# What is a convincing explanation of how Russell's "golden mountains" argument is logically fallacious?

Here is the now famous passage in his book on Western philosophy where Bertrand Russell explains why Aristotle's position that the universal affirmative "All Greeks are men" implies the particular affirmative "Some Greek is a man" was mistaken:

The statement "all Greeks are men" is commonly interpreted as implying that there are Greeks: without this implication, some of Aristotle's syllogisms are not valid. Take for instance: "All Greeks are men, All Greeks are white, therefore some men are white." This is valid if there are Greeks, but not otherwise. If I were to say: "All golden mountains are mountains, all golden mountains are golden, therefore some mountains are golden," my conclusion would be false, though in some sense my premisses would be true. — Bertrand Russell, History of Western philosophy (1947)

Russell's argument seems dodgy and ultimately unconvincing to me but I can't seem to be able to pinpoint where he goes wrong.

I cannot see any problem with Aristotle's view that "All S's are P" implies "Some S is P".

So my question is:

What is a convincing explanation of how Russell's argument about the "golden mountains" syllogism is logically fallacious? (last edited Feb. 12, 2023)

Thank you to provide a reference, if any.

EDIT As to the previous question, "Does this syllogism by Russell show that Aristotelian logic doesn't work?", it is about Aristotle's syllogistic, while this question is whether there is any convincing argument that Russell's argument is wrong.

• The issue is with the so-called Existential import. In modern logic the statement "All unicorns are white" is vacuously true and from it we cannot derive the (false) "Some unicorn are white". Feb 7, 2023 at 14:34
• Having said that, I agree with you that Russell's example is not the best one: he intend to use an empty universal (the golden mountain). It is like "All square-circle are square. All square-circle are circular. Therefore: Some square is circular" Feb 7, 2023 at 14:36
• IF All does not imply Exist the argument is NOT valid, that's all. Feb 7, 2023 at 17:20
• There is nothing unsound about Russel's argument, but you have to understand what he is arguing for. He isn't arguing that Aristotle was wrong; he is arguing that Aristotle's formalization of the concept "all" can and ought to be generalized to apply to empty sets. Feb 7, 2023 at 17:57
• It's not only the same category of formal defect; it's arguably the very same formal defect--a formalism that doesn't deal with the null case. And I think Russell's argument is obviously not fallacious. Feb 8, 2023 at 18:01

Perhaps a translation from Aristotelian categorical logic to Fregian predicate logic will help clear up the matter.

Categorical logic Predicate logic
All golden mountains are mountains IF there exists golden mountain then they're mountains
All golden mountains are golden IF there exists golden mountains then they're golden
Some mountains are golden There exists at least one golden mountain

It becomes quite clear that the premises don't assert the existence of golden mountains (note the IF), but the conclusion does i.e. the inference is invalid.

I don't know that I'd call Russell's argument directly fallacious, in the sense of fallacy-as-invalidity, but I think that the absence of modal qualifiers makes it either indirectly invalid (on some level) or perhaps just unsound. Consider the difference between the following propositions:

1. All actual/existent golden mountains are golden.
2. All possible-but-nonexistent golden mountains are (or would be, if actualized) golden.

If we work with (1), then we get the "bad" conclusion that there are some golden mountains in the "real world" (except that (1) is already "messed up" in that it is tantamount to a universal quantification and an existential quantification all in one, but so the latter dimension of the proposition is already false). But (2), at worst, just gives us abstract objects that sustain "being golden" (perhaps by a Zaltaesque encoding relation), and the eventual inference to, "Some possible mountains are golden," could be reformulated in line with actualist sensitivities as, "There is some actual mountain that could have been, or could become, golden," or, "It is possible to take all the gold in the world and make a mountain out of it" (although note that the USGS website says that only 244,000 tons of gold have been found on Earth to date, whereas an average mountain's weight is on the order-of-magnitude of millions of tons).

Another way to look at the problem is by noting the difference between universal generalization and universal instantiation. So consider:

1. {All golden mountains in general are golden.} → {Generally, if there are golden mountains, then there are mountains that are golden.}
2. {Every particular golden mountain is golden.} → {There is a particular mountain, which is golden, somewhere.}

(3) comes across as somewhat pointlessly tortuous, like it's a kind of sentence that we would rarely have any pragmatic reason to say, write, or explicitly believe, but otherwise it doesn't seem "bad." (4) also sounds kind of "silly," or then false (as far as the eventual conclusion goes). But so still, it is important to keep track of the generality/particularity distinction, here, since the one condition leads us towards the "bad" idea that there really are golden mountains somewhere, while the other leads us to a merely redundant fact about what would be true if there were golden mountains here or there.

Actually, it's not absolutely obvious that there aren't golden mountains anywhere, after all, at least if our definition of the word "mountain" is flexible enough. For perhaps one might think that an asteroid with a lot of gold in its composition is mountain-like enough to count, or more generally that in our incredibly vast universe, there could well be at least one planet with so much gold, shunted about and somewhere to the surface, such that there is at least one mountain on that planet that is effectively "made of gold."

Postscript: it might be worth mentioning that the prevailing theory of modality in much older times was the "statistical" theory, where, "X is possible," comes out to, "X has existed or will exist at some point in time," and, "X is necessary," comes out to, "X exists at all times." There, actuality has logical priority over possibility. So maybe one aspect of Russell's opposition to Aristotelian logic involved a "paradigm shift" in prevailing theories of modality. I don't know that Aristotle himself defined possibility and necessity in the "statistical" manner, but if he did, then the nature of the problem of existential import, there, must be rather different from the manner of this problem in a modern context. So while Russell objects to, "All golden mountains are golden," for leading, Aristotelianism-wise, to the existence of particular golden mountains, yet if Aristotle thought that golden mountains (or whatever) were possible, he would have meant to say that there have been or will be golden mountains somewhere, someday. So again, maybe there is a problem with Russell's argument in that he is holding an older system of (implicit) modal logic to a standard that is not quite proper to that logic's semantics?

• Thanks for your answer, you offer an interesting angle. Still, I assume you understand you are not really addressing the question, which is not about Aristotle or his syllogistic, but about Russell's argument about the golden mountain syllogism.. May be in your own time? Feb 9, 2023 at 8:13
• @Speakpigeon I will edit my answer to frame it more in relation to Russell. I'm on break at work so it will take a while. Sorry for the misunderstanding, sometimes (a lot of the time) I have an elliptical sense of what I'm saying and my schizophrenia-spectrum mental health condition presents an obstacle to me communicating effectively, but I will see if I can make my answer more relevant to your question. Feb 9, 2023 at 9:37
• @Speakpigeon I have rewritten my answer, albeit pursuant to the same theme of my first attempt at an answer. Hopefully it is now more relevant to your concerns in the OP. Feb 9, 2023 at 17:14
• Yes it is more relevant and there is a number of interesting observations. Feb 10, 2023 at 17:06

It depends on how one defines "All greeks are men".

Let's say we consider it equivalent to "No greek exists who is not a man", which seems reasonable, then it is perfectly compatible with "No greek exists". Which makes "No greek exists who is not a man, No greek exists who is not white, therefore some men are white" a non-sequitur.

On the other hand if we define "All greeks are men" as "There is at least one greek and they are all men", which is equally reasonable, "There is at least one greek and they are all men, There is at least one greek and they are all white, therefore some men are white" becomes sound.

From the quote provided by OP it seems this is the point Russell is making.

• The issue is called Existential Import and has been clarified since 19th Century. According to Aristotle's original works, all universals (i.e. terms, like Human, Greek) are instantiated: he does not consider "empty terms" and thus we cannot find in his work an example like "All unicorn are green". Thus, if the term are always instantiated, we can infer from "All Greeks are Humans" the conclusion "Some Greek is Human". Modern (since 1800) logic remove the limitation about "empty" terms; [1/2] Feb 8, 2023 at 15:01
• Thus, when we symbolize "All Unicorns are Green" with ∀x (Ux → Gx), that is True (vacuously) exactly because there are not unicorns, we are no more entitled to derive ∃x(Ux & Gx), that is False (for the same reason). Feb 8, 2023 at 15:04
• It seems that the OP does not agree that this is the issue discussed into Russell's quote above... Feb 8, 2023 at 15:04
• @Speakpigeon because it can't be "adressed", it is perfectly correct, as everybody's telling you. It's even more obvious if you use some kind of ensemble notation or even a Sven diagram. The ambiguity of Aristotle's natural language just let this slip. Just admit it and get on with your day. Feb 8, 2023 at 22:45
• @MauroALLEGRANZA thanks for the reference and the formal notation. I don't think there is any ambiguity in the quote's first sentence "The statement "all Greeks are men" is commonly interpreted as implying that there are Greeks: without this implication, some of Aristotle's syllogisms are not valid." Feb 8, 2023 at 23:15

Formal rules can never do justice to all of the niceties of natural language, so there is always some tradeoff between simplicity and computability on the one hand, and complexity and expressiveness on the other. Aristotle and Russell are trading off in different places.

Russell's argument is not fallacious. He is choosing to treat statements of the form "all Fs are Gs" as lacking existential import, because this simplifies the logic without loss of expressiveness. Aristotle's logic is actually quite difficult to make sense of, and has several different interpretations, as I mentioned in my answer to this question.

In defence of Russell, the fact is that sometimes "All Fs are Gs" entails "Some Fs are Gs" and sometimes it does not. Sometimes we wish to proceed from an assumption "all Fs are Gs" without committing in advance to whether there are any Fs. Sometimes we proceed from "all Fs are Gs" in order to prove by reductio that there are no Fs. Sometimes we proceed from "all Fs are Gs" to learn that there are no Fs. For example:

• Premise 1. All unicorns are animals with the body of a horse and a single horn on their heads.
• Premise 2. There are no animals with the body of a horse and a single horn on their heads.
• Conclusion. There are no unicorns.

This argument is valid and sound. We know premise 1 to be true because that is how 'unicorn' is defined, and how we would recognise a unicorn if ever we saw one. Premise 2 has been established as true by extensive exploration of the Earth. The conclusion follows. But if "all Fs are Gs" is understood always to have existential import then premise 1 would be inconsistent with the conclusion, so there could be no sound instances of this argument form.

If we were to try to defend the rest of Arisototle's square of opposition, it becomes even more difficult, since it is even less plausible to accept that "no Fs are Gs" has existential import. The proposition, "there are no unicorns on Mars" is not rendered false by adding the rider, "...or anywhere else for that matter".

The fact that "all Fs are Gs" usually has the consequence "some Fs are Gs" is commonly handled using the theory of conversational implicature.

• 1. "it is even less plausible to accept that "no Fs are Gs" has existential import." All Aristotle's syllogistic means is "All S's are P" implies "Some S is P". He didn't talk of anything like "existential import". This is a modern interpretation. Reproaching him a claim he never made is fallacious. 2. Can I trust that your answer covers the state of the art on the validity of Russell's argument? Are you sure no scholar convincingly invalidated Russell's argument? 3. "using the theory of conversational implicature." Any author or paper particularly? Feb 13, 2023 at 17:00
• The term 'existential import' may be modern, but the concept is not. Most interpreters of Aristotle understand him to be taking at least the A and I propositions as requiring the existence of the subject: this goes back at least as far as Ibn Sina and was well established by the time of Ockham and Buridan. The Aristotelian square does not mean only that "All S is P" entails "Some S is P". It also requires that "No S is P" entails "Some S is not P" and this is what I claim is implausible. It requires O propositions to mean, "Either there are no S, or if there are, at least some are not P". Feb 13, 2023 at 21:11