Topics Logic

Question Is it true that anything can be concluded from a contradiction? Can you explain? It's seems like its a tautology if taken figuratively because we can indeed conclude anything if we suspend the rules of reasoning, but there is nothing especially interesting in that fact in my humble opinion.

Accepted:
December 20, 2012

Comments

The topic is controversial (as I indicate below), but the inference rules of standard logic do allow you to derive any conclusion at all from any (formally) contradictory premise. Here's one way (let P and Q be any propositions at all):

1. P & Not-P [Premise: formal contradiction]

2. Therefore: P [From 1, by conjunction elimination]

3. Therefore: P or Q [From 2, by disjunction introduction]

4. Therefore: Not-P [From 1, by conjunction elimination]

5. Therefore: Q [From 3, 4, by disjunctive syllogism]

Those who object to such derivations usually call themselves "paraconsistent" logicians; more at this SEP entry. They typically reject step 5 on the grounds that disjunctive syllogism "breaks down" in the presence of contradictions. I confess I've never found their line persuasive.

It's not just that disjunctive syllogism breaks down, but that the conclusion Q is, in general, irrelevant to the premise, which only talks about P (and Not-P). So, in Bertrand Russell's famous version, given a contradiction about, say, arithmetic ("2+2=4 & 2+2=5"), you can use the derivation given by Maitzen to prove that Russell (a famous atheist) is the Pope.

For interesting (and amusing) arguments in favor of the importance of relevance, see the early chapters of Anderson, Alan Ross, & Belnap, Nuel D., Jr. (eds.) (1975), Entailment: The Logic of Relevance and Necessity, Vol. I (Princeton, NJ: Princeton University Press). Relevance logics, a form of paraconsistent logic, have found important applications in artificial intelligence, where it is desirable, in devising a computational model of a mind, to have it use a system of logic that does not lead to irrelevancies. For discussion on that topic, see Shapiro, Stuart C. & Wand, Mitchell (1976), 'The Relevance of Relevance', Technical Report 46 (Bloomington, IN: Indiana University Computer Science Department) and Shapiro, Stuart C., 'Relevance Logic in Computer Science', in Anderson, Alan Ross; Belnap, Nuel D., Jr.; & Dunn, J. Michael (eds.) (1992), Entailment: The Logic of Relevance and Necessity, Vol. II (Princeton, NJ: Princeton University Press), Sect. 83, pp. 553-563.

@William Rapaport: Unless disjunctive syllogism or one of the other two rules used in the derivation fails, the "irrelevance" of the conclusion to the premise is irrelevant to whether the conclusion follows from the premise. Relevance logic has to give up at least one of those rules, none of which is easy to give up.

Stephen Maitzen has given a syntactic response, showing how formal rules of logic can be used to derive any conclusion Q from a contradiction P & not-P. It might be worthwhile to point out that one can also give a semantic explanation of why Q follows from P & not-P--that is, an explanation based on the truth or falsity of the premise and conclusion, rather than on rules for manipulating logical symbols.

The semantic definition of logical consequence is this: We say that a conclusion follows from a collection of premises if, in every situation in which the premises are all true, the conclusion is also true. To put it another way: the conclusion fails to follow from the premises if (and only if) there is some situation in which all the premises are true, but the conclusion is false. For example, the conclusion "It is snowing" does not follow from the premise "It is either raining or snowing," because there is a situation in which the premise is true but the conclusion is false--namely, if it is raining.

Now, let's apply this definition to Stephen's example. The only way that the conclusion Q could fail to follow from the premise P & not-P is if there is some situation in which P & not-P is true but Q is false. But there is no such situation, for the simple reason that there is no situation in which P & not-P is true. So the conclusion Q does follow from the premise P & not-P, according to the semantic definition of logical consequence.

By the way, in your question you suggest that this kind of reasoning involves some sort of suspension of the rules of reasoning. But that is not true. The point of Stephen's response is to show that according to the rules of reasoning, Q follows from P & not-P. No rules of reasoning have been suspended or changed in any way. Similarly, the point of my explanation is to show that according to the usual semantic definition of logical consequence, Q is a consequence of P & not-P.