Lewis Carroll’s Infinite Premises

The British writer, mathematician and logician Charles Lutwidge Dodgson (which was Lewis Carroll’s real name) worked in the fields of geometry, matrix algebra, mathematical logic and linear algebra. Dodgson was also an influential logician. (He introduced the Method of Trees; which was the earliest use of a truth tree.) For some time Dodgson was Mathematical Lecturer at Christ Church, Oxford University.

And, of course, Dodgson (under the name Lewis Carroll) also wrote Alice in Wonderland, Alice Through the Looking Glass and many other books and poems.

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Lewis Carroll (I’ll use Dodgson’s better-known pen name from now on) formulated his premises paradox in 1895. (The argument was advanced in Carroll’s paper ‘What the Tortoise Said to Achilles’, which was published by the journal Mind.) This paradox refers to the possibility of infinite premises being required in order to reach a single conclusion.

Firstly, two or more premises are usually linked to a conclusion in a logical argument.

So how are they linked?

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

To put this simply: in order to justify (or explain) how any given premises are related to a conclusion (or how the premises entail or imply the conclusion), then a further premise will need to be brought into the argument in order to do so. Moreover, another premise will be required in order to tell us (or show us) how (or why) it’s the case that if the premises are true, then the conclusion must follow and also be true.

Now that added explanation (or justification) will itself be a premise within the argument.

Of course, the same problem will repeat itself.

Just as we brought in a premise to link two, three or more premises to a conclusion, now we have to say how this new premise is itself linked to those previous premises. Or, alternatively, we’ll need to know how all the premises (when taken together) are linked to the conclusion.

A solution has been offered to this logical paradox.

That solution is simply to argue that no added premises are needed in the first place. That is, the link between the premises and the conclusion doesn’t need to be explained (or justified) by a further premise.

So why is that?

The (or one) answer is to say that the premises and conclusion are simply linked by a rule of inference. That rule itself explains (or shows) the relation between the premises and the conclusion. Nonetheless, that seems (at least at a prima facielevel) like a cop-out. To simply say that premises are linked to a conclusion by a rule of inference doesn’t appear to be saying anything… much. Surely that rule -again! — will need to be explained by a further premise..

It can also be argued that stating that premises are linked to a conclusion by an inference rule is itself a further premise.

However, that’s unless that rule of inference somehow works without any justification or explanation.

In other words, it’s not a justification: it’s just a rule. It’s something that (to use Wittgenstein’s much-quoted phrase) “can be shown but not said”, just as the nature of the logical constants can’t be said, only shown.