In 1895, Lewis Carroll formulated his well-known logical paradox about the possibility of infinite premises being required to reach a single conclusion.
Firstly, two or more premises are usually linked to a conclusion in a logical argument.
How are they linked?
Now that can often be a question of the logical — or 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 (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 into the argument in order to do so. Moreover, a further 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. And that added explanation (or justification) will itself now be a premise within the argument.
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 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 simply by a rule of inference. That rule itself explains (or shows) 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 a rule of inference doesn’t appear to be saying anything… much. Indeed that rule, again, will need to be explained by a further premise. It can also be said that saying that “premises are linked to a conclusion by an inference rule” is itself a justification or a further premise.
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. Perhaps it’s something that “can be seen but not said” (to use Wittgenstein’s phrase); just as the logical nature of the logical constants can’t be said, only seen.