After developing three figures of the categorical syllogism, Aristotle bombastically claims, in Prior Analytics A23:
It is clear from what has been said that the deductions in these figures are made perfect by means of the universal deductions in the first figure and are reduced to them. That every deduction without qualification can be so treated, will be clear presently, when it has been proved that every deduction is formed through one or other of these figures (40b17-22, emphasis mine).
Contrary to this, Augustus De Morgan argues that there are deductions that cannot be reduced to a syllogism.
There is another process which is often necessary, in the formation of the premises of a syllogism, involving transformation which is neither done by syllogism, nor immediately reducible to it. It is the substitution, in a compound phrase, of the name of the genus for that of the species, which the use of the name is particular (FL, p. 114).
The most notorious example is, the horse head argument (HHA): ‘horse is animal, therefore the head of a horse is the head of an animal’. I should clarify that De Morgan uses ‘man’ rather than ‘horse’ in his example, but otherwise, the argument is the same. Now, our predicate logic is quite powerful and can handle compound substitutions ably. Here is an indirect proof that seems to comport to the demands of HHA:
Hx – x is a horse
Ax – x is an animal
Cxy – x is the head of y
1. (∀x)(Hx ⊃ Ax) (premise)
2. ~(∀y)(∀x)[(Hx & Cyx) ⊃ (Ax & Cyx)] (IP)
3. (∃y)~(∀x)[(Hx & Cyx) ⊃ (Ax & Cyx)](2 QN)
4. (∃y)(∃x)~[(Hx & Cyx) ⊃ (Ax & Cyx)] (3 QN)
5. (∃x)~[(Hx & Cux) ⊃ (Ax & Cux)] (4 EI)
6. ~[(Hv & Cuv) ⊃ (Av & Cuv)] (5 EI)
7. ~[~(Hv & Cuv) ∨ (Av & Cuv)] (6 Impl)
8. ~~(Hv & Cuv) & ~(Av & Cuv) (7 DeM)
9. (Hv & Cuv) & ~(Av & Cuv) (8 DN)
10. Hv & Cuv (9 Simp)
11. Hv (10 Simp)
12. Hv ⊃ Av (1 UI)
13. Av (11, 12 MP)
14. ~(Av & Cuv) (9 Simp)
15. ~Av ∨ ~Cuv
16. ~~Av (13 DN)
17. ~Cuv (15, 16 DS)
18. Cuv (10 Simp)
19. Cuv & ~Cuv 17, 18 Conj)
20. (∀y)(∀x)[(Hx & Cyx) ⊃ (Ax & Cyx)] (2-19 IP)
So, the inference seems to be valid, given the rules of first order predicate calculus. Is it really the case, though, that a parallel proof cannot be rendered in a Categorical syllogism? A categorical syllogism has three terms, and two premises, yet the above argument has one premise, which leads directly to the conclusion. So we need to identify the terms that would operate in a syllogistic version of HHA. And we need to allow that the reduction will contain two premises.
We must be cautious in how we articulate this syllogism, as Aristotle warns:
It is not the same, either in fact or in speech, for A to belong to all of that to which B belongs, and for A to belong to all of that to all of which B belongs; for nothing prevents B from belonging to C, though not to every C: e.g. let B stand for beautiful, and C for white. If beauty belongs to something white, it is true to say that beauty belongs to that which is white; but not perhaps to everything that is white. If then A belongs to B, but not to everything of which B is predicated, then whether B belongs to every C or merely belongs to C, it is not necessary that A should belong, I do not say to every C, but even to C at all. But if A belongs to everything of which B is truly said, it will follow that A can be said of all of that of all of which B is said. If however A is said of that of all of which B may be said, nothing prevents B belonging to C, and yet A not belonging to every C or to any C at all. If then we take three terms it is clear that the expression ‘A is said of all of which B is said’ means this, ‘A is said of all the things of which B is said’. And if B is said of all of a third term, so also is A; but if B is not said of all of the third term, there is no necessity that A should be said of all of it (APr 49b14-31).
So we don’t want to say that because all horses are animals, everything that a horse has, like a head, is something that every animal has. Some animals, after all, could be headless! And what we really mean to say is that, since ‘animal’ is the genus of ‘horse’, and since a horse has a head, an animal has a head. Perhaps, then, we should formulate the argument as follows:
21. All horses are animals.
22. All horses are those that have heads.
∴23. Some of those that have heads are animals.
By making the conclusion particular, we do not run the risk of affirming that all animals have heads to the consternation of amoebas and sponges. The preceding argument is Darapti, and it is a valid syllogism, barring any objections on the grounds of existential import.1 However, it is not quite what HHA demands. Recall that we need to conclude that ‘the head of a horse is the head of an animal’, since ‘horse is animal’. To approximate the conclusion more closely, we might use repetition. Aristotle mentions the use of repetition in the syllogism, but stipulates how it is to be used. We find in Prior Analytics A38:
A term which is repeated in the propositions ought to be joined to the first extreme, not to the middle. I mean for example that if a deduction should be made proving that there is knowledge of justice, that it is good, the expression ‘that it is good’ (or ‘qua good’) should be joined to the first term. Let A stand for knowledge that it is good, B for good, C for justice. It is true to predicate A of B. For of the good there is knowledge that it is good. Also it is true to predicate B of C. For justice is identical with a good. In this way an analysis of the argument can be made (APr 49a11-18).
So, Aristotle sets down that A is ‘knowledge that it is good’, B is ‘good’, and C is ‘justice’. Formally, the proof would be:
24. AaB (Knowledge, that it is good, belongs to all good.)
25. BaC (Good belongs to all justice.)
∴ 26. AaC (Knowledge, that it is good, belongs to all justice.)
Let us set down that D is ‘animal, qua horse’, E is ‘horse’, and F is ‘head’. Still making use of Darapti, the argument would then be:
27. DaE (Animal, qua horse, belongs to all horse.)
28. FaE (Head belongs to all horse.)
∴29. DiF (Animal, qua horse, belongs to some head.)
Or, in a more readable English prose:
30. Every horse is an animal in virtue of being a horse.
31. Every horse is that which has a head.
∴32. Some of those which have heads are animals, in virtue of being horses.
Now one might protest that the conclusion reached here is particular, whereas in predicate calculus one reaches a universal conclusion. But what does that universal conclusion really say? It says that, for all things x and y, if x is a horse and y is the head of x, then x is an animal and y is the head of x. In effect, it is not saying that the head of a horse is an animal head, but that if something is a horse and it happens, also, to have a head, then it is an animal that happens to have a head. Is this the same as HHA? There is no real sense in which the deduction formed by predicate logic has anything to do with the relationship between genus and species, as De Morgan indicates. But the categorical syllogism that we have formed does have this information, in that a horse belongs to its genus in virtue of belonging to its species.
I grant that the conclusion of the categorical syllogism is syntactically divergent from HHA. Nonetheless, I think it captures a similar, if not the same, sense. Perhaps this is the best we can do.
1For those who are particularly bothered by the “existential fallacy”, we could run a similar argument on Datisi.
Aristotle. 1995. “Prior Analytics.” In The Complete Works of Aristotle: The Revised Oxford Translation. Trans. A.J. Jenkinson. Ed. J. Barnes. Vol. I. Princeton, New Jersey: Princeton University Press.
De Morgan, A. 1847. Formal Logic. London: Taylor and Walton Booksellers and Publishers.