Monday, May 25, 2015

not either, not both, and not even between.

Let ‘T’ abbreviate ‘true’; consider:
(1)         (x)(Tx v ~Tx)
(1) says that everything is either true or not true. I say (1) is false for an infinite number or reasons.  Here are five: it’s not true that my cat is either true or not true. It’s not true that my nostril is either true or not true. It’s not true that a banana is true or not true.  It’s not true that this italicized sentence is either true or not true. If you’re not convinced that (1) is not true, I suggest that you consider each and every natural number n, and you’ll see that it’s not true that n is either true or not true.
    But here’s my problem. If I deny (1) by affirming
(2)      ~(x)(Tx v ~Tx)
I thereby affirm
(3)       (x)(~Tx • Tx)
for (2) and (3) are logically equivalent if quantifier negation and De Morgan’s law are valid, which they are. So here’s my dilemma: affirm (1) or affirm a contradiction. Because (1) itself will eventually lead to a contradiction, my choice is really between affirming a contradiction now or affirming one later. I choose never. So I say that though (1) and (2) appear to be more than a string of symbols, they are not.


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