Sunday, April 01, 2007

This Sentence Is False. True Or false? (Conclusion)

For convenience, I’ll repeat the sentence in question, immediately below this one.


This sentence is false.


The sentence in bold-face immediately above this one is not true, and is not false. It is “undecidable,” and has no “truth value.” The reason is rooted in its self-referential nature.

Here it is:

According to classical Aristotelian logic, what we call “common sense,” a statement cannot be both true and false. It must be one or the other.

If the sentence we’re considering is true, then it is false (after all, that’s what it directly states, that it is a false sentence). It's impossible for a sentence to be both true and false, and so we can rule out the possibility that the sentence is true.

So, the sentence in question must be false (which aside from "true," our initial assumption, is the only remaining alternative).

However, the sentence in question cannot be false either. If the sentence is false, then it is a lie, and the opposite of what it states must in fact be true. Since the sentence in question states that it is a false sentence, the opposite must in fact be the case – i.e. the sentence must be true. Again, no statement can be false AND true, so we must also throw out the possibility that the sentence in question is false.

The sentence in question is therefore demonstrably not true AND also not false – according the rules of “common sense” logic.

Weird. Apparently, logic is sometimes illogical (another strangely self-referential idea)!

If you like this kind of thing, you’ll love Hofstadter’s book (see part 1 of this investigation, here). However, be forewarned. It’s easily the "thickest” book I’ve ever attempted to read. I’m not just referring here to the length of this prodigious tome, but mainly to the difficulty of grappling with the ideas presented within it. Reading that book is like trying to step on your right foot with your right foot, nearly every page of it! It’s about radical, "outside the box" observation and expansion of the human mind – literally thinking hard about thinking hard. An intellectual challenge of the first order, this meta-logical journey will be truly maddening for most mere mortals, but is a sublimely rewarding experience for those brave and stout enough to make the trip.

For further study or additional masochism, try a google search on “self reference” or “fuzzy logic.”

Good luck!

(Click here to go to Part 1.)


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