I went to Stanford yesterday with a bunch of my colleagues for a relatively informal workshop put together by Johan van Benthem. In addition to the Berkeley and Stanford students, there were visitors from Amsterdam and Paris in town, so it was quite a nice chance to meet people working in formal areas of philosophy in a variety of locations. Because the people presenting were mostly Johan’s students and Branden Fitelson‘s students, there was an interesting mix of talks on dynamic epistemic logic and talks on probability. Future work to synthesize these two approaches to representation of uncertainty should be quite interesting.

It’s a shame that there haven’t been more interactions like this between the Berkeley and Stanford philosophy departments, but I guess it’s because we’re extremely far apart for two universities in the same metropolitan area. Anyway, it sounds like more such things will go on in future – and this was a great warmup for FEW!

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Lurker(16:42:15) :Random logic question.

Let “S” stand for the Sheffer stroke. I know that “p & q is” equivalent to “(pSq)S(pSq),” but I’m not entirely sure as to why.

Is it merely that the main connective of a proposition containing only “p” and “q” has the same truth table as conjunction?

But the conjuncts are not “p” and “q.” They are “pSq.” Why doesn’t that matter? Or do I have it completely wrong?

If you could supply an answer or point me in the right direction (say a source in the literature), I’d be appreciative.

Kenny(17:26:12) :I think this discussion would probably work better over e-mail rather than in comments. But you should look at the truth-tables you get for various formulas of this sort. The one for “and” is true in the case where both inputs are true, and false everywhere else. Conversely, for “or” it’s false if both inputs are false, and true everywhere else. For the Sheffer stroke, I believe it’s true where both inputs are false, and false everywhere else.

Thus, “pSq” is false when both p and q are true, and true anywhere else. So “(pSq)S(pSq)” is false where both “pSq” and “pSq” are true, which is anywhere that not both p and q are true. And it’s true everywhere else, which is just where boht are true. This is exactly the same as for “and”.

This should be covered in any standard logic textbook in the chapter on truth-tables and truth-functions. (Though they don’t necessarily mention the Sheffer stroke in particular.)