Seminar in Semantics: Gradation & Modality Winter 2014

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1 Seminar in Semantics: Gradation & Modality Winter 2014 Dan Lassiter 1/15/14 Handout Contents 1 Sign up to present a reading and lead discussion 1 2 Finishing up Kratzer Klein Orders and measurement theory (Reading: Lassiter 2014: 2) 8 1 Sign up to present a reading and lead discussion 2 Finishing up Kratzer 1991 Definition 1. (Modal base) A modal base is a function f from worlds to sets of propositions [f W P(P(W ))]. Some interpretations: If f is epistemic, f(w) is the set of propositions known at w. f(w) is the set of worlds compatible with everything that is known at w (so w f(w)). (With caveats) If f is doxastic, f(w) is the set of propositions believed at w. f(w) is the set of worlds compatible with everything that is believed at w, and w may or may not be in f(w). If f is circumstantial, f(w) is a subset of what is known at w (so w f(w)). Epistemic modals typically get epistemic or doxastic modals bases; root modals (deontic, ability,...) get circumstantial modal bases. Definition 2. (Ordering source) An ordering source is a function f from worlds to sets of propositions [g W P(P(W ))]. Modal bases and ordering sources are distinguished only by how the definitions of lexical items make use of them. 1

2 Definition 3. (Kratzerian world-ordering) A set of propositions A induces a preorder 1 A on W as follows: w w {p p A w p} {p p A w p} Gloss: w is at least as close as w to the set of ideals given by the propositions in A iff w every proposition in A that is true in w is also true in w. Example Let W = {w 1,..., w 6 }. Consider a modal base and ordering source which assign to world w 3 two pieces of background knowledge and two ideals (norms, rules, expectations,...). f(w 3 ) = {{w 1, w 2, w 3, w 4, w 5 }, {w 1, w 2, w 3, w 4, w 6 }} g(w 3 ) = {{w 1, w 2 }, {w 1, w 3, w 6 }} Then we have the set of accessible worlds f(w 3 ) = {w 1, w 2, w 3, w 4 } As for g(w3 ) focusing on f(w 3 ), which is where the action is: For all w f(w 3 ), w g(w3 ) w (i.e., g(w3 ) is reflexive because is) w 1 g(w3 ) w 2 w 1 g(w3) w 4 w 2 g(w3) w 4 w 1 g(w3 ) w 3 w 2 g(w3 ) w 3 w 3 g(w3 ) w 4 But we do NOT have w 2 g(w3 ) w 3. (Why?) In this model f(w), g(w3 ) has the structure of a Boolean algebra (with g(w3 ) restricted to f(w 3 )). This does not hold in general: There may be two worlds in f(w 3 ) which fulfill exactly the same g(w 3 )-propositions, in which case g(w3 ) will be a preorder rather than a partial order For some p, q g(w 3 ) no w f(w 3 ) is in both p and q. (Then g(w3 ) will be gappy ) However, the induced ordering on g(w3 )-equivalence classes does form a Boolean algebra whenever all the propositions in g(w 3 ) are logically independent, and For every p, q g(w 3 ), there is some w f(w 3 ) such that w p q. 1 A preorder is a reflexive and transitive binary order. Kratzer calls this a partial order, but standard mathematical usage applies this term only to preorders which are also antisymmetric. 2

3 Definition 4. (Comparative Possibility) Relative to w, f, and g, a proposition p is at least as good a possibility as a proposition q iff: for every world u that is in both f(w) and q, there is a world v that is in both f(w) and p such that u g(w) v. (Space-saving abbreviation: p s w q.) Examples In the model described above, {w 2, w 3 } s w 3 {w 3 } {w 2, w 3 } s w 3 {w 3, w 4 } {w 2, w 3 } s w 3 {w 4 } {w 2 } s w 3 {w 3 } {w 3 } s w 3 {w 2 } {w 2, w 4 } s w 3 {w 3, w 4 } {w 1 } s w 3 {w 2, w 3, w 4 } {w 1 } s w 3 {w 1, w 2, w 3, w 4 } Indeed we have {w 1 } s w 3 W if there is any world in f(w) that satisfies all the relevant norms/expectations/etc., then the unit set containing that world is at least as good a possibility as the entire set of possible worlds i.e., as likely as any tautology, or any arbitrary proposition. Definition 5. (Interpretation of at least as likely as) φ is at least as likely as ψ w,f,g = 1 iff φ w,f,g s w ψ w,f,g, for appropriate f and g. Definition 6. (Interpretation of more likely than) φ is more likely than ψ w,f,g = 1 iff φ w,f,g < s w ψ w,f,g for appropriate f and g. [with < s w defined from s w in the obvious way] Definition 7. (Interpretation of likely) φ is likely w,f,g = 1 iff φ w,f,g < s w φ w,f,g, for appropriate f and g. Though she does not mention it explicitly, presumably Kratzer (1991) intends these definitions to suffice for good/better, with different restrictions on f and g. Definition 8. (Simplified interpretation of must) must φ w,f,g = 1 iff all of the worlds in f(w) that are not strictly dominated in g(w) by any f(w)-worlds are in φ w,f,g. 2 In the model above, any proposition which is true at w 1 must be the case; we can basically ignore the rest of the model and focus exclusively on facts about the optimal world(s). 2 w strictly dominates w in the relevant order is here a way of saying w < w in English, where the directionality of dominance is implied by the use to which we are putting the model. This definition given here is equivalent to Kratzer s more complicated definition unless both (a) there is an infinite number of propositions in g(w), and (b) (f(w) g(w)) =. 3

4 Definition 9. (Simplified interpretation of might) Might is must s dual: must φ w,f,g = 1 iff some of the worlds in f(w) that are not strictly dominated in g(w) by any f(w)-worlds are in φ w,f,g. Definition 10. (Kratzer conditional) If q is explicitly modalized, If p, q w,f,g = q w,f+p,g If q is not explicitly modalized, If p, q w,f,g = must q w,f+p,g where f + p adds p to the modal base, i.e., returns f(z) {p} for any z W. In our model, both must {w 1 } and If not {w 1 }, {w 2, w 3 } come out true. Kratzer argues that this is an improvement on usual theories of modality and conditionals based on conflicts between norms: Example Modify the model above so that the ordering source is inconsistent: g(w) = {{w 1, w 2 }, {w 2, w 3 }, {w 1, w 3 }} g(w) =, but we don t have an explosion: must not {w 4 } is true and must {w 4 } false. As I mentioned last week, I don t find the goat-owners example very convincing, since it seems to be best represented as uncertainty about what the law is (i.e., uncertainty about which function g is) rather than conflicting laws. Anyway, later in the seminar I ll argue that Kratzer s solution gets the facts wrong for conflicting obligations of a more familiar type. 3 Klein 1980 How should we interpret positive-form adjectives in sentences like Sam is tall and Mary is clever, and how do their relate to those of comparative constructions like Sam is taller than Mary and Mary is more clever than Sam? And of course we want to know how this relates to It s at least as like to rain as it is to snow Giving money to charity is better than using it to buy lottery tickets noting in particular that Kratzer does not give a compositional account of these locutions. Can we use her theory to provide one? To see how, we ll start by considering qualitative models of comparison and degrees that resemble Kratzer s in some respects. Klein begins by arguing that we shouldn t take degrees as a primitive, as degree semantics does, but instead should respect the principle of compositionality and the primacy of surface syntactic constituency by deriving the meaning of the comparative from the meaning of the positive form. That is, we should have logical forms like these tall LF [ AP tall] taller LF [ AP tall er] 4

5 rather than these, where only part of the positive form occurs in the comparative. tall LF [ AP pos tall] taller LF [ AP tall er] Notably, no language seems to pronounce pos. Klein concludes: As far is I can tell, there is no independent justification for introducing pos; it is merely a device for fixing up the semantics. [Not sure if the situation really is as dire as all this, but let s run with it for the moment.] How can we do better? Let s start with the assumption that the predicative use of adjectives is fundamental, as in (1). We ll ignore for the moment attributive uses like (2). (1) a. Sam became happy. b. Mary seems content. c. Bill is bewildered. (2) a. Sam is a happy guy. b. Content people are unproductive. c. Bill gave us a bewildered stare. 2 kinds of adjectives: Definition 11. (Linear adjective) For all contexts c, and all x and y in the sortal range of A, x is more A than y is either true or false. (e.g., tall, heavy) Definition 12. (Non-linear adjective) For some context c, and some x and y in the sortal range of A, x is more A than y is neither true nor false. (e.g. clever) The former exhibit graduality ; the latter also exhibit indeterminacy of criteria for establishing a comparison. Definition 13. (Partial interpretation of adjectives) Suppose that D e is the set of entities in the model and C is the set of possible contexts of use. Then adjectives denote context-sensitive, (possibly) partial functions from D e to {0, 1}. In the usual case, for some c and x, x is A c = 1 for some c and x, x is A c = 0 for some c and x, x is A c = # (is undefined) If x is A c = 1, we say that x is in the positive extension of A in c. If x is A c = 0, x is in the negative extension of A in c. If x is A c = #, x is in A s extension gap in c. Definition 14. (Precisification (Kamp 1975; Fine 1975)) A precisification of A in c is an interpretation relative to a new context c + which agrees with the interpretation of A in 5

6 c on the positive and negative extension of A. That is, a precisification function I assigns to each context c and adjective A a set of new contexts such that: (Classicality) For all c + I(c, A), A has no extension gap at c +. (Agreement) For all x in A s positive (resp. negative) extension at c, x is in A s positive (resp. negative) extension at c +. (No Reversal) If there is a context c + I(c, A) such that x is A c+ = 1 and y is A c+ = 0, then there is no context c ++ I(c, A) such that x is A c++ = 0 and y is A c++ = 1. Definition 15. (Supervaluation) φ is supertrue at c iff φ c+ = 1 for all c + I(c, A). φ is superfalse at c iff φ c+ = 0 for all c + I(c, A). φ is super-undefined at c if it is neither supertrue not superfalse at c. A supervaluation will agree with the partial interpretation from which the precisified contexts are derived in all judgments about applications of adjectives to specific individuals: sentences that are super-undefined at c are also undefined at c. However, we obtain interesting behavior with respect to certain complex formulae of interest. For example, φ φ is a super-tautology supertrue at all contexts c even when φ is undefined, and so true at some contexts and false at others. The no reversal property allows us to recover a notion of degree by considering whether the precisifications in which x is A are a subset of the ones in which y is A. Definition 16. (Kamp-style semantics for comparatives) x is more A than y c = 1 iff {c + I(c, A) x is A c+ = 1} {c + I(c, A) y is A c+ = 1}. Klein s objection: if x and y are both in the positive or negative extension, the comparative is trivially false. This is clearly wrong: Yao Ming is taller than Shaq, even though both are clearly tall. Klein s solution: relativize interpretations not only to contexts, but also to comparison classes. Intuitively, the idea is that the context-sensitive meaning is sensitive not only to objective features of the context, but also to which items we happen to be paying attention to. Extensions are resolved in more or less the way we ve seen, except that supervaluations are induced and extended into the positive and negative extensions of the adjective relative to a big comparison class by considering ever smaller comparison classes. In this way, we ll be able to assign truth to Yao Ming is taller than Shaq by 6

7 Requiring that tall have a non-empty positive and negative extension relative to any comparison class (Non-Triviality) Enforcing the No Reversal requirement we saw above above Giving the comparative a kind of modal interpretation, operating over contexts and comparison classes rather than worlds: x is taller than y is true at c and CC if there is some subset CC + of the comparison class CC such that x is tall and y is not tall comes out true relative to c and CC +. Noting that there is a comparison class on which Yao Ming is taller than Shaq will then come out true: the one that contains just these two individuals. Many formal definitions and interesting syntactic and semantic details ensue; here I ll give just a few key points. Klein defines an ordering c,a (a strict weak order) for any adjective A, relative to context c, essentially as: x c,a y iff there is a subset CC + of the comparison class appropriate to c such that x is in the positive extension of A relative to c and CC +, and y is in A s negative extension relative to the same. This order is used to provide truth-conditions for comparatives: x is more A than y iff x c,a y. Klein shows that, given the constraints he imposes on comparison classes, c,a is a strict preorder (transitive and asymmetric) if A is a non-linear adjective, and a strict weak order (all that plus almost-connected) if A is linear. Proof (p.23-24): Asymmetry follows from No Reversal Transitivity is proved as follows: if x c,a y then x is A and y is not A in CC {x, y}. Likewise if y c,a z then y is A and z is not A in CC {y, z}. It follows by No Reversal and Non-Triviality that x is A and z is not A in the CC {x, y, z}, so x c,a z. For linear adjectives, almost-connectedness is definitional. Measure phrases such as foot: Klein suggests further constraining his orders by adding a concatenation relation, as in Measurement Theory (see below). We then pick some arbitrary object f (something one foot long) as the unit, and make sure that x is six feet tall comes out true just in case x is at least as tall as the concatenation of 6 items exactly as tall as u. (compositional details suppressed) Klein discusses an important methodological point (p.15-16): It might be objected at this point that adjectives turn out to be no less relational on my approach than they are in the degree theory that I criticised earlier...; the main difference is that the extra argument is a comparison class rather than 7

8 a degree, and it has been shunted out of the logical structure into a contextual coordinate. Related question: isn t this just a laborious way of defining a degree semantics? Klein responds by pointing to some arguments involving ellipsis. I m unsure if they re really compelling, but either way, the point is that there are deep connections between degree-free and degree-ful theories of scalar adjectives. The choice between probably can t be made on the basis of expressiveness, as some have claimed (e.g., von Stechow (1984))). Rather, it involves some fairly intricate issues about our choices in setting up a compositional semantics and its interactions with other empirical and theoretical issues in syntax and semantics. 4 Orders and measurement theory (Reading: Lassiter 2014: 2) Let s have a little talk about binary orders. Definition 17. (Transitivity) R is transitive iff xry yrz xrz for all x, y, z. Definition 18. (Reflexivity) R is reflexive iff xrx for all x. Definition 19. (Antisymmetry) R is antisymmetric iff xry yrx x = y for all x, y. Definition 20. (Connectedness/completeness) R is connected (complete) iff xry yrx for all x, y. Definition 21. (Asymmetry) R is asymmetric iff there are no x, y such that xry yrx. Kratzer s (1991) w order on worlds is reflexive and transitive, but not generally connected. That makes it a preorder (a.k.a. quasi-order). Here s this and some more useful terminology defined: Definition 22. (Preorder/quasi-order) R is a preorder (quasi-order) iff R is reflexive and transitive. Examples: Kratzer s w and s w relations. Definition 23. antisymmetric. (Partial order) R is a partial order iff R is reflexive, transitive, and Example: The powerset algebra generated by a set with two or more elements. Definition 24. (Weak order) R is a weak order iff R is R is reflexive, transitive, and connected. Examples: The at least as tall as relation height on a set of objects with vertical extent The powerset algebra generated by a 1-element set. 8

9 Definition 25. (Equivalence relation) R is an equivalence relation iff R is reflexive, transitive, and symmetric. Examples: The exactly as tall as relation height defined from the at least as tall as relation height above, setting x height y iff both x height y and y height x. An accessibility relation in the modal logic S5, where worlds are arranged into fully connected clusters. Definition 26. (Total order/linear order) R is a total (linear) order iff R is R is transitive, connected, and antisymmetric (and so reflexive). Examples: (a) The at least as great as relation on the real numbers (b) The at least as great as relation on numerical measurements of height (c) the order on equivalence classes of the at least as tall as order above, induced by the rule EC [x] is at least as great as EC [y] iff every member x of [x] is at least as tall as every member y of [y]. Finally, we define the strict counterpart of a partial, weak, or total order by setting x y = df x y (y x). Suppose we have a binary order representing comparative judgments of height among some set X, together with perhaps with some further relations and operations. We ll call S = X,,... a structure. The fundamental question of Measurement Theory (Scott & Suppes 1958; Krantz, Luce, Suppes & Tversky 1971) is: Taking structure S = X,,... to be the basic object, which numerical representations faithfully preserve the properties of? Less intuitively, but more precisely: what is the class of homomorphic embeddings of into the real numbers? Definition 27. (Homomorphism) A homomorphism is a structure-preserving map between two structures. For example, µ is a homomorphism from X, into R, iff x y implies µ(x) µ(y). µ is a homomorphism from X,, into R,, + iff x y implies µ(x) µ(y), and x y = z implies µ(x) + µ(y) = µ(z). 9

10 (Note: nothing special about + here, but that s the operation we ll see the most of.) Intuitively, a homomorphic embedding of S 1 into S 2 preserves all of the information in S 1, while possibly also adding more information. For instance, consider the following homomorphisms from X, into R, (with the latter interpreted as usual): X = {a, b, c}; = {(a, a), (a, b), (b, b), (a, c), (b, c), (c, c)} µ 1 (a) = 2; µ 1 (b) = 1; µ 1 (c) = 0 µ 2 (a) = 10; µ 2 (b) = 1; µ 2 (c) = 0 µ 3 (a) = ; µ 3 (b) = 0; µ 3 (c) = All of these functions maintain the property that x y implies µ i (x) µ i (y). But, notice that other mathematical properties of the reals are not constant across these homomorphisms (e.g., addition, multiplication, division). Definition 28. (Admissible measure function) µ is an admissible measure function on X, relative to scale S, iff µ is a homomorphism from S into R,.... Definition 29. (Interpretability) A statement which makes reference to a numerical measurement derived from a structure S is interpretable (RTM: meaningful ) iff it has the same truth-value under all homomorphisms from S into R,.... The reason for adding this constraint is this: we don t want to say that a has twice as much of the property in question as b, simply because µ 1 assigns it a number twice as great. This information is not carried in the source structure, and so should be prevented from coming out true simply because we happened to look at µ 1, which would make it true. So, we make sure that we only assign this sentence truth if it is true in all admissible µ, which it isn t. Definition 30. (Ordinal scale) An ordinal scale is a structure X,, where is a weak order. Examples: Air quality measurement on a 1-5 scale (pp. 33-4). Measuring pain on a 0-10 scale (?) Definition 31. (Ratio scale) A ratio scale is a structure X,,,..., where is a weak order and X 2 X is a concatenation operation. 3 Examples: extensive measurements : height, width, length,... 3 Usually required to meet very strong constraints when ratio scales are under discussion: associative, monotonic, and Archimedean. We can get away with a weaker formulation that s useful for finite X and for other kinds of scales, though. 10

11 Probability Theorems (Scott & Suppes 1958; Krantz et al. 1971): If S = X,, is a ratio scale then, for all admissible µ, all x X, and all α > 0: µ (x) = µ(x) α is also an admissible measure function. [This reflects the arbitrariness of the unit of measurement.] Ratio scales are additive: for all admissible µ and all x and y, µ(x y) = µ(x) + µ(y). Statements about ratios of measurements are interpretable, even though absolute quantities are not. Definition 32. (Interval scale) An interval scale is a structure X, Y,, where Y X 2 and is a binary relation over Y satisfying some additional conditions (weak monotonicity, solvability, Archimedean). The relation (a, b) P (c, d) can be read a exceeds b with respect to property P by more than c exceeds d. Theorems (Krantz et al. 1971): If S is an interval scale then, whenever µ is admissible for S, for all µ : If µ (x) = α µ(x) + β for any α R + and β R, then µ is also admissible for S P. α represents the fact that interval scales have no fixed zero point; β represents the fact that the unit of measurement is arbitrary. (a, b) P (c, d) iff [µ(a) µ(b)] [µ(c) µ(d)] Ratios of differences are interpretable on interval scales, but absolute ratios are interpretable only in the trivial case where µ(x) = µ(y). Examples of interval scales: Conversion from Celsius to Fahrenheit: α = 9 5 and β = 32 Expected utility Common-sense theories of temperature and clock time Some reasons to care about these distinctions: (3) Sam grew from 2 feet to 3 feet, and Harry grew from 4 feet to 6 feet. a. So, Harry grew twice as much as Sam did. b. So, Harry is now twice as tall as Sam is. (4) My car is 1.5 times as wide as yours. 11

12 (5) I ran from 2PM to 3PM, and you ran from 4PM to 6PM. a. So, you ran for twice as long as I did. b. # So, you started running twice as late as I did. (6)?? Atlanta is twice as hot as New York. These contrasts can be explained if height and width are ratio scales, but length of time and temperature are (mere) interval scales. Further points in ch. 2, to be discussed as they become important later on: relationship between concatenation and the join operation of algebraic semantics intermediate scales: µ(x) > µ(y) µ(x) > µ(x y) > µ(y) (e.g., danger, temperature, expected utility) partial independence of boundedness and scale type References Fine, K Vagueness, truth and logic. Synthese 30(3) Kamp, Hans Two theories about adjectives. In E. Keenan (ed.), Formal semantics of natural language, Cambridge University Press. Klein, Ewan A semantics for positive and comparative adjectives. Linguistics and Philosophy 4(1) Krantz, David H., R. Duncan Luce, Patrick Suppes & Amos Tversky Foundations of Measurement. Academic Press. Kratzer, Angelika Modality. In von Stechow & Wunderlich (eds.), Semantics: An international handbook of contemporary research, de Gruyter. Lassiter, Daniel Measurement and Modality: The Scalar Basis of Modal Semantics. Oxford University Press (to appear). Scott, Dana & Patrick Suppes Foundational aspects of theories of measurement. Journal of Symbolic Logic 23(2) von Stechow, Arnim Comparing semantic theories of comparison. Journal of Semantics 3(1)