1. 1 Lecture 3 The Languages of K, T, B and S4

2. 2 Last time we extended the language PC to the language S5 by adding two new symbols ‘□’ (for ‘It is necessary that’) and ‘  ’ (for ‘It is possible that), and allowing two new types of wff – ‘□  ’ and ‘  ’. Interpreting S5 is more involved than interpreting PC. We specify a set of possible worlds, W, singling out one of them as the actual world, @, and for each sentence letter we specify a proposition as its meaning – a function from W to the truth- values {T, F}. For convenience, rather than saying ‘the proposition that  means is true at a world w’ we simple say ‘  is true at w’. This should cause no confusion.

3. 3 We defined the meanings of ‘□’ and ‘  ’ in such a way that: ‘□  ’ is true at a world w iff  is true at every world. ‘  ’ is true at a world w iff  is true at some world. In particular: ‘□  ’ is true (at the actual world) iff  is true at every world. ‘  ’ is true (at the actual world) iff  is true at some world.

4. 4 Giving the semantics of S5 in this way renders the following semantic sequents correct:  A╞  A (and vice-versa)  A╞    A (and vice-versa) ╞  A (if ╞ A) □A, □[A  B]╞ □B □A╞ A A╞ □  A □A╞ □□A  A ╞ □  A If we want our deductive system to be complete then we need to add extra rules of deduction to the language so that the corresponding syntactic sequents are correct.

5. 5 We get a sound and complete deductive system by adding the following rules: Def  From   deduce  (and vice-versa) From  deduce    (and vive-versa) RN From  [  ] deduce   (  not in the scope of an assumption in  […]) RK From   and  […] deduce  […  ] (   not in the scope of an assumption) RT From   deduce  RB From  and  […] deduce  […  ] R4 From   and  […] deduce  […   ] R5 From  and  […] deduce  […  ]

6. 6 Here is a proof of □[A  B], □  A ├ □B:

7. 7 Here is a problem for the idea that S5 captures the logic of ‘It is necessary that’ and ‘It is possible that’ that: It is a consequence of the account that anything that is possibly true is necessarily possibly true. This can be seen by noting that ‘  ╞ □  ’ is a correct semantic sequent of S5. But what about this: it is possible that I will get testicular cancer, but it might not have been possible – I might have been a woman. This seems to be a counterexample to the result that anything that is possibly true is necessarily possibly true.

8. 8 Saul Kripke proposed that we give a more general semantics for the language: As well as specifying a set of possible worlds, W, we also specify an accessibility relation on W – that is, a set of ordered pairs where w 1, w 2 are members of W. Intuitively, w 2 is accessible from w 1, or w 2 is possible relative to w 1. Then we define the meanings of ‘□’ and ‘  ’ in such a way that: ‘□  ’ is true at a world w iff  is true at every world that is accessible from w. ‘  ’ is true at a world w iff  is true at some world that is accessible from w.

9. 9 The following semantic sequents are still correct: □A╞  A (and vice-versa)  A╞  □  A (and vice-versa) ╞ □A (if╞ A) □A, □[A  B]╞ □B But these semantic sequents are now incorrect: □A╞ A A╞ □  A □A╞ □□A  A ╞ □  A

10. 10 What we actually do is define five different languages, K, T, B, S4, and S5, all sharing the same syntax but differing slightly in their semantics: For K, any accessibility relation is allowed. For T, the accessibility relation must be reflexive. For B, the accessibility relation must be reflexive and symmetric. For S4, the accessibility relation must be reflexive and transitive. For S5, the accessibility relation must be reflexive, symmetric and transitive.

11. 11 An accessibility relation R is reflexive iff for all w in W, is in R. An accessibility relation R is symmetric iff for all w and w' in W, if is in R then is also in R. An accessibility relation R is transitive iff for all w, w' and w" in W, if and are both in R then is also in R.

12. 12 We sometimes use subscripts on the turnstiles to indicate which language we are talking about: ‘  ╞ T  ’ means that there is no interpretation of T (that is, on which the accessibility relation is reflexive) on which each wff in  is true and yet  is false. We say that  T-entails . If  is empty, we say that  is T-valid. Similarly for: ‘  ╞ K  ’ ‘  ╞ B  ’ ‘  ╞ S4  ’ ‘  ╞ S5  ’

13. 13 KTBS4S5 □A╞  A □A╞ A  A╞ □  A  □A╞ □□A   A╞ □  A  Some semantic sequents and whether or not they are correct in each of the five languages:

14. 14  A╞ □  A A╞ □  A □A╞ A □A╞ □□A □A╞  A

15. 15 Since each of the five languages yields different sets of correct semantic sequents, each has to have a different set of deductive rules in order to be sound and complete. It turns out that each can be made sound and complete by adding deductive rules as follows: K Def  + N + RK T Def  + N + RK + RT B Def  + N + RK + RT + RB S4 Def  + N + RK + RT + R4 S5 Def  + N + RK + RT + R5, or Def  + N + RK + RT + RB + R4

16. 16 Why are we interested in so many different languages? Because although S5 might capture the logic of logical necessity and possibility, it does not seem to capture the logic of physical necessity and possibility (e.g. the example above), but one of K, T, B, or S4 might. Also, later we will use ‘□’ and ‘  ’ to translate ‘It is obligatory that’ and ‘It is permitted that’. Since we do not want ‘□A╞ A’ to come out as a correct sequent, S5 does not capture the logic of obligation and permission. We will also use ‘□’ and ‘  ’ to translate ‘It will always be the case that’ and ‘It will be the case that’. Again, S5 does not capture the logic of these expressions.