Contradictions entail any proposition: A Proof and a Question in need of answer at the the Foundations?

[mynameischristo1651]

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)

[JCS]

There are some confusions in your question1) That from a contradiction anything follows is not a reductio ad absurdum inference; it is, in fact, a property of most classical logical systems that is known since the Medieval Age (and a valid inference rule in Natural Deduction systems for these logics), and it's also still referenced by its latin name: ex falso quodlibet (it's not accepted by some modern currents: the so-called paraconsistent logics, but I don't find them convincing). A proof by reductio doesn't start by a contradiction; it starts with the premises and the negation of the conclusion, and ENDS with a contradiction (there are a few authors that distinguish between proofs by contradiction and proofs by reductio, a distinction that makes some sense in intuitionistic logic, but that's a different story).(2) Your philosophical argument is flawed because you made a confusion with the (classical) notion of Logical Consequence: an argument is sound iff the truthness of the premises implies the truthness of the conclusion. You cannot refute an argument in logical form because one of the premises is false; there are lots of logical flawless arguments with patently false premises. You can only refute an argument LOGICALLY if you find a state of affairs (a model) in which the premises are true and the conclusion false. Valid logical inference doesn't care about the truth of the premises; that's an issue outside Logic. (Before you ask: this notion of Logical Consedquence is compatible with a contradictory premise, because it has the form of an implication; if a premise is contradictory, then there is no model in which the premises are true, so the argument is vacuously true).(3) This being said, there is another unsavory mix in your question: beliefs with questions about logic. The fact that someone believes in something is logically irrelevant; Logic can only pass judgement about the consistency of those beliefs, and the consequences that follow from them, by sound rules of inference, IF they are true.