Malink’s book proposes a formal system which, if consistent, shows that Aristotle’s judgments about syllogisms form a consistent set. But because of the artificiality of the system, and its lack of fit with Aristotle’s reasoning and language, the book will not put aside worries that “Aristotle’s modal syllogistic is incoherent”.1
There is a helpful introduction, outlining the entire argument of the book, technical appendices and, in between, sections for each main branch of Aristotelian syllogistic (assertoric, apodeictic, and problematic), each section presenting a novel idea regarded as essential by Malink for rendering Aristotle’s judgments within that branch consistent.
The appendices are the substance of the book. There Malink first provides tables which list exhaustively Aristotle’s judgments of the validity, invalidity, or inconcludence of some 400 syllogisms or premise pairs. Next, Malink sets down an axiomatic formal system, the “predicable semantics”. Finally, he shows that, within that system, each syllogism judged valid by Aristotle may be vindicated with a proof; each judged invalid may be shown to be so with a countermodel; and each premise pair judged inconcludent may be shown to have a like status in the formal system. That is, Malink treats Aristotle’s 400 or so judgments like data points and his own formal system like a theory which successfully predicts all those data points. Thus Malink’s main conclusion: “the set of Aristotle’s claims of validity, invalidity, and inconcludence is consistent” (269). Malink is the first researcher within the program of studying Aristotelian syllogistic “from the standpoint of modern formal logic” to have achieved this.
Yet what does this result establish, as a matter of interpreting Aristotle? It would be open, of course, to a logician to give us a formal system which has this unusual property of coinciding with all of Aristotle’s judgments and leave it at that. But Malink’s stated goal is to understand and interpret Aristotle, and so his technical result must be evaluated as a contribution to interpretation. Here, as Malink concedes, everything depends on the naturalness versus artificiality, and the closeness of fit to Aristotle, of the formal system (270). To the extent that the system is contrived and the fit is poor, then the worry about the incoherence of the modal syllogistic remains. Indeed, if only a highly contrived system can render Aristotle’s judgments consistent, then the system actually serves as further evidence of Aristotle’s own lack of consistency.
The bulk of Malink’s book is devoted to motivating the formal system and discussing its merits as an interpretation, typically in relation to the “orthodox” interpretation of Aristotelian syllogistic as a branch of first-order logic with modal operators. Malink frankly acknowledges various “limitations” of his approach and tallies them up in the last chapter: the formal system cannot account for ten proofs in the apodeictic and problematic syllogistic; it actually requires that two such proofs be rejected as leading to inconsistent results; some of Aristotle’s counterexamples must be rejected as well; and in the problematic syllogistic it must rely upon constructions and definitions which are “too artificial to be attributed to Aristotle” (268-70).
These are simply the aspects of the formal system which uncontroversially diminish its appeal as an interpretation. In addition, the novel ideas behind the formal system can be regarded as limitations too, not strengths, if one finds them unconvincing—as, if only something implausible can render Aristotle consistent, then Aristotle is implausibly consistent.
The Problematic Syllogistic. Malink insists that Aristotle’s claims here can be rendered consistent only if we take Aristotle to be denying principles of modal opposition which are intuitively plausible, which he appears committed to in various arguments, and which he actually seems to affirm (204-5). Perhaps, but let us count the cost. To follow Malink, we must interpret Aristotle as in effect denying the principle of double negation—that, although “possibly not” in the relevant propositions means precisely “not of necessity”, nevertheless “not possibly not” does not mean “of necessity” (204-5). Again, we must believe that Aristotle invents ad hoc a new type of categorical: “when Aristotle appeals to contradictories of negative M-propositions [“possibly not” propositions] in indirect proofs, these contradictories are not among his sixteen kinds of categorical propositions. They are merely contradictories of negative M-propositions. They occur in indirect proofs of moods that have an M-conclusion, but not elsewhere in the modal syllogistic” (205). Finally, to deal with Aristotle’s plain language to the contrary, Malink would have us think that Aristotle is confused as between the object language and the metalanguage: “within the language used to talk about the modal syllogistic, Aristotle helps himself to principles of modal opposition that are not valid in the modal syllogistic itself” (205). So Malink’s treatment of the problematic syllogistic, depending upon one’s comfort with the degree of fit, will appear either ingenious or desperate.
The Assertoric Syllogistic. Here Malink’s novel idea is his interpretation of what it is to predicate one term of all of another (κατὰ παντός), traditionally called the dictum de omni, viz.: “We say
Moreover, this “heterodox” interpretation has no advantages over the orthodox and several additional disadvantages. If the orthodox account erred by construing categorical statements as invariably about individuals, this one errs by never allowing that an individual can serve as a counterexample, since individuals do not have pluralities associated with them, in the required sense, and, for Aristotle, they are not predicated of anything. Note that in “Man is not predicated of all of animal, but animal is predicated of all of man” (25a25-6), presumably what corroborates the first claim is the possibility of finding a species of animal “of which man will not be said”, and the second, the impossibility of finding an individual man “of which animal will not be said”. And Aristotle clearly presumes that counterexamples can be individuals in his explication of the dictum in An. Post. I.1.
Neither does it give a good account of existential import. Malink defines an I-categorical, “Some A is B”, as: “Some member of some plurality associated with some term is such that both A is predicated of all of it and B is predicated of all of it.” He additionally holds that “A is predicated of all of A” is true for any term whatsoever. He ascribes this principle to Aristotle and makes it an axiom in the formal system. Moreover, Malink thinks that each term is a member of the plurality associated with that term. So from “A is predicated of all of B” it follows that “Something is both A and B”, because B itself is a member of a plurality associated with a term; B is predicated of all of it; and, by the initial assumption, so is A. Yet, once again, there is no good fit to Aristotle’s reasoning and language: it is essential to Malink’s way of underwriting existential import that a term be predicated of all of itself; but that claim simply does not appear in Aristotle’s own explication (25a17-19).
Furthermore, Malink’s account forces us to adopt bizarre views about the basis of existence claims. If I claim, for instance, “All the men are awake,” and there are five of them, on Malink’s account it is true that “Some man is awake”, not because of the five, but because man is man, and that man (the term) is awake. Again, because for Malink it is an axiom that a term is predicated of all of itself, then, for all terms, it always is the case that something is that term, and therefore “Nothing is A” is false always for all terms. Many absurdities follow, such as that no E-categorical is true when there happens to be a name for the concurrence of the terms.
The Apodeictic Syllogistic. Malink’s innovation here is meant to explain why Aristotle claims that the following syllogism is valid (“Barbara NXN”):2
1. A belongs of necessity to all B.
2. B belongs to all C.
3. Therefore, A belongs of necessity to all C.
—whereas Aristotle claims in the same passage that, if the modality of the premises is reversed (“Barbara XNN”), the syllogism becomes invalid. Aristotle’s position seems to offend against the idea that an argument’s conclusion can be no stronger than the weakest link, besides apparently leading to difficulties elsewhere.
Malink’s generally novel idea here is to draw from Aristotle’s theory of categories in the Topics.3 Malink says that, on that theory, some terms— “essence” terms—are never truly predicated universally of anything, except when they are predicated in view of something that holds of necessity. Basically, essence terms are those that would be used in a proper scientific taxonomy which reveals the essences and necessary connections of things. For example, “animal” would not be predicated of all of anything except of some species of animal. Thus “Animal belongs to all man” is not true except when “Animal belongs of necessity to all man” is also true. Thus, if it could be guaranteed that only an essence term could be understood for the term, B, in the first syllogism above, then the second premise would imply a hidden premise, which would be true of necessity: Barbara NXN would imply Barbara NNN, and the puzzle about Aristotle’s claims would be removed. However, Malink claims, only of an essence term could it be true that some term belong to all of it of necessity. Thus, the nature of the first premise in Barbara NXN guarantees that, if that premise is true, then B is an essence term. To capture this interpretation in his formal system, Malink makes it an axiom that, in his words: “If one term is predicated of necessity of all of a second, and that second is predicated of all of a third, then the first is predicated, too, of necessity of all of the third.” Serious difficulties strain the plausibility of this interpretation:
(1) It makes the validity of some syllogisms depend on the “matter” to which they are applied (essences arranged hierarchically), rather than their logical form simply. But Aristotle seems to say that Barbara NXN is true in virtue of its logical form (30a19-20). That there is an axiom in the formal system which builds in this restriction does not deflect the difficulty, since the axiom would not be a logical axiom. Besides, the axiom would be useless for types of necessity other than formal necessity, such as necessity of end, or necessity of compulsion (see Met. V.5) where, indeed, Barbara NXN seems to have an even more intuitive application. To see this, take as terms “compelled to enlist”-“able-bodied man”-“football player” for necessity of compulsion, and “take a boat”-“traveler to Corfu”-“foreign tourist” for necessity of end.
(2) It seems against Aristotle’s intentions to render Barbara NXN valid by reducing it to Barbara NNN. Aristotle does after all state as his view that “a syllogism can reach a necessary conclusion with only one necessary premise” (32a8). Again, presumably, as in other cases, the very same counterexample (moving-animal-man), which Aristotle gives to show the invalidity of Barbara XNN, should be applicable to Barbara NXN, where it should fail to show invalidity; but on Malink’s interpretation the counterexample is not even allowable, as the second premise, “man belongs to all moving”, would not involve an essential predication. Again, when Aristotle wishes to explain what an X-premise means in an apodeictic syllogism, he says that the premise asserts that something holds, but not that it holds of necessity (cf. 31b23). Again, Aristotle’s argument against the validity of the XNN form is that, if the conclusion were necessary, the first premise too would in fact be necessary, which, he says, “is false”: by parity, it would not simply be something unstated, but actually “false”, for the second premise in NXN to be or to amount to a necessary premise. True, it is consistent with all of these considerations that Aristotle believed that there was a kind of shadow N-premise, in addition to the explicit X-premise —but not plausible. Why does he not mention the shadow premise? At least, one would have expected Aristotle to draw attention to potentially misleading cases of non-apparent Barbara XXN validity, that is, where the first premise uses “essence” terms but leaves out any explicit statement of necessity.
(3) The gap between Malink’s formal system and Aristotle’s reasoning is especially yawning for the syllogistic form Darii NXN. In that syllogism, the second premise is particular (“B belongs to some C”); ostensibly, then, in that premise the term B is never applied to all of another term; ostensibly, then, Malink’s axiom cannot come into play, and “B belongs to some C” will not imply “B of necessity belongs to some C”, as would be required. Aristotle explains the syllogism very simply: that, as A holds of necessity of all B, and C, as it happens, is B, then A holds of necessity of C. But on Malink’s account, for the syllogism to be valid one must bring in the premise that “B belongs to all of B”, and one must bring in the definition of an I-categorical which was explained above, which construes it as containing implicitly two universal claims, and one must also introduce a certain odd construction of a necessary I-categorical, as “Either B belongs of necessity to some C, or C belongs of necessity to some B”. So why does Aristotle not appeal to a single one of these putatively essential logical moves? How would his perfect syllogisms be perfect, on Malink’s reconstructions? Note we are simply appealing to the same standard of a sound interpretation that Malink appeals to when criticizing the “orthodox” interpretation (111).
Malink’s book has perhaps put to rest “a notorious unsolved riddle in the history of philosophy”, as the blurb on the back claims, but inadvertently, through compelling a solution opposite to that intended.
Notes
1. See for example G. Striker, “Modal vs. Assertoric Syllogistic”, Ancient Philosophy 14 (1994), 39-51 at 39.
2. Where “N” indicates a premise as necessary, and “X” as simply being true, without the modality of necessity.
3. But see R. Patterson, Aristotle’s Modal Logic: Essence and Entailment in the Organon (Cambridge: Cambridge University Press, 1995).