Aristotle's definition of the syllogism -- a dialogical interpretation

By Catarina Dutilh Novaes
(Cross-posted at NewAPPS)

(I am currently finishing a paper on the definition of the syllogism according to Aristotle, Ockham, and Buridan. I post below the section where I present a dialogical interpretation of Aristotle's definition.)

Aristotle’s definition of ‘syllogismos’ in Prior Analytics (APri) 24b18-22 is among one of the most commented-upon passages of the Aristotelian corpus, by ancient as well as (Arabic and Latin) medieval commentators. He offers very similar definitions of syllogismos in the Topics, Sophistical Refutations, and the Rhetoric, but the one in APri is the one having received most attention from commentators. In the recent Striker (2009) translation, it goes like this (emphasis added):

A ‘syllogismos’ is an argument (logos) in which, (i) certain things being posited (tethentôn), (ii) something other than what was laid down (keimenôn) (iii) results by necessity (eks anagkês sumbainei)(iv) because these things are so. By ‘because these things are so’ I mean that it results through these, and by ‘resulting through these’ I mean that no term is required from outside for the necessity to come about.

It became customary among commentators to take ‘syllogismos’ as belonging to the genus ‘logos’ (discourse, argument), and as characterized by four (sometimes five) differentiae:

(i)            there are at least two premises which are posited

(ii)          the conclusion is different from the premises

(iii)         the conclusion follows necessarily from the premises

(iv)         the premises imply the conclusion by themselves; they are jointly necessary and sufficient for the conclusion to be produced.

My starting point is the idea that the best way to understand Aristotle’s project in the APri is as the formulation of a formal theory that could be suitably applied in particular in contexts of dialectical disputations. In other words, dialectical (or more generally, dialogical) considerations are always in the background in the development of the syllogistic theory (as also argued by (Kapp 1975)). True enough, he states at the very beginning of APri that the framework applies both to demonstrative and to dialectical syllogisms. But in both cases we may think of a multi-agent, dialogical situation (e.g. demonstration in the context of teaching), even if there are important differences between dialectical and demonstrative contexts. However, while the dialectical context is inherently dialogical and multi-agent, the demonstrative context need not be.

As Aristotle presents it in Chap. 1 of Book I, the distinction between dialectical and demonstrative syllogisms seems to pertain exclusively to the status of the premises: if known to be true, and more primary than the conclusion, then the syllogism will be demonstrative; if merely ‘reputable’ (endoxa), then the syllogism is dialectical. But with respect to the pragmatics of the two situations, there are other relevant differences. In particular, demonstrative syllogisms used in the context of teaching will presuppose an asymmetric relationship between the interlocutors (teacher and pupil), whereas in a dialectical context, although questioner and answerer have different roles to play, their statuses are usually comparable – they are peers. Indeed, the overall goals of a demonstration are quite different from the goals of a dialectical disputation, even though both can rely on syllogistic as a background theory of argumentation.

Be that as it may, each of the clauses formulated by Aristotle and numbered above can be given compelling dialogical, if not dialectical, explanations (on occasion I will also refer to demonstrative contexts). Let us discuss each of them in turn.

(i) Multiple premises. This requirement excludes single-premise arguments as syllogistically correct. Indeed, in the formal theory subsequently developed in APri, the arguments considered are almost exclusively those that we now refer to as syllogistic arguments, namely composed of two premises and one conclusion, all of which are categorical sentences of the A, E, I, O forms. But as often noted, this definition excludes for example the conversion rules (from AiB infer BiA and vice-versa; from AeB infer BeA and vice-versa), creating some difficulty to account for the nature of the validity of these rules. Moreover, consider the following description of the general enterprise by Striker:

Aristotle intended his syllogistic to serve as a general theory of valid deductive argument, rather than a formal system designed for a limited class of simple propositions. (Striker 2009, 79)

If we follow Striker (as I think we should!), the specific features of the theory later developed in APri should not be taken to explain the general definition at the starting point: this would amount to putting the cart before the horse. Indeed, it is the formal theory that is meant to offer a regimented account of the conceptual starting point, which is the general notion of a valid deductive argument. So this specific feature of the formal apparatus cannot be summoned to explain this aspect of the definition.

What could then explain the requirement that there be multiple premises? As noted by Striker (2009, 79), the verb ‘to syllogize’ originally meant something like ‘to add up’, ‘to compute/calculate’, and so it immediately suggests the idea of ‘putting things together’, of a fusion of more than one element (a point often made by the ancient commentators).

Plato already used the term ‘to syllogize’ in the sense of ‘to infer’ or ‘to conclude’, which Aristotle seems to have adopted. Indeed, from a dialectical/dialogical perspective as illustrated in Plato’s dialogues, the multiple premises requirement makes good sense. In a typical dialectical situation, the questioner (e.g. Socrates) elicits a number of discourse commitments from the answerer, and then goes on to show that they are collectively incoherent – for example, because they entail something absurd – thus producing a refutation. Typically, a refutation will not come about with only one discursive commitment: it is usually the interaction of multiple commitments that gives rise to interesting (and sometimes embarrassing!) conclusions.

Notice also the use of the terms ‘posited’ and ‘laid down’, which have a distinctive dialectical flavor. They introduce the dimension of a speech-act, of an agent actually putting forward premises to an interlocutor or audience, again suggesting multi-agent situations. Later authors such as Boethius will make the multi-agent dimension even more explicit, adding that the premises are not only laid down by the producer, but also granted by the receiver.

(ii) Irreflexivity. Aristotle’s requirement that the conclusion be different from the premises seems puzzling at first sight, since it entails that the consequence relation underlying syllogistic is irreflexive. This is in tension with the currently widely accepted view that reflexivity is a core feature of deductive validity.

However, here again, taking into account the various contexts of application of syllogistic arguments, irreflexivity makes good sense for each of them (as argued in (Duncombe 2014)). Indeed, in a demonstrative context, the function of a syllogism is to lead from the known to the unknown, and so obviously premises and conclusion should be different. In a dialectical context, it makes no sense to ask the opponent to grant as a premise precisely that which one seeks to establish as a conclusion; this would amount to an instance of petition principii. So the irreflexivity of the syllogistic consequence relation is exactly what one would expect, given the applications Aristotle seems to have in mind when developing the theory. (There are issues of propositional identity that arise in connection with this requirement (e.g. are logically equivalent propositions such as AiB and BiA ‘the same’?), but we will set those aside for the present purposes.)

(iii) Necessary truth-preservation. Aristotle distinguishes syllogistic arguments from those whose premises make the conclusion likely but not certain, such as induction or analogy. It is in this sense that his main target seems to be the notion of a valid deductive argument, but from the start necessary truth-preservation will be a necessary but not sufficient condition for deductive validity (in particular, in light of the three other clauses).

There is much to be said with respect to why the ‘results by necessity’ clause makes sense in the different contexts of application of syllogistic arguments, in particular demonstrative and dialectical contexts, but let us keep it brief for the present purposes. In a dialectical context, an argument having this property will force the opponent to grant the conclusion, if she has granted the premises, so it is a strategically advantageous property for the one proposing the argument. In a demonstrative context, Aristotle’s whole theory of demonstration is premised on the idea of deriving rock-solid conclusions from self-evident axioms, and thus again necessary truth-preservation becomes advantageous.

(iv) Sufficiency and necessity of the premises. This is perhaps the most obscurely formulated of the four clauses in the definition, and indeed Aristotle goes on to offer a gloss of what he means, which is however still not very illuminating. In the Topics, his phrasing is more transparent, as described by Striker:

The definition as given in the Topics is clearer in this respect: it has the clause ‘through the things laid down’ instead of ‘because these things are so’. In this passage, Aristotle adds the remark that this clause should also be understood to mean that all premises needed to derive the conclusion have been explicitly stated. (Striker 2009, 81)

This clause has been variously interpreted by commentators. It seems to amount to some sort of relevance requirement: it must be precisely in virtue of the premises that the conclusion comes about. To be sure, the premises may be false or uncertain (at least outside demonstrative contexts), but the conclusion must be produced through them. Some commentators, in particular in the Arabic tradition, have interpreted this clause as a requirement for an essential connection between premises and conclusion. But the requirement can also be interpreted logically as stating that no premise is redundant for the conclusion to come about; all of them are de facto needed for the conclusion to result of necessity. (This is indeed one of the two main formulations of the requirement of relevance in modern relevant logics, known as ‘derivational utility’ (Read 1988, 6.4).) This requirement is also often discussed in connection with the fallacy of False Cause, which we will discuss briefly below.

Moreover, as Aristotle’s gloss suggests, this clause can also be read as the requirement that everything that is needed for the conclusion to result of necessity has been explicitly stated; there are no hidden premises required (“no term is required from outside”). And so, this clause may be read as the requirement that the premises laid down are exactly those needed for the conclusion to come about; no more, no less.

In demonstrative contexts, this clause is very natural: for Aristotle, a demonstration is an explication unearthing the causes of a given phenomenon, and so both redundancy and lack of explicitness go against this desideratum. In dialectical contexts however, both these requirements are less straightforward: the participants may have a fair amount of endoxa in common, which could plausibly be taken for granted without being explicitly put forward; and redundancy may be advantageous in purely adversarial contexts, as asking for various redundant premises may serve the strategic purpose of confusing one’s opponent. But in the Topics, Aristotle wants to move away from the purely adversarial dialectical disputes (though he also gives advice on how to perform well in such cases – see also the Sophistical Refutations) and towards a more cooperative model – dialectic as inquiry, where two parties together consider what would follow from given assumptions (Topics VIII.5). In such contexts, redundancy would be out of place, and relevance comes out as a notion related to cooperativeness.