All A is B and all A is C, therefore some B is C
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"?
(?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"?
Comments (116)
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.
What do you mean by "empty terms"? Are you refering to arguments with undefined variables?
Quoting 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...
Terms without referents. No pegasi exist, so pegasus is an empty term.
Quoting 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.
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".
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.
No, that is invalid. It becomes more obvious if we reformulate the two propositions as follows.
The first proposition clearly does not entail the second.
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?
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".
Quoting Ben92
Yeah, I read it and i was kinda doubtful too. For me it doesn't make any sense.
Quoting 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?
Quoting 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?
Quoting MindForged
Well, actually he says it on Logic and Knowledge (p. 229), and I think it's kinda weird.
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.
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."
Correct.
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.
Quoting 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:
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.
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 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.
Quoting 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."
It does, though. It's the same as "All silver toasters are toasters. All silver toasters are silver. Therefore some toasters are silver."
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.
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.
Quoting 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."
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.
For all x, if x is A then x is B by virtue of?
We know that no animal is magical in what context?
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.
Quoting 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.
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.
Sorry, I do not understand this question.
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?
The problem with "yes" is that logic has nothing to do with claims about what actually exists.
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.
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.
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."
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.
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.
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.
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.
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.
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.
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.
Which isn't itself logic. Hence the wording.
You're not making sense. Previously you said this:
Quoting 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).
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.
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.
Quoting 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
Right, the problem is the form in that the form doesn't guarantee that the conclusion is true.
That doesn't mean that the conclusion can't be true. If All guitars are Gibsons and All guitars are Les Pauls, then some Gibsons are Les Pauls. So the conclusion is true in that case. But it's not impossible for the premises to be true and the conclusion false, because we can formulate a version of the argument where the conclusion is that some orange is bouncy.
Quoting MindForged
I was explaining this to aletheist, and I can explain it to you, but it will take a few steps. First, what is the domain you're dealing with in the example you have in mind?
It is a valid argument form in Aristotelian logic because statements of the form All A is B must have one or more instances in order to be true (just as with Some A is B).
To give an example using real things:
(1) All tall trees on my property are tall.
(2) All tall trees on my property are trees.
(3) Therefore some trees on my property are tall.
If I have no tall trees on my property then (1) and (2) are not true. And there are no cases where (1) and (2) are true and (3) is false, so it is a valid argument.
Wikipedia gives the rationale for Aristotle's choice.
Quoting Existential import - Wikipedia
But deductive validity requires that the form must guarantee deriving only true conclusions from true premises.
Quoting Andrew M
Right, but Aristotle stipulated that additional premise; as your Wikipedia quote states, it was "a thoughtful choice, not an inadvertent omission."
Sure but as I said empty terms show this to be improper. As your quotesaid, Aristotle stipulated that logic was to regard known existing things and thus empty terms were off the table by fiat, not by argument. But this is kind of ridiculous. It's just true that, for example, mathematicians both use formal logic and do not assume that every entity they quantify over is instantiated. Thus if we followed Aristotle we'd handicap mathematics in pretty ridiculous ways. Logic ought to work just just the same regardless of whether or not there are instances of the things referenced.
But that's exactly the point. An invalid argument doesn't mean the conclusion is false, it means the form of the argument is such that the truth of the premises does not necessarily entail the truth of the conclusion. And as Darapti does not guarantee it's rightly deemed invalid.
Right not necessarily, but the conclusion does follow in your example. Your example is actually a bit different structurally, because you're saying that all F that G can also be predicated of are F, and all F that G can also be predicated of are G . . .
Right. I wasn't arguing that it was valid, and I explained why it's not. What I argued is that people gave examples where the conclusion did follow. They didn't give examples where the conclusion doesn't follow.
I don't think he's stipulating an additional premise. He's instead saying that predication is only applicable when the subject term refers to something that exists.
That is, for Aristotle, vacuous statements are neither true nor false since the presupposition that the subject term refers fails.
Quoting MindForged
Can you give any examples where this issue would be important?
Quoting MindForged
To me it seems analogous to "The King of France is bald". Note that Russell and Strawson disagreed on how to treat this kind of statement, with Strawson defending the view that the presupposition fails (and thus the statement is neither true nor false).
Try that one mo 'gin in Engrish.
Any time one uses the universal quantifier I would think. "For each natural number n, "n x n" = "n + n". That does not assume there is some existing n, it's just a statement about how to define an abstract operation, whether or not that holds in the physical world. If quantifying did assume existence we'd expect to see, I dunno, every number be instantiated by some collection of elements or weird alegbraic models have mapped onto real things, or every kind of geometry have a corresponding universe modeled on it, wouldn't we?
Quoting Andrew M
Well OK, there is a debate here but I'm of the view that one ought to try and use as few different logics as possible. I'm not familiar with what Strawson said here but Russell's general route seems correct (use classical logic if it can give a good answer), though I don't think I actually accept his theory of descriptions to resolve these.
Why would you define an abstract operation, and moreover assign "true" to it (assuming we can even really make sense of that), if it can't be satisfied by anything we plug into the variable (in whatever domain you're working in)?
What? It seems to me that if the premises are true, then the conclusion must also be true. It logically follows.
Having read a little more of the discussion, I think I understand some of what has been said about Russell's view, and maybe he was indeed onto something, but if this is a consequence, it's very counterintuitive.
Quoting Terrapin Station
That's what I was thinking too. There seems to be a disconnection involved in his assessment.
But that wasn't in the argument, and it doesn't seem appropriate to interpret the argument in that way.
:lol:
It took me a while and several reads to figure out what he was saying, but I think I got it, and it's a lot clearer if you break it up:
No, the structure was:
[I]All F that E can be predicated of are E
All F that G can be predicated of are G[/I]
Unless you can point to where the winged horses are you cannot say it's valid. If the conclusion of an argument is false in spite of true premises, then the argument for is invalid. There is thus a model where truth is not preserved, that's the definition of an invalid argument.
Quoting S
"Therefore some horses have wings" has an existential operator, how is that inappropriate to point out when it creates a false conclusion from true premises?
I didn't say it couldn't be satisfied, what I said was that quantifying over all the elements of a set does not entail the set has members who exist (that's a separate set). Winged horses could exist but they do not. So given we know this we infer the invalidity of form. The reason I keep repeating this was because way back you seemed to be saying it was valid:
Quoting Terrapin Station
I'm not clear on how the example applies. There's one natural number that satisfies that identity (the number 2). But even if the result were an empty set, I don't see any predication of its members analogous to "all winged horses are horses".
Anyway, what I'm saying is there doesn't need to be anything that instantiates this for us to reason about it. For Aristotle, logic is supposed to be used for things known to exist. In the above, "For every" is just the universal quantifier yes? That's before the predication. But surely it's fine to reason in mathematics without assuming something in the physical world corresponds to this? (Obviously it does in this case but pure mathematics isn't guaranteed to)
That's not how validity works.
Quoting MindForged
Soundness is about actual truth or falsity. Validity is about assumed truth or falsity. In your example syllogism, under the assumption that the premises are true, it follows that the conclusion is true. Hence, the syllogism is valid.
Quoting MindForged
You're jumping ahead based on your own assumptions. I'm questioning these very assumptions of yours. You're begging the question.
Quoting MindForged
Why do you think that it must be interpreted in that way, as implying existence? In English, as opposed to symbolic logic, and worded as such, it is more ambiguous than you're making out.
I grant that Russell might have a good point. But you are not making a very convincing case.
In which case saying anything about winged horses puts us in the domain of things that we're imagining. If we change domains midstream we're equivocating.
That's fine, as far as I can tell. Aristotle was fine with abstractions and hypotheticals, as long as their use was intelligible. The only issue I've raised is with vacuous statements. But there doesn't seem to be a vacuous statement in your example.
um, no. Validity is defined as truth preservation over all cases. As we know the first two premises of the argument are true, yet the conclusion is false, we know the issue has to be with the form of the argument. It's invalid, in other words. As the SEP article in logical consequence says:
[quote='SEP']
The model-centered approach to logical consequence takes the validity of an argument to be absence of counterexample. A counterexample to an argument is, in general, some way of manifesting the manner in which the premises of the argument fail to lead to a conclusion. One way to do this is to provide an argument of the same form for which the premises are clearly true and the conclusion is clearly false. Another way to do this is to provide a circumstance in which the premises are true and the conclusion is false. In the contemporary literature the intuitive idea of a counterexample is developed into a theory of models. Models are abstract mathematical structures that provide possible interpretations for each of the non-logical primitives in a formal language. Given a model for a language one is able to define what it is for a sentence in that language to be true (according to that model) or not. So, the intuitive idea of logical consequence in terms of counterexamples is then formally rendered as follows: an argument is valid if and only if there is no model according to which the premises are true and the conclusion is not true. Put in positive terms: in any model in which the premises are true (or in any interpretation of the premises according to which they are true), the conclusion is true too.
[/quote]
Quoting S
It does not follow. If I assume all winged horses are horses and all winged horses have wings, we cannot infer that some horses have wings because we see in the actual world that the first two premises are true yet there are no winged horses. This argument form is known to be invalid in classical logic, it commits the existential fallacy. To make it valid you need a fourth premise that explicitly states that there is at least one existing winged horse. But then we see exactly why the original argument form was invalid.
Quoting S
What? How am I begging the question when I just stating the definition of semantic logical consequence?
Quoting S
Because "There is" is more or less always interpreted as an existence claim. We're talking about formal logic, not informal natural language reasoning.
That's more or less what I'm saying. That's what makes it invalid. The class can't be assumed to have members unless we state that it does. Aristotle didn't see this as a problem because he thought logic ought only consider classes with known existing members, but that's not assumed in mathematical reasoning nowadays. It's too limited.
Quoting MindForged
You're not even quoting the wording of the argument, which is funny, given that you're the one who wrote it. There is no "There is" contained in the argument. You're reading that into it, which is the problem.
I grant that there might be a version of the argument where what you're saying applies, but that's a different argument to the one that you presented, and I don't agree that your interpretation is the only possible way that the argument can be interpreted. The wording is ambiguous.
"A class having members" when we're doing mathematics is a matter of whether we're thinking about things in a particular way or not. If you're conceiving of some class with particular properties, especially so that you could utter a statement a la "All x are F" and assign "T" to it, then that class has members, because the domain is what we're imagining, and an xF exists by virtue of conceiving of it (it exists as a conception, which is the domain we're dealing with).
The problem is the premises are true. Are you seriously denying that all winged horses are horses or that they have wings? If so then it has to be a terminological disagreement. Otherwise you're flat out wrong because no winged horses actually exist. 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.
Quoting S
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.
Quoting S
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. 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. 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.
The truth-maker of any statement in logic is never going to be whether something obtains empirically.
The truth-maker for a conclusion is whether the conclusion follows from the premises. What's the case in the actual world has zilch to do with it.
Do you just object to model theory? It would be the same with just the syntax route.
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.
What you're quoting is about plugging values into variables, by the way. You're winged horse argument isn't variables.
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.
No.
Quoting 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.
Quoting 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.
Quoting 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?
Quoting MindForged
Then my queries would be regarding what's standardly done. Do you see that this is just kicking the can down the road?
Quoting 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?
Quoting 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?
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."
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'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.
Quoting 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.
Quoting 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.
Quoting 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.
Quoting 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.
Quoting 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:
Quoting 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.
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.
Quoting 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.
Okay. I think I've wrapped my head around that. So, going back to this:
Quoting 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 [i]winged horses[/I] is about the real world?
Wait, wasn't this basically Meingong's proposed solution? What am I getting myself into!? :scream:
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.
Yeah, that makes sense.
Quoting MindForged
Abrasive? No worries, lol. I'm not exactly known around these parts for being all cuddly and shit.
Nope, you've lost me there. Are you saying that I can't say "some unicorns are black" and still be understood?
And I think that it's fair to say that this isn't a normal context.
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.
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.
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.
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.
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.
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.
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.
(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.
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".