Modality

1. Modality Counterfactuals

2. Possible Worlds • A statement is something that is true or false. • A set of statements is consistent iff all of the members could be true together. • A set of statements is complete iff, for any statement S, either S or not-S is a member of the set. • A possible world is a consistent and complete set of statements. • The actual world (wa) is the possible world consisting of all true statements.

3. Possible Worlds • Predicates • ‘W’ = ‘is a possible world’ • ‘T’ = ‘is true in’ • Definitions • □ p = (x) (Wx  Tpx) • ◊ p = (x) (Wx & Tpx) • Colloquially • Necessity is truth in all possible worlds • Possibility is truth in some possible world

4. Indicatives and Subjunctives • If Oswald didn’t shoot Kennedy, then somebody else did. • If Oswald hadn’t shot Kennedy, then somebody else would have.

5. Indicatives and Subjunctives • Indicative Conditionals (p  q) • If I dropped the pen, it fell. • If I’m standing outside, I’ve got a good tan. • If I went to a different school than I actually did, then I didn’t meet my wife.

6. Indicatives and Subjunctives • Subjunctive Conditionals (p □ q) • If I had dropped the pen, then it would have fallen. • If I had been standing outside, then I would have gotten a nice tan. • If I had gone to a different school than I actually did, then I wouldn’t have met my wife.

7. Counterfactuals • (p □ q) is true iff the closest possible world in which p is true is a world in which q is true (i.e., the closest p-world is a q-world). • If I had dropped the pen, it would have fallen. • If I had been standing outside, I would have gotten a nice tan. • If I had gone to a different school than I actually did, then I wouldn’t have met my wife.

8. Counterfactuals • If I had dropped the pen, I would have failed first grade. • If I had dropped the pen, it would have floated up into the sky. • If were in the Antarctic right now, I would be really cold.

9. Counterfactuals • It is of first importance to avoid big, widespread, diverse violations of law. • It is of second importance to maximize the spatio-temporal region throughout which perfect match of particular fact prevails. • It is of third importance to avoid even small, localized simple violations of law. • It is of little or no importance to secure approximate similarity of particular fact, even in matters that concern us greatly. David Lewis, “Counterfactual Dependence and Time’s Arrow,” pp.47-8

10. Counterfactuals • (p □ q) is true iff the closest possible world in which p is true is a world in which q is true (i.e., the closest p-world is a q-world). • The closest world is the one that does the best overall job of matching the actual laws of nature and particular matters of fact.

11. Causation • If I hadn’t pushed the eraser, it wouldn’t have fallen. • My pushing the eraser caused it to fall.