Friday, July 9, 2010

Thoughts on the Liar's Paradox

I recently was looking at The Liar: An Essay on Truth and Circularity by Jon Barwise and John Etchemendy.

The Liar refers to the famous "paradox" of saying "This statement (meaning this very sentence) is false." If it is true then it is false, but if it is false it must be true. There are other very similar paradoxes out there. One is the Russell paradox which apparently crushed Gottlob Frege's hopes of axiomatizing mathematics. Russell's paradox was that some sets are members of themselves and some are not. For instance the set of forks is not a fork and therefore not a member of itself. On the other hand the set of ways in which to group things is also a way in which to group things and therefore a member of itself. But what of the set of those sets which are not members of themselves. If that set is not a member of itself, why then it is a member of itself, but if it is a member of itself it is no longer the set of sets which are not members of themselves.

It seems if I am correct that Barwise and Etchemendy attack these notions of self-reference and circularity from a number of directions, but that in some fashion they conclude that the alleged paradoxes do not occur if we are attentive to the particular in which situation a statement is made. They also do some work with hypersets that I did not look at to closely.

In any event after reading Popper's account of Tarski's correspondence theory of truth I was reminded of something I did almost four years ago. I googled the phrase "this phrase gets no hits on google." As recorded almost contemporaneously here I got no hits. But I assumed that my blogging about it would result in my later getting when I googled the phrase, thus creating as sort of paradox.

A comment made by Stolen Brown Evergreen seems to indicate that he got a hit, but I just recently googled the phrase and got a hit, but not my original blog entry rather I got this single result:

The Bible is Word of God! - Page 6 - JREF Forum
There is no "law from the birth of life" which is why this phrase gets no hits on Google. Quote: and law from information (increasing of information by ...

It occurred to me that if we, in line with the prevailing culture perhaps, substitute "gets a hit on google" for "is true" about a given statement, even this self referential statement, sometimes it is true and sometimes it is not depending on the situation. For originally it was true-- the phrase didn't get a hit on google. I attempted to make this a self-referential paradox by blogging it. This failed but google's crawlers which are the eyes for most of us about the cyber world came upon the phrase in another context which had been posted on 3rd June 2006, 08:32 A but was not picked up when I googled the phrase on November 14, 2006. Meanwhile the paradox producing blog entry has not been found. (Will this entry fare better. Perhaps I will put the phrase in the tags)

So if truth of a statement is like getting a hit with google, meaning it corresponds to something in the world, then how does this play out? That sometimes the statement "This statement is false" is false when there is no set of facts to which it corresponds, and sometimes it is true when there is a set of facts to which it corresponds. All of which sounds like there is something going on, but I can't quite figure it out. I think it may have to do with the alleged eternity of statements and logic. Maybe it says something about intersubjectivity. Thus the Liar becomes "this statement as I say it does not correspond to anything anyone sees or knows about, but I am putting it out there in such a fashion that it may, or maybe something out there will correspond to it once it is discovered."

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