Because empty terms show this argument form to fail and thus Aristotle was wrong to deem it a valid argument, hence Classical Logic was right to distance itself from Aristotle's logic. Following from Russell, take this argument:

All winged horses are horses,
All winged horses have wings,
Therefore some horses have wings.

Clearly the first two premises are true but the conclusion is clearly false, we know there are no horses with wings. So this ought not be regarded as a valid argument in the logical systems developed after Aristotle.

Because empty terms show this argument form to fail — MindForged

What do you mean by "empty terms"? Are you refering to arguments with undefined variables?

All winged horses are horses,
All winged horses have wings,
Therefore some horses have wings.

Clearly the first two premises are true but the conclusion is clearly false, we know there are no horses with wings. — MindForged

But why is the conclusion false? I mean, I know that horses doesn't have wings, but it's inductive, empirical constatation, isn't? It's not logically impossible that a winged horse exists, unless you define horse as something that doesn't have wings. But, if this the case, then both premises are nonsense, because you would be saying something like "all winged things that doesn't have wings have wings". I don't know if i understood...

What do you mean by "empty terms"? Are you refering to arguments with undefined variables? — Nicholas Ferreira

Terms without referents. No pegasi exist, so pegasus is an empty term.

But why is the conclusion false? I mean, I know that horses doesn't have wings, but it's inductive, empirical constatation, isn't? It's not logically impossible that a winged horse exists, unless you define horse as something that doesn't have wings. But, if this the case, then both premises are nonsense, because you would be saying something like "all winged things that doesn't have wings have wings". I don't know if i understood... — Nicholas Ferreira

It's false because we know it's true that winged horses are non-existent. Logical impossibility is irrelevant, this isn't a discussion about possibility or any other modality. This is about existence or non-existence. The first two premises are objectively true and the conclusion surely false. For logic the content or possibility of the premise is immaterial. What matters is that the truth of the premises does not entail the truth of the conclusion. If a counter example exists we know the argument form is not valid.

I always found this a bit confusing as well- the key thing for Russell's point of view (as opposed to Aristotle) is that statements like "All winged horses have wings" can be trivially true, i.e. true by default because the term doesn't refer to anything that exists.

It might help to also consider that Russell would also interpet "All winged horses are wingless" to be true. Since nothing is in the set "winged horses". It's a bit similar to how "a->b" is always true when "a" is false.

However, a statement like "there exists a winged horse that has wings" can't be trivially true in the same way. It's saying that "for the set of winged horses, there is at least one member, and that member is also a member of the set of things that have wings".

Bertrand Russell, in his Logic and Knowledge essay (p. 230), states that the argument "All A is B and all A is C, therefore some B is C" is a fallacy. The formalization would be:
(∀x)[Ax⊃Bx]
(∀x)[Ax⊃Cx]
∴(∃x)[Bx⊃Cx]
Why is this a fallacy? I thought it is because if the premises are universal (∀x) then the conclusion must be so, and not an existential one (∃x). But can't we imply "some B are C" from "all B are C"? — Nicholas Ferreira

The problem is that it is invalid to cross predicates in this way. A is B is to predicate B of the subject A. A is C is to predicate C of the subject A. B and C are predicates of the subject A. Until we convert either B or C into a subject, and predicate something of that subject, we have nothing to allow us to draw any conclusions about either B or C, because B and C have not been presented as subjects.

But can't we imply "some B are C" from "all B are C"? — Nicholas Ferreira

No, that is invalid. It becomes more obvious if we reformulate the two propositions as follows.

• For all x, if x is B, then x is C.
• There exists an x, such that x is B and x is C.

The first proposition clearly does not entail the second.

Um, what? 'All winged horses are horses' isn't trivially true because the term doesn't refer to anything. It's trivially true because the initial referent makes explicit reference to belonging to the category of the second referent (namely, a winged horse is clearly a kind of horse).

And I don't see how Russell would consider "All winged horses are wingless" to be trivially true. His description theory of names doesn't say sentence with empty terms are by default true, especially contradictory ones. They are deemed false in his theory because they must posit the existence of some thing (winged horses) but we know the thing to not exist. Nothing satisfies the condition "winged horse" so the translation of the previous argument into classical logic would have a suppressed premise, namely:

There exists at least one winged horse.

Which gets the value false, leading to a false conclusion. Have I misunderstood you or perhaps Russell's theory?

It's false because we know it's true that winged horses are non-existent — MindForged

Honestly, how do you know that winged horses are non-existent? I'm noy saying that they exists or that I believe that they exist, but you can't affirm that categorically only based on "no winged horse has ever been seen".

It might help to also consider that Russell would also interpet "All winged horses are wingless" to be true. Since nothing is in the set "winged horses". — Ben92

Yeah, I read it and i was kinda doubtful too. For me it doesn't make any sense.

The problem is that it is invalid to cross predicates in this way. A is B is to predicate B of the subject A. A is C is to predicate C of the subject A. B and C are predicates of the subject A. Until we convert either B or C into a subject, and predicate something of that subject, we have nothing to allow us to draw any conclusions about either B or C, because B and C have not been presented as subjects. — Metaphysician Undercover

Hmmm, this makes sense to me. I think that an argument like "All apples are red and all apples are sweet, therefore some red are sweet" would represent this, because both "red" and "sweet" are adjectives. So it's an invalid argument because there are some predicates that can't be set to the subject and simultaneously to each other, right?

No, that is invalid. It becomes more obvious if we reformulate the two propositions as follows.

For all x, if x is B, then x is C.
There exists an x, such that x is B and x is C.

The first proposition clearly does not entail the second. — aletheist

But it's wrong, the argument says "some B are C", not "all B are C". And even if it was correct, why the second doesn't follow from the first? I mean, if for any x, x is B and x is C, then there exists an x that is B and C. Why would it be wrong?

And I don't see how Russell would consider "All winged horses are wingless" to be trivially true. His description theory of names doesn't say sentence with empty terms are by default true, especially contradictory ones. They are deemed false in his theory because they must posit the existence of some thing (winged horses) but we know the thing to not exist. Nothing satisfies the condition "winged horse" so the translation of the previous argument into classical logic would have a suppressed premise, namely:

There exists at least one winged horse.

Which gets the value false, leading to a false conclusion. Have I misunderstood you or perhaps Russell's theory? — MindForged

Well, actually he says it on Logic and Knowledge (p. 229), and I think it's kinda weird.

"If it happened that there were no Greeks, both the proposition that "All Greeks are man" and the proposition that "No Greeks are men" would be true. The proposition "No Greeks are man" is, of course, the proposition "All Greeks are not-man". Both propositions will be true simultaneously if it happens that there are no Greeks. All statements about all the members of a class that has no members are true, because the contradictory of any general statement does assert existence and is therefore false in this case. This notion, of course, of general propositions not involving existence is one which is not in the tradictional doctrine of the syllogism." — Russell

All winged horses are horses,
All winged horses have wings,
Therefore some horses have wings. — MindForged

You're treating the premises in a purely logical manner, but assessing the conclusion with respect to whether it's contingently true in the actual world.

But it's wrong, the argument says "some B are C", not "all B are C". — Nicholas Ferreira

Your statement was, "But can't we imply "some B are C" from "all B are C"?" So the premise was "All B are C," which is equivalent to "For all x, if x is B then x is C"; and the conclusion was "Some B are C," which is equivalent to "There exists an x, such that x is B and x is C"; but this does not follow. An existential quantification cannot be derived from a universal quantification. "If something is a unicorn, then it is a horse with a single horn" is true, but does not entail "Something exists that is a unicorn."

Oh, i think i got it. The problem is that the existential quantification does affirm that something exists, while universal quantification only states that "for any x, if Px, then Qx", for instance, without assuming the existence of some x. Right?

Correct.

Honestly, how do you know that winged horses are non-existent? I'm noy saying that they exists or that I believe that they exist, but you can't affirm that categorically only based on "no winged horse has ever been seen". — Nicholas Ferreira

Based on this one either can't claim to know almost anything or else you have to change our understanding of biology and what creatures exist on Earth in order to credibly say we don't know them to not exist.

Well, actually he says it on Logic and Knowledge (p. 229), and I think it's kinda weird.

"If it happened that there were no Greeks, both the proposition that "All Greeks are man" and the proposition that "No Greeks are men" would be true. The proposition "No Greeks are man" is, of course, the proposition "All Greeks are not-man". Both propositions will be true simultaneously if it happens that there are no Greeks. All statements about all the members of a class that has no members are true, because the contradictory of any general statement does assert existence and is therefore false in this case. This notion, of course, of general propositions not involving existence is one which is not in the tradictional doctrine of the syllogism." — Nicholas Ferreira

Note the bit I bolded. "All Greeks are man" is a universal statement, while "No Greek are men" is a particular. One is about an abstract domain not involving existence while the other is explicitly about existence. Russell says this just before the bit you quoted:

I want to say emphatically that general propositions are to be interpreted as not involving existence. When I say, for instance, 'All Greeks are men', I do not want to suppose that that implies that there are Greeks.

This is more an issue of syllogistic logic not modern logic. Syllogistic is supposed to be used for things known to exist and so it's fairly limited in a few ways including inferences involving empty terms. That's why the argument form is deemed invalid in classical logic despite Aristotle's logic deeming it valid.

You're treating the premises in a purely logical manner, but assessing the conclusion with respect to whether it's contingently true in the actual world. — Terrapin Station

Ok? The point is the argument form is invalid because it can take one from definitely true premises to a definitely false conclusion. In modern logic it commits the existential fallacy, it's suppressed premise ("There exists at least one winged horse") is clearly false.

Logically, it's a matter of whether the conclusion follows from the premises, not whether the conclusion is true per our beliefs about the actual world.

But the conclusion does not follow from the premises. We know in the actual world that the conclusion is false so we have a counter example to the inference form to show us that. If the form we're valid there would be winged horses. I'm saying something about the actual world makes that inference invalid, but that it's a way we can know it to be invalid.

Based on this one either can't claim to know almost anything or else you have to change our understanding of biology and what creatures exist on Earth in order to credibly say we don't know them to not exist. — MindForged

But this is really a problem. Answer me: how do you know that a winged horse doesn't exist? Unless you define horse as being something wingless, you can't know if there is a horse with wings. You can induce from previous data that is improbable that something like this exist, but it's not a certainty. It's like you saying that black swans doesn't exists because no one has seen any before, but then one day someone sees one.
The problem is that if you define horse as being something wingless, then the proposition "all winged horses are horses" doesn't make any sense, even it being an analytical one, just like the second proposition and the third. They all would be self contradictory.

"All Greeks are man" is a universal statement, while "No Greek are men" is a particular. One is about an abstract domain not involving existence while the other is explicitly about existence. — MindForged

Why woudn't both be universal? Russell says that "No Greek are men" is the same of "All Greek are not-man". For me, it's clear that both propositions "all greeks are man" and "no greek are man (all greeks are not-man)" are universal ones. For it to be a particular one, it would need to use existential quantification and, therefore, assume the subject existence, woudn't? Thanks for answering :D

Right, and it is not logically valid to derive an existential proposition directly from a universal proposition with the same terms. "All A is B" does not entail "Some A is B."

But the conclusion does not follow from the premises. — MindForged

It does, though. It's the same as "All silver toasters are toasters. All silver toasters are silver. Therefore some toasters are silver."

Right, and it is not logically valid to derive an existential proposition directly from a universal proposition with the same terms. "All A is B" does not entail "Some A is B." — aletheist

If all A is B, then obviously some A is B.

So long as you're not dealing with an empty domain, and you know this a priori. The universal quantifier is equivalent to 'not for some x not (rest of expression)', and trivially when there are no x's, the statement to the right of the first not '(for some x not) is false because there are no x, thus the whole statement is true, since it is the negation of a falsehood.

No logical claim is a metaphysical claim about the actual world. Logic is simply about formal relationships per se. So whether there are really (in the actual world) any x's is always irrelevant.

No idea what you're talking about. See here. Usually however we are not talking about empty domains or inexistent objects.

Logically, "All cell phones in the room are turned off" has absolutely nothing to do with whether in the actual world there is any room, any cell phones in the room, etc. Logic has nothing to do with epistemology with respect to claims about the (contingent) actual world. Logic is about the formal relationships of statements to each other, re implication/inference. It's a matter of what follows or not given certain assumptions. The real world need not apply.

Truth value re the real world is pertinent to soundness versus validity, but logic itself has nothing to do with assigning those truth values.

"All A is B" does not entail the existence of any A, but "Some A is B" does; so it is not deductively valid to derive the latter from the former. Note that in this context, "existence" pertains to the universe of discourse, which is not necessarily the actual universe.

"All A is B" does not entail the existence of any A, — aletheist

Logic has nothing whatsoever to do with claims about whether anything exists in the actual world. It follows from all A is B that some A is B. Whether any A exist outside of that is irrelevant.

I guess you missed my second sentence.

Note that in this context, "existence" pertains to the universe of discourse, which is not necessarily the actual universe. — aletheist

I am not sure why we are having this debate at all; it is an uncontroversial principle of modern deductive logic that deriving "Some A is B" from "All A is B" is a fallacy, unless the universe of discourse is specified separately as including at least one member of A. In other words, it requires the additional premise, "Some A is A."

Re "All As are B" that is your universe of discourse by virtue of the stipulation that all As are B.

What you linked to is wrong on multiple fronts. I can explain everything it's getting wrong if you're interested in learning this.

No, a universal proposition does not establish the universe of discourse all by itself. I provided a link, so if you want to disagree with modern categorical logic, I suppose that is your prerogative.

Again, the error is more apparent if we make the quantifications explicit. "All A is B" is equivalent to "For all x, if x is A then x is B"; and "Some A is B" is equivalent to "There exists an x, such that x is A and x is B." Since the former is a hypothetical proposition, a second premise is required in order to derive the latter conclusion; namely, "There exists an x, such that x is A."

It does, though. It's the same as "All silver toasters are toasters. All silver toasters are silver. Therefore some toasters are silver." — Terrapin Station

No, it doesn't follow otherwise one could not posit a counterexample. The universal quantifier does not imply existence, this is a known fact about the theory of quantifiers. Just think about an obvious example.

All chimeras are animals. All chimeras are magical. Therefore some animals are magical.

But we know no animal is magical. If there is a known counterexample to an inference it cannot be a valid argument form. It doesn't preserve truth in all models.