Seminar in Semantics: Gradation & Modality Winter 2014

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1 1 Subject matter Seminar in Semantics: Gradation & Modality Winter 2014 Dan Lassiter 1/8/14 Handout: Basic Modal Logic and Kratzer (1977) [M]odality is the linguistic phenomenon whereby grammar allows one to say things about, or on the basis of, situations which need not be real. (Portner 2009: 1) The traditional empirical focus in the study of modality and so our focus for most of today is the semantics of the modal auxiliaries (must, ought, should, can, might), adjectives and adverbs (possible/-ly, necessary/-ily), a few verbs and semi-verbs (know, believe, want, have to). But many other kinds of expressions have modal semantics too! 2 Pep talk (excerpted from Lassiter 2014) Recent work on modality in formal semantics (Yalcin 2007, 2010; Portner 2009; Lassiter 2010) has highlighted the fact that many modal expressions are gradable; for example, they accept at least some degree modifiers and can take part in comparatives and equatives. Although these papers have discussed epistemic modal adjectives for the most part, gradability in the modal domain goes beyond adjectives and beyond epistemic modals; some examples are given in (1). (1) Gradability among Modals a. How necessary is it to marinade meat before making jerkies? 1 (Degree questions) b. Bill wants to leave as much as Sue wants to stay. (Equatives) c. I need to go on vacation more than I need to finish this work. (Comparatives) d. There are situations in which concerns of autonomy ought very much to matter. 2 (Degree modification) e. It is 95% certain that our team will win. (Measure Phrases) The modal expressions in these examples cut across syntactic categories, including adjectives necessary, certain, main verbs want, need, and the (quasi-)auxiliary ought. The examples in (1) closely resemble examples of gradability among the better-studied classes of non-modal gradable adjectives and verbs, for example: (2) Gradability among Adjectives a. How angry can I make my teacher and still get an A? (Degree questions)

2 b. The Gettysburg Memorial is as old as the Eiffel Tower. (Equatives) c. My child is cleverer than yours. (Comparatives) d. This carton of milk is almost empty. (Degree modification) e. Mary is six feet tall. (Measure Phrases) (3) Gradability among Verb Phrases a. How much do you like chocolate? (Degree questions) b. John likes chocolate as much as Mary does. (Equatives) c. I loathe Battlefield Earth more than any other movie. (Comparatives) d. Harriet has almost finished her art project. (Degree modification) e. We walked six miles before finding a gas station. (Measure Phrases) The standard approach in current formal semantics is to tie facts about gradability and comparison in the adjectival and verbal domain to scales that is, to abstract representations of measurement to which gradable expressions relate their arguments. Scales are assumed to be composed of degrees which are partially or totally ordered. Roughly, then, clever is an expression which relates people to their degrees of cleverness; loathe is an expression which relates pairs of individuals (x, y) to the degree to which x loathes y; and so on. This book considers gradability, comparison, and other evidence for scalar semantics in the modal domain. I develop a new semantics for epistemic, deontic, and bouletic modals which is very closely related to standard scale-based theories of the semantics of non-modal gradable expressions. In particular, modals are analyzed as expressions which relate their propositional arguments to points on a scale, just like gradable adjectives. I also argue that some modal expressions which do not show evidence of gradability notably the modal auxiliaries may, might, and must can be shown to have a semantics built around scales nonetheless. That is, their logical relations to expressions which are gradable, and the entailments that they license, are mysterious if these items have a quantificational semantics while other modals have a scalar semantics. Implementing this idea requires making a careful distinction between semantic scalarity and grammatical gradability,... As a general theory of modality, the approach developed here is novel and quite different from standard approaches to modal semantics which treat most or all modals as expressing quantification over possible worlds. In addition for providing a natural account of the gradability of many modals... which quantificational theories do not the logical behavior of modal expressions on the scalar alternative is quite different from the behavior of quantifiers. Most obviously, quantificational theories predict that virtually all modal expressions are upward monotonic. I give a variety of arguments for the conclusion that, while epistemic modals are indeed upward monotonic, deontic and bouletic modals are non-monotonic. Unlike quantificational semantics, the scalar approach makes it possible to model monotonic and non-monotonic modalities alike, and to explain the difference using a parameter of variation which is also reflected in the semantics of non-modal gradable adjectives. 2

3 3 Modal logic Let s get started by learning (or reviewing) some modal logic. Definition 1. (L ML ) The language of propositional modal logic L ML is the smallest set satisfying the following: For i N, p i L ML. For φ L ML, φ L ML. For φ and ψ in L ML, φ ψ, φ ψ, φ ψ are in L ML. For φ in L ML, φ and φ are in L ML. Definition 2. (Frame) A frame is a pair W, R, where W is a set of points of evaluation ( possible worlds ) and R is a binary relation on W. Remember: a binary relation on W is a set of pairs a subset of W W. Instead of w, w R, we ll often write wrw. All frames satisfy axiom K: (φ ψ) ( φ ψ). Definition 3. (Some interesting classes of frames) W, R is a frame iff... Reflexive: for all w W, wrw. Axioms: K plus T: p p. ( T ) Symmetric: for all w, w W, wrw w Rw. Axioms: K and T plus B: p p. ( B ) Serial: for all w W there is a w W such that wrw. ( No dead ends ) Axioms: K plus D: p p. ( D ) Transitive: for all w, w, w W, (wrw w Rw ) w Rw. Axioms: K and T plus 4: p p. ( S4 ) Equivalence: reflexive, symmetric, and transitive. Axioms: K and T plus E: p p. ( S5 ) Definition 4. (Model) A model for L ML is a pair M ML = F, V, where F is a frame 3

4 V w, p i {0, 1} is a valuation function mapping each pair of a world w W and an atomic sentence p i to a truth-value. Definition 5. (Truth) φ M,w = 1 ( φ is true in M at w ) if and only if φ is atomic and V (w, φ) = 1, or φ is not atomic and receives value 1 by the usual recursive definitions, plus two additional clauses for the box and diamond: φ M,w = 1 iff φ M,w = 0 φ ψ M,w = 1 iff φ M,w = 1 and ψ M,w = 1 φ ψ M,w = 1 iff φ M,w = 1 or ψ M,w = 1 φ ψ M,w = 1 iff φ M,w = 0 or ψ M,w = 1 φ M,w = 1 iff, for all w such that wrw : φ M,w = 1 φ M,w = 1 iff, for some w such that wrw : φ M,w = 1 Drawing pictures helps here; consider interpretations of φ and φ in models with different kinds of frames. Note that we could also have set up the semantics using V p i U, where a valuation is a function from atomic sentences to the set of worlds U W in which p i is true. Then we set φ M = W φ M φ ψ M = φ M ψ M φ ψ M = φ M ψ M φ ψ M = (W φ M ) ψ M φ M = {w W for all w W wrw w φ M } φ M = {w W for some w W wrw w φ M } This corresponds to the popular slogan propositions are sets of worlds. It s important to be able to move between the two ways of thinking/talking, since both are widely employed, but also remember which one is being employed. Here s a slight variant, essentially the one proposed by Kratzer (1977). Let f W P(P(W )) (a function from worlds to sets of propositions). Then set: φ W,f,V = W φ W,f,V φ ψ W,f,V = φ ψ W,f,V φ ψ W,f,V = φ W,f,V ψ W,f,V 4

5 φ ψ W,f,V = (W φ W,f,V ) ψ W,f,V φ W,f,V = {w W for all w f(w) w φ W,f,V } φ W,f,V = {w W for some w f(w) w φ W,f,V } As Kratzer (1991) notes, this is equivalent to the relational semantics given above except that we represent contextual variability in the interpretation of modals using a contextual parameter, rather than relying on the metalanguage notion of a relation. It s easy to modify the relational semantics to eliminate even this difference, e.g.: φ W,R,V = W φ W,R,V φ ψ W,R,V = φ ψ W,R,V φ ψ W,R,V = φ W,R,V ψ W,R,V φ ψ W,R,V = (W φ W,R,V ) ψ W,R,V φ W,R,V = {w W for all w W wrw w φ W,R,V } φ W,R,V = {w W for some w W wrw w φ W,R,V } The only way Kratzer makes use of f, even with her additional modalities, is to intersect its contents. The equivalence of the interpretations is derived by setting R = f(w). On to Kratzer References Kratzer, Angelika What must and can must and can mean. Linguistics and Philosophy 1(3) Kratzer, Angelika Modality. In von Stechow & Wunderlich (eds.), Semantics: An international handbook of contemporary research, de Gruyter. Lassiter, Daniel Gradable epistemic modals, probability, and scale structure. In N. Li & D. Lutz (eds.), Semantics & Linguistic Theory (SALT) 20, CLC Publications. Lassiter, Daniel Measurement and Modality: The Scalar Basis of Modal Semantics. Oxford University Press (to appear). Portner, Paul Modality. Oxford University Press. Yalcin, Seth Epistemic modals. Mind 116(464) doi: /mind/fzm983. Yalcin, Seth Probability Operators. Philosophy Compass 5(11)