The “Horse’s Head” argument

Aristotle claims that all necessary arguments are syllogisms, or reduce to them. One objection to this- which has bothered me for years- was the “horse’s head” objection, namely:

A horse is an animal.

therefore the head of a horse is the head of an animal

From a single premise, one draws an absolutely necessary inference from relations that belong to both terms in the premise.

I stumbled on the response to the objection yesterday in an introduction written by Richard Berquist to the Commentary on the Posterior Analytics. His claim, which I find convincing, is that though the horse’s head argument is a necessary inference, it’s not a proof or even an argument, for it provides no evidence or reason for the conclusion. If someone were in doubt about whether the head of a horse were the head of an animal, then to tell them that a horse was an animal would neither convince them of anything, nor provide any evidence. This becomes more clear if we appeal to a disputed case: someone who is in doubt about whether the stem cells of a blastocyst are the stem cells of a person, for example, receives no evidence or even argument from one who says “every blastocyst is a person”.   


  1. Brandon said,

    February 24, 2008 at 6:50 am

    I’m reading Berquist’s edition of Thomas’s commentary, too.

    The way I think of it is this: if you try to articulate why the horse’s head argument works, you find that the most natural way to do it is to frame it in a series of hypothetical statements having to do with species and genus. But the proper place for such statements in Aristotelian logic is not Post. An., i.e., demonstrations, but Topics, i.e., dialectical arguments. And such arguments can be necessary arguments in some sense, but not in the right sense. The general principles or rules (the ‘maximal propositions’, as Boethius calls them) admit of demonstration; so, for instance, the general and necessary principle governing parts of genus and species can be demonstrated, and any necessity the Horse’s Head argument gets it gets from that. But the argument itself isn’t necessary; we keep forgetting that we’re making all sorts of assumptions in navigating this reasoning. (If a horse’s head is detached from the horse it is still in some sense a horse’s head, but is it still in the relevant sense an animal’s head? This would, of course, depend on what you’re using the argument to do; the argument is plausible not only because of necessary principles about the relation between genus and species but also because it is very vague about the relation between horses and their heads, and animals and their heads.) This is all related, I think, to the point made by Berquist that you note above, and his discussion of modus ponens and modus tollens in the previous section of the Introduction.

  2. a thomist said,

    February 25, 2008 at 3:34 am

    That’s right. I don’t think one can fully dislodge the Horse’s head argument from the previous two sections of the introduction and preseve what Berq. is saying about the sufficiency and necessity of the syllogism for necessary “why” proofs- i.e. demonstrations.

  3. Brandon said,

    February 25, 2008 at 10:14 am

    I just thought of an argument for the point about the assumptions about genus and species and their parts that we have to make in order to treat the argument as necessary. Tell me what you think. Suppose we give the argument,

    A horse is an animal, therefore a horse’s head is an animal’s head.

    And suppose someone denies it because he has a weird view of horses: that is, he holds that horses have two distinct natures, and a head for each. Thus the horse has an animal’s head and an XYZ’s head; by a sort of figure of speech one can hold that the latter is an ‘animal’s head’, i.e., it’s the head of something that is also (but distinctly) an animal, although the head itself is not an animal’s head; but that’s it. The only response to this is in fact to say something, in terms of genus and species, about the nature of horses; there is no purely formal, purely syntactical foundation for rejecting the argument.

    This would seem trivial except for arguments like the following:

    A human being is a finite creature;
    therefore a human being’s cognition is a finite creature’s cognition;
    Christ is a human being;
    therefore Christ’s cognition is a finite creature’s cognition.

    The conclusion carries a number of ambiguities that can only be kept track of if we keep clear about natures.

    The point, of course, is not that the horse’s head argument isn’t a good inference (since obviously it is), but that the foundation for its necessity is something other than itself. It lacks the intrinsic necessity of a demonstration, which, one might say, draws the conclusion from the light of reason itself.

  4. a thomist said,

    February 25, 2008 at 10:33 am

    Yes! That’s it!

    I had a suspicion that the argument secretly was appealing to something from the matter all along. The argument was, in my experience, invoked to show the superiority of formal systems in being able to deal with relations, and so it was taken for granted that there was merely formal relation in play. This argument from Christ’s two natures is very good at showing that there is an appeal to the matter- and perhaps even an intrinsic inability to deal with divine things in this sort of conception of logic- I think of the many times that the Mav. Philosopher. talks about the logical possibility of the Trinity- there is an implicit materialism in the logic itself which is invisible because people don’t realize that there is a reference to the matter in the symbols themselves.

  5. Brandon said,

    February 25, 2008 at 3:25 pm

    I think that it may result from a sort of flattening of logic into just one part: everything is (so to speak) treated as Prior Analytics, or, perhaps more accurately, as a conflation of Prior Analytics and Topics. Thus the sort of indirect necessity that can characterize the both general and special topics are treated as if it were really just formal; and genuine demonstration becomes virtually lost, as I think you’ve noted before somewhere, swallowed up in deduction from principles that are just (as far as the logic goes) taken hypothetically.

    If the argument I gave is right then it means that the contemporary logic of relations is really much more like contemporary modal logic — where it is overwhelmingly obvious that the different logical systems are borrowing from the matter — than is usually thought. It can be a useful and worthwhile thing — but it has limits that perhaps aren’t being recognized. That will certainly give me food for thought for a while.

  6. Kyle Brooom said,

    July 7, 2012 at 5:48 pm

    This argument can be proven using only truth-functional logic, the quantifiers, and polyadic predicates. The only assumption being made is that the premise (A horse is an animal) is true, and since the conclusion follows truth-functionally, using substitutional predicate schemeta, no assumptions about “the matter” are being made in order to draw the conclusion. This is made evident by the fact that any two-place relation can be substituted for ‘O’ in the the concluding schema, so: Anyone who rides a horse rides an animal, owns a horse, sells a horse, talks about a horse, makes a horse fly with pixie dust, etc.

    A horse is an animal.
    THEREFORE, a horse’s head is an animal’s head.


    ∀x(x is a horse ⊃ x is an animal)
    ∀x(Hx ⊃ Ax)

    THEREFORE, ∀x(x is the head of a horse ⊃ x is the head of an animal)
    ∀x(∃y(y is a horse . x is the head of y) ⊃ ∃y(y is an animal . x is the head of y))
    ∀x(∃y(Hy . Oxy) ⊃ ∃y(Ay . Oxy))


    [1] (1) ∀x(Hx ⊃ Ax) P
    [2] (2) ∃y(Hy . Oxy) P
    [2,3] (3) Hu . Oxu (2)u EII
    [1] (4) Hu ⊃ Au (1) UI
    [1-3] (5) Au (3)(4) TF
    [1-3] (6) Au . Oxu (3)(5) TF
    [1,2] (7) ∃y(Ay . Oxy) (6) EG; [3] EIE
    [1] (8) ∃y(Hy . Oxy) ⊃ ∃y(Ay . Oxy) [2](7) D
    [1] (9) ∀x(∃y(Hy . Oxy) ⊃ ∃y(Ay . Oxy)) (8) UG

    • July 8, 2012 at 1:19 pm

      The only assumption being made is that the premise (A horse is an animal) is true, and since the conclusion follows truth-functionally, using substitutional predicate schemeta, no assumptions about “the matter” are being made in order to draw the conclusion

      No, this is incorrect; an additional assumption being made, namely, that there is no modal or intensional component that would require the regimentation of predicates from each other. This is the point in my comment above. In fact this seems already assumed once you mentioned “truth-functional logic” — the ‘truth-functional’ in ‘truth-functional logic’ is equivalent to saying that modality or intensionality are not relevant, since once modality and intensionality are added the logic can no longer be assumed to be strictly truth-functional. Thus the entire argument as you present it is that, if one assumes that the matter is irrelevant to the inference, then the argument can proceed in a purely formal manner. Which is undeniably true; and also doesn’t get anyone anywhere.

    • July 8, 2012 at 1:44 pm

      To put the same matter in other words: We know that there are non-truth-functional logics and that some contexts actually require them. Thus we cannot know that the applicable is a truth-functional logic until we have ruled out these contexts. But these contexts concern the matter, not the form. (For instance, whether one needs a tense logic or an alethic logic depends on what is said, and not merely how it is structured.) There is also no known purely truth-functional way of determining whether a context requires a non-truth-functional logic. Therefore no one can determine whether truth-functional logic is the right logic until they have made certain assumptions about the matter of the inference.

      • Kyle Brooom said,

        July 9, 2012 at 3:51 pm

        Insofar as I understand your counterargument, I’d describe it as a “but you forgot the kitchen sink” argument. The reason I claim that no additional assumption is being made to the premise that ‘All horses are animals’ is that I prove the conclusion using only that premise and truth-functional transformation rules. The additional assumption you suggest I make does not appear in my proof.

        But you’ve claimed I’ve made the further assumption that more apparatus is not needed. This isn’t an assumption so much as something that’s been demonstrated – I’ve show I don’t need it by getting to the conclusion without it. There are lots of other things besides truth-functional logic in heaven and earth, like green grass, sunday picnics and the kitchen sink, but I hardly think it should count as an additional assumption on my part that I don’t need to appeal to these things or knowledge of these things in making an argument. The way I demonstrate I’m not appealing to them is simply not to appeal to them.

        In this sense, it’s not true that I assume the matter is irrelevant and proceed from there. We find out that it’s irrelevant by being able to complete the proof using only truth-functional transformation rules. You seem to think that it’s possible to just assume truth-functional logic is adequate and then proceed to make a proof from there, in effect, begging the question that truth-functional logic is adequate. But this is not the case, because not all cogent arguments can be proven using truth-functional logic. In cases where the matter is relevant, it is not possible to prove the conclusion using only truth-functional transformation rules, whatever assumptions I might care to make. So, for example, the argument

        Jim is Tom’s nephew.
        Therefore, Tom has a sibling

        Is a cogent one, but it cannot be proven using only truth-functional transformation rules. If I schematize the argument,

        Therefore, St

        The argument is not truth-functionally valid. This is because the inference depends on our understanding the content (“the matter”) of the predicate ‘has a sibling’ and the relation ‘is nephew of’. Not all arguments depend on an understanding of the content, the horse’s head argument is one of these; I know this not because I assume it, but because I can prove it using only formal schemata truth-functional transformations.

  7. July 9, 2012 at 5:56 pm


    You seem not to understand what the question at hand is. We are not doing schoolboy exercises, but simply looking a bit more critically at certain kinds of assumptions that are usually accepted uncritically, as, indeed, you accepted them uncritically in your original comment. Consider a parallel situation to this one involving relevance logic. You know, someone looks at an inference like,

    The moon is made of green cheese.;
    therefore London is in England or it is not,

    and says, “Obviously there’s a straightforward sense in which this is a good inference, but I think there are problems with considering this a proof of the conclusion, or even maybe an argument for it,” and someone else says, “Well, perhaps this is because proof requires that the premise and conclusion be related in a certain non-truth-functional or broadly modal way.” And then someone swoops in and says, “No, because here is a proof using classical truth-functional logic that you can get the conclusion just starting with innocuous assumptions; and it’s a proof, ’cause I proved it, so we don’t need to assume any non-truth-functional components anywhere!”

    Given the original argument,

    A horse is an animal.
    Therefore a horse’s head is an animal’s head.

    The question is, what presuppositions actually underlie this, and in what way can this inference really be considered a proof of this conclusion (not some proxy) from this premise (not some proxy). Notably, your original comment justified neither the logic used nor the accuracy of its translations. Your original ‘proof’, for instance, included a bald assertion that the correct translation for “A horse is an animal” is ∀x(Hx ⊃ Ax). This is certainly the way it would usually be translated, but ‘That’s the way it’s usually done’ doesn’t contribute in any way to clarifying in what way this can be considered a proof, if at all. Likewise, it included a bald assertion that the translation for “A horse’s head is an animal’s head” is ∀x(∃y(Hy . Oxy) ⊃ ∃y(Ay . Oxy)). OK, are we just supposed to take your word for it? Is it supposed to be self-evident that this is the right translation and that (for example) the possessive in this context does not require a model involving, say, modal operators? No way to say, and you insisted that you needed no content-based assumptions for your ‘proof’ at all. Now, this is just silly and bespeaks a hocus-pocus notion of logical form in which it magically just floats around unaffected by any concerns of what is actually meant. Given, however, that your ‘proof’ required already two translations of content and a deliberate rejection of a modal logic, which takes certain kinds of content into account, it was clearly a no-go from the beginning.

    And here again we see you doing the same sort of thing. You didn’t show that you proved anything “using only formal schemata truth-functional transformations”. I saw nothing showing that your translations were accurate and, by the same token, I saw nothing ruling out hidden modal operators that, if there are any that need to be considered, would make your use of a non-modal logic absurd. And there is, in fact, reason to think that we should be cautious in ruling out hidden modal operators, for the reasons given in my above comments, namely, that there are apparent wacky metaphysics in which the inference would fail. It’s nonsense to suggest that you have a proof of anything without any consideration, given that what is at hand is what has to be the case for something to be properly a proof. (And, frankly, you should know better than to commit the rookie mistake of assuming that because you can get the right answer on a set of a logical assumptions that you have proven that your logical assumptions are the most proper to apply in a given case. I have no clue what you’re thinking on this point.)

    Consider a basic example. Take the following inference:

    A circle is a possible shape.
    Therefore, a circle’s having four straight sides is a possible shape’s having four straight sides.

    Let H=circle, A=possible shape, and O=having four straight sides. Your argument proceeds exactly as before. So you’ve proven the conclusion. But is the inference here actually a proof, especially given that we’re not starting with a premise and seeing what we can get but looking at a complete inference and examining the underlying logical character? We start with a necessary truth, but the conclusion in the original argument is unclear at best. A circle is a possible shape, but a circle with four sides is not a possible shape at all, and there’s a straightforward sense in which the conclusion is necessarily false: a circle’s having four sides is not any property of a possible shape, because circles are constituted by their geometrical features, which rule out having four straight sides, and so the inference has actually introduced an incoherence. Now, of course, this is a child’s puzzle, and it’s not difficult to come up with resolutions; but it again raises the question of what we are assuming, and whether, for instance, we are in the horse’s head case being misled by the mundaneness of the example into thinking the inference less complicated than actually is, or whether we are actually appealing to implicit assumptions about what logical predication represents in the real world in order to shortcut a process that in other circumstances could be considerably more complicated, or perhaps not even possible. And when we go back to look at the original question, “Do we actually end up proving the conclusion from the premise at all, and if so, what contributes to making this the case, and if not, what is missing?” we keep running into the question of the matter of the inference. We know for a fact that before we can even apply a formal system, we must have made certain content-based assessments (to rule out equivocation or hopeless vagueness, to rule out other logical systems like default logics or modal logics as being more suitable models of the inference, etc.). So do these contribute nothing to its being a proof? But at the same time, we run into the problem that any attempt to handle these inferences in purely truth-functional terms already requires not just one but two translations between original content and the formal language used. For us to actually have a proof, do these translations have to have certain characteristics, e.g., self-evidence, or particular logical characteristics? &c. &c.

  8. Kyle Brooom said,

    July 9, 2012 at 7:28 pm

    Hi Brandon,

    Thanks for taking the time to reply. I gather from some of your language that I have somehow perturbed you; if this is correct, I regret it. I’m sorry if you find my remarks uncritical or if they remind you of schoolboy exercises.

    I agree that the green cheese/London argument is a strange one, in part, because, as you say, the premise and conclusion are unrelated. They’re even unrelated truth-functionally, since the conclusion, truth-functionally, is logically true, so the premise is irrelevant. Since this is the case, it’s not clear what that has to do with me or my position.

    Regarding my original translations of the horse’s head premise and conclusion, I believe they’re correct. But, please don’t take my word for it; if you would like to show that they are not correct, you could provide a description of a scenario where ‘A horse is an animal’ and ‘∀x(Hx ⊃ Ax)’ differ in truth-value. Alternatively, you could provide a description of a scenario where ‘the head of a horse is the head of an animal’ and ‘∀x(∃y(Hy . Oxy) ⊃ ∃y(Ay . Oxy))’ differ in truth-value. Were you to do one of these things, you would have shown that this way of translating the original argument into a truth-functional logic is somehow illegitimate; I don’t believe you can.

    There are three rejoinders to your third argument, the circle argument. First, you’ve been suggesting that there might be some hidden modal properties in the horse’s head argument. Therefore, it’s not quite legitimate to replace the original argument with one that exhibits explicitly modal predicates (‘possible shape’). Second, as my first post made explicit, the schema ‘O’ substitutes for a relation, or a two-place predicate (‘___ is the head of …’). You have proposed to substitute ‘having four straight sides’ for ‘O’, which is a one-placed predicate. Therefore, whatever the correct translation of your circle argument into truth-functional schemata would be, it’s clear that it would not be the same translation as mine, of the original horse argument. Assuming we ignore the first two rejoinders, a third, more serious one is available. The supposed paradox of your circle conclusion, correctly interpreted within truth-functional logic isn’t a paradox at all; rather it’s true. Since ‘a circle’s having four straight sides’ isn’t really an ordinary English construction, I would paraphrase your conclusion as follows:

    A circle that has four straight sides is a possible shape that has four straight sides.

    If this is an unfaithful paraphrase, then please let me know; as the sentence stands currently, I don’t know what else it could mean. Interpreted using the truth-functional operators, this would be true just because the antecedent of the conditional would always be false. Just to be as explicit as possible, I would schematize the conclusion in the following way:

    ∀x((Cx&Fx) ⊃ (Px&Fx))

    Where ‘C’ substitutes for ‘___ is a circle’, ‘F’ for ‘___ has four straight sides’, and ‘P’ for ‘___ is a possible shape’

    The antecedent of the conditional is true of all and only things which are both circles and have four straight sides. But there are no such things. Since there is no thing that makes both ‘C’ and ‘F’ true, the conjunction is always false. Since the conjunction is always false the whole conditional is true; conditionals with false antecedents are true, since the only way a conditional can be false is if its antecedent is true and its consequent is false.

    Since you seem to think that your circle argument introduces incoherence, I am led to believe that you are reading the conclusion to somehow say that four-straight-sided circles are possible. As I understand it, it does not say this – it only says that such are possible if there are any four-side circles. Well, of course four-straight-sided circles would be possible if there were some four-sided circles around – they’d be actual!

    The incoherence you’re claiming only arises when the conclusion is something like ‘There is a possible circle with four straight sides.’ I agree that this would be false, but it’s not relevant to my position both because it bears little resemblance to the original horse’s head argument and because it is not implied truth-functionally by the premise ‘A circle is a possible shape.’

  9. July 11, 2012 at 11:13 am


    The question at hand is not “Is the horse’s head inference a good inference?” since everyone here agrees that there is some straightforward sense in which it is, but “What makes a proof, or, perhaps a bit more broadly, a genuine argument for something?” Your comments keep not being relevant to this question. This is why the green cheese argument — which is truth-functionally valid, since a logical truth follows from any premise on truth-functional principles in classical logics — is analogous. When relevance logicians reject these kinds of inferences, or those based on the various implication paradoxes, they aren’t arguing that the inferences are bad inferences in classical logic, nor that you can give no defense of them in the context of classical logic, but that classical logic overlooks something relevant to proper assessment of these as arguments in a more fundamental sense. This is, as you say, relevance, but relevance is a non-truth-functional notion — once you’ve introduced it, you’ve introduced a modality, broadly construed, as a necessary element in whether something counts as an argument of the right type, and your inferences have to meet something other than a truth-functionality condition. Likewise, the problem raised by Berquist, and noted in James’s post, is whether the way in which one can regard the horse’s head inference as a good inference is lacking something required for proof, or being an argument-for-something. When I said that the cases were parallel, I meant it quite seriously. The two cases are exactly similar up to the point considered here: what’s being looked at here is not whether you can have a truth-functional structure linking the premise with the conclusion, since this is clearly the case, but whether this truth-functional structure can be a proof or argument-for in a proper sense. (After this point, there is a divergence: the discussion doesn’t go any farther, while relevance logicians go on to look at logical systems that avoid the kinds of irrelevance raised by the paradoxes; this is in part because for the purposes of discussion here we already have such logical systems — Aristotelian ones — and the analogous question is not whether they exist but what motivates them.)

    Similarly, one of the important secondary issues is that we are not working with an inference from within the predicate calculus: we are working with a natural language inference. Thus the question of whether the proper model for this inference is classical or something else is live and relevant from the get-go. Thus, I don’t think the circle inference is incoherent or not susceptible of some strained truth-functional interpretation; this was precisely my point: it is not incoherent, but it is an apparently similar natural language inference that at least suggests that conclusions of the kind found in the horse’s head inference can involve modal ambiguities in meaning that have to be set aside or eliminated in order even to get to the truth-functional interpretation, and that are usually overlooked in the horse’s head argument due to its mundaneness; that there might well be a hidden modal component, ‘modal’ being broadly construed, in the conclusion, “A horse’s head is an animal’s head,” as it would be generally understood is also suggested by the strange-metaphysics and reduplication argument in the above thread. I do not in fact know what the modal component is: I can only point to the signs that suggest that we should at least be looking for it. But we’re getting a weird situation here, in which you deal with my point (that similar natural language inferences may involve modal components, and that this suggests we can’t assume that the proper model for the horse’s head is truth-functional until we have eliminated the possibility that there is such a modal component in this case) by saying that the two are different for no other reason than that the circle inference may involve a modal component, and the proper model for the horse’s head inference is truth-functional and therefore can’t have any modal component. This is obviously question-begging as a response to my argument.

    The problem with validity in a truth-functional context in general is that it is merely a logical structure linking propositions, independently of any questions of how those propositions function in reasoning or argument; merely showing that an argument is valid in a truth-functional context does not seem to be sufficient for showing that it is a proof or genuine argument-for-the-conclusion (begging the question, the reasoning paradoxes that inspire relevance logic, and other things all suggest this independently of this case). Thus the question you really need to be answering is why your truth-functional account should be considered to show that in the horse’s head inference the premise is genuinely an argument for the conclusion, genuinely a proof of the conclusion. This would be the case if you could show that, appearances aside, there is in fact no hidden modal component in the natural language inference, or at least that there is likely not. Possibly it could still be the case if there were a modal component but it turned out in this context somehow to be trivial or irrelevant. This would show that regarding the original inference as a proof would not require smuggling in assumptions about the matter of the inference, just as you originally claimed. Other than something like that, you’re seem to be talking past everyone here.

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