Aristotle on Universal Quantification

I was rereading the Posterior Analytics in preparation for my lecture today and I was struck by the following passage from Book I chapter 4 (72b 28-30)

Now I say that something holds of every case if it does not hold in some cases and not others, nor at some times and not others; e.g. if animal holds of every man, then if it is true to call this a man, it is true to call him an animal too; and if he is now the one, he is the other too;

Here Aristotle seems to be defining ‘all A’s are B’s’ in terms of a universally quantified conditional statement (for any thing (and/)or for any time, if that thing is an A then that thing is a B). This sounds surprisingly modern (indeed, by the end of the chapter he seems to be talking about universal instantiation), since most of us were told in our logic classes that rendering universal statements in terms of a quantified conditional is supposed to correct an error in Aristotle’s logic (i.e. the error of thinking that ‘all’ implies ‘some’). But if we take Aristotle at face value here the way he formally defines ‘all’ will give us perfectly good truth conditions for ‘all A’s are B’s’ even if there aren’t any A’s at all.

So it doesn’t seem that Aristotle’s logic is committed to the existential import of universal affirmative statements (though I know that this isn’t Artistotle’s position since he is clear that No A are B is the contrary of all A are B (i.e. they both can’t be true. He gives as examples ‘all men are just’ and ‘no men are just’)). I wonder if Aristotle had thought explicity about empty categories if he would have rejected the contrary bit from On Interpretation

UPDATE:

Thinking about this a bit more it occurs to me that what this shows is the implicit truth-conditional definition of the conditional Aristotle is using. ‘If p then q’ From what he says we can see that the sentence will be true when p is true and q is true and it will be false when p is true and q is false (cf his evidence in Post. A. 72b 30). He does not say anything about the case when p is false, but we can infer a bit about this condition by his claim about contraries. Since when All A’s are B’s is true No A’s are B’s must be false we know that the conditional cannot be counted as true when teh antecedant is false (that would render both of these statements true and so not contraries). So, in the F F and F T combinations the conditional must be counted as false. That satisfies the requirement that the two cannot be true together. So we can see a kind of operator being defined here; let’s call it ‘xxx>’. ‘xxx>’ is defined truth functionally as

P         Q     P xxx> Q

t          t        T

t          f         F

f          f         F

f          t          F

Is the ‘XXX>’ connective a connective from relevance logic? No, it is just the ‘&’ of classic first-order logic…this fits very nicely with the metaphor of universal quantification as a giant conjunction…

5 thoughts on “Aristotle on Universal Quantification

  1. I’m not sure how Aristotle should be interpreted on this point, although it does seem you are on to something in your Update. But that there is a link between A propositions and conditionals has been known a very long time, independently of any questions about subalternation. Which is not an error, but that’s another story. And it’s false that universal statements are always put in conditional form even in modern logic, but that’s another story again. (I agree we’re usually told these things in logic classes, though.)

  2. Hey Brandon,

    Do you know if the connection between A propositions and conditionals was discussed in the Port Royal Logic? I would be interested to see how they are handled. Are conditionals truth-functionally defined in these earlier treatments?

    Do you really think that Subalternation should be preserved in modern predicate logic?

  3. I don’t recall it discussed in the Port Royal; like a lot of logic in the early modern period it is more focused on the practical task of improving reasoning in natural language than on looking at the theory behind it. and truth-functional definitions are fairly late, although there are plenty of scattered cases in the history of logic in which people were clearly discussing truth-functional properties of propositions. Some medievals say things that show that they at least recognize the connection between A and conditionals; Leibniz the same; but as far as I am aware it only begins to be discussed explicitly in its own right in the nineteenth century starting with Jevons.

    We do have subalternation in the modern predicate logic; it’s just subalternation to an instance rather than a particular and is called universal instantiation. You can, when you can combine it with existential generalization, get subalternation in the straightforward sense. (This, I think, is why proponents of free logic tend to make arguments against standard predicate logic that are very similar to arguments made by proponents of the latter against Aristotle. They’re actually the same argument, with slight adjustment.) You can also get the same effect more directly if you simply take as an axiom “Some S is S”; that can be done if you don’t assume that particular statements have existential import.The reason it’s usually rejected in modern predicate logic has nothing to do with the formalism (IMHO) and a great deal to do with the fact that modern logicians conflate particular statements and existential statements. Prior to the twentieth century this was highly controversial, and no one took it as obvious; and the reality is more complex. Roughly: If you take A to be conditional-like, and to have existential import, and you allow subalternation, the result is like concluding “P and Q” from “If P then Q”, because I with existential import is conjunction-like. If you reject the view that I has existential import, you can keep your conditional-like A, allow subalternation, and simply refuse to assign existential import to A; but I ceases to be conjunction-like. Likewise, A can be conditional-like, and I have existential import, if you reject subalternation. The formalism is indifferent to which; it’s just a matter of what interpretation-based constraints you wish to put on the application of it. And there are advantages and disadvantages to each move. In any case, whichever one we prefer, subalternation is not an error; if a logically consistent system has a feature that disagrees with the logically consistent system we happen to use, that is not a logical error; at the very most it is a practical inconvenience for doing particular kinds of things.

  4. Thanks Brandon for the informative comment!

    “You can, when you can combine it with existential generalization, get subalternation in the straightforward sense. “

    You can? Given,

    All men are mortal = (for all x) If (Man (x) then Mortal (x))
    Some men are moratl = (there is an x such that) (Man (x) and Mortal (x)

    it doesn’t seem like it goes through.

    1. (x) (Man(x) then Mortal(x))
    2. If Man(a) then Mortal (a) [universal instantiation]
    3. ~ ( Man(a) and ~Mortal(a) [definition of material implication]
    4. (~Man(a) or Mortal(a)) [DeMorgan]
    5. (Ex) (~Man(x) or Mortal (x))

    5 does make an existence claim, namely that something is either not a man or is a mortal, but that isn’t worrisome, is it? At least not in the way that subalternation is thought to be. That was the question of whether it was true that if ‘all men are mortal’ is true must it then be the case that ‘some men are mortal’ is true, not the question of whether there is existential import to logic in general. And surely whether this is an error or not there is no denying that Russell and company thought that it was an error (at least the Russell of Logical Atomism).

    But maybe I am missing what you had in mind?

  5. I’m not sure why you think (5) is not worrisome “in the way that subalternation is thought to be”; subalternation is usually said to be worrisome in the sense that it concludes that something exists from a proposition that doesn’t imply that anything exists. Certainly Russell, et al., who simply assume one particular view of existential import without question, usually bring it up as the issue — it’s the reason why you can’t conclude the particular from the universal. What other problem would there be?

    You are assuming the standard conditional interpretation of A propositions; but this is not read off of the formalism. (1) also admits of the interpretation “Everything is an if-man-then-mortal thing”; then the deduction can be interpreted as:

    (1) Everything is an if-man-then-mortal thing.
    (2) a is an if-man-then-mortal thing.
    (3) a is a not-both-man-and-not-mortal thing.
    (4) a is an either-not-man-or-mortal thing.
    (5) Something is an either-not-man-or-mortal thing.

    And that, of course, is the same as “Something is an if-man-then-mortal thing” and we have concluded an I proposition from an A proposition. The point is general. My point is not that this is the best way to interpret it, but that there is nothing about the formalism that forbids it. If you can UI+EG there is an interpretation under which you can subalternate for those propositions for which you can UI+EG; and if you can’t subalternate for those propositions under any interpretation, that rules out UI+EG as well.

    In any case, part of my point was that universal instantiation is logically parallel to traditional subalternation; they are functionally similar, structurally similar, and they have been subjected to structurally analogous criticisms, so much so that universal instantiation can be considered simply a modern form of subalternation.

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