Infinite regress
Philosophers speak of an infinite regress in a series of propositions when the truth of the first proposition in the series necessitates the truth of a second proposition, and the truth of this second proposition necessitates the truth of a third, and so on indefinitely, i.e., the series cannot be brought to an end.
That a claim generates an infinite regress will often, though not always, serve to show that the claim is unacceptable and must be rejected, and so philosophers typically appeal to the fact that a claim generates an infinite regress in order to justify rejection of the claim. When an infinite regress justifies rejection of the claim which generates it, it may be described as a vicious infinite regress.
A famous example of an appeal to an infinite regress to justify rejection of the claim which generates it may be found in Plato’s dialogue Parmenides 131e-133a, where Plato raises several possible objections to his own Theory of Forms. The theory holds, inter alia, that what makes it possible for us to apply one and the same word (e.g., the adjective ‘large’) to many different things (a large town, a large dog, a large debt, a large crowd, etc.) is that all these large things resemble the single (abstract) Form of large, i.e., it assumes that when a single word is applicable to a range of items, this must be in virtue of their resemblance to a single entity apart from themselves. But, the objection runs, the things we originally called large (the large town, the large dog, etc.) and the Form of large can all be described as large – after all, the theory holds that particular large things resemble the Form of large – and so we must recognise the existence of a second Form of large in virtue of which we can call all the things in this new class large. And this move can be repeated: if we consider the class containing the things we originally called large, the first Form of large, and the second Form of large, we need to recognise the existence of a third Form of large. And so on ad infinitum. In brief, the argument used to establish the existence of a single Form of large necessitates the existence of an infinite number of Forms of large. (Whether Plato considered this a valid objection to the Theory of Forms is unclear and, needless to add, its validity has been a matter of dispute amongst philosophers from Plato’s time to the present day.)
For discussion of another infinite regress (generated by the claim that for us to know a proposition to be true we must be able to deduce it from another proposition or propositions also known to be true), see Aristotle, Posterior Analytics I 3, 72b1-15.
