4

Relevance logic takes a closer look at the implication operation in first-order logic. It suggests that implications such as:

p and not p -> q

cannot hold; in ordinary English, an example of this is:

'Socrates is a man and Socrates is not a man, hence the capital of England is London'.

The conclusion although true, appears to be irrelavnt to the premises under discussion.

Now, Meyer in the 70s, looked at Peanos Axioms (PA) in Relevance Logic, and showed that contra Godel that it was provably consistent.

If the proof of the consistency of PA is important and vital, could not one say that relevant PA is perhaps the correct PA? Or that at least first-order PA is not right PA to look at?

Of course, not all of the theorems of traditional PA holds in relevant PA; but is anything essential lost, say to mainstream number theory or physics, rather that the exotic outer reaches of what is possible in traditional PA?

(Another way to examine this result in the light of Godel, is to consider his theorem that a theory cannot be both complete and consistent; if one desires completeness, one must embrace inconsistency; and in fact relevant logic is paraconsistent).

4

"Is anything essential lost" or are the shortcomings of Meyer's system in "the exotic outer reaches of what is possible in traditional PA"?

PA proves that formula if p > 2 is prime, then there is a positive integer y which is not a quadratic residue mod p; that is,

∃y ∀z: ¬(y ≡ z^2 (mod p)).

That looks like a pretty unexotic bit of number theory to me. Fails in Meyer's system though. Bad news?

See R. Meyer and H. Friedman, Whither Relevant Arithmetic?, JSL 1992, 824–831.

  • [Thanks for the edit. Isn't it annoying, the lack of LaTeX markup here!] – Peter Smith Sep 20 '13 at 12:49
  • That does seem like a bit of fairly straight-forward traditional number theory to lose. On the other hand, can relevant arithmetic saying that all integers are quadratic residues be seen as a simplification - unless of course its number theory is simplified into a triviality. – Mozibur Ullah Sep 20 '13 at 16:11

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