# Regarding Logic and Reductio Ad Absurdum

If I know that proposition A is true, can I perform a reductio ad absurdum this way (I know it sounds redundant but humour me for a moment):

Assume ~A.

This leads to the contradiction ~A & A, because we know A is true.

Therefore, conclude the negation of the assumption: A

I have a specific case in mind. Let A = "God exists." Let B = "It's possible that God exists."
Assume ~A. Contradiction: ~A & B (because B is true (we are not hard atheists here)). Therefore A.

In fact it seems anything that is unknown can be proven this way, so there must be a problem with it.

I'm thinking that the propositions used in the contradiction must arise from calculations derived from the assumption. They cannot arise from anywhere external. Do you reckon that's right?

• NO; you do not know that A is true: you want to prove it. Sep 7, 2017 at 6:02
• We assume not-A and we derive a contradiction whatever; thus, we are licensed to conclude that the assumption is wrong. Sep 7, 2017 at 6:03
• Ok so you're saying that if you know that A is true, you cannot use a reductio ad absurdum in this way? You cannot assume anything that contradicts A? Sep 7, 2017 at 6:15
• If A is true, no sound logical system would derive a contradiction from it. If so, with the same inference rule as above, we would be licensed to conclude not-A. But we know that not-A is false ! Sep 7, 2017 at 6:23
• But you are saying "if you know that A is true, you cannot use a reductio ad absurdum in this way? You cannot assume anything that contradicts A ?" YES, we can assume not-A and we have immediately a contradicition. Thus, we conclude with the negation of the assumption, i.e. not-not-A that (in classical logic) it is again A. Sep 7, 2017 at 6:45

The first argument simply proves A from the assumption of A. It is not surprising that one can do that or that it is valid! Philosophers call it "begging the question."

The second argument is not the same argument. It tries to employ not-A and B as a contradiction. But that is not a formal contradiction, as that must be some sentence and the negation of that same sentence.

Neither can be used to prove any old sentence. The first argument proves any A, yes, but only given the assumption that A is true!

It is not a restriction of standard first order logic that the sentences come from anywhere in particular. For example, if an argument has a contradiction in its premises, that contradiction can be employed in reasoning from them toward a conclusion.

• For the second argument, what about this way ~A -> ~B (If god doesn't exist, it's not possible that God exists).... so the contradiction could be ~B and B which leads to the conclusion that A. I can't see where the problem with this is. I tried looking at modal logic, but that didn't help. Took A = God exists, Took 'it is possible that God exists = God exists in some possible world, and the logic of the argument worked out, but I don't think 'it is possible that God exists' is the same as 'God exists in some possible world'. Sep 7, 2017 at 2:20
• You would have to spell out what the premises and conclusions are in this new formulation, as opposed to what is proof strategy. I am not at all sure I'm understanding your suggestion. Sep 7, 2017 at 2:37
• A=God exists B=it is possible that God exists ~A->~B (if God does not exist, then it is not possible that he exists) Then formulate an absurdity from the assumption that God does not exist ~A Since ~A->~B, we can say ~B but then at the same time, B is true because it is possible that Gor exists, so B&~B is a contradiction, therefore A, if The God example doesn't suit, just think of anything that is possible, like 'it will rain tomorrow' and 'it is possible that it will rain tomorrow', hopefully this clears it up Sep 7, 2017 at 5:05
• @RobHv - Assumptions: (i) ¬A → ¬B; (ii) B; (iii) ¬A; thus, by Modus Ponens with (iii) and (i) we have: (iv) ¬B: contradiction ! Thus, by Reductio, we conclude with (v) ¬¬A, i.e. with A: "God exists". The proof is formally valid; to say that it concludes... it depends on our "belief" in the assumptions (i) and (ii). In particular, what about (i) "if God does not exist, then it is not possible that he exists" ? Sep 7, 2017 at 7:33
• The limitation of working from a premise that is intuitive to you but not generally accepted is that your argument will be persuasive at most to those who agree with that premise, and others will shrug and ignore. If it helps you see an interesting implication of a view you hold, however, it's worthwhile work! Sep 7, 2017 at 16:56