Dialectical refutations and reductio ad absurdum

By Catarina Dutilh Novaes

(Cross-posted at NewAPPS)

This is the fifth installment of my series of posts on reductio ad absurdum from a dialogical perspective. Here is Part I, here is Part II, here is Part III, and here is Part IV. In this post I discuss a closely related argumentative strategy, namely dialectical refutation, and argue that it can be viewed as a genealogical ancestor of reductio ad absurdum.

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Those familiar with Plato’s Socratic dialogues will undoubtedly recall the numerous instances in which Socrates, by means of questions, elicits a number of discursive commitments from his interlocutors, only to go on to show that, taken collectively, these commitments are incoherent. This is the procedure known as an elenchus, or dialectical refutation.

The ultimate purpose of such a refutation may range from ridiculing the opponent to nobler didactic goals. The etymology of elenchus is related to shame, and indeed at least in some cases it seems that Socrates is out to shame the interlocutor by exposing the incoherence of their beliefs taken collectively (for example, so as to exhort them to positive action, as argued in (Brickhouse & Smith 1991)). However, as noted by Socrates himself in the Gorgias (470c7-10), refuting is also what friends do to each other, a process whereby someone rids a friend of nonsense. An elenchus can also have pedagogical purposes, in interactions between masters and pupils.

There has been much discussion in the secondary literature on what exactly an elenchus is, as well as on whether there is a sufficiently coherent core of properties for what counts as an elenchus, beyond a motley of vaguely related argumentative strategies deployed by Socrates (Carpenter & Polansky 2002). (A useful recent overview is (Wolfsdorf 2013); see also (Scott 2002).) For our purposes, it will be useful to take as our starting point the description of the ‘Socratic method’ in an influential article by G. Vlastos (1983) (a much shorter version of the same argument is to be found in (Vlastos 1982), and I'll be referring to the shorter version). Vlastos distinguishes two kinds of elenchi, the indirect elenchus and the standard elenchus:

Here [in the indirect elenchus] Socrates is uncommitted to the truth of the premise-set from which he deduces the negation of the refutand. This mode of argument is a potent instrument for exposing inconsistency within the interlocutor's beliefs. But it cannot be expected to establish the truth or falsehood of any particular thesis. For this Socrates must turn to standard elenchus. (Vlastos 1982, 711)

He then goes on to describe the alternative, standard elenchus, in the following way:

I argue that this is a search for moral truth[1] through two-part question-and-answer adversative argument, which normally proceeds as follows:

1. The interlocutor, "saying what he believes", asserts p, which Socrates considers false, and targets for refutation.

2. Socrates obtains agreement to further premises, say q and r, which are logically independent of p. The agreement is ad hoc: Socrates does not argue for q or for r.

3. Socrates argues, and the interlocutor agrees, that q and r entail not-p.

4. Thereupon Socrates claims that p has been proved false, not-p true. (Vlastos 1982, 712)

This scheme can be viewed as a dialogical version of traditional (monological) reductio ad absurdum, where one and the same agent makes the initial hypothesis, goes on to show that it leads to absurdity, and then concludes the contradictory of the initial hypothesis. In the scheme above, by contrast, we have what could be described as a division of labor between the different participants: the interlocutor claims p, and then it is Socrates who shows that commitment to p and to other premises q and r is incoherent, given that q and r entail not-p. (Notice also that commitment to q and r is elicited by Socrates in Vlastos’ description.) It is then Socrates again who concludes not-p, and thus not the interlocutor who had committed to p in the first place.

Vlastos’ account of the ‘standard elenchus’ has been criticized by a number of scholars (e.g. (Benson 1987, 1995)), who point out that the Socratic elenchus can only have a negative function, that is, to show an interlocutor not that p is false, but merely that p is inconsistent with their other beliefs. In other words, it is step 4 in Vlastos’ reconstruction that is particularly contentious, as up to step 3 what has been achieved is merely to show the incoherence of the interlocutor’s simultaneous commitment to p, q, and r. (Step 4 is precisely what Fabio describes as an ‘act of faith’.) At this point, the consensus in the literature is essentially that Vlastos’ account is incorrect as an account of Socratic elenchus, as found in Plato’s dialogues. But one compelling feature of Vlastos’ account is his emphasis on the adversarial (‘adversative’ is the term he uses above) component of a dialectical refutation, even if adversariality is combined with cooperative elements (e.g. the reference to friendship in the Gorgias).

As noted above, on Vlastos’ account, an elenchus comes out as virtually equivalent to a reductio ad absurdum, given that the culmination is the establishment of the truth of a given thesis, namely the contradictory of the interlocutor’s initial assertion, through an intermediate stage of incoherence/absurdity (simultaneous commitment to p, q, and r). And if his account of elenchus were accurate, then the dialogical nature of reductio ad absurdum would emerge in a rather straightforward way: reductios ad absurdum and dialectical refutations are essentially the same thing; dialectical refutations are essentially dialogical; thus, reductios ad absurdum are essentially dialogical.

But of course, it has been convincingly argued that Vlastos’ account of elenchus is not correct; numerous scholars have argued that his attribution of a positive function to refutations is unfounded. However, I still want to claim that dialectical refutations are properly viewed as the genealogical ancestors of reductio ad absurdum arguments. The relations of historical priority between logic (philosophy) and mathematics in ancient Greece are contentious (Mueller 1974), and given the scarce availability of textual material we may never come to know for sure how exactly these developments took place. But it is reasonable to assume that there were extensive contacts between philosophers (and more generally those engaging in the practice of dialectic) and mathematicians. Moreover, Lloyd (1996) and Netz (1999) have persuasively established the role of orality and debating for the emergence of classical, Euclidean mathematics. Thus, it is not completely unreasonable to suppose that the practice of dialectical refutation may have had a significant influence in the development of the technique of reductio arguments, which then became pervasive in Greek mathematics.

A case in point is the ambiguous status of Zeno’s paradoxes. Indeed, they can be interpreted either as straightforward reductios, or as dialectical refutations. On the first interpretation, the result achieved is the establishment of the truth of Parmenides’ theses to the effect that there is no plurality, there is no change, no movement etc., once Zeno shows that assuming that there are such things leads to incoherence. On the second interpretation, by contrast (and this is the position that Plato attributes to Zeno in Parmenides 128a-e), what is achieved is merely to show that the positions of the opponents of Parmenides can lead to absurdity, and thus that they are not obviously correct despite the apparent strangeness of Parmenides’ views. On this interpretation, an elenchus would function above all as a ‘dialectical silencer’ (Castagnoli 2010), when it becomes apparent that the position of one of the interlocutors is incoherent.

In short, there is a fundamental distinction between a reductio argument and a dialectical refutation, namely that the former aims at establishing the truth (or falsity) of a given thesis, whereas the latter can only show that a certain number of claims, when taken collectively, lead to incoherence, without thereby singling one of them out as false. However, it still seems fitting to consider dialectical refutations as the genealogical ancestors of reductios. And if this is right, then reductio arguments would have clear dialogical origins, and arguably would have retained dialogical aspects.

References

Hugh H. Benson, "The Problem of the Elenchus Reconsidered," Ancient Philosophy 7 (1987) 67-85

Hugh H. Benson, "The Dissolution of the Problem of the Elenchus," Oxford Studies in Ancient Philosophy 13 (1995) 45-112

Thomas C. Brickhouse and Nicholas D. Smith, "Socrates' Elenctic Mission," Oxford Studies in Ancient Philosophy 9 (1991) 131-60

Michelle Carpenter and Ronald M. Polansky, "Variety of Socratic Elenchi," in Scott (2002) 89-100

Castagnoli, Luca (2010). Ancient Self-Refutation: The Logic and History of the Self-Refutation Argument From Democritus to Augustine. Cambridge University Press.

B. Castelnerac & M. Marion 2009, ‘Arguing for Inconsistency: Dialectical Games in the Academy’. In G. Primiero & S. Rahman (eds.), Acts of Knowledge: History, Philosophy and Logic. London, College Publications.

G.E.R. Lloyd 1996, ‘Science in Antiquity: the Greek and Chinese cases and their relevance to the problem of culture and cognition’. In D. Olson & N. Torrance (eds.), Modes of Thought: Explorations in Culture and Cognition. Cambridge, CUP, 15-33.

Mueller, Ian, 1974, "Greek Mathematics and Greek Logic." In John Corcoran (ed.), Ancient Logic and its Modern Interpretations (Dordrecht, Reidel), 35-70.

R. Netz 1999, The Shaping of Deduction in Greek Mathematics: A study in cognitive history. Cambridge, CUP.    

Gary Allan Scott, ed. (2002) Does Socrates Have a Method? Pennsylvania State University Press.

Gregory Vlastos (1982). The Socratic Elenchus. The Journal of Philosophy, Vol. 79, No. 11, 711-714

Gregory Vlastos (1093). The Socratic Elenchus. Oxford Studies in Ancient Philosophy 1, 27-58

Wolfsdorf, David (2013). Socratic philosophizing. In John Bussanich & Nicholas D. Smith (eds.), The Bloomsbury Companion to Socrates. Continuum.

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[1] Notice that Vlastos restricts the scope of action of elenchi to moral truths, which is an important element of his account, but it will not play any significant role for the present purposes.