| Contradictions entail any proposition: A Proof and a Question in need of answer at the the Foundations?
Many of us emply the reductio inference in mathematics (and thereby logic) and in ever-day life: A semi-formal proof of the acceptance of a contradiction shows that it entails any proposition whatever. This is odd, but here is how -1. P & ~P2. P Conjunction Elim 1.3. P v Q Disjunction Intro 2.4. ~P Conjunction Elim 1.C. Q Disjunction Elim 3,4.Now, I wan't to discuss a philosophical issue: I definetely have contradictory beliefs, unbeknownst to me (as most ppl do). Say I refute this argument - Q doesn't follow from the premises because premise (1) is false. Why is (1) false? Because one cannot knowingly believe a contradiction - but the claim (a) 'ppl can't knowingly believe in contradictions' would need a proof - yet, and proof of (a) would need to appeal to a proof by contradiction; that is, they would employ what is in question - hence, begging the question in some sense. How can we resolve this problem? Ideas (I must be wrong)
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