## Non-reflexive consequence relations

Among the many notes that I scribbled on handouts at the AAL a few weeks ago, one of them (on the handout for Colin Caret’s talk “Prospects for Non-Reflexivity” reads “Medieval non-reflexivity? Examples where $A\nvdash A$“. I had a vague feeling that because the Aristotelian syllogistic requires that the major, minor, and middle terms all be distinct, one could argue that it is an example of a non-reflexive consequence relation simply because something of the form $A,B\vdash A$ would be malformed. It turns out, a friend has recently published on exactly this topic: Matthew Duncombe, “Irreflexivity and Aristotle’s Syllogismos”, Philosophical Quarterly 64, no. 256 (2014): 434-452:

Abstract: Aristotle’s definition of syllogismos at Prior Analytics 24b18–20 specifies syllogistic consequence as an irreflexive relation: the conclusion must be different from each premise and any conjunction of the premises. Typically, commentators explain this irreflexivity condition as Aristotle’s attempt to brand
question-begging syllogismoi illegitimate in argumentative contexts. However, this explanation faces two problems. First, it fails to consider all the contexts in which Aristotle thinks syllogismoi are deployed. Secondly, irreflexivity rules out only some arguments that Aristotle considers question begging. Here I address these problems. First, I examine all the contexts in which Aristotle thinks syllogismoi can be
used. Secondly, I argue that, for each context, irreflexivity makes sense as a condition, but for different reasons. Assuming that a condition which holds in each context is a condition on syllogistic consequence tout court, this explains why Aristotle holds syllogistic consequence to be an irreflexive relation.

Here I want to highlight a couple of important points that Duncombe makes in this article.

First: “irreflexivity may seem strange, because if truth preservation is necessary and sufficient for a consequence relation, then that relation ought to be reflexive” (p. 434): Thus, if a consequence relation is irreflexive, this entails that there is more to that notion of consequence than mere truth preservation. If one things of consequence as “truth preservation” plus something more (which of course needs to be fleshed out, then one needn’t take the route that Beall and Restall take, denying logics with irreflexive consequence relations the status of being real logics, instead calling them “logics by courtesy and by family resemblance” (p. 437).

Second: irreflexivity cannot be (merely) a way to block question-begging, since there are examples which Aristotle considers question-begging which are not reflexive (p. 435; also, sec. II.2). Duncombe notes that five different types of question beginning can be identified in Aristotle, of which “the second and third types of question begging do not seem to be violations of irreflexivity” (p. 441).

Duncombe’s conclusion is that there is no uniform explanation for irreflexivity, but rather that irreflexivity is a consequence of different factors in different types of argumentative contexts: “Although in different contexts irreflexivity is a condition on syllogismoi for different reasons, in any pragmatic context in which Aristotle envisions syllogismoi being used, it is a plausible condition
for giving a syllogismos” (p. 435), and “it is not necessary that there is one single notion of validity across all contexts, but, as it turns out, the same notion applies across all contexts, for diverse reasons” (p. 441). A crucial feature of the various pragmatic contexts is that they are all ones which involve “logic in action”, in demonstrative, dialectic, peirastic or eristic contexts (p. 443). In each of these contexts, there is a reason why irreflexivity is desirable, even if it is a different reason in each context. Such a view of logic, with a focus on the application and use of logic, can be considered either surprisingly modern — or it can be considered a case where modern logicians are (unknowingly) returning to the original roots of logic as a dialectical tool. Thus, it should come as no surprise that just as modern interest in irreflexive logics tracks a growing interest in “logic in action”, so too irreflexivity comes hand-in-hand with Aristotle’s view of the application(s) of logic.

Though Duncombe identifies different reasons why irreflexivity is desirable in each of the different contexts, he argues that there may in fact be a general principle which underlies these different reasons:

But there may be interesting reasons why syllogismoi in different contexts converge on irreflexivity. For example, in mono-agent demonstrations the premises must be more acceptable than the conclusion (Smith 1989: 64b31–2; cf. Smith 1997: 159b8–9). When Aristotle comes to discuss multi-agent dialectic, particularly the use of dialectic for training (Smith 1997: 159a25–6) and inquiry (Smith 1997: 101a34–7), he formulates a corresponding condition for multi-agent contexts: the premises the answerer grants should be more acceptable than the conclusion (Smith 1997: 131)…it makes sense to specify that such an answerer should not admit a premise, unless it is more acceptable than the conclusion, since this condition corresponds to how to give a good demonstration…Such training dialectics would need irreflexivity as a condition (p. 451).

So not only is irreflexivity justifiable in this context, it may even be necessary!

Does anyone have any other examples of ancient or medieval irreflexive consequence relations? Please share in the comments!