Like the befuddled weaver in Wittgenstein's Philosophical Investigations, Kerry was sitting at an empty loom, he was going through the motions of weaving, and was thinking that he was weaving.  But he was not weaving. (p. 49)

Jolley's book is a deceptively thin volume, and a deceptively unassuming one.  For all that its contents may be intimidating to non-philosophers, it's worth taking the time to read, and to read carefully, for its target is nothing less than the nature of structured thought itself.

Jolley begins with a quick primer on Frege's theory of objects and concepts.  For Frege, the distinction between objects and concepts (which predicate on objects) is absolute.  Objects name (pick out) referents.  Concepts name (pick out) properties of referents.  Objects can be described/defined/predicated about.  Concepts cannot.  They and their nature can only be hinted at.  ("Hint" is, as Jolley notes, a word that Frege uses a lot.)  They can be used but not discussed.  Objects are, in Frege's terminology, "saturated":  complete unto themselves ("John").  Concepts are "unsaturated", incomplete:  ("__ is male").

In standard grammatical terms, objects are subjects, concepts are predicates.  An object and a concept together make a proposition (although, to Frege, this is the wrong way around:  it's never the case that propositions are derived from objects and concepts, but only that objects and concepts are derived from propositions; never, says Frege, ask for the meaning of a word except in the context of a proposition).  Concepts, for Frege, can never be objects nor objects concepts.

On the face of it, this may seem a not unreasonable requirement.  But a lot of potentially awkward consequences follow.  For starters, there can be no concept "horse", nor any other concept X, because that would imply that one could say things about the concept.  ("The concept 'horse'" would be an object like "the planet Neptune", and objects, by their nature, have properties.)  "The concept 'horse'" picks out something, to be sure, but what it picks out is not a concept.

Of course there can be no meta-concept like the concept of concept (i.e., our concept of what a concept itself is).  For the philosophers, it means that there can be no theory of concepts.  As a doctoral student studying theories of concepts, I would be out of a PhD thesis.

The concept "horse" paradox arises — if it arises at all — if you initially accept Frege's absolute distinction but then find yourself treating concepts as objects anyway (as, Jolley acknowledges, Frege himself found all but irresistible to do).  "The concept 'horse' is a concept easily attained", said Benno Kerry; to which Frege replied, "the concept 'horse' is not a concept at all".  Assuming that Frege was not intending to state a simple contradiction, what one gets is a very paradoxical proposition.  What does Frege mean?

To Frege, Kerry is trying to make an unsaturated concept-expression (e.g., "__ is male") stand in for a saturated object-expression (e.g., "John").  If we refer to concepts in the way that Frege wants us to, the problem seems obvious:  "'__ is a horse' is a concept easily attained."  Try as Kerry might, he can't fill in the blank (or he can, but it will only be to create another blank!).  "Why does Kerry fail to see this?" Jolley asks (p. 13).  "Because he transgresses… [Frege's] Context Principle.  Kerry thinks about the words "the concept 'horse'" outside the context of a proposition and thinks they must be a concept-expression.  They must be – because the words seem to him to assign that role to themselves."

Jolley is not, he says, attempting to give a definitive account of the concept "horse" paradox.  He does not, after all, even believe that there is a paradox, because Kerry is merely confused into thinking he's saying something when really he's not.  Rather, Jolley uses the debate between Frege and Kerry as a way of explicating the relationship between Wittgenstein's Tractatus and Philosophical Investigations, the bookends to Wittgenstein's philosophical career, often seen (incorrectly, Jolley believes, and after reading him I am inclined to agree) to be in irresolvable conflict with each other.  As he elegantly writes (p. xi), "to see the two books as warring with each other is confused, unless the war is a philosophical Civil War, the combatants brothers, the blood shed (if any) consanguine."  In both the Tractatus and the Philosophical Investigations, Jolley sees Wittgenstein as rejecting Kerry, embracing and refining Frege.  What changes between the two is the argument, not the conclusion.

Jolley may not be interested in resolving the concept "horse" paradox, but perhaps we should give it some further thought, to understand some possible limitations both in Wittgenstein's philosophy, and in Jolley's analysis of it and of Frege.

Once an absolute distinction has been made between concepts and objects, any self-reference exploiting this distinction is to be avoided:  an object cannot legitimately purport to declare itself a concept.  In general, any self-referential construction that allows the interpretation "I (by my form an X) am not X" is  verboten, because it leads to variations on Russell's Paradox:  logically possible constructions (in the case of Russell's Paradox, the set of all sets that do not contain themselves as members) that are nonetheless logically impermissible (because such a set both must contain itself and not contain itself).  Russell's solution was his Theory of Types (to which Jolley devotes much of a chapter).  Wittgenstein felt he could do one better by preventing the illicit construction from ever occurring in the first place – in effect, making it ungrammatical.

Here's why there might be a problem: consider X such that X must be y xor z (I.e., either y or z but not both).  For most instances of X, that's going to be unproblematic.  But what if you have an instance of X whose form (syntax) makes it y but whose only possible reading (semantics), form aside, seems to make it z (i.e., not y)?  In Jolley's terms, X appears to assign itself a role inconsistent with the role it should, because of its form, take. 

Consider Boolean statements, which by their form imply the truth of a proposition that may in fact be true or false:  "all birds fly", for example, or "the statement 'George Bush is the prime minister of the UK' is true", both of which are false.  Any statement of the form "all Xs y" or "x is y" can be expected to be implying the truth of some proposition.

(I wish to make a statement/proposition distinction here.  The words, by being in a certain structural arrangement, form a statement.  The statement expresses a proposition, as its meaning.  A proposition is something that may, but need not be, expressed or even expressible in words of a language.)

But what does one make of a statement like "this statement is false"?  As a statement (by its form), it lays claim to be true.  But as a proposition about a statement (in fact itself), it must (by its own admission) be false.

If the proposition is not permitted to refer to itself in the illegitimate way, the problem does not arise.  In that case there are no self-referential paradoxes, only simple contradictions ("night is day", "true is false", "concepts are [ or can be] objects"), and simple contradictions can straightforwardly be taken to be false.

Or consider adjectives, which by their form state some property or provide description of some entity.  All adjectives can be divided into those that are self-descriptive (e.g., "polysyllabic") and those that are non-self-descriptive (e.g., "green" or "monosyllabic").  Prima facie, adjectives must be one thing or the other.  We can call self-descriptive adjectives autological and non-self-descriptive adjectives heterological.  Question:  is the adjective "heterological" itself heterological?  "Heterological" appears to violate the usual heterological/autological distinction!

Translated into the context of the Frege/Kerry debate, the Russelian-style paradox arises as follows.  Suppose there are two types of sub-propositional entities, objects and concepts, and the one can never be the other.  We could name an object like "Tony" or a concept like "(__ is) male".  But what should we make of an (apparent) object like "the concept 'horse'" (of which we might predicate, as Kerry does, "the concept 'horse' is a concept easily attained!").  Is "the concept 'horse'" a concept (as it declares itself) or an object (as its role in various possible propositions would seem to indicate)?  It seems that it can't be either.

If Kerry is a bit naive in thinking that the absolute distinction between concepts and objects can be so easily disproved, then Frege is arguably naive in thinking that the distinction can ultimately be maintained.  In insisting that concepts are derived from propositions but that propositions can never be composed from concepts, Frege may not be buying himself nearly so much as he thinks.

What if, pace Frege (and by extension, Wittgenstein and Jolley), words themselves are or can be (implicit) propositions?  Then it will be possible to ask for the meaning of a word "only within the context of a proposition" and still produce the very situation Frege is trying to avoid.

(Frege thought the whole reason Kerry believed he could make an object of a concept in the first place is because he took the words "the concept 'horse'", outside of the context of any proposition, to name a concept; then he put them back into a proposition, in the object's place, and thought that, by doing so, he'd make the concept an object.  "Fully to recognize the unitary sense of a proposition is to recognize the hopelessness of asking for the meaning of a word in isolation from the context of a proposition." (p. 37))

Something Douglas Hofstadter wrote in Gödel, Escher, Bach comes to mind, when he is talking about how Gödel located paradox at the very heart of the Principia Mathematica (and, by standard interpretation, much of the rest of mathematics besides):

Mathematical statements – let us concentrate on number-theoretic ones – are about properties of whole numbers.  Whole numbers are not statements, nor are their properties.  A statement of number theory is not about a statement of number theory; it just is a statement of number theory….

Gödel had the insight that a statement of number theory could be about a statement of number theory (possibly even itself) if only numbers could somehow stand for statements.  The idea of a code, in other words, is at the heart of his construction.  …This coding trick enables statements of number theory to be understood on two different levels:  as statements of number theory, and also as statements about statements of number theory. (Hofstadter, p. 18)

It's the capacity of the part to refer to the whole that allows one to conclude that "all consistent axiomatic formulations of number theory include undecidable propositions" (Hofstadter, p. 17), the equivalent of "I (by my form X) am not X."

We need an absolute distinction between concept and object in order to conceptualize the world, an absolute distinction between subject and predicate in order to form and express propositions – fine.  But at some point — and Kerry is, I think, ultimately right here — the absolute distinction breaks down.

Given any apparent paradox, there are three obvious ways to respond:  First, one can deny the intuition of meaningfulness in the paradox and so reduce it to a simple contradiction.  The paradoxical statement purports to be saying something more than simple contradiction, but it's not.  By slightly different routes, this is the solution Frege, Wittgenstein and Jolley all, I think, wish to take.

Second, one can deny the contradiction:  the contradiction is only apparent, because our understanding is incomplete.  If we had but a wider perspective, we would understand how the contradiction resolves itself.

Either of these approaches resolve the paradox by denying it.  But there is a third possibility, which Hofstadter alludes to:  the contradiction at the heart of the paradox may be essential to the structure of our thought.  If the paradox really is unavoidable – if, after all our attempts to remove the paradox, the paradox persists – then this might seem the only option left to us.

The concept/object distinction may not ultimately be maintainable, but it may nevertheless be necessary in order to form and employ concepts in the first place.  The insight here, perhaps, is that, for all concepts may guide and simplify our interaction with our environment, there is a point where conceptual understanding breaks down.

The philosopher and the religious mystic who argue that there comes a point where the self/other distinction breaks down must be driven by a similar intuition.  That we need to distinguish the self from the not-self in order to engage with our environment and be part of our various societies does not mean that that distinction exists anywhere other than in our minds.

So with concepts and objects, and perhaps other absolute distinctions we feel compelled to make.  At least, we cannot logically rule it out.  This is, to me, the insight that Kerry reaches for but never quite grasps, and which, for all Jolley's denials, is, I think, lurking in this thought-provoking book.

References

Hofstadter, Douglas R. (1979)  GÖDEL, ESCHER, BACH:  An Eternal Golden Braid.  Basic Books, New York.

© 2008 Joel Parthemore

Joel Parthemore is a third-year DPhil student studying theories of concepts at the University of Sussex in Brighton, UK. He is a member of the Philosophy of AI and Cognitive Science research group in the Department of Informatics. In his spare time he plays with Linux computer systems. You can find him online at http://www.parthemores.com/research/.