Infinite regress

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See also: Infinite Regress (Voyager episode)

An infinite regress in a series of propositions arises if the truth of proposition P₁ requires the support of proposition P₂, and for any proposition in the series P_n, the truth of P_n requires the support of the truth of P_n+1. There would never be adequate support for P₁, because the infinite series needed to provide such support could not be completed.

Distinction is made between infinite regresses that are "vicious" and those that are not. One definition given is that a vicious regress is "an attempt to solve a problem which re-introduced the same problem in the proposed solution. If one continues along the same lines, the initial problem will recur infinitely and will never be solved. Not all regresses, however, are vicious." [1]

[edit] Aristotle's answer

Aristotle argued that infinite regress resulted from the fact that some knowledge can't be demonstrated:

“	Some hold that, owing to the necessity of knowing the primary premisses, there is no scientific knowledge. Others think there is, but that all truths are demonstrable. Neither doctrine is either true or a necessary deduction from the premisses. The first school, assuming that there is no way of knowing other than by demonstration, maintain that an infinite regress is involved, on the ground that if behind the prior stands no primary, we could not know the posterior through the prior (wherein they are right, for one cannot traverse an infinite series): if on the other hand-they say-the series terminates and there are primary premisses, yet these are unknowable because incapable of demonstration, which according to them is the only form of knowledge. And since thus one cannot know the primary premisses, knowledge of the conclusions which follow from them is not pure scientific knowledge nor properly knowing at all, but rests on the mere supposition that the premisses are true. The other party agree with them as regards knowing, holding that it is only possible by demonstration, but they see no difficulty in holding that all truths are demonstrated, on the ground that demonstration may be circular and reciprocal. Our own doctrine is that not all knowledge is demonstrative: on the contrary, knowledge of the immediate premisses is independent of demonstration. (The necessity of this is obvious; for since we must know the prior premisses from which the demonstration is drawn, and since the regress must end in immediate truths, those truths must be indemonstrable.) Such, then, is our doctrine, and in addition we maintain that besides scientific knowledge there is its originative source which enables us to recognize the definitions.	”
	— Aristotle, Posterior Analytics (Book 1, Part 3)