Transitivity of Predication?

I dedicate this post to London Ed, who likes sophisms and scholastic arcana.

Consider these two syllogistic arguments:

A1. Man is an animal; Socrates is a man; ergo, Socrates is an animal.
A2. Man is a species; Socrates is a man; ergo, Socrates is a species.

The first argument is valid.  On one way of accounting for its validity, we make two assumptions.  First, we assume that each of the argument's constituent sentences is a predication.  Second, we assume the principle of the Transitivity of Predication: if x is predicable of y, and y is predicable of z, then x is predicable of z.  This principle has an Aristotelian pedigree.  At Categories 3b5, we read, "For all that is predicated of the predicate will be predicated also of the subject." So if animal is predicable of man, and man of Socrates, then animal of Socrates.  

Something goes wrong, however, in the second argument.  The question is: what exactly?  Let's first of all see if we can diagnose the fallacy while adhering to our two assumptions.  Thus we assume that each occurrence of 'is' in (A2) is an 'is' of predication, and that predication is transitive.  One suggestion  -- and I take this to be the line of some Thomists -- is that (A2) equivocates on 'man.'  In the major, 'man' means 'man-in-the-mind,' 'man as existing with esse intentionale.'  In the minor, 'man' means 'man-in-reality,' 'man as existing with esse naturale.'  We thus diagnose the invalidity of (A2) by saying that it falls afoul of quaternio terminorum, the four-term fallacy.  On this diagnosis, Transitivity of Predication is upheld: it is just that in this case the principle does not apply since there are four terms.

But of course there is also the modern Fregean way on which we abandon both of our assumptions and locate the equivocation in (A2) elsewhere.  On a Fregean diagnosis, there is an equivocation on 'is' in (A2) as between the 'is' of inclusion and the 'is' of predication.  In the major premise, 'is' expresses, not predication, but inclusion: the thought is that the concept man includes within its conceptual content the subconcept species.  In the minor and in the conclusion, however, the 'is'  expresses predication: the thought is that Socrates falls under the concepts man and species.  Accordingly, (A2) is invalid because of an equivocation on 'is,' not because of an equivocation on 'man.'

The Fregean point is that the concept man falls WITHIN but not UNDER the concept animal, while the object Socrates falls UNDER but not WITHIN the concepts man and animal.  Man does not fall under animal because no concept is an animal.  Animal is a mark (Merkmal) not a property (Eigenschaft) of man.  In general, the marks of a concept are not its properties.  But concepts do have properties.  The property of being instantiated, for example, is a property of the concept man.  But it is not a mark of it.  If it were a mark, then man by its very nature would be instantiated and it would be a conceptual truth that there are human beings, which is false.

Since on the Fregean scheme the properties of concepts needn't be properties of the items that fall under the concepts, Transitivity of Predication fails.  Thus, the property of being instantiated is predicable of the concept philosopher, and the concept philosopher is predicable  of Socrates; but the property of being instantiated is not predicable of Socrates.