Does this syllogism by Russell show that Aristotelian logic doesn't work?

Question

I have been reading through Hurley's A Concise Introduction to Logic, and I just finished reading the chapters about categorical syllogisms. While prodding around Wikipedia looking for some interesting reading material about Aristotelian logic, I happened by a quote by Bertrand Russell:

"All golden mountains are mountains, all golden mountains are golden, therefore some mountains are golden."

I decided for fun to write this out as (what I hope is) a standard-form syllogism.

P1) All mountains that are made of gold are made of gold.

P2) All mountains that are made of gold are mountains.

C) Some mountains are made of gold.

Which checks out with the four rules both Boole and Aristotle use. For Rule 5, this would constitute an existential fallacy in Boole's system. But in the Aristotelian system, it only commits the existential fallacy if no instances of the critical term exist.

Mountains do exist though, so it seems like I am supposed to be left with the conclusion that there exists at least one mountain made of gold.

Does this show that Aristotle's formulation of logic doesn't work?

Answer 1

The short answer

The syllogism doesn't refute Aristotle; its correctness hinges on whether we read the premises with existential import (others have pointed out this). There seems to be a misconception about what the critical term is on your side. You regard mountain as critical term, whereas in fact, mountain that is made of gold is the critical term.

The long answer

Categorical statements (A, E) seem unproblematic unless we stumble upon one whose subject term is empty. What do we make of those? A key observation behind the question about existential import is that the validity of some syllogisms hinges on just how we interpret those statements.

(A): All S are P.

(E): No S is P.

So what if there is no S? There are two traditional standpoints.

• According to the Boolean standpoint, both statements are true. One way to motivate this is to think in terms of counterexamples. If there is no S, then a fortiori there is no S that is not P; now any counterexample to "All S are P" must give an S that is not P; hence, there is no counterexample, so the statement is true.
• On the other hand, the Aristotelian standpoint has it that categorical statements are true only if their subject term is not empty. In this case, if there is no S, both statements are false. So on this view, counterexamples are not the only way to make a statement turn out false.

(Which of the standpoints is true is a difficult question we cannot decide here; each one has some sentences in its favor. For instance, "All golden mountains are located in India" intuitively seems false, whereas "All golden mountains are golden" seems to be true by definition.)

So if we want to evaluate a syllogism from the Aristotelian standpoint, we shall read all categorial premises with existential import; that is, such that their subject term must not be empty. Since a syllogism may have two categorical statements, this may involve two terms. Now here's another key observation: It will always suffice only to read one of the categorical premises with existential import. The subject term of that statement then is the critical term, because the validity of the syllogism hinges on whether the critical term is empty or not.

In Russell's syllogism, both premises have the same subject term, viz. mountain that is made of gold. So this one must be the critical term here. The critical assumption then is that there is a mountain that is made of gold. In a way, this seems trivial, because the conclusion is just that. Anyways, the reason is that the premises also are rather trivial.

As per Hurley's book, it seems that your misconception might in part be his fault. This is his "definition" in the 12th edition:

This fact provides the basis for a definition of critical term. The critical term is the term in a categorical syllogism which, when it denotes at least one existing thing, guarantees that the subject of the conclusion denotes at least one existing thing. (p. 281)

In Russell's syllogism, there are two terms that satisfy this condition. One is mountain that is made of gold. Quite trivially, however, mountain also is such a term, because that just is the subject term of the conclusion. What we need to add is that the critical term must also be the subject term of one of the premises.

Hurley's criteria, however, deliver the right result. If you draw a Venn diagram, you will notice that the circle for mountain that is made of gold contains exactly one empty area. Also, in his table on p. 267, you find that for an AAI-3 syllogism, the required condition is that M exists, where M, in this case, is golden mountain.

Answer 2

The syllogism is correct, according to Aristotle's doctrine, exactly because of Existential import:

Aristotle's logic system does not cover cases where there are no instances.

See The Traditional Square of Opposition:

‘Some S is P’ is a subaltern of ‘Every S is P’. A proposition is a subaltern of another iff it must be true if its superaltern is true, and the superaltern must be false if the subaltern is false."

Thus, according to A, we nay say that the said syllogism is valid, but it has a false premise.

See also Hurley, page 262: "[see] table of conditionally valid forms. If it does, the syllogism is valid from the Aristotelian standpoint on condition that a certain term (the “critical” term) denotes actually existing things. The required condition is stated in the last column."