Modal Logic 2

1. Modal Logic 2

2. VALIDITY IN BROUWER A formula is B-valid just in case it comes out true in all possible worlds under all interpretations of its atomic constituents when the relation of accessibility is taken to be both reflexive and symmetrical.

3. Remember that Brouwer contains the axioms of M: • □ (A ⊃ B) ⊃ (□ A ⊃ □ B) • □ A ⊃ A And adds the following formula: • A ⊃ □ ◊ A

4. You want A ⊃ □ ◊ A to come out true in all possible worlds under all truth-values of A, and it only comes out false in one case: □ ◊ A is false in W if only if there is a world accessible to W for which there is no accessible world in which A is true.

5. But since A is true in W, □ ◊ A is only false if W is not accessible to all the worlds accessible to it. To ensure that □ ◊ A never comes out false, we require that accessibility be symmetrical.

6. As an example, consider that A reads: • Towson’s student philosophy club is cool. Thus, □ ◊ A reads: • Necessarily, it’s possible that Towson’s student philosophy club is cool. Without symmetry, accessibility relations are “one-way”, and the truth of □ ◊ A would require something like this:

8. So, without symmetry we need to posit additional worlds accessible to W1 and W2 in which A is true if the statement □ ◊ A can be said be to true in W; in this case, those additional worlds are W3 and W5. • But what if we said this:

10. But if accessibility relations are symmetrical, then there IS at least one world accessible to W2 in which A is true – world W.

11. In summary, we ensure validity in Brouwer, that is, we ensure that □ ◊ A never comes out false in a possible world, under any allocation of truth-function to A, by requiring that the accessibility relation be symmetrical.

12. VALIDITY IN S-4 • An S-4 model structure is like an M model structure except that accessibility is stipulated to be both reflexive and transitive.

13. Remember that S-4 takes as its axioms the theorems of M: • □ (A ⊃ B) ⊃ (□ A ⊃ □ B) • □ A ⊃ A And that it adds: • □ A ⊃ □□ A

14. Again, we want □ A ⊃ □□ A to come out true in every possible world on every assignment of truth value to A. When does it come out false? Only in one case, the case where □ A is true at W, and □□ A is false at W; this requires that A be true in all worlds accessible to W but false in some world accessible to a world accessible to W.

15. If we insist that accessibility relations be transitive, this can’t happen. Transitivity says that every world accessible to any world accessible to W will be accessible to W itself. Thus, with transitivity □ A will be true in W only if □□ A is true in W as well.

16. VALIDITY IN S-5 • S-5 contains the Brouwer system and S-4, and accomodates them both by adding the axiom ◊ A ⊃ □◊A.

17. If the possibility of something is itself a matter of necessity, you want this to come out true at all possible worlds on all assignments of truth-value to A.

18. It only comes out false in one case – the case where all worlds accessible to a world W don’t have at least one accessible world where A is true.

19. In the S-5 model structure, relative possibility is taken to be an equivalence relation; it is a relation that is reflexive, symmetrical, and transitive. This means ◊ A can’t be true unless □◊A is as well.

20. SUMMARY Validity and truth-function were established by Kripke for modal systems M, B, S-4, and S-5 by restricting accessibility relations: M – must be reflexiveB – Must be symmetricalS-4 – Must be reflexive-transitive S-5 – Must be reflexive, transitive, and symmetrical.