[Authors and chapter titles are listed at the end of the review.]
The title The Cambridge Companion to Ancient Logic might lead one to expect both more and less than this volume delivers. On the one hand, one might expect contributions on the Nyāya theory of inference in classical Indian philosophy, or a discussion of the theory of analogical reasoning developed by the Mohists in ancient China. But the contributions in this book instead restrict themselves to the European tradition (15-16). Fair enough: to cover all logical traditions of antiquity with the detail and rigor that this volume achieves would probably render the book too unwieldy for the companion genre. But adding the word ‘Greek’ to the title would have been a simple way to manage expectations.
On the other hand, this Companion covers a range of topics significantly broader than today’s logical curriculum. These diverse subjects are unified by the fact that they are all in some way related to Plato’s ‘logical agenda,’ a range of broadly methodological topics of central concern in Plato’s dialogues: the demarcation of philosophical reasoning from sophistry, eristic and rhetoric; the relationship between dialectical arguments and geometrical reasoning; and the requirements for, and method of discovering, real definitions. An important throughline is the idea that reflection on and reactions against Plato’s inchoate observations on a range of such broadly ‘logical’ topics helped to prompt the mature contributions to these subjects by later thinkers.
The first part of the book, which covers the development of Greek logic, begins with a fascinating overview by Nicholas Denyer of rudimentary logical observations in a variety of ancient authors: cases in which the author does not just reason, but reflects on the nature of reasoning. Paolo Fait then details Aristotle’s discovery of logic against the background of his gradual departure from a Platonic understanding of dialectic, and Karlheinz Hülser accounts for certain key innovations in Stoic logic as interventions in a debate around Diodorus Cronus’ Master Argument, according to which nothing is possible that neither is nor will be true. Finally, Ben Morison offers a lucid account of the logical projects that led to modest innovations (and to certain missteps) in late antiquity, a relatively understudied period in the history of Greek logic. These chapters do a real service to a topic in ancient Greek philosophy that has been described by one of its chief enthusiasts as ‘repulsively technical’ and ‘dry as dust’.[1] These authors bring to light some of the genuinely gripping philosophical concerns that likely animated these thinkers’ allegedly arid logical discoveries.
Despite the role that Plato’s logical interests play in organizing its contents, and despite the near-ubiquitous presence of Platonic concerns throughout its chapters, the Companion does not contain a historical chapter dedicated specifically to Plato. The editors, Luca Castagnoli and Paolo Fait, instead provide a helpful but brief treatment of Plato’s logical agenda in the introduction, in which they memorably summarize Plato’s role by writing, ‘if ancient logic was the promised land, Plato was its Moses. He never set foot in it, but enabled others to see the destination’ (1). The authors are not explicit about what Plato lacks, but they probably have in mind what Denyer describes as an ‘abstract, systematic and general’ (24) theory about how to reason well. If we restrict ourselves to ‘logic’ in the narrow sense of the study of valid reasoning, this is hard to dispute: Plato often reasons validly, but he offers very little in the way of abstract, general reflections on the nature of valid inference. But according to the Companion’s generous conception of ‘logic,’ this characterization seems to underestimate Plato’s achievements. For instance, Plato offers general, systematic reflections on the requirements for real definition (128-34) and on what is involved in simple sentences being true or false (150-1). Moreover, several contributions in this volume draw attention to Plato’s striking claim at Sophist 230b that genuine refutations involve showing that an interlocutor’s beliefs conflict with each other ‘at the same time, on the same subjects, in relation to the same things and in the same respects’ (4; 33; 240-1). These qualifications seem to specify general, topic-neutral requirements on the nature of the contradictions produced through refutations. We in fact sometimes see characters in Plato’s dialogues appealing to these very qualifications when confronted with refutations that result in merely apparent contradictions (see esp. Euthydemus 293b-298b).[2] On such broadly ‘logical’ topics as definition, truth and contradiction, Plato is arguably more of a Joshua than a Moses.
In addition to central logical notions like syllogism, truth, fallacy and modal logic, the second major section of the Companion, which is dedicated to ‘key themes,’ contains contributions on real definition in Plato and Aristotle, on their respective accounts of terms and propositions, on demonstration in Aristotle, the Stoics and Epicureans, and on the relationships that ancient Greek logic bears to rhetoric and mathematics. These chapters will prove very useful both for advanced students and for professional philosophers unfamiliar with, or looking for a refresher in, these key topics. Here authors often adopt interpretations that they, or others, have defended more fully in previous publications, but experts are also likely to learn and benefit from these synoptic treatments of central themes across multiple thinkers. For instance, even those steeped in the details of Aristotle’s theory of demonstration will likely read with interest Alexander Bown’s chapter showing that both Stoics and Epicureans share with Aristotle the basic idea that demonstration is an inference from the more to the less familiar, although they operate with radically different conceptions of familiarity. Experts will also find in these chapters several novel proposals with which to engage.
One such proposal concerns the cornerstone of Aristotle’s syllogistic theory in the Prior Analytics. Aristotle proves the validity of the so called ‘perfect’ syllogisms by appealing to a principle known as the dictum de omni et nullo (APr. 1.1, 24b28-30):
we speak of ‘being predicated of all’ whenever none of those of the subject can be taken of which the other will not be said, and likewise with ‘[being predicated] of none’.
In their chapter on validity, Castagnoli and Fait argue that this is a version of the Principle of Non-Contradiction (PNC) on grounds that this principle ‘is formulated as the negation of the possibility of a conjunction’ (176). There are, however, several obstacles to this suggestion. The authors apparently understand the dictum as the claim that
it is impossible that (A belongs to all B & A does not belong to some B)
which one might take to be an instance of PNC.[3] However, the idea that modal language in the dictum takes scope over two conjuncts is not the only or the most obvious way of understanding Aristotle’s statement. Because Aristotle appeals to the dictum as warranting the validity of perfect syllogisms (APr. 1.4, 25b39-40; 25b40-26a2; 26a24, 27), it is attractive to think of it as an inference rule whose modal language marks the necessity of the consequence: if A belongs to all B, then for any C that one might take from the Bs, it is not possible that A does not belong to C.[4] Moreover, Posterior Analytics 1.11—the only passage in Aristotle that discusses the role of PNC in proofs—seems to present a greater obstacle to their reading than Castagnoli and Fait suggest (176-7). For here Aristotle contends that PNC is only assumed in the case of arguments whose conclusion is a complex proposition with the form
(3) Animal belongs to Callias and it is not the case that animal does not belong to Callias.
According to Aristotle, such a conclusion ‘is proved by assuming that [to assert] the first [term] of the middle [term] is true and that to deny [this] is not true’ (APo. 1.11,77a10-13), as for example in the major premise
(1) Animal belongs to all human and it is not the case that animal does not belong to all human.
This, together with the minor premise that (2) human belongs to Callias, allows one to infer (3). Aristotle claims that the argument (1, 2 ⊢ 3) assumes PNC in assuming 1. Although every demonstration assumes PNC in some remote and implicit sense—for it is the principle of all other principles (Metaph. Γ.3, 1005b33-4)—unusual arguments such as this are supposed to be the only ones that rely on PNC in a direct way.[5] But importantly, Aristotle’s proofs of the validity of perfect syllogisms, however exactly they should be reconstructed, do not seem to involve unusual conclusions like 3 and are thus unlikely to be exempt from Aristotle’s claim that PNC is only assumed as a premise in proofs with conclusions in this non-standard form.
The Companion’s final three chapters, on the legacy of ancient logic, cover the reception of Aristotle’s logic in medieval Latin and Arabic philosophy, the influence of both Aristotelian and Stoic logic on figures from the renaissance to Frege and, finally, the relevance of these ancient logics today. These chapters record and explain the gradually waning influence of Aristotelian and Stoic logic on the working logician. But, although these final chapters suggest that many of the ancient contributions to this discipline have become obsolete from the systematic perspective, this book as whole is a testament to the fact that scholarship on ancient Greek logic is a live and accomplished branch of the history of philosophy. If, as the late John Woods suggests in the conclusion of his paper ‘Ancient Logic Today,’ the great works of Greek logic are like the edifices of classical antiquity, whose ‘beauty and importance’ is not diminished by the fact that they are ‘no longer in service’ (359), then the Companion will serve as an apt guide to these splendid ruins.
Authors and Titles
Introduction (Luca Castagnoli and Paolo Fait)
I. The Development of Logic in Antiquity
1. The Prehistory of Logic (Nicholas Denyer)
2. Aristotle and Theophrastus (Paolo Fait)
3. Megarians and Stoics (Karlheinz Hülser)
4. Late Antiquity (Benjamin Morison)
II. Key Topics
5. Truth as a Logical Property & Laws of Being True (Walter Cavini)
6. Definition (Michael Ferejohn)
7. Terms and Propositions (Paolo Crivelli)
8. Validity and Syllogism (Luca Castagnoli and Paolo Fait)
9. Demonstration (Alexander Bown)
10. Modalities and Modal Logic (Marko Malink)
11. Fallacies and Paradoxes (Luca Castagnoli)
12. Logic in Ancient Rhetoric (Christof Rapp)
13. Ancient Logic and Ancient Mathematics (Reviel Netz)
III. The Legacy of Ancient Logic.
14. Ancient Logic in the Middle Ages (John Marenbon)
15. Ancient Logic from the Renaissance to the Birth of Mathematical Logic (Mirella Capozzi and Leila Haaparanta)
16. Ancient Logic Today (John Woods)
Notes
[1] J. Barnes (2012), ‘Galen, Christians, Logic’ in J. Barnes, Logical Matters: Essays in Ancient Philosophy II, Oxford: Oxford University Press, pp. 1-21 at 3.
[2] See I. J. Campbell, ‘Plato, the Eristics and the Principle of Non-Contradiction,’ Apeiron 54 (2021), 571-614.
[3] For the view that the formulation of PNC does not include modal language as part of its content see I. J. Campbell and G. Shapiro, ‘Can You Deny the PNC? (Metaphysics Γ. 3, 1005b11–34)’ Oxford Studies in Ancient Philosophy 63 (2022): 89-133 at 110-14.
[4] For this view see B. Morison, ‘What is a Perfect Syllogism?’ Oxford Studies in Ancient Philosophy 48 (2015): 107-66 at 135-43.
[5] The precise way in which this argument assumes PNC in assuming 1—which is not an instance of the principle—is not immediately clear. For discussion see R. D. McKirahan, Principles and Proofs: Aristotle’s Theory of Demonstrative Science, Princeton University Press, 1992), 157-8 and Campbell and Shapiro 2022, 112-14 (cited in n. 3 above).