## Abstract

This paper sets out to evaluate the claim that Aristotle’s Assertoric Syllogistic is a relevance logic or shows significant similarities with it. I prepare the grounds for a meaningful comparison by extracting the notion of relevance employed in the most influential work on modern relevance logic, Anderson and Belnap’s *Entailment*. This notion is characterized by two conditions imposed on the concept of validity: first, that some meaning content is shared between the premises and the conclusion, and second, that the premises of a proof are actually used to derive the conclusion. Turning to Aristotle’s *Prior Analytics*, I argue that there is evidence that Aristotle’s Assertoric Syllogistic satisfies both conditions. Moreover, Aristotle at one point explicitly addresses the potential harmfulness of syllogisms with unused premises. Here, I argue that Aristotle’s analysis allows for a rejection of such syllogisms on formal grounds established in the foregoing parts of the *Prior Analytics*. In a final section I consider the view that Aristotle distinguished between validity on the one hand and syllogistic validity on the other. Following this line of reasoning, Aristotle’s logic might not be a relevance logic, since relevance is part of syllogistic validity and not, as modern relevance logic demands, of general validity. I argue that the reasons to reject this view are more compelling than the reasons to accept it and that we can, cautiously, uphold the result that Aristotle’s logic is a relevance logic.

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## Notes

- 1.
We could add Normore (1993, p. 448) who claims that ‘[a]ncient logics were all in some sense relevance logics.’

- 2.
The development of modern relevance logic happened against the background of a dominant logical tradition (called ‘the Official view’ by Anderson and Belnap) that, for Anderson and Belnap, was problematic insofar as it took no account of the notion of relevance and that, in particular, sanctioned as valid the so-called paradoxes of implication. At the same time, Anderson and Belnap considered their intuition that relevance should be a matter of logic to be widely shared: most textbooks on logic, they observe, emphasize the importance of relevance in their informal treatments but advocate classical validity in their formal treatments; a good example of this is Lemmon (1968). These two factors, historically speaking, provided the impetus for the development of relevance logic. Aristotle, on the other hand, in all likelihood did not work against a dominant logical tradition, at least not against a formalized and generally accepted logic that occupied the position of an ‘Official View’. However, he still might have been prompted by the moves of certain Sophists to shield his logic from the dangers of irrelevant premises. Cf.

*Topics*VIII.11, in particular 161b24–33 and 162a12–5. - 3.
I will not provide a defence of Anderson and Belnap’s relevance logic. Given the considerable literature on the debate between relevance logicians and advocates of classical as well as of other strains of logic, this is well beyond the scope of this paper.

- 4.
- 5.
Anderson and Belnap’s system is not the only one that claims the title of being a relevance logic (cf. Dunn and Restall 2002, pp. 1–2; Priest 2008, pp. 188–220), but since we are only interested in the essential ideas and concepts, what we will extract from

*Entailment*will also be true for other members of the family of relevance logic (cf. the criteria for membership proposed by Mares and Meyer 2001, p. 286).—I should point out that the focus on the pure implicational fragment in my presentation of Anderson and Belnap should not be understood as an affirmation of the view that syllogisms are implications (cf. Łukasiewicz 1957; Corcoran 1974a). The aim of this section is to extract the concept of relevance employed in the logic of Anderson and Belnap and for that the pure implicational fragment is the best choice. - 6.
Whether the so-called paradoxes of implication are real paradoxes or not depends on how we understand ‘paradox’. Some authors, understanding ‘paradox’ as something that leads to logical contradictions, do not think that the paradoxes of implication are real paradoxes because they do not lead to logical contradictions, even though they might regard them as highly counter-intuitive and possibly problematic for that reason; cf. (Sainsbury 2009, pp. 1–2).

- 7.
A basic set of rules contains: (1) hypothesis introduction (hyp): hypotheses may be introduced at any time; (2) repetition (rep): a formerly introduced hypothesis may be repeated in the same proof at any time; (3) reiteration (reit): a formerly introduced hypothesis may be repeated in a sub-proof; (4) entailment introduction (\(\rightarrow \!\hbox {I}\)): if a proof concludes B from hypothesis A, an implication \(\hbox {A}{\rightarrow }\hbox {B}\) may be introduced; (5) entailment elimination (\(\rightarrow \!\hbox {E}\)): given A, B may be deduced from \(\hbox {A}{\rightarrow }\hbox {B}\).

- 8.
The implication connective is, for Anderson and Belnap (1975, pp. 3, 5, 12–13), the ‘heart of logic’, because of its most intimate connection to logical consequence.

- 9.
More precisely, Anderson and Belnap’s reformed rules of the natural deduction system used before are: ‘(1) one may introduce a new hypothesis \(\hbox {A}_{\{\mathrm{k}\}}\), where k should be different from all subscripts on hypotheses of proofs to which the new proof is not subordinate; (2) from \(\hbox {A}_{\mathrm{a}}\) and \(\hbox {A}{\rightarrow }\hbox {B}_{\mathrm{b}}\) we may infer \(\hbox {B}_{\mathrm{a}\cup \mathrm{b}}\); (3) from a proof of \(\hbox {B}_{\mathrm{a}}\) from the hypothesis \(\hbox {A}_{\{\mathrm{k}\}}\) we may infer \(\hbox {A}{\rightarrow }\hbox {B}_{\mathrm{a}{\text {-}}\{\mathrm{k}\}}\), provided k is in a; and (4) reit and rep retain subscripts (where a,b,c, range over sets of numerals)’ (1975, p. 22).

- 10.
When we turn to Aristotle’s investigation of the syllogism we will see that he too first argues for a condition equivalent to

*variable*-*sharing*and then turns to the question whether or not a valid syllogism requires that all the terms and premises are used. - 11.
For a similar set of conditions for relevance logic, cf. Mares and Meyer (2001, p. 286).

- 12.
Of special interest in this context are passages in which Aristotle uses expressions such as ‘

*ex anagkēs sumbainei*’, ‘*ei*X,*anagkē*Y’ or simply the word ‘*anagkaion*’. - 13.
*Syllogismos*can, of course, just mean ‘argument’ or ‘reasoning’ in an everyday sense (Liddell et al. 1996,*ad loc.*) besides its technical logical meaning, which we could translate as ‘deduction’. I will use the word ‘syllogism’ in its technical meaning, i.e. as a technical term of logic as contrasted with an everyday use of the word ‘argument’ throughout the paper and disambiguate when necessary. - 14.
- 15.
I will argue that the way in which syllogisms enforce relevance is peculiar to syllogistic form. The superproperty validity, applying to arguments different from syllogisms, cannot therefore incorporate relevance in the same way. This, however, does not prevent validity from being relevant, as one kind of logic might incorporate the concept of relevance in a way appropriate to that logic, while another kind of logic might have to take a different route to ensure that the same concept is part of its notion of validity. Consider, for instance, the case of a relevant predicate logic; cf. (Dunn 1987; Anderson et al. 1992, §74).

- 16.
It is, however, unclear whether the individual statements are well-formed. Aristotle’s remarks on the form of premises up to this point of the

*Prior Analytics*do not expressly forbid statements such as ‘All A is A’ (cf. 24a16–7 and footnote 19). In my view, Aristotle ultimately rejects such statements, but this question does not affect my point here. See Corcoran (1974b, pp. 96 and 99). - 17.
In this particular example, since the conclusion is necessarily true, one could say that the validity does not depend on the particular premises used. But even if this is not the case, it becomes obvious that the validity depends (partly) on the premises, for an argument may never lead from truth to falsity. Hence, if we have an argument with a false conclusion, its validity depends on the falsity of its premises and therefore its validity could be said, from a classical viewpoint, to be

*dia tauta*the premises. - 18.
Aristotle does not state these conditions explicitly. As a matter of fact, he investigates premise-pairs; but from his actual responses to a particular premise-pair we see that he also has certain conditions in mind relating to the conclusions (e.g. he does not consider syllogisms of the form: all A is B, all B is C, therefore all D is E.)

- 19.
Aristotle, in I.1, 24a16–b12, stipulates that the propositions that serve either as premises or as conclusion have to be statements that affirm or deny something of something (

*tinos kata tinos*) either universally (*katholou*) or particularly (*kata meros*). The phrase*tinos kata tinos*does not preclude self-predications, but since it is very likely that we can ignore this case, the different kinds of statements that are possible are: universal affirmative (All A is B), universal negative (No A is B), particular affirmative (Some A is B), and particular negative (Some A is not B). - 20.
- 21.
There is a dispute over the question whether Aristotle’s letters are variables or not. Łukasiewicz (1957, pp. 7–8), Corcoran (1974b, p. 100), and others believe that they are variables; Barnes (2012, p. 99, n. 139), Frede (1974, p. 19) and others believe that they are not or that it is at least questionable whether they are. Cf. also Kirwan (1978, pp. 3–8) and (Barnes (2007), pp. 337–359).

- 22.
In what follows, I will use different notations depending on the purpose of the argument as well as to stay as close to Aristotle’s text as possible. In Aristotle, letters can stand either for propositions or for the terms the propositions relate to each other. In my notation, Latin uppercase letters always stand for terms and I will usually indicate the kind of relation that obtains between two such terms, designated by the Latin lowercase letters a, e, i, and o (see footnote 19). I will use lowercase x as a variable ranging over these letters, hence AxB is either AaB, AeB, AiB, or AoB. I may also omit the x and write AB when it is of no importance whether the statement is AxB or BxA. Greek uppercase letters, on the other hand, stand for the propositions that can occupy premise- or conclusion-position in an argument: \(\Pi , \Psi \vdash \Theta \) can stand, e.g., for AxB, BxC, AxC.

- 23.
Circular arguments are classically valid; in the field of relevance logic, there are some systems in which circular arguments are also valid and some in which they are not. Cf. Mares (2012).

- 24.
At any rate, the idea that in a circular argument no

*deduction*has taken place might be thought to be an intuitive judgment. - 25.
Alexander,

*in APr.*, 257.8–13. Smith (1989, p. 140) thinks that this might be the best explanation we can get. - 26.
Of course, the two parts might not be independent of each other. In fact, it is precisely Aristotle’s aim in this passage to show that if only one premise is assumed, there will be no syllogism. If we follow Alexander, we have to accept that syllogisms are multi-premised because of Aristotle’s stipulation in the definition of the syllogism. Aristotle’s own explanation, however, indicates that the demand that syllogisms have to be multi-premised is a consequence of the demand that syllogisms necessitate a conclusion. Also see Sect. 4 below on how this passage relates to the question whether or not Aristotle accepted a validity different from s-validity.

- 27.
Ross reads

*eipomen,*which would imply that Aristotle has already stated such a theorem (Ross 1949, p. 371, thinks it is implicitly stated in I.4–6). But since this is really the conclusion of the argument Aristotle has just presented, reading*eipōmen*with the first hand of ms. A and ms. B, as Ebert and Nortmann (2007, p. 742 n. 1) suggest, seems to make more sense. Aristotle’s explanation (*gar*) immediately following the theorem seems to do no more than summarize the steps of the argument he has just given. - 28.
In the argument I have presented, Aristotle only considers direct arguments. He then (41a21–b5) adds an argument to show that the same applies to

*reductio*arguments, which I will not present here. - 29.
Aristotle’s formulation of the theorem to be established in this chapter demands a short explanation. Given the undeniable fact that Aristotle allows for syllogisms with more than two premises, we should not understand the theorem to say that there will be no syllogisms with more than two premises, but rather that there are no deductions (in the sense of the conclusion, i.e. the result, of a syllogism) which require more than two premises: any given conclusion, the theorem says, can be stated as the conclusion of a syllogisms with two premises and no more. The occurrence of the word

*apodeixis*(demonstration) instead of*sullogismos*is curious but should not give rise to any concerns about the theorem: Aristotle will talk of*sullogismoi*in the remainder of the chapter and in any case he has stated before that every demonstration is a syllogism (but not*vice versa*), so the result of his discussion will apply to syllogisms. - 30.
In I.25 the letters stand for syllogistic statements (i.e. propositions), so either for a premise or for a conclusion. For the notation, see footnote 22.

- 31.
For an extensive analysis, see Ebert and Nortmann (2007, pp. 756–759).

- 32.
Cf. also I.32, 47a16–9, where Aristotle, in addition to calling premises

*matēn*uses the phrase*ti periergon*to describe them as ‘superfluous’ assumptions. - 33.
Analysability is an answer (or part of an answer) to the question: what makes a complex argument valid? In natural deduction systems, in order to show that a complex argument is valid, it is necessary to show that each step of the argument is sanctioned by rules of which we know that they preserve validity. Analysis allows us to reduce a complex argument into simpler parts of which we know that they preserve validity. It is minimal syllogisms for which Aristotle shows whether or not they are s-valid and it is the structure of the minimal syllogism for which he presents an argument concerning the conditions of its s-validity. Clearly, he needs some formal way of linking the simple parts (the s-validity of which he presents an argument for) with more complex arguments, the possible s-validity of which he accepts. It should be noted that analysability is not an alternative to a natural deduction system (or, for that matter, to an axiomatic system), but rather a different way to express the same underlying idea.

- 34.
For presentations of the order of the

*Organon*see, for instance, Simplicius,*in Cat*., 14,33–15.8 or David the Invincible,*Commentary on Aristotle’s Prior Analytics*, II,2–3. Due to the nature of the transmission of Stoic views, it is difficult to determine the latter’s position regarding our present question, but it has been observed (Sorabji 2005, p. 250) that the Stoics and the Aristotelians rejected each other’s examples of valid arguments. This, however, is not specific enough for our purposes. - 35.
We will see below how Alexander employs this thought in his commentary on

*Prior Analytics*I.32, an important chapter for the issue under consideration. - 36.
Cf.

*APo*I.14, 79a17–32. See Mendell (1998). - 37.
*Met*. \(\Delta \) V, 1015a20–b15. - 38.
Cf.

*APo*I.4, 73a21–b5. The fact that the necessity of the principles of demonstrations is not the necessity of logical consequence has nothing to do with the difference between syllogism and demonstration. It is, of course, possible to set up a syllogism that has as its conclusion a proposition that could serve as a principle in a demonstration. But the logical necessity this proposition receives from the fact that it is a conclusion of a valid argument is not the necessity that makes the proposition suitable for a demonstration. The kind of necessity that allows us to use a proposition as a principle in a demonstration cannot be, as Aristotle points out several times in the*Posterior Analytics*, that of a logical consequence. - 39.
I.32, 47a14–8. Aristotle in particular mentions the following cases: (ia) stating a universal premise but omitting to state the premise contained in it; (ib) stating a premise, but omitting the premises from which they are derived; (ii) asking for superfluous premises.

- 40.
Alexander understands

*to endees*not merely as inadequacy with respect to syllogisms, but as ‘error’ or ‘failure’ of arguments in general. Glossing 47a22 he says:*rhaion to phōrasai**tēn kata tous logous hamartian*(346,26). - 41.
Cf. Aristotle’s discussion of enthymemes in II.27, where Aristotle explicitly says that in enthymemes some premises are not stated (

*ou legousi*) while others are formally assumed (*lambanousin*), 70a19–20. - 42.
- 43.
It is remarkable, however, that

*sumbainei/ gignetai anagkaion*always seems to be used in sentences with negations signifying that the case under investigation does not allow for a logical consequence to be derived. Cf. 26a4–5; 26a7–8; 27a16–8; 27a25; 29a19–21. - 44.
68b15–37. Cf. also McKirahan (1983).

- 45.
The difference between syllogism and induction is explained in 68b30–37. This explanation does not create problems for the claim that inductions rely on syllogisms, but it is difficult to make sense of it if

*syllogismos*is understood as ‘argument’. - 46.
Aristotle contrasts

*sēmeion*(sign or indication) with*syllogismos*. A*syllogismos*can, of course, be an argument;*sēmeion*, on the other hand, can hardly signify ‘argument’. But if we understand the words to signify the outcome of an argument and*sēmeion*is ‘sign’ or ‘indication’,*syllogismos*should be ‘consequence’. This is close to Smith’s suggestion to understand*syllogismos*as ‘deduction’. - 47.
Given the nature of the A-view, which ascribes to Aristotle a notion of validity as an unanalysed primitive, and the Aristotelian texts we have, it is doubtful whether a conclusive argument against the A-view can be made at all. It is hard to argue against a view, when it is based in part on the idea that there is no textual evidence for what it claims.

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## Acknowledgments

The first version of this paper was written during a DAAD funded stay at King’s College London and has been read at the Philosophisches Kolloquium in Cologne. I am grateful for the helpful remarks I received from the participants of this colloquium, especially Nicholas White and Marius Thomann. Since then, various versions of the paper have been read by Peter Adamson, Matthew Duncombe, Luca Gili, Jan Heylen, Peter Larsen, Jan Opsomer and Marius Thomann. Any remaining mistakes are my own. I would also like to thank two anonymous referees for their helpful comments on the paper in general and in particular on the issue discussed in Sect. 4.

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Steinkrüger, P. Aristotle’s assertoric syllogistic and modern relevance logic.
*Synthese* **192, **1413–1444 (2015). https://doi.org/10.1007/s11229-014-0631-y

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### Keywords

- Aristotle
- Syllogism
- Relevance logic
- History of logic
- Logic
- Relevance