The first two premises are about a conception; they're a priori claims about how you're using terms. They're not about the external world. — Terrapin Station

Well yes,that doesn't make them untrue in the external world, because then they'd be false which sounds incorrect. The terms have to have a definition.. And of course the argument uses variables. I'm just taking the general form of the argument and replacing the variables with consistent values across the different propositions.

I must be missing something obvious here or helse we just have different views about logical consequence. Cheers.

The problem is the premises are true. Are you seriously denying that all winged horses are horses or that they have wings? — MindForged

No.

Otherwise you're flat out wrong because no winged horses actually exist. — MindForged

That only contradicts the conclusion if the conclusion implies that winged horses actually exist. But that alleged implication is exactly what I'm calling into question. It certainly isn't explicit anywhere in the argument, as worded. It's your interpretation. Is your interpretation the only possible interpretation? No. It doesn't take a genius to think of other ways of interpreting the conclusion, as it is worded, which do not necessarily imply actual existence.

The article I quoted does support me because the real world is a model in which the argument is proven to not be truth preserving. It goes from true premises to a false conclusion. The idea that the actual world doesn't count is absurd. We use logic to come to true conclusions about the actual world all the time. — MindForged

Only, it seems, if you equivocate between the premises to the conclusion. Like Terrapin said, you switch domains partway through the argument. You interpret the premises to be about an abstraction and about categories or sets, yet you interpret the conclusion to be about actual flying horses existing in the real world. It seems to me that it's your interpretation that's the problem, not the argument itself. You aren't interpreting it charitably.

Read it again. "Therefore some horses have wings" uses existential quantification, that's what "some" is translated as in formal logic. I'm quoting myself correctly. There's no argument here, the argument is considered invalid by logicians for exactly this reason. It does not preserve truth in all models. — MindForged

I don't need to read it again, you need to pay closer attention to what I'm saying. I told you that there is no "There is" contained in the wording of the argument, and that's true. I also said that you're reading that into the argument, which is also true. You can't fault me here.

That you can show me systems of logic where "some" is interpreted as an existential quantifier doesn't address the issue. Does it have to be interpreted in that way? If so, why? Is that the best or most charitable way to interpret the argument? If so, why?

The wording is only ambiguous if you don't interpret the logical terms as they standardly are done. "All" is universal quantifying, "some" is existential quantifying. — MindForged

Then my queries would be regarding what's standardly done. Do you see that this is just kicking the can down the road?

You cannot validly move from quantifying over a set to saying the set has members who satisfy the conditions to be part of the set. — MindForged

Why not? My understanding is that you say that this causes problems if you go by an interpretation that necessitates actual existence. But could it not be that the problem is with this interpretation?

That has to be an extra premise otherwise it commits the existential fallacy. Unless I'm much mistaken, this is the exact argument Russell gives to show why modern logic does not admit this as a valid form. — MindForged

It is possible that Russell is wrong, unthinkable as that might seem, yes? Maybe we could avoid the fallacy altogether with a different interpretation. Is Russell's the only interpretation? Are there no competing interpretations?

The truth-maker for a conclusion is whether the conclusion follows from the premises. — Terrapin Station

Right; and in modern deductive logic, the conclusion "Some B is C" does not follow from the premises "All A is B" and "All A is C," since a universal proposition does not entail that the categories corresponding to its terms each have at least one member. An additional premise is required--"Some A is A." In Aristotelian deductive logic, that additional premise is effectively stipulated from the outset by the rule that "All A is B" is only true if "Some A is A" is also true.

More perspicuously, the conclusion "There exists an x, such that x is B and x is C" does not follow from the premises "For any x, if x is A then x is B" and "For any x, if x is A then x is C," since a hypothetical/conditional proposition does not entail the existence of anything in the universe of discourse. An additional premise is required--"There exists an x, such that x is A."

More perspicuously, the conclusion "There exists an x, such that x is B and x is C" does not follow from the premises "For any x, if x is A then x is B" and "For any x, if x is A then x is C," since a hypothetical/conditional proposition does not entail the existence of anything in the universe of discourse. An additional premise is required--"There exists an x, such that x is A." — aletheist

So, here, an additional premise about existence is required for a conclusion about existence, yes? But what if the conclusion in the horsey argument isn't about existence? Mind=blown? :grin:

That only contradicts the conclusion if the conclusion implies that winged horses actually exist. But that alleged implication is exactly what I'm calling into question. It certainly isn't explicit anywhere in the argument, as worded. It's your interpretation. Is your interpretation the only possible interpretation? No. It doesn't take a genius to think of other ways of interpreting the conclusion, as it is worded, which do not necessarily imply actual existence. — S

That's literally the standard theory of quantifiers used in virtually every modern deductive logic, whether classical, intuitionist, paraconsistent, or many-valued. That lends way more credence to the standard formalization of these terms than to an unspecified one which on its face leads to false conclusions. "Some" implies existence, because natural language terms like "some" are translated as "There is at least one blah blah such that blah blah". It's not that you cannot interpret these terms another way, it's that you cannot do so in a way that will actually capture a consistent formalization of how these linguistic features are used in reasoning. If I say "Some apples are delicious" everyone is going to agree I'm talking about actually existing apples and that there is at least one apple that is delicious because otherwise the assertion would be rendered false.

Only, it seems, if you equivocate between the premises to the conclusion. Like Terrapin said, you switch domains partway through the argument. You interpret the premises to be about an abstraction and about categories or sets, yet you interpret the conclusion to be about actual flying horses existing in the real world. It seems to me that it's your interpretation that's the problem, not the argument itself. You aren't interpreting it charitably. — S

I don't see where the hang -up is. Yes I'm switching domains, that's the point. That's what makes it fallacious. And how do we make it (the argument form) valid? As myself and aletheist (above) have said, you have to add a fourth premise asserting the existence (the actual existence) of a member of the set. Talking about sets of abstractions and then immediately concluding objects outside that domain of discourse populate that set does not follow in standard logic systems.

I told you that there is no "There is" contained in the wording of the argument, and that's true. I also said that you're reading that into the argument, which is also true. You can't fault me here. — S

"Some" is translated as "there is". There's only two kinds of quantifiers, and that's one of them. I can't be reading that into the argument if that's how virtually every logician is going to translate that argument.

That you can show me systems of logic where "some" is interpreted as an existential quantifier doesn't address the issue. Does it have to be interpreted in that way? If so, why? Is that the best or most charitable way to interpret the argument? If so, why? — S

The reason is has to be that way is because sets are not by default populated by members (sans the empty set). The only way for a set to be populated is the assert that it is so. The problem is the Darapti argument goes from talking about sets to concluding the set actually has members. It would be like making a conditional statement and then concluding the consequent without asserting or denying either of them first, e.g. If the Sun is out, then it is hot. Therefore it is hot. It's the same kind of mistake.

Then my queries would be regarding what's standardly done. Do you see that this is just kicking the can down the road? — S

You're not articulating an alternate theory (nor its pitfalls) or even showing that the standard theory comes out wrong. So I don't see how this is a real possibility to consider.

Why not? My understanding is that you say that this causes problems if you go by an interpretation that necessitates actual existence. But could it not be that the problem is with this interpretation? — S

Existential quantification asserts existence (or at least set membership; I'll skate by the latter). If you believe something different ought to be done then you'd need a new logic, or more likely, you'd just add a new quantifier but ti would be useless since it's not adding any new kind of way of talking about things. As the SEP says on it's page on quantifiers:

Much of contemporary ontology builds on the assumption that existence is to be understood in terms of quantification: in a slogan, to exist is to be something. Ontology is largely concerned with the domain of the existential quantifier. This assumption can be traced back to the work of Frege and Russell, both of whom analyzed quantification in terms of predication, and plays a crucial role in Quine’s admonition to transform ontology into the study of the ontological commitments of our global theory of the world regimented in the language of quantificational logic and identity.

It is possible that Russell is wrong, unthinkable as that might seem, yes? Maybe we could avoid the fallacy altogether with a different interpretation. Is Russell's the only interpretation? Are there no competing interpretations? — S

It's not just Russell, but Russell gave the clearest statement as to why. Russell can be wrong, but since virtually every logic understands quantifiers the same or in very similar ways that don't differentiate on this point, I don't see the critical error in just being careful how you use quantifiers.

Right; and in modern deductive logic, the conclusion "Some B is C" does not follow from the premises "All A is B" and "All A is C," since a universal proposition does not entail that the categories corresponding to its terms each have at least one member. An additional premise is required--"Some A is A." In Aristotelian deductive logic, that additional premise is effectively stipulated from the outset by the rule that "All A is B" is only true if "Some A is A" is also true. — aletheist

Hmm. What I'm wondering is which system makes more sense? Presumably the more modern one, as it's supposed to be an improvement.

So, with Aristotelian logic, if it's true that all unicorns have a horn, then that entails that at least one unicorn has a horn? I don't see a problem with that unless you bring existence into it. If you think that it implies a unicorn when there aren't any, then I can understand why you'd think that that's a problem. I gather that this was Russell's view, but I'm not sure I agree with Russell.

But what if the conclusion in the horsey argument isn't about existence? — S

I don't see a problem with that unless you bring existence into it. — S

In both modern and Aristotelian logic, every particular proposition (such as "Some B is C") is about existence in the universe of discourse. In Aristotelian logic, every universal proposition is also about existence in the universe of discourse, since "All A is B" is only true if "Some A is A" is true. The universe of discourse is usually understood to be the actual world, but a different one can be stipulated.

Returning to the example, "Some horses have wings" is true only if the universe of discourse is a fictional world that includes winged horses. The additional premise, "Some winged horses exist," serves the purpose of stipulating just such a universe of discourse.

In both modern and Aristotelian logic, every particular proposition (such as "Some B is C") is about existence in the universe of discourse. In Aristotelian logic, every universal proposition is also about existence in the universe of discourse, since "All A is B" is only true if "Some A is A" is true. The universe of discourse is usually understood to be the actual world, but a different one can be stipulated.

Returning to the example, "Some horses have wings" is true only if the universe of discourse is a fictional world that includes winged horses. The additional premise, "Some winged horses exist," serves the purpose of stipulating just such a universe of discourse. — aletheist

Okay. I think I've wrapped my head around that. So, going back to this:

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. — MindForged

Setting aside the premises, which I accept are true, and which, in accordance with modern logic, are not about existence in the universe of discourse, can we not avoid the problem that MindForged seems to have gotten himself into if we simply think of the conclusion as relating to a possible or fictional world where it's true that some horses have wings? I mean, isn't that more charitable than assuming that an argument about winged horses is about the real world?

Wait, wasn't this basically Meingong's proposed solution? What am I getting myself into!? :scream:

Alright, thanks. I'm beginning to come around more to your way of seeing things now, but I need to mull it over.

MindForged seems to have gotten himself into if we simply think of the conclusion as relating to a possible or fictional world where it's true that some horses have wings? I mean, isn't that more charitable than assuming that an argument about winged horses is about the real world? — S

If one doesn't stipulated what domain of discourse one is speaking in (or if the argument doesn't make it obvious) then the assumption is that they're presenting a model of the real world. Like if I say "If Clinton had won the election, then x, y, z" I'm clearly talking about counterfactuals that might have happened. But since horses exist one has to be careful how they shift about the terms they're talking about because it can lead to this problem.

I was probably a bit abrasive. I'm just not a good communicator probably. I'll leave it to aletheist, lol.

If one doesn't stipulate what domain of discourse one is speaking in (or if the argument doesn't make it obvious) then the assumption is that they're presenting a model of the real world. Like if I say "If Clinton had won the election, then x, y, z" I'm clearly talking about counterfactuals that might have happened. But since horses exist, one has to be careful how they shift about the terms they're talking about because it can lead to this problem. — MindForged

Yeah, that makes sense.

I was probably a bit abrasive. I'm just not a good communicator probably. I'll leave it to aletheist, lol. — MindForged

Abrasive? No worries, lol. I'm not exactly known around these parts for being all cuddly and shit.

If I say "Some apples are delicious" everyone is going to agree I'm talking about actually existing apples and that there is at least one apple that is delicious because otherwise the assertion would be rendered false. — MindForged

Nope, you've lost me there. Are you saying that I can't say "some unicorns are black" and still be understood?

Unless you explicitly say you're speaking about some type of fictional scenario, no otherwise the statement is clearly false. It's understood as saying "There exists at least one unicorn and it is black". I didn't say it couldn't be understood but that it couldn't be rendered true under how we understand terms like "some" to work in normal contexts.

I think that that's covered by the first quote in the comment above yours. I would charitably assume that you were talking about a fictional or hypothetical or conceptual domain.

Hold up, you previously made the point that if the language makes it obvious that it's not the real world which is the domain of discourse, then you shouldn't assume that it is. What does the term, "unicorn", suggest to you? Does he really need to explicitly say that he's talking about fiction?

And I think that it's fair to say that this isn't a normal context.

Unless you explicitly say you're speaking about some type of fictional scenario, no otherwise the statement is clearly false. It's understood as saying "There exists at least one unicorn and it is black". I didn't say it couldn't be understood but that it couldn't be rendered true under how we understand terms like "some" to work in normal contexts. — MindForged

But isn't that an empirical statement? What evidence/authority are you referring to when you say "It's understood as saying "There exists at least one unicorn and it is black""? I'm older than I care to remember and in my life experience people tend to guess the context from the language game. Unless you've got a longer life experience than me (unlikely, I'm ridiculously old), or some large sample evidence, I don't really see how you're in a position to say how an expression is 'normally' understood.

I think that that's covered by the first quote in the comment above yours. I would charitably assume that you were talking about a fictional or hypothetical or conceptual domain. — S

Yes, I thought for a while that the discussion immediately below the quote I highlighted actually answered my query, but on reflection I don't think it quite does, pretty much for the reason you've just posted. It seems to me that we have little trouble determining parameters of meaning from context and so I don't really see a justification for some system of logic to claim it represents our 'usual' understanding of the expression in this manner.

Hold up, you previously made the point that if the language doesn't make it obvious that it's not the real world which is the domain of discourse, then you shouldn't assume that it is. What does the term, "unicorn", suggest to you? Does he really need to explicitly say that he's talking about fiction? — S

What I said was that saying there is something that is such and such does not follow from talking about a category of things. Of course you can tweak what you mean but that's why I said one has to be careful or you'll make an invalid argument. Yes we know that people nowadays don't believe in unicorns and pegasi and that in normal speech (usually...) it's assumed to be fictional. But then if you formalized the argument from before on this basis, it would essentially be done in the way that affirmed what I'm saying, like so:

All fictional winged horses are horses
All fictional winged horses have wings.
There is at least one fictional winged horse.
Therefore some fictional horses have wings.

That's valid, but only because we've established a domain of things where it's clear the state of these existing things are different (they're fictional) and we've assumed there is some fictional pegasi. If you use an existential quantifier you should do so in a way that's clear about what you mean. It's an issue that exists, some have suggested adding an explicitly fictional quantifier for these situations. But the main issue is the argument form, not whether people believe the things they're talking about are real.

Maybe a real world example is needed so the fiction thing isn't a hang-up.

All forest people live in the forest.
All beasts live in the forest.
Therefore some forest people are beasts.

Even if the conclusion might be true today or at some point in the past, the issue is really how this is structured. We need to assert that there really is at least one such beastial person first. And that's the premise that will show whether or not the conclusion follows.

But isn't that an empirical statement? What evidence/authority are you referring to when you say "It's understood as saying "There exists at least one unicorn and it is black""? I'm older than I care to remember and in my life experience people tend to guess the context from the language game. Unless you've got a longer life experience than me (unlikely, I'm ridiculously old), or some large sample evidence, I don't really see how you're in a position to say how an expression is 'normally' understood. — Isaac

I believe "There exists" is usually pretty unambiguous. Obviously a context can change that but it doesn't make it a valid argument to move from a category of things to saying there is some existing thing that's part of that category.

I believe "There exists" is usually pretty unambiguous. — MindForged

But I wasn't asking about "there exists". You said that "some x is y" is normally parsed as "there exists some x that is y". I'm saying that's an empirical claim. I'm not at all convinced that "some x is y" is 'normally' parsed as anything at all, but rather parsed differently depending on the context, which itself cannot even be determined from the sentence alone.

You said that "some x is y" is normally parsed as "there exists some x that is y". I'm saying that's an empirical claim. — Isaac

It is normally parsed that way in formal logic, both modern and Aristotelian, which is the universe of discourse for this thread. In fact, the OP explicitly stipulated modern formal logic as explicated by Bertrand Russell.

Yes I'm aware of that, I only commented because the thread seems to have moved to a claim that this somehow reflects normal use.

Maybe a real world example is needed so the fiction thing isn't a hang-up. — MindForged

Here's a familiar real world example that follows Darapti:

(1) All black swans are black
(2) All black swans are swans
(3) Therefore some swans are black

Each of those premises are true, so it is not a counterexample to the validity of Darapti.

However suppose this argument were considered prior to the discovery of black swans in Australia (or, alternatively, if the color was red).

In modern logic, the first two premises would be considered true by definition. The third premise would be considered false. So the argument would be considered invalid.

However in Aristotelian logic, since the subject (black swans) had not been observed and thus thought to be non-referring, all three premises would be considered false. So it would not be a counterexample to the validity of Darapti.

It wouldn't be a counterexample to Darapti in Aristotelian logic, but that's because it makes the assumption that non-referring terms are to be disallowed in logic. And that seems strange nowadays since logic is viewed as being a very broad (maybe the broadest) abstract generalization and the properties we find in this 'follows from' relation. It seems like logic in the modern sense ought to work just fine whether or not the things actually exist, much in the way that mathematics is taken to be fine whether or not it corresponds to any sort of physical phenomenon.

OK, but how about the following example:

(1) All Alice's children have three legs
(2) All Alice's children are children
(3) Therefore some children have three legs

If Alice has no children then (1) is true (vacuously). Is that not also strange?

That seems to me a reason to treat non-referring terms separately.

In the example:
All winged horses are horses,
All winged horses have wings,
Therefore some horses have wings.

If it is assumed that there are winged horses, then the conclusion is a tautology (which is disconnected from whether you have seen or imagined a winged horse or not).

If it is assumed that there is no winged horse, then the conclusion is false.

Without either assumption, one might say that the conclusion doesn't follow since an additional assumption is required to decide the truthness of the conclusion. Or one might use a logic where the conclusion is true from the two premises. Fundamentally logic is just a tool, what matters is whether the system of logic used is helpful to make sense of everything else.

But there is a difference between the title of this discussion and the example about winged horses.
The title is: All A is B, all A is C, therefore some B is C
The example is: All A is B, all A have C, therefore some B have C

Following the title, an example would be:

All winged horses are horses,
All winged horses are animals,
Therefore some horses are animals.

Or

All winged horses are horses,
All winged horses are imaginary creatures,
Therefore some horses are imaginary creatures.

There, regardless of whether there are winged horses or not, the conclusion is easier to accept as true. Then one might use a system of logic where "Some B is C" is a logical consequence of "All A is B" and "All A is C".