Transitivity of Predication?

Here Bill Vallicella asks us to consider the following two 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.

All six statements are to be taken as predications and the rule of inference that justifies the conclusions is transitivity of predication.   Bill says that the first argument is valid but that something goes wrong with the second, for which he offers 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.'

I don't think this can be right.  Bill says that 'the concept man includes within its conceptual content the subconcept species'.  Another way of putting this is that the concept Man is subordinate to the concept Species.  The analogy here is with Man and Animal, say.  The defining characteristics of Man include all the defining characteristics of Animal.  There is a route from the root of the Porphyrean tree through the branch Animal leading to the branch Man.   But is Man subordinate to Species?  I think not.  If it were then some branch of the Porphyrean tree would be labelled 'Species', and I don't think this is the case.

My own diagnosis is this.  In both A1 and A2 we can see 'man' as a proper name---the name of a concept.  The major premiss of A1 says that this concept is an animal.  This is clearly false---no concept is an animal.  So A1 is unsound.  The major premiss of A2 says that the concept man is a species.  For the sake of argument let's accept this.  Any concept appearing in the Porphyrean tree is a species concept.  But A2 is invalid.  If x is an instance of concept A and concept A has the property B it does not follow that x has the property B.  Bill gives the example

...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.

 Where does this leave the notion of  transitivity of predication?  It must be approximate to some rule of inference, surely?  My answer is that Frege would rename it transitivity of subordination: 

 if concept A is subordinate to concept B, and concept B is subordinate to concept C, then concept A is subordinate to concept C.

An example of the application of transitivity of subordination:

A man is a mammal; a mammal is an animal; ergo, a man is an animal.

But this principle is not applicable to our example arguments.  The closest we can get is to reword the major premisses as follows:

B1. A man is an animal; Socrates is a man; ergo, Socrates is an animal.
B2. A man is a species; Socrates is a man; ergo, Socrates is a species.

B1 now says that Socrates is an instance of the concept Man, the concept Man is subordinate to the concept Animal, ergo Socrates is an instance of the concept Animal.  This is valid, being an inference of the form

if x is an instance of concept A, and concept A is subordinate to concept B, then x is an instance of B

B2 is valid, having the same form as B1, but is unsound, the major premiss being false as explained above:  Man is not subordinate to Species.