Volume 10, Issue 2 - Infinite Regress and Non-Contradiction in Ancient Greek Philosophy Guest editors: Matthew Duncombe and Luca Pitteloud

In Metaphysics Γ, Aristotle argues against those who seem to accept contradictions. He distinguishes between the Sophists, who deny the principle of non-contradiction through arguments, and the Natural Philosophers, whose physical investigations lead to the acceptance of objective contradictions. Heraclitus’ name appears throughout the discussion. Usually, he is associated with the discussion against the Sophists. In this paper, I explore how the discussion with the Natural Philosophers may illuminate both the interpretation of Heraclitus by Aristotle and Heraclitus’ own worldview. To refute the Natural Philosophers, Aristotle proposes a general reconstruction of their reasoning. Roughly, relying on sensory evidence (A1), they see that the same thing changes from one opposite to another (A2). Such a change appears to characterize a generation out of non-being, which a Natural Philosopher does not accept (A3). To solve the problem, despite their different worldviews, Natural Philosophers hint at a state in which opposites co-occur, characterizing an objective contradiction (C). Looking at the discussion in Metaphysics Γ and Heraclitus fragments, sections 1–3 show how assumptions A1, A2, and A3 easily apply to Heraclitus. The case of the conclusion is more challenging. In the case of the Pluralists, the co-existence of opposites characterizes a state in which there is no generation. Such a view does not fit Heraclitus’ mobilism. To argue that Aristotle’s argument is general enough to encompass dynamic views, I examine his problematization of accepting the change of change in Metaphysics K and Physics V. There, after re-stating several points that appear in Metaphysics Γ , Aristotle argues that accepting the becoming of another becoming leads to a state of contradiction in which the becoming is perishing. Heraclitus’ B8, cited in Nicomachean Ethics , gives evidence that, for Aristotle, Heraclitus puts a process at the origin of an opposite process. Moreover, after examining the expression ‘living the death/dying the life’ in B62, I argue that Heraclitus was aware that his worldview implied a dynamic objective contradiction. Finally, an analysis of elemental changes in B36 proves that accepting objective contradictions does not make Heraclitus’ worldview less attractive.

This paper discusses two broadly logical issues related to Protagoras’ measure doctrine (M) and the self-refutation argument (SRA). First, I argue that the relevant interpretation of (M) has it that every individual human being determines all her own truths, including the truth of (M) itself. I then turn to what I take to be the most important move in the SRA: that Protagoras recognises not only that his opponents disagree with him about the truth of (M), but also that they hold that (M) is false simpliciter . By recognising that his opponents do not make the relativising concession he makes for them, he is forced to accept that (M) is false. I go on to argue that several other defenders of the SRA end up with a regress which is difficult to end and might not favour anti-Protagoreans. On my reading, by recognising what his opponents believe, Protagoras is barred from adding qualifiers, and the possible regress doesn’t get off the ground. I conclude with brief discussions of how Protagoras might try to avoid the result of the SRA and the argument’s role in this part of the Theaetetus.

On Malcolm Schofield’s highly influential reading of the Similarity Regress in Part I of the Parmenides , the problem that the Regress poses is explanatory. Socrates posited the Similarity Form in order to explain why similar things are similar: similar things are similar because they participate in the Form Similarity as copies of the same original. Yet, the Similarity Regress generates an infinite series of Similarity Forms such that explanation is deferred ad infinitum. Schofield provides a philosophical incentive for adopting his reading. He argues that the treatment of similarity in Part II of the dialogue yields a complete explanation of similarity. If we adopt this account, we can avoid the Similarity Regress altogether since a Form of Similarity is not needed in order to explain why similar things are similar. Thus, his interpretation has a hugely important philosophical pay-off. However, there is a different way to read the argument. Socrates claims that each Form is only one. Yet, the Similarity Regress is an argument that generates an infinite series of Similarity Forms. This results in a violation of the principle of non-contradiction: there is both only one Similarity Form and infinitely many Similarity Forms. Yet, anything that is incompatible with the principle of non-contradiction is surely absurd. Nobody, as far as I am aware, has explored whether this reading also has philosophical pay-off if we look at it together with similarity in Part II. However, should this interpretation pay off, it would be a viable alternative to Schofield’s. In this paper, I explore both views in the context of the treatment of similarity in Part II of the Parmenides . I argue for an account of similarity that differs from Schofield’s. Although my account is not wholly dismissive of Schofield’s, it renders the pay-off of Schofield’s account less appealing than he suggests. Furthermore, I show that the account of similarity in Part II also yields important lessons for the proponent of the alternative reading of the Similarity Regress: similarity as treated in Part II simply leads to further infinite regresses, thereby pushing us to consider rejecting the account of similarity in Part II too and to look for some other account intead.

Infinite regress arguments are a powerful tool in Aristotle, but this style of argument has received relatively little attention. Improving our understanding of infinite regress arguments has become pressing since recent scholars have pointed out that it is not clear whether Aristotle’s infinite regress arguments are, in general, effective or indeed what the logical structure of these arguments is. One obvious approach would be to hold that Aristotle takes infinite regress arguments to be per impossibile arguments, which derive an infinite sequence. Due to his finitism, Aristotle then rejects such a sequence as impossible. This paper argues that this obvious approach does not work, even for its most amenable cases. The paper argues instead that infinite regress arguments involve domain-specific infinities, and so there is not a general finitism which underpins infinite regress arguments in Aristotle, but rather domain-specific reasons that there cannot be an infinite number of entities in each domain in which Aristotle invokes an infinite regress argument.

in this paper, I examine how the Mode of Infinite Regress functions in Pyrrhonian scepticism. I argue that it is used both to generate an infinite regress of reasons and to show that such infinite regresses are epistemically defective. I clarify precisely how this occurs while examining the Mode’s efficaciousness and whether ancient philosophers might have accepted infinite regresses of reasons. I ultimately argue that they would not for reasons which have hitherto not been adequately appreciated and which shed further light on how the Mode of Infinite Regress functions, its efficacy, and the relevant notions of justification at stake.

For Damascius, any attempt to grasp the first principle of all things, the Ineffable, implies the rejection of the principle of non-contradiction (PNC). The reasoning soul, using aporia, is forced to admit contradictory statements as true when it comes to cognising what lies beyond any intelligible being. Damascius shows that it is necessary to postulate a completely transcendent and unknowable Absolute which is the uncoordinated cause of all things beyond the One. This paper examines how Damascius relates the rejection of the PNC and regressive arguments by distinguishing two types of reversal ( peritropê ) of arguments. This paper begins by analysing: i) Damascius’ first aporia, ii) his characterisation of the first principle as ineffable and iii) its unknowable nature. This leads to some comments about Damascius’ use of predicates and their negations. Finally, the paper distinguishes between regression and reversal to suggest that Damascius’ reversal of the predicates implies that the first principle lies beyond the PNC but does not involve an infinity of first principles.