According to the Boolean standpoint, it rejects the notion that a universal statement implies existence. For instance, the statement "all S are P" does not tell us whether or not any members of "S" actually exist or not but rather that if any members of "S" did exist, they would also be contained in "P."

Does the Aristotelian standpoint have the same interpretation of universal statements when talking about non-existent things? I know that from the Aristotelian perspective a universal statement does imply existence as long as the things being talked about actually exist in the world, but would universal statements for non-existent things have the same interpretation as a universal statement in the Boolean perspective?

For instance, from the Aristotelian perspective, the statement "all unicorns are one-horned creatures" being true would not imply that unicorns actually exist but rather that if any unicorns did exist, then they would also be a one-horned creature, right?

  • rather that if any members of "S" did exist, they would also be contained in "P." The Boolean perspective doesn't even make this claim (which would require some sort of modal logic), it just says "all S are P" is a vacuous truth when there are no members of S. The wiki article gives the example of a childless person saying "all my children are goats", which is true even though in modal logic, if we were talking about the "closest possible world" where the person has children, it would be false.
    – Hypnosifl
    Jun 23 at 18:13
  • I'm confused then because the wiki article on existential fallacy (en.wikipedia.org/wiki/Existential_fallacy) states "'Every unicorn has a horn on its forehead.' It does not imply that there are any unicorns at all in the world, and thus it cannot be assumed that somewhere there is a unicorn in the world. The statement, if assumed true, implies only that if there were any unicorns, each would definitely have a horn on its head." Does this not agree with the notion if any members of "S" did exist, they would also be contained in "P," or am I mistaken?
    – Slecker
    Jun 23 at 19:30
  • I can understand your confusion. First off why would any valid logic system ALLOW false statements in reasoning in the first place? If we know all unicorns are sky scrapers is false why allow it as a premise? This means validity is a bit shaky as how am I supposed to know if the argument applies to reality or not? Clearly some valid arguments might not be realistic! This is an issue with the modern interpretation. Aristotelian logic assumed if the proposition was true its particular MUST also be true otherwise the proposition is false. If false then the conclusion is not absolute anymore.
    – Logikal
    Jun 23 at 19:56
  • @Logikal, "All unicorns are skyscrapers" is not false. The reason it is peculiar is due the extensional nature of modern logics. Maybe it will be easier to understand if you read it in the explicitly extensional language of set theory: "forall x.p(x)->q(x)" can be translated to "p is a subset of q". If p is the empty set, then this is clearly true. You might want to look up the paradox of material implication which is a related problem. Jun 23 at 20:32
  • @David Guderman, what do you mean "all unicorns are Skyscrapers" is not false? In the real world there are zero unicorns in existence. You are speaking from model theory & not reality. This also separates validity from Soundness. Because an argument is valid doesn't mean it applies to reality whatsoever. Soundness is when the premises must be true while the argument also must be valid simultaneously. Which one handles reality better? In epistemology the focus would be on soundness --not validity.
    – Logikal
    Jun 23 at 20:38

As you say in your question, modern logic (really a combination of Boole, Brentano and Frege) treats universal statements of the form, "all A's are B's" as hypothetical. They are understood as something like: for any thing, if it is an A then it is also a B. If we then understand the 'if' to be material implication, then because material implications with a false antecedent are true, the statement "all A's are B's" comes out as trivially true if there are no A's.

In modern logic, statements of the form, "some A's are B's" are understood to mean: there are some things that are both A and B. This is therefore false in the event that there are no A's. Hence, it follows that "all A's are B's" does not entail "some A's are B's" since in the event that there are no A's the former is true and the latter is false.

So, at this point modern logic disagrees with Aristotle's, under which "all A's are B's" entails "some A's are B's". Whether you think one is more plausible than another is irrelevant. They are different logics and have different uses. According to Grice, the fact that "all A's are B's" does appear to imply "some A's are B's" can be explained using the theory of conversational implicature, but that is perhaps a topic for another question.

The usual explanation of the difference between Aristotelian logic and modern logic is that Aristotle's understanding of "all A's are B's" has existential import. In other words, it is false if there are no A's. From a modern point of view, this is unsatisfactory, since sometimes we don't know whether things of a certain kind exist and we wish to reason hypothetically from an assumption that they might. It makes sense to argue that (P1) all unicorns are animals with the body of a horse and a single horn on their forehead, (P2) there are no animals with the body of a horse and a single horn on their forehead, therefore (C) there are no unicorns. But this argument can only be sound if both premises are true, so we need P1 to come out true.

On this understanding of Aristotle, reasoning about non-existent things is simply not possible. Arisotle assumes that we speak only of things that exist, perhaps because he is not interested in making statements about things that don't exist.

However, there is some disagreement among medieval logicians on this point. William of Ockham appears to follow this interpretation, and he detects an asymmetry between the left and right sides of the Aristotelian square of opposition. There is one way to prove "all A's are B's", namely show that all A's are indeed B's, but there are two ways to prove it false, viz, either show that some things that are A are not B, or show that there are no A's. Similarly, there is one way to demonstrate "no A's are B's" is false, namely find an A that is a B, but two ways to prove it true, namely, show that no A is also a B, or show that there are no A's.

Other medieval logicians differed and treated existence as a property. On this understanding, neither "all A's are B's" nor "some A's are B's" has existential import. We would have to make an additional claim, such as "some A's are" or "some A's exist" to assert their existence. On this view, "some dragons breathe fire" might come out true if some hypothetical dragons breathe fire while allowing that some might not, but "some dragons exist" would come out false. "All unicorns are one-horned animals" would come out true on this account, even if there are none.

As to what Aristotle himself would have said, it is hardly possible to say. There are interpreters who go both ways on this issue, but it seems pointless to speculate. So, unfortunately there is no categorical answer to your question.

The significance of the Boole-Brentano-Frege approach is that it treats existence as a quantifier and not as a predicate. There are many advantages of doing this, though if we wish to reason about non-existent objects in modern logic, we can still do so using free logic with an existence predicate.

  • On why modern logic defines things the way it does, I talked in this answer about how part of the usefulness of material implication's truth table is that it can be used in conjunction with the universal quantifier to "translate" statements of the "all A are B" form in a way that's equivalent if there are some As. As you said, this truth table says A -> B must be true if A is false, so if you still want "all As are Bs" to translate to "for all x, A(x) -> B(x)" when there are no As, you have to define "all As are Bs" to be true in this case.
    – Hypnosifl
    Jun 24 at 17:34
  • That's true, but even so, arguably material implication is not the best choice for this role. It only works in the special case of the universal quantifier. There are many other quantifiers, such as most, many, a lot, a few, hardly any, etc., and if you try to use these with material implication you get the wrong result. E.g., if you try to represent "most penguins fly" as (most x)(penguin(x) -> flies(x)) then this comes out true, just because most things are not penguins. Similarly for all the other quantifiers. Using generalised quantifiers is probably a better approach.
    – Bumble
    Jun 25 at 6:09

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