2. Overview

3. I. Whither modal semantics?

4. II. A. Modal syntax

7. II.D-E. World Diagrams Interpretation: W = {w 1, w 2, w 3 } R = {,, } v w1 (~p) = v w1 (~q) =1 v w2 (p) = v w2 (q) =1 v w3 (p) = v w3 (~q) =1

8. World Diagrams So, it’s possible thatq in w 1. Why? v w1 (  q) = 1 since, for some world w’  W w 1 Rw’, v w’ (q) = 1; 0 otherwise. (see w 2 ) Similarly, v w1 (  q) = 0, since for some world w’  W w 1 Rw’, v w’ (q) = 0 (see w 3 )

9. II. F. Validity An inference is valid if it is truth- preserving at all worlds of all interpretations, i.e.: Φ ╞ Ψ iff for all interpretations and all w  W: if v w (Φ) = 1 then v w (Ψ) = 1

10. Modal Tableaux: First Step First, let’s imagine that we have possible world semantics but no  and . Then, we’d still have to make the following modifications to the tableaux methods for propositional logic: We’d need to index every proposition with the world on which it is true. We’d start at world 0, asserting the premises and denying the antecedent.

11. Easy Example: P→Q, P ├ Q 1.P →Q, 0A 2.P, 0A 3.~Q, 0Neg Concl /\ 4. ~P, 0 Q, 01, → 5.X 2,4 X 3,4