It is a standard argument of classical logic that any proposition can be logically derived from a contradiction (See Lewis and Langford, Symbolic Logic, page 250). This argument has been used by almost everybody, including Karl Popper (in the article “What is Dialectic?” republished in his book Conjectures and Refutations), to emphasize that permitting contradictions in any context would be a very bad thing. Popper says, “Without contradictions, without criticism, there would be no rational motive for changing our theories: there would be no intellectual progress. . . . Dialecticians say that contradictions are fruitful, or fertile, or productive of progress, and we have to admit that this is, in sense, true. It is true, however, only so long as we are determined not to put up with contradictions, and to change any theory which involves contradictions . . . (Conjectures and Refutations, page 424 – 425).” Popper then goes on to repeat a version of the Lewis and Langford “proof”:
Grant the following “valid modes of inference”:
1) From A and B to infer A
2) From A and B to infer B
3) From A to infer A or B, and
4) From A or B and not A to infer B
a) Assume a contradiction, A and not A are both true (premiss)
b) A is true (from a by 1)
c) Not A is true (from a by 2)
d) A or B is true (from b by 3)
e) B is true (from 4)
This is not the exact form of the argument that Lewis and Langford or Popper use, but I propose that it amounts to the same thing. The form in which I have presented it comes from the book, Entailment The Logic of Relevance and Necessity (1975) by Alan Ross Anderson and Nuel D. Belnap, Jr. (with assistance from many others). The argument purports to show that any proposition at all (B) can be derived from a contradiction (A and not A). But is this argument valid?
Anderson and Belnap show convincingly (to this reader, at least) that it is NOT universally valid and that “it is immediately obvious where the fallacious step occurs: namely, in passing from c and d to e. The principle 4 (from A or B and not A to infer B), which commits a fallacy of relevance, is not a tautological entailment. We therefore reject 4 as an entailment and as a valid principle of inference (p 165).” Their “principle 4” is otherwise known as the “disjunctive syllogism” or “material detachment.”
Anderson and Belnap do not deny that the disjunctive syllogism is valid all contexts, just in this context. The issue is the meaning of “or”; whether “or” is taken “truth functionally” or “intensionally.”
The sense in the first case comes from the use of truth tables. For example, if A is “Napoleon was born in Corsica” and B is “The number of the beast is 666”, the truth table (in classical two-valued logic) would be as follows:
“A or B” is false only if both A and B are false. But this is not a universally valid implication, argue Anderson and Belnap.
The intensional sense of “or” would be “if it isn’t one, then it is the other.” For example, the case of The Dog:
“And according to Chrysippus, who was certainly no friend of non-rational animals, the dog even shares in the celebrated Dialectic. In fact, this author says that the dog uses repeated applications of the fifth undemonstrated argument-schema when, arriving at a juncture of three paths, after sniffing at the two down which the quarry did not go, he rushes off on the third without stopping to sniff. For, says this ancient authority, the dog in effect reasons as follows: the animal went this way or that way or the other; he did not go this way and he did not go that; therefore he went the other (Translation of Sextus Empiricus, Outlines of Pyrrhonism, Book 1, line 69, by Benson Mates in his book, The Skeptic Way).”
The dog makes no error, because his use of “or” is intensional, not truth-functional. Anderson and Belnap say of this, “Finding truth in this way is perfectly all right with us provided the “or” is . . . not truth functional . . . and hence not such as to allow the inference from A to A-or-B. . . . It is at any rate apparent that we need charge neither The Dog of Chrysippus nor that of the Beastiarist (a medieval author repeating the story of The Dog – my parenthesis) with thinking that his “or” was truth functional; it remained for our human forebears to make this error (page 297).”
Anderson and Belnap say of the Lewis argument, “Furthermore, in rejecting the principle of the disjunctive syllogism, we intend to restrict our rejection to the case in which the “or” is taken truth functionally. In general and with respect to our ordinary reasonings this would not be the case; perhaps always when the principle is used in reasoning one has in mind an intensional meaning of “or,” where there is relevance between the disjuncts. But for the intensional meaning of “or,” it seems clear that the analogues of A -> (A or B) are invalid, since this would hold only if the simple truth of A were were sufficient for the relevance of A to B; hence, there is a sense in which the real flaw in Lewis’s argument is not a fallacy of relevance but rather a fallacy of ambiguity. The passage from b to d is valid only if the “or” is read truth functionally, while the passage from c and d to e is valid only if the “or” is taken intensionally.”
So, if the Lewis (and Popper) argument is bogus, where does that leave us? Can we no longer use reductio ad absurdum? Is reasoning in peril, as Popper claimed? Graham Priest (in his books In Contradiction and Beyond the Limits of Thought as well as in dozens of articles in the philosophical literature starting with “The Logic of Paradox”) argues that it leaves us just where we were, but with a few adjustments to the canon of classical logic. Most contradictions are false, but some, like the famous Liar Paradox, are true. For more on this, see Priest’s voluminous output or, perhaps, start with my short post here.