*Keith Douglas*

No answers on the conventions discussion from last time, which was admittedly a bit of a repeat.

*Three Classic Paradoxes*

This time I would like to offer several classic paradoxes. Each one could fill an entire column, but I’d rather leave them as short and sweet (last column was a bit long!) and perhaps expand on them later.

1) The uselessness of identity claims.

Consider the statement form “A is the same as B.” Suppose first that A and B are different; then the statement is false and hence not informative. If, by contrast, A and B are identical, then this statement seems to say “A is the same as A,” which is uninformative too, because it seems to be a tautology.

Hint: What is the role of identity in mathematics?

2) How do we learn, anyway?

If we don’t know about X, how do we recognize that we’ve come to true belief about X, since we don’t know. If, on the other hand, we already know about X, we don’t come to a true belief about X, because knowledge is already true belief about X.

Hint: This is the less contentious part of Plato’s *Meno*. The leading questions to get what is wanted, and the doctrine of recollection, are more so.

3) The odd conditional. (This one is for those who understand classical logic.)

Objective patterns are often reconstructed in thought in terms of conditionals. One might be “If I take an aspirin, I will get relief from my headache.” That seems plausibly true enough. This is of the form “if *p* then *q*.“ If you are familiar with classical logic you will remember that if that’s true, then “if *p* and *r* then *q*” is also true (because for “*p* and *r*” to be true, *p* must be true). Yet, then it would seem that “if I dip an aspirin in cyanide and take the aspirin I will get relief from my headache” is also true but this seems implausible.

Hint: Some people think the material conditional is at fault; I am not sure as I write this if that’s actually plausible.

I was busy getting ready to go on vacation at the time, so this comment is belated.

“If I take cyanide then I will NOT get relief from my headache” seems true (unless we are willing to count being dead as getting relief, but lets not). If that is r, then we have both “If p then q” and “if r then not-q”, so it’s not obvious from logic alone what the consequent of “If p and r then….” should be. It seems that “If p and r then q” holds only if one of the conjuncts is irrelevant to the consequent.