Friday, March 30, 2007

What Ifs?

Kevin W. Saunders, Michigan State University College of Law: "What if every 'if only' statement were true?"

Counterfactuals are complicated animals. People do a pretty good job on them, but they can’t explain why.

In standard propositional logic, “p implies q” is true every time p is false, because q’s state doesn’t contradict “p implies q” whether q is true or false. But that’s not good enough for counterfactual reasoning. Also, if p were other than it is, other things would be different: If I were the Queen of France, my husband would be the King.

Nested spheres of possible worlds. Some propositions would be true in every possible world, others wouldn’t. We can look at a curve intersecting those nested spheres as worlds in which p is true – if it were true, the curve would encompass the center. The parts of the curve closest to the center are the closest states of the actual world in which p is true:

Strong theory of strict implication: in any world in which p is true, q would be true. But that’s generally too strong (in a world where I was Queen of France, I might have a different husband). Instead, look at the region where p is true and closest to our world. If q is true in that most similar world, then “p implies q” is right in the counterfactual:

Counterfactual logic creates a failure of transitivity: If J. Edgar Hoover had been born in Russia, he’d have been a Communist. If J. Edgar Hoover had been a Communist, he’d have been a traitor. But if J. Edgar Hoover had been born in Russia, he wouldn’t have been a traitor.

Also, a failure of contraposition: if p implies q, not-q implies not-p. This is true regularly, but not true for counterfactuals.

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