Understanding Necessity and Possibility in Modal Logic with Different Modals and Context -, Study notes of Linguistics

Download Understanding Necessity and Possibility in Modal Logic with Different Modals and Context - and more Study notes Linguistics in PDF only on Docsity! 1 Semantics 1 Fall, 2007 Instructor: Satoshi Tomioka E-mail: [email protected] 2 Modality (1) Necessarily / It must be the case that 5 is smaller than 7. (2) Possibly / It can be the case that I will sneeze within 24 hours. The symbols in Modal Logic 9p: Necessarily p, it must be the case that p p: Possibly p, it can be the case that p 9p is true iff for all w0W, p is true in w. p is true iff for some w0W, p is true in w. 5 Then, ƒmust "„ = 8w. for all w’ such that w’0 CB(w), ƒ"„ (w’)=1CB (CB) ƒcan "„ = 8w. for some w’ such that w’0 CB(w), ƒ"„ (w’)=1 CB (CB) Then, the difference among (2)-(6) is just what kind of CB is used in the interpretation. This is really not far from the customary analysis of modality with the notion of accessibility relations. ƒmust "„ = 8w. for all w’ such that wRw’, ƒ"„ (w’)=1R For all w, w’, wRw’ iff w’ is epistemically/deontically/circumstantially accessible from w. There are several challenges to be overcome... 6 Inconsistencies (From Kratzer 1977) Scenario: Let us imagine a country where the only source of law is the judgments which are handed down. To make the matter simple, imagine that the country has had exactly three judgments handed down. Judgment 1: Murder is a crime. Judgment 2: Abortion is a crime. Judgment 3: Abortion is not a crime. Our intuition says that (1acd) should be true and (1b) false under this scenario. (1) In view of what the law provides... a. It must be the case that murder is a crime. b. It must be the case that murder is not a crime. c. It could be the case that abortion is a crime. d. It could be the case that abortion is not a crime. What does our theory say? 7 (1ab) are true, but (1cd) are false! Why? (1a-d) mean: (1') a. For all w’ such that all three propositions (= judgments 1-3) are true in w’, a murder is a crime in w’. b. For all w’ such that all three propositions (= judgments 1-3) are true in w’, a murder is not a crime in w’. c. For some w’ such that all three propositions (= judgments 1-3) are true in w’, abortion is a crime in w’. d. For some w’ such that all three propositions (= judgments 1-3) are true in w’, abortion is not a crime in w’. Because CB in this scenario is inconsistent. If you take all three propositions and intersect the possible worlds in which they are true, we get i. Then, i is a subset of all sets, so (1ab) are vacuously true. If you intersect any set with i, the result is i. Therefore, (1cd) are false. 10 Strength Let’s admit it: (5a) is a stronger, more certain statement than (5b). (5) a. Jerry has left. b. Jerry must have left. Why should this be? (5a) basically says that it is true in the actual world that Jerry has left. (5b) says that in all possible worlds epistemically accessible from the actual world, Jerry has left. But assuming that the actual world is accessible from itself, (5b) should entail (5a), but not vice versa. This would make (5b) a stronger statement than (5a). What went wrong? 11 Modality Graded Modality (1) a. You absolutely have to leave soon. b. You sorta have to leave soon. c. You more or less have to leave soon. (2) a. It is barely possible that Jerry has left already. b. It is highly likely that Jerry has left already. c. It is easily possible that Jerry has left already. d. It is almost impossible that Jerry has left already. We want to derive these without generalized quantification over possible worlds. We have already seen how we relativize modal statements with Conversation Background (CB). Kratzer’s (1990) proposal is to make CB doubly relative by splitting CB into two parts: Modal Base and Ordering Source. Modal Base is basically what we have been assuming with in view of X. What is Ordering Source? 12 Ordering Source tells you how close to or how far fetched a certain possible world is from the normal course of events. AA set of proposition A induces a partial ordering # on W in the following way. For all w, w’0W, for any A f Pow(W): Aw # w’ iff {p: p0A and w’0p} f {p: p0A and w0p} Translating ... w is equally close to or closer to the ‘ideal’ worlds that w’ is, with respect to A, iff among the propositions in A, those that are satisfied in w’ are a subset of those that are satisfied in w. Using Ordering Source... 15 A proposition p is a better possibility than q wrt a modal base f and an ordering source g iff: p is at least as good a possibility as q wrt a modal base f and an ordering source g but not vice versa. A proposition p is a weak necessity wrt a modal base f and an ordering source g iff: p is a better possibility than ~p wrt a modal base f and an ordering source g. Etc. 16 Strength (3a) is a stronger, more certain statement than (3b). (3) a. Jerry has left. b. Jerry must have left. Now, (3b) means something like: (3b) is true in w wrt f and g iff for all accessible worlds w’ in which all propositions in f(w) are true and which come closest to the ideal established by the ordering source g(w), Jerry has left in w’. Because of this ‘cautionary notes’ by the ordering source, (3b) is weaker because there is not guarantee that no strange things happen in the actual world (the evaluation world). It is a little like the contrast found in (4). (4) a. Jerry works well with others. b. Under normal circumstances, Jerry works well with others. 17 Inconsistencies (From Kratzer 1977) Scenario: Let us imagine a country where the only source of law is the judgments which are handed down. To make the matter simple, imagine that the country has had exactly three judgments handed down. Judgment 1: Murder is a crime. Judgment 2: Abortion is a crime. Judgment 3: Abortion is not a crime. Our intuition says that (5acd) should be true and (b) false under this scenario. (5) In view of what the law provides... a. It must be the case that murder is a crime. b. It must be the case that murder is not a crime. c. It could be the case that abortion is a crime. d. It could be the case that abortion is not a crime. But...