A second installment from the Ostrich of London.
Another difficulty with the function-argument theory is staring us in the face, but generally unappreciated for what it is. As Geach says, the theory presupposes an absolute category-difference between names and predicables, which comes out in the choice of ‘fount’ [font] for the schematic letters corresponding to name and predicable. For example ‘Fa’, where the upper case ‘F’ represents the predicable, as Geach calls it, and lower case ‘a’ the name. As a direct result, there is only one negation of the proposition, i.e. ‘~Fa’, where the tilde negates whatever is expressed by ‘Fa’. But ‘F’ is a function mapping the referent of ‘a’ onto the True or the False, so ‘~Fa’ says that a does not map onto the True. The object a is there all right, but maps to a different truth-value. Thus ‘Fa’ implies ExFx, ‘~Fa’ implies Ex~Fx, and excluded middle (Fa or ~Fa) implies that something, i.e. a, does or does not satisfy F. The function-argument account has the bizarre consequence that the name always has a referent, which either does or does not satisfy the predicable. There is no room for the name not being satisfied. Indeed, the whole point of the function theory is to distinguish the idea of satisfaction, which only applies to predicables, from reference, which is a feature of proper names only. As Frege points out here:
The word 'common name' is confusing .. for it makes it look as though the common name stood under the same, or much the same relation to the objects that fall under the concept as the proper name does to a single object. Nothing could be more false! In this case it must, of course, appear as though a common name that belongs to an empty concept were as illegitimate as a proper name that designates [bezeichnet] nothing.
The scholastic two-term account, by contrast, allows for the non-satisfaction of the proper name. ‘Frodo is a hobbit’ is true if and only if something satisfies both ‘hobbit’ and ‘Frodo’. It is essential to Aristotle’s theory of the syllogism, as Geach notes, that the middle term (the one which appears in both premisses) can be subject in one premiss, predicate in another. The notion of ‘satisfaction’ or ‘supposition’ applies to both subject and predicate, even if the subject is a proper name like ‘Frodo’. Thus the negation of ‘Frodo is a hobbit’ can be true in two ways. Either some individual satisfies ‘Frodo’ but does not satisfy ‘hobbit’. We express this in English by so-called predicate negation ‘Frodo is not a hobbit’, where the negative is placed after the copula. Or no individual satisfies ‘Frodo’, which we can express by placing the negation before the whole proposition, ‘it is not the case that Frodo is a hobbit’. So the scholastic theory neatly accounts for empty proper names. Not so for the function-argument theory, a difficulty which was recognised early on. Frege developed a complex and (in my view) ultimately incoherent theory of sense and reference. Russell thought that proper names were really disguised descriptions, which is actually a nod to the scholastic theory.
Of course there is a separate problem for the two term theory, of making sense of a proper name not being ‘satisfied’. What concept is expressed by the proper name that is satisfied or not satisfied, and which continues to exist as a concept even if the individual ceases to exist? Bill and I have discussed this many times, probably too many times for his liking.
BV: What is particularly interesting here is the claim that Russell's theory of proper names is a nod to to the scholastic theory. This sounds right, although we need to bear in mind that Russell's description theory is a theory of ordinary proper names. Russell also allows for logically proper names, which are not definite descriptions in disguise. The Ostrich rightly points out that that for Frege there there is an absolute categorial difference between names and predicables. I add that this is the linguistic mirror of the absolute categorial difference in Frege between objects and concepts (functions). No object is a concept, and no concept is an object. No object can be predicated, and no concept can be named. This leads directly to the Paradox of the Horse: The concept horse is not a concept. Why not? Because 'the concept horse' is a name, and whatever you name is an object.
This is paradoxical and disturbing because it imports ineffability into concepts and thus into logic. If concepts cannot be named and objectified, then they are not wholly graspable. This is connected with the murky notion of the unsaturatedness of concepts. The idea is not that concepts cannot exist uninstantiated; the idea is that concepts have a 'gappy' nature that allows them to combine with objects without the need for a tertium quid to tie them together. Alles klar?
Now it seems to me that Russell maintains the absolute categorial difference between logically proper names and predicates/predicables. ('Predicable' is a Geachian term and it would be nice to hear how the Ostrich defines it.) Correct me if I am wrong, but this presupposition of an absolute categorial difference between logically proper names and predicates/predicables is a presupposition of all standard modern logic. It is 1-1 with the assumption that there are atomic propositions.
Here is one problem. On the Russellian and presumably also on the scholastic theory, an ordinary proper name stands to its nominatum in the same relation as a predicate to the items that satisfy it. Call this relation 'satisfaction.' Socrates satisfies 'Socrates' just as he and Plato et al. satisfy 'philosopher.' Now if an item satisfies a term, then it instantiates the concept expressed by the term. But what is the concept that 'Socrates' expresses? One candidate is: the unique x such that x is the teacher of Plato. Another is: the greatest philosopher who published nothing.
Notice, however, that on this approach singularity goes right out the window. 'Socrates' is a singular term. But 'the greatest philosopher who published nothing' is a general term despite the fact that the latter term, if satisfied, can be satisfied by only one individual in the world that happens to be actual. It is general because it is satisfied by different individuals in different possible worlds. Without prejudice to his identity, Socrates might not have been the greatest philosopher to publish nothing. He might not have been a philosopher at all. So a description theory of names cannot do justice to the haecceity of Socrates. What makes Socrates precisely this individual cannot be some feature accidental to him. Surely the identity of an individual is essential to it.
If we try to frame a concept that captures Socrates' haecceity, we hit a brick wall. Concepts are effable; an individual's haeceity or thisness is ineffable. Aristotle says it somewhere, though not in Latin: Individuum ineffabile est. The individual as such is ineffable. There is no science of the particular qua particular. There is no conceptual understanding of the particular qua particular because the only concepts we can grasp are general in the broad way I am using 'general.' And of course all understanding is conceptual involving as it does the subsumption of particular under concepts.
Some will try the following move. They will say that 'Socrates' expresses the concept, Socrateity, the concept of being Socrates, or being identical to Socrates. But this haecceity concept is a pseudo-concept. For we had to bring in the non-concept Socrates to give it content.
There are no haecceity concepts. As the Ostrich appreciates, this causes trouble for the scholastic two-name theory of predication according to which 'Socrates' and 'wise' are both names, and the naming relation is that of satisfaction. It makes sense to say that the concept wise person is uninstantiated. But it makes no sense to say that the concept Frodoity is uninstantiated for the simple reason that there cannot be any such concept.
It looks like we are at an impasse. We get into serious trouble if we go the Fregean route and hold that names and predicates/predicables are radically disjoint and that the naming/referring relation is toto caelo different from the satisfaction relation. But if we regress to the scholastic two-name theory, then we have a problem with empty names.