Thursday, November 17, 2016

Implying Bottom

Stephen Maitzen, "The Problem of Magic":

Instead, any theistic explanation of the operation of the laws of logic must say at least this: If God didn’t exist, then the laws of logic wouldn’t hold. But no sense at all can be attached to the consequent of that conditional. What could it mean for the laws of logic not to hold? Would it “mean” that the laws of logic never hold and yet sometimes hold? Would it “mean” that the laws of logic sometimes hold, never hold, and neither sometimes nor never hold? If it wouldn’t have either of those pseudo-meanings, why not? Presumably not because the laws of logic would prevent it! No one can make any sense of what would be implied by the failure of the laws of logic, and therefore no one can make any sense of the supposition that the laws of logic might not hold. In the face of that senselessness, one might retreat to the claim that without God only some rather than all of the laws of logic would fail. But which laws would fail, and why only those? I can’t see any plausible answer to those questions.

This is a truly baffling line of argument. Surely the obvious point being made by the position being criticized is that, in fact, the denial of "the laws of logic hold" is incoherent. Nor is there in fact any problem with this, if we can make sense of "The laws of logic hold" -- which Maitzen's argument requires. There is no logical or philosophical problem with a claim implying what is usually represented by the Bottom operator (incoherence, or contradiction, or impossibility, usually represented as a ⊥, and also called falsum); that just establishes, assuming the conditional is true, that the opposite of the antecedent is a necessary truth. In fact, in logical systems that are made specifically for dealing with necessary truths, it's fairly standard for negation to be defined as the implication of Bottom; anything implying Bottom is false. You can create analogous conditionals for any necessary truth.

The tricky thing would be establishing the truth of the conditional, of course, but this is not something that can be seen by looking at its consequent. Rather, one would have to look at the conditions for the contradictory of the consequent to be true -- in this case, looking at what it means for "The laws of logic hold" to be true. Which is, of course, precisely what common sense would suggest.

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