Friday, March 18, 2011

Curry's paradoxes

The Liar paradox concerns utterances such as ‘what I’m saying isn’t true,’ which is, if true, not true, and which seems true if not. Another way of saying the same thing would seem to be to say ‘if what I’m saying is true, then pigs fly.’ Yet that utterance is paradoxical in a way so different that not only has it a different name – Curry’s paradox – it’s debatable whether its resolution should even resemble that of the Liar.
......Suppose I say ‘if what I’m saying is true, then P,’ where ‘P’ stands for any proposition. For simplicity, let’s say that C is the assertion that C implies P. Suppose, just suppose, that C is true. We are thereby supposing that C implies P. So we would also have P. That much is simple enough. Given C, and that if C then P, we get P. And yet that much is too much. If it’s true that by supposing C we also get P, then C really is true, and hence P is true, even though P could be asserting anything at all (even that pigs fly).
......That seems quite unlike the Liar. E.g. there was no ‘not’ in the previous paragraph: We didn’t consider C being either true or else not, and find both possibilities inadequate; nor did we see C seeming to say that it wasn’t something that C did indeed seem not to be. Rather, just by wondering what C was – in particular, whether C might be true – we seemed to get P. And so by deriving actual being from mere possibility, Curry’s paradox seems to be more like the Ontological Argument than the Liar.
......On the other hand, if it follows from the meaning of ‘not’ that either A or not-A, where ‘A’ stands for any proposition (e.g. that it’s raining), then it seems that if A implies B (e.g. that I’m carrying an umbrella), then either B or not-A (either I’m carrying an umbrella or it’s not raining). And conversely, if it’s the case that either not-A or B, then if it’s A (if it isn’t not-A), it must be B. So in short, C could seem to be asserting that either not-C or P. And then P effectively disappears if it’s false, leaving C asserting not-C.


Danny said...

Wow I got lost in all the C and Ps. I thought this was about paradox was about food lol.

I can't seem to find your email information anywhere. I was hoping we can do a link exchange.

A friend and I write a webcomic weekly. Nothing as insightful as your blog, but I try to jam in some philosophical thought once in a while.

Check out my webcomic some time.

Danny said...

I can't seem to find the darn edit button for the above post. I forgot to link it, must remember to hit preview this time first :D

Anonymous said...

Getting lost in all the 'C's is how the paradox works, I think. 'C' names both 'if this is true then P' and 'if that was true then P'. Those must be saying different things, as otherwise P must be true.