The wikipedia page on dialetheism contains* the following, from the section on "Formal Consequences," immediately after a brief run-through of the standard proof that contradictions explode in classical logic:

"Any system in which any formula is provable is trivial and uninformative; this is the motivation for solving the semantic paradoxes. Dialethesists solve this problem by rejecting the principle of explosion, and, along with it, at least one of the more basic principles that lead to it, e.g. disjunctive syllogism or transitivity of entailment, or disjunction introduction."

Now, on a nit-picky level, I object to say that "this is the motivation for solving the semantic paradoxes," as if there weren't any other motivations, but I suppose that's debatable. (It's certainly not *my* motivation for wanting to solve them, but one could maybe argue that I'm just eccentric that way and that the use of the definite article there is still basically accurate.) More importantly, though, I'm pretty sure that the last bit is flat-out wrong. Certainly, there are paraconsistent logics (e.g. the ones most used for the computer database applications) where disjunction introduction is eliminated, but (a) there seems to be no obvious reason why dialetheism being true would mean that disjunction introduction wasn't universally truth-preserving, and (b) to the best of my knowledge, there aren't any dialetheists who reject disjunction introduction (much less the transitivity of entailment), whereas all the ones I know about reject disjunctive syllogism, for obvious reasons...if a statement P can be both true and false, then P could be true, P v Q would also be true (since one of its disjuncts would be true), ~P could be true and Q could just be false. On the face of it, it seems hard to see how disjunctive syllogism *could* be valid given the assumption that there are true contradictions, or, given this, what motivation there would be for a dialetheist to reject disjunction introduction. In fact, even if some pragmatically useful formal systems disregard it, I don't know of the existence of s*any* logical monist, dialetheist or otherwise, who don't think that disjunction introduction isn't present in whatever they think the One True Logic is, or any logical pluralists who think that there aren't any logics adequate for at least some contexts that contain disjunction introduction.

Anyone have any information otherwise? Is this just a gap in my knowledge of the field? Any dialetheists out there who reject disjunction introduction? Anyone know about any that reject it?

*'Contains,' in this context, of course means 'contains on Monday, August 11th, 2008.' We are talking about Wikipedia here, so it could be edited to say something entirely different at any mnute.

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## 1 comment:

Hi, thanks for putting up an interesting blog. I'm a mathematician rather than a professional logician, but I am very interested in trying to model mathematics using a paraconsistent logic in which disjunction introduction and the "rule of weakening" are not accepted. The main reason I want to exclude these logical axioms is that they are 'unnatural' from the point of view of 'natural' reasoning, such as mathematics. They are unnatural in the sense that they do not appear (or appear very rarely, does anyone know any examples?) in normal mathematical discourse.

If I remember correctly, disjunction introduction is also responsible for Gettiers counter-example to the classical definition of knowledge, so it could be argued that it is questionable from several 'pragmatical' points of view.

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