Lecture 24: Motivating Modal Logic, Translating into It

Transcription

1 Lecture 24: Motivating Modal Logic, Translating into It 1

2 Goal Today The goal today is to motivate modal logic, a logic that extends propositional logic with two operators (diamond) and (box). We do this by examining how we talk and reason about words like might, possible, can (which we translate by ) and words like must and necessary (which we translate by ), and their formal similarities with words like knows and believes. 2

3 A Problem Case Consider (1) It might rain, but it might not rain. Can we compositionally translate this into propositional logic in a way that preserves truth-conditions? Our usual method seems to deliver incorrect results. 3 For, usual method suggests: Translate it might rain as p Translate but as Translate 2nd conjunct as p So our usual method suggests translating (1) as p p. But this seems wrong, since contradictions are never true. And it seems like (1) is true a lot of the time.

4 Some Implausible Responses (1) It might rain, but it might not rain. So our usual method suggests translating (1) as p p. One implausible response is: move to a 3-valued logic, where p p has third truth-value. If idea is to always give (1) the third truth-value, then it s implausible because sometimes (1) seem definitely true. 4 Another implausible response would involve moving to predicate logic. This is implausible because there don t seem to be any uses of for all or there is in the statement. But maybe it contains the kernel of a plausible response: the idea should be to enrich our language to handle sentences like (1), just like we enriched our logic with,.

5 Generalizing the Example Such enrichment would be uninteresting if this was an isolated example. But consider other examples much like this one: (2) Anne can attend the meeting, but she does not. (3) Claire must do her work, but she does not. 5 Again, if we employed the method used so far, it seems like we may be tempted to translate these by p p. But clearly when someone says (2), they are not contradicting themselves, i.e. they are not saying that Anne can and Anne can t do something.

6 A Guiding Idea Consider the way we talk about our beliefs and our other attitudes like knowledge. An obvious way to translate these sentences would be to introduce operators K for knowledge and B for belief. Then we would translate as: (4) Anne believes that skidiving is safe, but it isn t. (5) If the earnings report is low, then Claire knows it. (6) Clark flies, but Lois doesn t know that. 6 (4 ) B(p) p (5 ) ℓ K(ℓ) (6 ) c K(c)

7 More on the Idea So we introduce two operators, K for knowledge and B for belief. And we translate (4) Anne believes that skidiving is safe, but it isn t by (4 ) B(p) p, with the key: p = skidiving is safe Bq = Anne believes q 7 A key feature of this translation, as revealed by how we write out the key, is that for each formula q, there is another formula Bq and another formula Kq. Obviously if we wanted to get more elaborate and subscript, we could translate rather as (4 ) Ba(p) p with key p = skidiving is safe Ba(q) = Anne believes q

8 An Obvious Distinction We introduce two operators, K for knowledge and B for belief. Introducing these operators gives us a way to mark a very intuitive and obvious distinction. Consider 7) Anne does not know that it is raining. 8) Anne knows that it is not raining. 8 These are describing very different situations. (7) is true when Anne is ignorant of weather facts, while (8) is true if Anne knows certain weather facts. This distinction is mirrored in our translations: 7 ) K(p) 8 ) K( p)

9 Importing the Guiding Idea Our original problem case was: (1) It might rain, but it might not rain. Let s first note that (1) seems equivalent to: Let s introduce a symbol for might. Then we translate it might be the case that p by p. This operator is pronounced diamond and p is pronounced as diamond p. Hence we see that we can translate as (1) and (1*) as 9 (1*) It might be the case that it rains, but it might be the case that it does not rain. (1 ) p p with key: p = it rains

10 Seeing Might Distinctions Original English Sentence: It might rain Original EnglishSentence: It might not rain Equivalent English Sentence: It might be the case that it rains Equivalent English Sentence: It might be the case that it does not rain Translated Sentence: p Translated Sentence: p Original English Sentence: It s false that it might not rain Original English Sentence: It s false that it might rain. Equivalent English Sentence: It s not the case that it might be the case that it does not rain. Equivalent English Sentence: It s not the case that it might be the case that it rains Translated Sentence: p Translated Sentence: p 10

11 Seeing Might Distinctions p p p p 11

12 From Might to Know When we write in terms of the symbols and, we automatically see the distinctions. Part of the difficulty in the case of might is usually we first have to write the English sentences involving might as it might be the case that before we can translate into the symbols like and. When we do the analogous set of distinctions in the case of know there s just not this intermediary step, as the following examples show. 12

13 Seeing Knows Distinctions Original English Sentence: Anne knows that it is raining Original EnglishSentence: Anne knows that it is not raining Translated Sentence: Kp Translated Sentence: K( p) Original English Sentence: Anne does not know that it is raining Original English Sentence: Anne does not know that it is not raining Translated Sentence: K(p) Translated Sentence: K( p) 13

14 Seeing Knows Distinctions K(p) K( p) K(p) K( p) 14

15 Recipe + examples: Might First, replace each instance of it might.... with the more cumbersome it might be the case that... Second, translate each instance of Anne/Bill/Claire might yadayada with It might be the case that Anne/Bill/Claire does yadayada Example: Bill might attend the meeting. So we write the equivalent sentence: It might be the case that Bill attends the meeting. Third, translate it might be the case that p by p. 15 Then we translate as p with key: p = Bill attends the meeting.

16 Recipe + examples: Might First, replace each instance of it might.... with the more cumbersome it might be the case that... Second, translate each instance of Anne/Bill/Claire might yadayada with It might be the case that Anne/Bill/Claire does yadayada Example: Anne might attend and Bill might not attend. First we write equivalent It might be the case that Anne attends, and it might be the case that Bill does not attend. Third, translate it might be the case that p by p. 16 Then translate by: a b, with key: a = Anne attends, and b = Bill attends meeting.

17 Recipe + examples: Possible The locution it is possible that seems very similar to it might be the case that. For, if it is possible that p, then it might be the case that p. Likewise, if it might be the case that p, then it s possible that p. However, unlike might, it doesn t seem that possible needs to be expanded, and hence it s easier to translate. So there is simply one step in the recipe: Translate it is possible that p by p. Example: if it is possible that Anne gets the job, then it is possible that Anne is wealthy. 17 Translation: p q, with key: p = Anne gets the job, q = Anne is wealthy.

18 Recipe + examples: Possible So there is simply one step in the recipe: Translate it is possible that p by p. Example: If Anne attends the meeting, then it s possible that Bill attends the meeting. Translation: a b, with key: a= Anne attends the meeting, b= Bill attends the meeting. 18 Example: It s not possible that Anne attends the meeting [she is traveling!], and it s not possible that Bill does not attend the meeting [he is in town and definitely coming!] Translation: a b Example: it is possible that if Anne attends, then Bill attends. Translation: (a b).

19 General Discussion: Can vs. Might (1) Consider Bill can attend the meeting. If we say this after looking at Bill s calendar and seeing that he is available, then this seems very close to Bill might attend the meeting. But there s a use of can that doesn t fit well with might. (2) Bill can read Chinese. It seems I m ascribing a capacity or skill to Bill, and not just saying that it might happen. 19 Turns out modal logic doesn t have much to say about capacities. But since things like Bill can attend the meeting do occur naturally, we just agree to translate this as b, where of course b = Bill attends the meeting. In essence, this is borne of a tacit agreement to focus on examples more like (1) and less like (2).

20 General Discussion: Must In the past slides, we ve seen that might and it is possible that and can have closely related ranges of application, and so we just translate with a single symbol. Seems that (1) and (2) could be false while (1 ) and (2 ) true: (1 ) Bill might attend (2 ) It might be the case that it rained today in Irvine Hence, might means something different from must. It s so different that we need a new symbol. But what about must? (1) Bill must attend. (2) It must be the case that it rained today in Irvine. 20

21 Basic recipe + examples: Must In short, translate it must be the case that p by p. This is pronounced box p. Example: if it rains then it must be the case that the sidewalks are wet. Translation: r w, with key: r= it rains, and w = sidewalks are wet. 21 It must be the case that Bill attended the meeting. If Bill did not attend the meeting, then people would have noticed. Translation: b. b p with key: b = Bill attends the metting, p = people would have noticed.

22 General recipe+examples: Must First, translate each instance of Anne/Bill/Claire must yadayada with It must be the case that Anne/Bill/Claire does yadayada Second, translate it must be the case that p by p. Example: Anne must attend, and if Anne must attend, then Bill must attend. 22 First we replace by the equivalent It must be the case that Anne attends, and if it must be the case that Anne attends, then it must be the case that Bill attends. Second, translate by a a b with key: a = Anne attends b = Bill attends

23 General recipe+examples: Must First, translate each instance of Anne/Bill/Claire must yadayada with It must be the case that Anne/Bill/Claire does yadayada First we replace by the equivalent it must be the case that Anne attends or it must be the case that Bill attends, and it must be the case that it s not the case that Anne attends and Bill attends. Second, translate as ( a b) ( (a b)) Second, translate it must be the case that p by p. Example: Anne must attend or Bill must attend, but it must be the case that not both attend (They don t like each other at all!). 23

24 General Discussion: Necessity So as we saw earlier, there s a close connection between might and possible Similarly, there s a close connection between must and necesssary One difference was might occurs both in it might and Anne might locutions, while possible only occurs in it is possible that locutions. Hence, we translate it is necessary that p as p. Examples: It is necessary that Bill attends. If Bill does not attend, then Bill loses his job. Translation: b. b ℓ 24

25 Paradigm Examples Original Sentence Equivalent Sentence Translation It might snow It might be the case that it snows p, key: p = it snows Anne might attend It might be the case that Anne attends p, key: p = Anne attends It s possible that Anne attends p, key: p = Anne attends Anne can attend p, key: p = Anne attends It must have rained It must be the case that it rained p, p = it rains Bill must attend It must be the case that Bill attends p, p = Bill attends It is necessary that Bill attends p, p = Bill attends 25

26 Goal Today The goal today is to motivate modal logic, a logic that extends propositional logic with two operators (diamond) and (box). We do this examining how we talk and reason about words like might, possible, can (which we translate by ) and words like must and necessary (which we translate by ), and their formal similarities with words like knows and believes. 26

27 Ω 27