"All A is B" is equivalent to "For all x, if x is A then x is B" — aletheist

For all x, if x is A then x is B by virtue of?

But we know no animal is magical. — MindForged

We know that no animal is magical in what context?

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. — Nicholas Ferreira

On this basis one can never ever at all give a truth value to any proposition that does not contradict itself. In which case you've lost the ability to use logic for anything useful. We have abundant evidence that not only would a winged horse be biologically silly, but that they don't exist. We've been everywhere such a creature could be on Earth and it's not here. Unless you redefine what a horse is (which would lose the argument) then this is known to be true unless you think knowledge is impossible, which would be another problem. It doesn't make sense to me to be skeptical to the degree that the normal mode of discourse is obscured.

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? — Nicholas Ferreira

"No Greek is man" is just to say that it is not the case that some Greek exists and is a man. That's an existential quantifier, not a universal one.

We know that no animal is magical in what context? — Terrapin Station

I'm not sure I even understand you here. Magical in the sense that it exists or acts in some manner inconsistent with the laws of the physical world. The content is really irrelevant though. The argument does not come out valid whether you do logical metatheory semantically or syntactically.

For all x, if x is A then x is B by virtue of? — Terrapin Station

Sorry, I do not understand this question.

I'm not sure I even understand you here. — MindForged

When you say that you know that no animal is magical, you're talking about real animals, real properties, etc., right?

What makes the claim the case that if x is A then x is B?

Yes. As I said, we know the argument form is invalid but Darapti was regarded by Aristotle and the medieval logicians as valid. You cannot validity make an inference from a domain of things to that domain having members without stating it as a separate premise.

The problem with "yes" is that logic has nothing to do with claims about what actually exists.

But logic is used to analyze what actually exists and infer things about them. If I am taller than Terrapin, and Terrapin is taller than aletheist, then I am taller than aletheist. I am (for arguments sake) taller than Terrapin, therefore I am taller than aletheist. That's a valid inference.

But if what is inferred to exist or be true is done on a basis which yields a false conclusion from true premises, that means the argument form was not a valid one. That's the very definition of (semantic) logical consequence.

What makes the claim the case that if x is A then x is B? — Terrapin Station

I still do not understand the question. We are discussing formal logic, what true conclusions we can--or rather, cannot--derive from that proposition, assuming that it is true.

"No Greek is man" is just to say that it is not the case that some Greek exists and is a man. That's an existential quantifier, not a universal one. — MindForged

It's a negative existential one, but any universal proposition can be transformed into an existential one.
(∀x)[Px] ↔ ¬(∃x)[¬Px]

Correct--we can derive "It is not the case that some A is not B" from "All A is B." However, we still cannot derive "Some A is B" from either of these without the additional premise, "Some A is A."

Correct--we can derive "It is not the case that some A is not B" from "All A is B." However, we still cannot derive "Some A is B" from either of these without the additional premise, "Some A is A." — aletheist

But isn't implicit on any argument that some A is A? I mean, why do we need to put the identity law in a premise? And I really can't understand why "Some A is B" cannot be infered from "All A is B". :(

The law of identity is "All A is A." We cannot derive "Some A is A" from that, either. Again, in modern deductive logic it is always a fallacy to derive a particular conclusion from universal premises; such an inference is not necessarily truth-preserving.

Ok, I understood, thanks for the explanation and the link :D
Also, do you have any recomendation of book to read specifically about this? I'm reading Introduction to Logic by irving M. Copi, and he talks about predicative logic, but doesn't says anything about this kind of fallacy.

No particular book recommendation, sorry. Needless to say, there is a lot of helpful material online.

So it might just be that I need to go back and read more Russell, but I think he would view a sentence like 'unicorns have wings' as true because unicorn is an empty set rather than because of qualities about the unicorns. It might not be a great example for meto use since I know Russell ran into a lot of problems around so called 'fictional truths'.

The real point I was trying to get atwas that logically a sentence of the form 'all Ps are Qs' is always true if there are no Ps (so in effect 'Q' here cloukd be anything without affecting the truth value of the sentence). This doesn't hold for 'a P is Q' since it requires a P to exist.

(∀x)[Ax⊃Bx]
(∀x)[Ax⊃Cx]
∴(∃x)[Bx⊃Cx

1. (x)(Ax -> Bx)
2. (x)(Ax -> Cx)....prove (Ex)(Bx & Cx)
3. Ad -> Bd................1 UI
4. Ad -> Cd................2 UI
We can't conclude (Ex)(Bx & Cx)

Now, if the additional information is Ae i.e. at least one thing, e, is an A then
3. Ae
4. Ae -> Be.............1 UI
5. Ae -> Ce.............2 UI
6. Be.......................3, 4 MP
7. Ce.......................3, 5 MP
8. Be & Ce..............Conj
9. (Ex)(Bx & Cx)......8 EG


The problem is universal quantifiers don't make existential claims while (Ex) claims that "at least one thing" exists.

I still do not understand the question. We are discussing formal logic, what true conclusions we can--or rather, cannot--derive from that proposition, assuming that it is true. — aletheist

Either we're imagining As with property B in a domain, or there possibly are As in a domain, independent of our imagining.

If we're imagining As, we're imagining them to have property B, then it makes no sense to also imagine that there are no As in that domain. For one, we've already imagined As in that domain in order to imagine them having property B. If we imagine a domain with no As (if that's even really possible, it might not be), then there no As in that domain to have any property whatsoever.

If we're talking about a domain where As can obtain independent of our imagining, then we can't--especially logically--say what properties the independent As would have at all, as we could always be contingently wrong about that.

You could say "I'm only going to call x 'A' if x has property 'B'," but then we're talking about something we're imagining (in other words, these sorts of statements, statements about "essences" and the like, are statements about how we think about something), and we're stuck with the same problem as above. Our imagining is the domain in question, in which case we've imagined As with property B, and it doesn't make sense to say that we've both imagined that and imagined that same domain without any As.

But logic is used to analyze what actually exists and infer things about them. If I am taller than Terrapin, and Terrapin is taller than aletheist, then I am taller than aletheist. I am (for arguments sake) taller than Terrapin, therefore I am taller than aletheist. That's a valid inference. — MindForged

Logic is about the relationships of the statements qua statements. It can't tell you what's true of the actual world. It can suggest what's true of the actual world just in case such and such is true, but it can't tell you that such and such is true. You have to look outside of logic for that. Logic is only about relational structure per se, and really only about how we think about that on an abstract level.

Either we're imagining As with property B in a domain, or there possibly are As in a domain, independent of our imagining. — Terrapin Station

Whether we are imagining them or not, the issue is whether there are any As at all. The proposition "All A is B," or equivalently "For all x, if x is A then x is B," takes no position on this. It simply states that if there are any As, then all of them are Bs. Hence it is not deductively valid to derive the proposition "Some A is B," or equivalently "There exists an x, such that x is A and x is B," since this entails that there is at least one A--a conclusion that was not included in the premise.

Either you're ignoring what I just explained re what it is for there to be any As at all in a domain, or you didn't understand what I wrote . . . which could be my fault, because maybe I wasn't very clear, but it doesn't help to just proceed so that I'd have to basically write the same thing again.

Logic is about the relationships of the statements qua statements. It can't tell you what's true of the actual world. It can suggest what's true of the actual world just in case such and such is true, but it can't tell you that such and such is true. You have to look outside of logic for that. Logic is only about relational structure per se, and really only about how we think about that on an abstract level. — Terrapin Station

That's not right. I am looking outside the logic, so I am deriving "this is true in the actual world because such and such is true". Winged horses do not exist. Despite the truth of the two premises, the conclusion does not follow. Ergo there is at least one model where the inference does not preserve truth. So it cannot be a valid inference. Semantic logical consequence is not a new thing I'm making up.

I am looking outside the logic — MindForged

Which isn't itself logic. Hence the wording.

You're not making sense. Previously you said this:

It can suggest what's true of the actual world just in case such and such is true, but it can't tell you that such and such is true. — Terrapin Station

Which means that given the truth of the first two premises (All winged-horses are horses; All winged-horses have wings) and the falsity of the conclusion (Some horses have wings) that the inference is invalid. The logic cannot admit the inference because it does not preserve truth in all models.

A conclusion is true or false because of the structure of the argument. You don't determine if the conclusion is true or false by doing an empirical examination re whether there are any winged horses (in whatever domain).

No no no, the premises ARE true. That's what you keep ignoring. All winged horses are horses. So too do all winged horses have wings. An argument's structure is valid or invalid bases on whether it *preserves* the truth of the premises to the conclusion. So with the Darapti argument (All As are Bs; All As are Cs; Therefore some Bs are Cs) we go from true premises to a false conclusion. That is the very definition of an invalid inference. It's not a valid argument in classical logic, we know it fails to preserve truth in particular models.

So with the Darapti argument (All As are Bs; All As are Cs; Therefore some Bs are Cs) we go from true premises to a false conclusion. — MindForged

The reason that that argument wouldn't be valid would be that Bs and Cs can both be properties (or in other words, things "predicated of A") where it doesn't make sense to say that some of property B is (or has) property C. For example, if B is "orange" and C is "bouncy" (and As are bouncy orange balls) it doesn't make sense to say that some orange is bouncy.

That's not the case with your winged horse example. In the winged horse example, we're not positing properties where it doesn't make conceptual sense to say that one property somehow is or has the other property.

For example, if B is "orange" and C is "bouncy" (and As are bouncy orange balls) it doesn't make sense to say that some orange is bouncy. — Terrapin Station

What? The problem with that is the argument is just the form, not the truth value of the two premises. All bouncy orange balls are bouncy, all bouncy orange balls are bouncy, but that does not imply "some orange are bouncy", it doesn't follow. Of course the sentence doesn't make sense but the relevant logical issue is just the use of an invalid argument form.

In the winged horse example, we're not positing properties where it doesn't make conceptual sense to say that one property somehow is or has the other property. — Terrapin Station

I'm not sure I understand you here. How does it not make sense to say that a winged horse is a horse, that a winged horse has wings, or that some horse might have wings? I don't see the conceptual issue here, these seem like perfectly comprehensible properties some object might have even if they do not in fact have them