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Archive for February, 2015

What is reductio ad absurdum? It will take a while to get to an answer to that question, and it ends up being beyond the scope of this post to get the answer. But the way to an answer is interesting.

Let’s get some background. Classical logic has a number of features. Relevant to this discussion are the Law of Non-Contradiction (LNC) and the Law of Excluded Middle (LEM). Speaking extremely loosely1, and speaking only about statements, sentences, propositions, or whatever can possibly be true or false (and so not talking about incomplete sentences, ill-formed statements, nonsense words or phrases, etc.), LNC says that nothing is both true and false, and LEM says that everything is either true or false. An interpretation assigns truth or falsity (at least one, by LEM, and no more than one, by LNC!) to the smallest “parts” in the language, called “propositions”. By rules for combining propositions, compounds are also assigned truth-values. Thus, if “p” and “q” stand for propositions, an interpretation assigns truth-values to “p” and to “q”, and a rule will tell us, for instance, that “p and q” is true if and only if “p” is assigned truth by the interpretation and “q” is assigned truth by that interpretation. The utility of this way of looking at the truth-values of all that can be said in a formal language is that truth-value assignment is defined only for the atomic parts of the language – anything more complex than that can have its truth-value derived from the truth-values of its constituent parts by analysis of the rules of the connective words or phrases in the complex statement. Thus, “if p then q”, “p and q”, “p or q”, “not-p” are true or false by virtue of the truth or falsity of “p” and “q” as well as whatever the rules for “if…then”, “and”, “or”, and “not-” are. Because the connectives take the truth-values of the smaller parts as inputs and map to exactly one of true or false as output, you sometimes see this logic called “truth-functional” – the truth-value of a compound is a function of the truth-values of its parts.

We need a little more background on the classical picture. “Valid inference”, again speaking extremely loosely, means something like this: B can be validly inferred from A if and only if all interpretations that make A true also make B true. A is the premise, B the conclusion (again, a rather loose way of speaking). If there are multiple premises, then let Σ be the set of those premises. Then B can be validly inferred from Σ if and only if every interpretation that makes each element of Σ true also makes B true.

There is one simple, but tedious, way to check whether an inference is valid – list every possible assignment of truth-values to the atoms of every premise in the set of premises, and check if all those assignments that make each premise true also make the conclusion true. As this is sometimes laborious, other methods exist, including natural deduction. Any classical method worth using exhibits the properties of Soundness and Completeness; it’s beyond the scope of this post to explain them, but essentially if a proof method is Sound, it will never mistakenly “prove” a conclusion that doesn’t have to be true, given a certain set of premises; if a proof method is Complete, anything that would be true under all interpretations that make a given set of premises true can be derived from the premises as a conclusion using the rules of the proof method.

Two deduction rules of classical logic are relevant here: ex falso quodlibet and reductio ad absurdum. Let’s add one bit of notation: ⊥ will stand for a proposition that’s true under no valuations. Ex falso quodlibet says that one can infer anything from ⊥; since no interpretation makes ⊥ true, in every interpretation that makes ⊥ true, A is true, for all A whatsoever. This is so because “every interpretation that makes ⊥ true” is a grand total of 0 interpretations; in all 0 of those, A is true! When would we have ⊥ as a premise from which to infer anything, though? Well, “⊥” stands in for whatever simply cannot be true, and by LNC, we know that something like “A and not-A” (assuming “and” and “not-” are truth-functions behaving in the expected way) comes out true if and only if A is both true and false. This will never happen. Thus, “A and not-A” and “⊥” can be substituted for each other.

Why would this be useful? Well, imagine we want to infer B from a certain set Σ of premises. By LEM, with p some proposition, p is true or false. If I can prove that p’s truth implies the truth of B, and that p’s falsity, together with the truth of each premise in the set of premises, implies a contradiction, then I can prove that p’s falsity and the truth of each premises in the set of premises also implies B! Once I get ⊥ inferred, I can infer B, while (in my hypothetical) B can be inferred from p’s truth as well, and since one of “p is true” or “p is false” must be true, I get to B one way or the other. Note that I had to use another rule – proof by cases – to make this work, but that detail isn’t important (here).

Reductio ad absurdum says that if one can infer ⊥ from not-A, then one can infer A. If all interpretations that make not-A true make ⊥ true, then none make not-A true. Since not-A must be true or false, it must be false. Given a standard interpretation of “not-“, A therefore must be true.

This is just the classical picture. Classical logic is not the only logic. There is a philosophical position called “dialetheism” that denies (in some sense) the LNC (in some form). In other words, the same thing may be true and false at the same time, although not everything is both true and false (the latter position is called “trivialism”).

Dialetheism is “tolerant” of contradictions and thus it cannot regard “⊥” and “A and not-A” as equivalent – if “⊥” just stands in for something that is true under no interpretations, but a contradiction can be true in some (dialethetic) interpretations, then “⊥” won’t represent a contradiction…or the rules above have to be rejected.

The thought that started my current interest in reductio ad absurdum was my imagining an opponent of dialetheism saying, “If dialetheism were true, one would have to give up reductio ad absurdum. But to give up reductio is to give up an essential part of our inferential machinery. Many mathematical results are achieved by reductio. Since reductio works, and dialetheism says it doesn’t, dialetheism must be wrong.”

One can of course directly argue for dialetheism, but there is a possibly strategically better way to argue against this point – intuitionistic logic rejects reductio ad absurdum but has a much better reputation, not being as radical as dialetheism and having the inferential machinery necessary to prove important results in mathematics2. Intuitionistic logic rejects LEM; thus, reductio doesn’t work because showing that the negation of something leads to a contradiction shows only that the negation isn’t true; with LEM rejected, showing that the negation of something is not true, and thus that that thing isn’t false, doesn’t mean it has to be true, as there’s a third option – neither true nor false. Note that this doesn’t commit the intuitionist to thinking that anything actually is suspended between truth and falsity – it merely means that a negative proof showing that one option must be rejected is not yet a positive proof that the other option must be accepted (my “speaking loosely” caveat is necessary here, as I need to beg a great deal of charity in the interpretation of this – it does not, to put it mildly, give an entire clear and accurate picture of intuitionism).

Another thing came to mind, though – does dialetheism require rejecting reductio ad absurdum…at least of a sort? Dialetheism definitely rejects ex falso quodlibet, because, since some interpretations (may) make contradictions true, an arbitrary sentence is not vacuously true in all (i.e., all 0) interpretations that make a contradiction true. If “⊥” still stands for something that’s never true, it has to be “unhooked” from its equivalence to a contradiction, but even this may not be good enough – as ex falso quodlibet allows anything at all to be derived from “⊥”, the inferred statement may have nothing to do with the inferences that led to ⊥. The thought that premises and conclusion need to be related to each other takes us too far afield – into relevant (or relevance) logic, something I have to skip both because it’s not relevant (!) and because I’m largely ignorant (!!) of its details.

Come back to what I said in the previous paragraph – the meaning of “⊥” has to be unhooked from contradiction in a dialethetic logic. Reductio doesn’t fail in a dialethetic logic because dialethetic logics reject LEM – Graham Priest’s Logic of Paradox (LP) validates LEM, in fact. In LP, everything is either true, false, or both. Imagine we want to use reductio in LP. If “not-A” leads to “⊥”, which is now not a contradiction (because this would be fine in LP, and being able to infer ⊥ from not-A would not show not-A to lead to anything problematic, so it would be perfectly consistent with the falsity of A, which means we can’t prove A this way) but something else true in no interpretations, then, as this makes not-A true in no interpretations, but A must be true or false (and if not-A is true and negation works as we expect, A is not false), then A must be true.

This looks like reductio ad absurdum, but it’s obviously not the same, as ⊥ has had its meaning changed. It seems to me as if the natural language description of reductio is as follows: “If the non-obtaining of a state of affairs would imply an unacceptable state of affairs, the former state of affairs must in fact obtain.” In classical logic, “unacceptable state of affairs” means a contradiction. But I don’t think it needs to in all contexts.

In short, I’m not really sure what reductio ad absurdum is.

1. The Law of Non-Contradiction: New Philosophical Essays is relevant here, especially Grim’s chapter. What “the” LNC is is at least contentious. It doesn’t end up mattering here; I’m not talking about LNC (and LEM) themselves, but about some particular restrictions on logical interpretations they (seem to) enforce. If it’s not the LNC enforcing the restriction, so be it; what I care about in this specific context is the extent of the restriction. As an aside, fantastic book.

2. And I just learned while doing this research to minimize (oh, I hope) the number of stupid things I say that Euclid’s proof of the infinitude of primes was not by reductio but was “constructive” – in other words, it met the additional requirements imposed on proof in intuitionistic contexts vis-a-vis classical logic.

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