Perhaps the most commonly used account of validity is modal; it is the idea that an argument is valid when, its premises being true, the conclusion cannot be false. This account of validity makes the following argument valid:
Socrates is human;
therefore Socrates is an animal.
However, it would often be said that this is not valid in terms of its form; it is not formally valid but materially valid. Since anything formally valid that does not equivocate would certainly have also to be materially valid, the question arises as to what the principle is identifying an argument as formally valid. The usual principle suggested is some variant of Buridan's idea that it has to hold for all terms, keeping the form common to all of them the same. Thus people would usually say that the above argument is not formally valid because it would not be invariant under a consistent but arbitrary substitution of terms:
New Orleans is a city;
therefore New Orleans is a river.
But I'm not sure we should let this pass so easily. For one thing, 'keeping the form the same' has to apply to the actual logical principles used in drawing the conclusion, and it seems clear enough that the New Orleans argument does not use the same logical principles to draw its conclusion that the Socrates argument does. The conclusion in the Socrates argument 'works' because it simply moves the predicate from specific to general; that's what pretty much anyone making the argument would be doing. But this is not involved at all in the New Orleans argument.
One could perhaps argue that this is somehow not part of the argument's form, but it's difficult to see how it wouldn't be. It certainly would be if I did something like this:
Socrates is an animal that is rational;
therefore Socrates is an animal.
But there's not any obvious way in which this is different from the other. Logical terms are not words but meanings, so if 'human' includes 'animal' as part of its definition, the movement from 'human' to 'animal' should be as formal as the movement from 'animal that is rational' to 'animal', if they both proceed on the same principles, which they appear to do.
One could perhaps argue that the original is an enthymeme:
Socrates is human;
(everything human is an animal);
therefore Socrates is an animal.
This is certainly formally valid. But it's unclear that the implicit premise actually adds anything that is not already in the first premise, for exactly the reason just noted.
If one took each logical term to be a 'slot', so to speak, in a classification system, then an argument's form relates these 'slots' to each other; but then one would expect species-to-genus inference to be as formal as subalternation, universal instantiation, or inference involving the dictum de omni et nullo.
It has been noted by others that, while we can fairly easily handle the notion of logical form in particular logical systems, we don't have any general account of logical form. This is a related issue, I think, since there appears to be no useful account, applicable to the full range of arguments people would want to consider formally valid, that also obviously rules out the possibility that the classification of the terms can be part of the formal structure of an argument.
