratched, it would have lighted. The first problem [in the interpretation of counterfactuals] is... to specify what sentences are meant to be taken in conjunction with the antecedent as a basis for inferring the consequent. Goodman, 1947

4 Basic Apparatus I A model M is a pair < W,V > consisting of a non-empty set of worlds W and a valuation function V. Definition 1 [[p ]] M w = 1 iff w V(p). 2 [[φ ψ]] M w = 1 iff [[φ]] M w = [[ψ]] M w = 1 3 [[φ ψ]] M w = 0 iff [[φ]] M w = [[ψ]] M w = 0 4 [[φ ψ]] M w = 0 iff [[φ]] M w = 1 and [[ψ]] M w = 0 5 [[ φ]] M w = 1 iff [[φ]] M w = 0

5 Basic Apparatus II Prem w (φ) is the set of premise sets associated with φ at world w. Definition (would-counterfactual) [[φ ψ]] w = 1 iff X Prem w (φ) Y Prem w (φ)[x Y \ Y W ψ] The would counterfactual φ ψ is true at w if and only if every set in PREM w (φ) has a superset in PREM w (φ) which entails ψ.

6 Basic Apparatus III Definition (might-counterfactual) [[φ ψ]] w = 1 iff X Prem w (φ) Y Prem w (φ)[x Y \ Y ψ W /0] The might-counterfactual φ ψ is true at w if and only if there is a set in PREM w (φ) all of whose supersets in PREM w (φ) are consistent with ψ.

7 Basic Apparatus IV Proposition φ ψ iff (φ ψ)

8 Naive premise semantics I Definition (Naive premise set) Prem n w(φ) = {X f(w) {φ} : T X /0 and φ X}, where f(w) = {p (W) : w p}. (2) a. If Paula weren t buying a pound of apples, the Atlantic Ocean might be drying up. b. (If Paula isn t buying a pound of apples ) n (the Atlantic Ocean is drying up).

9 Naive premise semantics II Definition (Kratzer s might-counterfactuals) A might -counterfactual is true in a world w if and only if not every way of adding propositions which are true in w to the antecedent while preserving consistency reaches a point where adding the consequent would result in an inconsistent set. (3) a. Paula is buying a pond of apples. b. The Atlantic Ocean isn t drying up. c. Paula is buying a pound of apples or the Atlantic Ocean is drying up.

10 Naive premise semantics III Proposition 1 φ n ψ (φ ψ) ( φ (φ ψ)) 2 φ n ψ (φ ψ) ( φ (φ ψ))

11 Naive premise semantics IV If ψ is a false sentence that is consistent with the negation of a true sentence φ, then φ n ψ is true.

12 Partition semantics I Definition (Partition function) A function f : W ( (W)) is a partition function if and only if for every w W, T f(w) = {w}. Definition (Partition premise set) let f be a partition function. Then Prem p w(φ) = {X f(w) {φ} : \ X /0 and φ X}

13 Partition semantics II Each premise set corresponds to an equivalent ordering frame. Proposition (Lewis, 1981) Equivalent frames evaluate counterfactuals alike, at least in the finite case.

14 Partition semantics III (4) Let a world w be such that a. a zebra escaped; b. it was caged with another zebra; c. a giraffe was also in the same cage. (5) a. If a different animal had escaped, it might have been a giraffe. b. (a different animal escaped) p (it was a giraffe)

15 semantics: Motivation I Pedant: What did you do yesterday evening? Paula: The only thing I did yesterday evening was paint this still life over there. Pedant: This cannot be true. You must have done something else like eat, drink, look out of the window. Paula: Yes, strictly speaking, I did other things besides paint this still life. I made myself a cup of tea, ate a piece of bread, discarded a banana, and went to the kitchen to look for an apple.

16 semantics: Motivation II Lunatic: What did you do yesterday evening? Paula: The only thing I did yesterday evening was paint this still life over there. Lunatic: This cannot be true. You also painted apples and you also painted bananas. Hence painting this still life was not the only thing you did yesterday evening.

17 semantics: Motivation III There is an aspect of the actual world that makes the proposition that Paula painted a still life true. And that very aspect of our world also makes the proposition that she painted apples true. [... ] Let us say that the proposition that Paula painted a still life lumps the proposition that she painted apples in the actual world. Kratzer, 1989, p 609.

18 semantics: Motivation IV The proposition that Paula painted apples does not lump the proposition that she painted a still life in the actual world. If Paula had painted only apples and no bananas then the proposition that Paula painted apples would lump the proposition that she painted a still life in the actual world.

19 semantics: Formalization I Definition (Situation Model) A situation model is a triple M =< S,,V >, where S is a non-empty set of situations. is a partial order on S satisfying the following condition: For all s S there is a unique s S such that s s and for all s S, if s s then s s. V is a function mapping propositional variables to subsets of S. Definition (Truth) A proposition p is true in a situation s S if and only if s p.

20 semantics: Formalization II Definition (Worlds) For each s S, let w s S be the maximal situation such that s w s. The set of worlds in M is the set W = {w s : s S}. Definition (Consistencys) A set of propositions A (S) is consistent if T A W /0. Definition (Logical consequence) A proposition p (S) logically follows from a set of propositions A (S) if and only if T A W p.

21 semantics: Formalization III Definition (Persistence) A proposition p S is persistent if and only if for all s,s S, if s p and s s, then s p. Definition [[φ ψ]] s = 1 iff [[φ]] s = 1 and [[ψ]] s = 1 [[φ ψ]] s = 1 iff [[φ]] s = 1 or [[ψ]] s = 1 (6) a. φ is persistent. b. {φ, φ} is inconsistent; hence, by persistence, for all w W, if w φ then there is no s w such that s φ. c. W φ φ.

22 semantics: Formalization IV Definition (Relevance function) A function f : W ( (S)) is a relevance function iff for all w W, f(w) {p (S) : w p and p is persistent}. Definition () For all p,q S and w W, p lumps q in w if and only if w p and p {s : s w} q. We write p w q for p lumps q in w.

23 semantics: Formalization V Proposition Let p and q be persistent propositions. If w p and w q, then (p q) w q. Definition (Closure under lumping) A set of propositions A (S) is closed under lumping in w (relative to f(w)) if and only if for all p A and all q f(w), if p lumps q in w, then q A. Definition (Closure under logical consequence) A set of propositions A (S) is closed under logical consequence (relative to f(w)) if and only if for all p f(w), if p follows logically from A, then p A.

24 semantics: Formalization VI Definition ( premise set) Let f be a relevance function. Prem l w(φ) = {X f(w) {φ} : T X W /0 φ X p X q f(w)[p w q q X] p f(w)[ T (X f(w)) W p p X f(w)]}.

25 Kratzer s solution I (7) If Paula weren t buying a pond of apples, the Atlantic Ocean might be drying up. (8) a. Paula is buying a pond of apples. b. The Atlantic Ocean isn t drying up. c. Paula is buying a pound of apples or the Atlantic Ocean is drying up. Proposition (8-c) lumps proposition (8-a) in the actual world. (8-a), however, is not compatible with the antecedent of the counterfactual (7). Therefore we cannot add the proposition expressed by (8-c) to the antecedent of the counterfactual. Therefore (7) is not true in the actual world.

26 Kratzer s solution II (9) If a different animal had escaped, it might have been a giraffe. (10) a. A zebra escaped b. A striped animal escaped c. A black and white animal escaped d. A male animal escaped e....

27 premise set: Kratzer s version I Definition Let f be a relevance function. Prem l w(φ) = {X f(w) {φ} : T X W /0 φ X p X q f(w)[p w q q X] p f(w)[ T (X \ {φ}) W p p X \ {φ}]}.

28 premise set: Kratzer s version II (11) a. If Paula were buying a pound of apples, she would be buying a pound of Golden delicious. b. (Paula is buying a pound of apples) l (Paula is buying a pound of Golden Delicious) (12) a. If Paula were buying a pound of apples, she might be buying a pound of Golden delicious. b. (Paula is buying a pound of apples) l (Paula is buying a pound of Golden Delicious)

29 Preliminary lemmas Lemma Suppose φ is true at w. Then φ l ψ is true at w iff f(w) {φ} is consistent with ψ. Lemma If φ and ψ are true at w, then φ l ψ is true at w. Lemma Suppose φ and ψ are true at w, and ψ f(w). Then φ l ψ is true at w. Lemma Suppose { ψ} {p (S) : w p} f(w). If φ is true at w, φ l ψ is true at w iff ψ is true at w.

30 Triviality of lumping semantics I Proposition Suppose that {W,{w}} f(w). Then φ l ψ is true at w iff φ ψ is true at w. Proposition Under the same assumption, φ l ψ is true at w iff φ ψ is true at w.

31 Triviality of lumping semantics II Proposition Suppose that {φ φ, φ, ψ} {p (S) : w p} f(w). Then φ l ψ is true at w iff φ ψ is true at w, and φ l ψ is true at w iff φ ψ is true at w.

32 Triviality of lumping semantics III (13) a. If Paula were not buying a pound of apples, she might not be buying a pound of apples. b. (Paula is buying a pound of apples) l (Paula is buying a pound of apples)

33 Triviality of lumping semantics IV Proposition Suppose that (i) f(w) contains no propositions that are true in all possible worlds; (ii) whenever p is in f(w), (p φ) (p φ) is in f(w) and moreover, if φ is true at w, p φ is also in f(w); and (iii) if ψ is true at w, ψ is in f(w). Then φ l ψ is true at w iff φ n ψ is true at w, and φ l ψ is true at w iff φ n ψ is true at w.