Saturday, 24 May 2014

Lewis Carroll's Premises Paradox

In 1895, Lewis Carroll formulated his well-known logical paradox about the possibility of infinite premises being required to reach a given conclusion.

Firstly, premises are linked to a conclusion in a logical argument (or form).

How are they linked?

Now that can be a question of the logical and indeed metaphysical nature of the link between premises and conclusion: whether it's an example of entailment, consequence, implication or whatnot. However, that wasn't the point that Lewis Carroll was making. He was making a purely logical point (i.e., not a philosophy-of-logic point).

To put this simply: in order to justify or explain how the premises are related to the conclusion (or how the premises entail or imply the conclusion), other premises will need to be brought in order to do so. Moreover, a further premise will be required in order to tell us (or show us) how or why if the premises are true, then the conclusion must also be true.

That added explanation (or justification) will itself now be a premise within the given argument-frame.

Of course now the same problem will repeat itself. Just as we brought in a premise to link two or three premises to a conclusion: now we have to say both how this new premise is itself linked to those two or three premises. Or, alternatively, how all three or four premises (when taken together) are linked to the conclusion.

A solution has been offered to this logical paradox. That solution is simply to say that the added premise isn't needed in the first place. That is, there's a link between premises and conclusion which doesn't need to be explained (or justified) by a further premise. 

Why is that?

The simple solution is to say that the premises and conclusion are linked by an inference rule. That is, an inference rule itself explains the relation between the premises and the conclusion.

Nonetheless, that sounds (at a prima facie level) like a cop-out. To simply say that premises are linked to a conclusion by an inference rule doesn't appear to be saying anything... much. Indeed that inference rule, again, will need to be explained by a further premise... surely. 

That is unless that inference rule somehow works without any justification or explanation. Perhaps (as with Wittgenstein’s phrase) it's something “seen but not said”; just as the logical nature of the logical constants cannot be said; only seen.

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