Attention to variant readings and to modern developments in logic, especially modal and temporal logic, have helped scholars to recapture classical authors’ views of necessity, possibility, time, and the like. Both are on display in this collection.
Part I, “Inference and syllogism,” contains three papers: “Syllogism and Deduction in Aristotle’s Logic,” which argues that Aristotle’s definition of syllogism includes only those deductions traditionally called syllogisms; “Expository Proofs in Aristotle’s Syllogistic,” on reduction by ἔκθεσις; and “The Stoic themata,” in which Mignucci claims to find only two of the four mentioned by Galen for use in reducing imperfect syllogisms to indemonstrables.
Part II, “Identity, predication, and quantification,” offers four papers. In “Remarks on Aristotle’s theory of predication,” the author addresses the “often-mentioned problem of the logical square and the absence of singular inferences in the theory of syllogisms” among the “many difficulties that the modern rendering of Aristotelian propositions entails” (65). His alternative is to treat predication as involving extensions which are parts of other extensions (72), from which standpoint universal propositions and singular propositions may be represented in the same way (74-75). Conceding that his interpretation “goes far beyond the texts,” he claims that it explains “some features of Aristotle’s logical theory, which would otherwise be considered either wrong or nonsensical (77).”
In “Puzzles about Identity: Aristotle and his Greek commentators,” Mignucci focuses on Leibniz’s Law (if two things are the same, they both share all the same properties, 79), which Aristotle considers when he understands ταὐτόν as numerical identity—Aristotle’s example being cloak and mantle. The author addresses this concept in terms of the inclusion relation and the membership relation, suggesting that ‘is predicated of’ in Aristotle can mean either relation (83-84) according to what is substituted for the variables (85). Mignucci interprets Aristotle’s analysis of fallacies of accident in the Sophistici Elenchi as showing that Aristotle did not admit exceptions to Leibniz’s Law (96). Alexander, Simplicius, and Michael of Ephesus, according to the author, offered different interpretations of Aristotle on identity.
“Aristotle’s Topics and Contingent Identity” is based largely on Topics 1.7, which distinguishes three senses of ταὐτόν and explains what is meant by one in number. After a discussion of definite descriptions (expressions which indicate the one item which has a certain character—“the unique x which F-s,” 116), Mignucci concludes that “identity in Aristotle’s view is always strong in the sense that it is necessary, a notion of contingent identity being outside his conceptual framework” (124).
In “Aristotle on universals and particulars,” the author considers the opening lines of De Interpretatione 7, which eventually gave rise to the theory of universals. His conclusion (at 152) is:
What is an object of our discourse, and in particular a subject of predication, does not necessarily exist. More generally, we are not compelled to claim that whatever falls in the universe of discourse is something that exists. However, whatever is in the universe of discourse is so structured that it can be qualified as universal or singular.
He finds that Aristotle’s ideas “find a more adequate systematization in the present interpretation than in reference to the standard semantics of modern logic” (153).
Part III, “Modality, time, and future contingents,” begins with “Aristotle’s conception of the modal operators,” which addresses the necessary and the possible as interdefinable (159). Though recognizing that Aristotle sometimes has δυνατόν or ἐνδεχόμενον where one might expect the other (160), Mignucci uses the diamond for Aristotle’s one-sided possibility (δυνατόν, not impossible) and epsilon for his two-sided possibility (ἐνδεχόμενον, neither necessary nor impossible). He goes on to explore, as Aristotle’s definition of one-sided possibility, that a proposition is possible this way if and only if it does not imply an absurdity (163). The author argues that “Aristotelian necessary propositions cannot be confined to theorems of logic” (169) and that the definitions of the modal operators are limited to “standard Aristotelian propositions” (171).
In “Logic and omniscience: Alexander of Aphrodisias and Proclus,” the author considers indefinitely tensed statements, such as ‘Socrates is sleeping’ (175). These statements can change their truth value over time because they contain only an implicit temporal reference and are contingent statements (176). That is to say, “one and the same statement can change its truth value” and “true and false are accidental properties of indefinitely tensed statements” (178)—thesis (T). The connection with omniscience poses a theological challenge: “if the gods know the world, they must change, because they must modify their knowledge in order that it remain true. The immutability of the gods, their knowledge of changing individuals, and thesis (T) are inconsistent” (179). For Alexander, apparently, “thesis (T) has an essential function in limiting the domain of divine knowledge,” leaving the gods not omniscient but immutable (184). According to the author, Proclus takes a different tack: “He does not give up the immutability and omniscience of the gods, but rather he prefers to modify thesis (T)” (187). For him, “the meaning of the proposition is not given by a single truth value, namely the truth value that the proposition takes at the time of its utterance, but is determined by the sequence of the truth values that the proposition takes in time,” so that the proposition “becomes a function that associates truth values to instants” in “ordered pairs” (188). A divine mind outside time would be “able to know every pair . . . altogether” (189). Proclus’s adoption of “a new semantics for indefinitely tensed statements” helps to explain “the extraordinary success that this view had in Christian culture” and shows that the “thesis that logic, because of its formalism, has nothing to do with ontology is false not only from a conceptual but also from a historical point of view” (191).
In “Ammonius on future contingent propositions,” the author argues that, for Ammonius, the principle of bivalence (a proposition is either true or false) holds unconditionally, so that both definitely true and indefinitely true propositions, especially future contingents, are true (197-198). Alluding to Aristotle’s example of the sea battle, Mignucci observes that, “before the decision [to engage is taken], the future of the battle is still open, and in this sense it is contingent that the battle will take place” (199). And again: “Contingent propositions about the future are indefinitely true or false not because the future is hidden or unknown to our mind, but because the ontological status of the facts they refer to is not yet settled” (201) or “not yet causally determined” (202). In Mignucci’s analysis, the predicates indefinitely true and indefinitely false are three-place predicates which link the time at which a proposition is used, the state of the world, and the proposition itself, for example, “A [a proposition] is indefinitely true at [time] t [and] with respect to [the situation of the world] Si”(202). Thus Ammonius can maintain that some future events are not necessitated but contingent, that provident gods know future contingent events, and that predictions of such events are possible (208).
In “Truth and modality in late antiquity: Boethius and future contingents,” Mignucci argues that Boethius also professes that the principle of bivalence holds for future contingents, which are indefinitely true or false (222). Drawing mainly on Boethius’s commentary on Aristotle’s De Interpretatione, Mignucci confirms his view with a passage from Consolatio 5, where “Boethius is considering the way in which god foreknows future contingent events” (240); the Consolatio allows for true predictions of contingent events, divine foreknowledge, and providence (241).
Part IV, “Paradoxes,” contains “The Stoic analysis of the Sorites,” according to which, for some Stoics at least, the paradox is resolved by recognizing that soritical predicates “admit of degrees of truth” (260) so that we must “give up some of our commonsense beliefs” (261); and “The Liar paradox and the Stoics,” which presents Cicero’s Chrysippus arguing that ‘I am speaking falsely’ is neither true nor false although it is a proposition and although the Stoics believed that every proposition is either true or false (273).
Part V, “Relatives,” contains three papers. In “Relatives in Plato,” Mignucci sets out to show that in Plato we find “many of the ideas that will not only constitute the starting point of the subsequent debates on the nature of relations in Greek philosophy but also the main ideas that will be further elaborated in the medieval tradition and in at least part of the modern one” (279). Using examples initially from the Phaedo, the author argues that “immanent properties like the one to which the expression ‘the largeness of Simmias’ refers must be treated as relational properties” (288). He will infer from this example that the author of the Phaedo “is operating not only with [a] clear and firm distinction between relative and absolute properties but also with a clear and firm distinction between absolute and relative predicates” (290). The difference from the modern logic of relations is that in, say, ‘Simmias is larger than Socrates’, for modern logicians both Simmias and Socrates function as logical subjects of the relational predicate ‘being larger than’ while, in the Phaedo, only Simmias is a logical subject in this sentence (296).
In “Aristotle’s definitions of relatives in Categories 7,” the author considers the definitions at Categories 6a36-37 and 8a31-33, interpreting the latter by lines 35-37, which state that “if someone knows any relative definitely he will also know definitely that in relation to which it is spoken of” (305). Mignucci takes knowing here as “minimal knowledge” (306) and “definitely” as “simply reinforcing” (314) this sense of knowing—that is, knowing that the item to which something is spoken of as related exists and no more, as, in the case of double, one might know that there is a number of which a given number is double without having calculated exactly what that other number is, as with very large numbers (305). With respect to substitutivity in cognitive contexts (whether, for example, ‘n knows that Tully denounced Catiline’ follows from ‘n knows that Cicero denounced Catiline’, 308), which modern logicians are accustomed to exclude, Aristotle, while being aware of the restrictions on substitutivity, “is more interested in isolating cases in which substitutivity can be safely applied” (319).
“The Stoic Notion of Relatives” begins with SVF 2.403, the report of Simplicius in his commentary on Aristotle’s Categories that the Stoics enumerate “two kinds of relatives rather than one”—τὰ πρός τι (the sweet and the bitter) and τὰ πρός τι πως ἔχοντα (on the right of, father of)” (324). He argues further that, as the argument proceeds, τὰ πρός τι πως ἔχοντα “no longer form a proper subclass of the class of τὰ πρός τι” (332). The reason is that τὰ πρός τι πως ἔχοντα do not change their internal state when one of the relata is changed—the so-called Cambridge change, as when a, which once was to the right of b, is not there anymore; the relation has changed, but a has not changed its internal state (335). After reviewing the fragmentary material in additional authors, Mignucci concludes that we do not have any evidence “that warrants attributing Simplicius’s distinction to Chrysippus or any other of the older Stoics” but that “the doctrine was held by some Stoics earlier than, or contemporary with, Boethius of Sidon (second half of the first century BC)” (368).
As the reader will see from this brief account, Mignucci challenges some traditional interpretations of classical thought by exploring them until a position is found which is more fully justified than its predecessors. Having both the evidence and the arguments presented makes it possible to evaluate Mignucci’s suggestions as he goes along. While the book seems to be designed for fairly advanced students of ancient thought and modern logic, less advanced students will learn a great deal from investing their effort in it. Although the conscientious reader should be able to cope with the typographical mishaps, it is to be hoped that they will be corrected in a second printing.