In a subsequent criticism of my argument, “Jacquette on Grelling's Paradox” (Ketland 2005), Jeffrey Ketland argues that my method of explicating the concept of heterologicality is faulty because it violates the above-mentioned syntactical constraint on definitions. The crux of the matter, according to Ketland, is that I fail to recognize the need for including in the definiens whatever variables appear in the definiendum. If Ketland's objections are correct then they offer a more general moral for the proper formal definition of terms in philosophical logic. Thus, there is more at stake in understanding the formal requirements of definitions than simply the question of whether Ketland's criticism of Jacquette is well founded, or for that matter of whether Grelling's paradox is sustainable despite type theory.
I argue in what follows that Ketland is mistaken both in the substance and in the application of the syntactical constraints on definitions he proposes, and that accordingly his critique provides no justification for overturning my conclusions concerning the possibility of resurrecting Grelling's paradox within type theory.
3a. One- or Two-Place Heterologicality
Ketland begins by misunderstanding the logical structure of my definitions. He writes: “To see what is wrong with Jacquette's argument, reconsider the ‘definition’ (1). In writing the argument dependency (i.e. the free variables which occur) of the definiendum ‘Hn+1’ as ‘Hn+1(Fn)’, Jacquette wishes to represent Hn+1 as a one-place relation. But the definiens is ‘¬Fn+1(Fn)’ and thus contains two distinct variables, ‘Fn+1’ and ‘Fn’. So, Hn+1 must be a two-place relation” (Ketland 2005, 259).
Although an argument is lacking and the exposition is obscure on this crucial point, Ketland seems to believe that my formalization of Grelling's concept of heterologicality needs to be a two-place relation or relational predicate as a consequence of the syntactical constraint he mentions, by which variables appearing in a properly formulated definition's definiens must also appear in its definiendum. I consider this part of Ketland's objection at length in section 3c below. I find it intuitively implausible, independently of the merits of my definition, to consider heterologicality as a two-place relation. If this were true, then the original untyped formulation of heterologicality in Grelling's paradox, ∀F(H(F) ↔ ¬F(F)), as it is usually explicated, would also have to be two-place. By Ketland's reasoning, it would then follow that the original concept of heterologicality was improperly defined in the historical formulation of Grelling's paradox, that the original paradox was logically misconceived, and hence that it represents no motivation from that quarter for the development of type theory in the first place. Yet it is obvious at a glance that there is only one argument place within the scope of the predicates “H(__)” (untyped) and “Hn+1(__)” (typed).
Perhaps what Ketland means is that heterologicality should be construed as a one-place 1-adic predicate (or 1-ary property), and that in defining the property as I do I de facto wrongly make it into a two-place relation. This is also clearly not the case. The definiendum and definiens in my definition are both one-place. Thus, sans quantifiers, we have: Hn+1(Fn) defined as ¬Fn+1(Fn), and hence, sans arguments, Hn+1(__) defined as ¬Fn+1(__). With structural parity on both sides of the equivalence perfectly preserved, there is no reason, as far as Ketland's criticism is concerned, why I could not have proceeded constructively to the same effect:
- Hn+1(__) ↔ ¬Fn+1(__)
- Hn+1(Fn) ↔ ¬Fn+1(Fn)
- ∀Fn(Hn+1(Fn) ↔ ¬Fn+1(Fn))
- ∀Fn+1∀Fn(Hn+1(Fn) ↔ ¬Fn+1(Fn))
- ∀Fn(Hn+1(Fn) ↔ ¬Hn+1(Fn))
We could, in other words, leave the whole question of definitions and formal constraints on proper definitions entirely out of account, and simply defer to the two-stage syntactical constructibility of the paradox sentence within type theory, in order to see that type-theoretical restrictions are inadequate to forestall Grelling's paradox.
As a result, it seems compulsory to reject Ketland's conclusion (Ketland 2005, 259) that my definition (1) should be rewritten instead as:
Indeed, it is hard even to make sense of such a syntax combination; nor is it possible to recognize in Ketland's formalization what Grelling means by heterologicality or how I could attach any meaning to his (5).
Second, Ketland is simply wrong to assert, again without attempt at justification, that if the definiens in a properly definitional biconditional is two-place, then the corresponding definiendum must also be two-place. An obvious counterexample is seen in the use of λ-abstraction to define, for example:
The syntactical permissibility of such constructions is further in evidence when we write:
Again, if my typed definition of heterologicality involved a one-place definiendum and two-place definiens, that, in itself, and contrary to Ketland's interpretation, would still not disqualify my strengthened two-step syntactical construction of Grelling's paradox within type theory. On inspection, moreover, the definition involves no distinction whatsoever in the number of argument places on the two sides of the definition's biconditional that Ketland attributes to it.
Ketland correctly maintains that the argument dependency “Hn+1(Fn+1,Fn) is not type-theoretically acceptable” (Ketland 2005, 259). True as far as it goes, but also beside the point, since anyone can look at the syntax in my constructions to see that I am not committed to a two-term analysis for a two-place relation of the ultimate predication objects of heterologicality on either side of the definition. Ketland, therefore, refutes only a strawman when he concludes: “In short, what is wrong with Jacquette's argument is that (according to his own intentions) the defined heterologicality relation Hn+1 is a two-place relation, with arguments Fn+1 and Fn. When this is clarified, the proposed definition (i.e. (5) above) violates type-restrictions” (Ketland 2005, 260).
3d. Syntax Versus Meaning Content as Paradox Source
Ketland next tries to show how things can get even more out of hand when theorists venture to define logical concepts in violation of the formal constraint. His “amusing” (his term) “proof” “that Peano arithmetic (indeed, any theory) is inconsistent” (Ketland 2005, 260) is not a proof of any such conclusion at all, nor does it turn on the occurrence of free variables in a definition's definiens that do not also appear in the definiendum. The paradoxical undermining of Peano arithmetic is supposed to be brought about by the offending definition:
From (7), in two steps of universal instantiation, Ketland now derives:
And thus, since ‘1 < 2’ is provable,
But also, from (7), we may infer,
And thus, since ‘1 < 0’ is refutable,
Does this remarkable simple argument show that arithmetic is inconsistent? No, it merely shows that ‘definitions’ like (7), and, similarly, Jacquette's (1), violate the aforementioned formal constraint. (Ketland 2005, 260)
It is true that from the definition of “P” in (7) we can prove P(1) ∧ ¬P(1). This, however, as Ketland rightly observes, should not be construed as showing that arithmetic is logically inconsistent—that is merely the absurd alternative—but only that something is wrong with definition (7). The difficulty then is to diagnose exactly how (7) fails. If, as Ketland admits, Peano arithmetic + predicate “P” as defined in (7) is logically inconsistent, and if Peano arithmetic is not logically inconsistent, then the logical inconsistency in Peano arithmetic + (7) can only be rightly attributed to (7). Suppose, then, that (7) implies a proposition that is logically inconsistent with Peano arithmetic. The same effect can be achieved in many ways, and with considerably less fuss, as in the example cited in the second paragraph of this essay above, and now labeled:
It still does not follow that (7) or (7*) is faulty by virtue of violating Ketland's syntactical constraint on allowing only variables appearing in a definition's definiens that also appear in its definiendum. We have already seen that Ketland's constraint is not sufficient to forestall definition-based paradoxes. Nor is it necessary.
There is no obligation to impose Ketland's constraint and thereby disavow as genuine the unlimited numbers of normal intelligible and useful technical and colloquial definitions in which at least some predicate terms do not appear in both definiendum and definiens. We would have no choice but to violate Ketland's constraint if our purpose were to reductively explain an individual's identity conditions, in writing, truly or falsely, a = F, where a is an object and F is all of a's properties. Does that mean that Ketland's syntactical constraint contradicts the logical possibility of Leibnizian identity reductions and expansions? Does Ketland's syntactical constraint also contradict the above-mentioned definition of a circle, of a bachelor, and so on? Or is there perhaps another way to recapture their logic? I do not think that this is what either Tarski or Haack had in mind.
Clearly, (7) and (7*) violate Ketland's constraint. The question is whether by doing so the definition contributes to its inconsistency with Peano arithmetic. This is something that desperately needs to be shown, but for which Ketland does not try to argue. For there may be other reasons having to do with the content of the predicate that account for its logical inconsistency with arithmetic that Ketland does not consider. Ketland's argument does not obligate us to reconsider the practice of defining predicates by means of bound variables that appear in the definiens but not in the definiendum, in this instance, because the procedure does not always or inevitably result in antinomy. It seems to depend again on the exact content of the definiens. Ketland's (7) and our (7*) are evidently paradoxical. The syntactical constraint as a result is not necessary to avoid paradox or similar inconvenience in many harmless applications. Other definitions, of the same general structure, however, are altogether logically innocuous, as we see in the sentences:
The syntactical constraint on formalized definitions that Ketland undercritically invokes accordingly throws out the baby with the bath water.
Nor, significantly, is it sufficient to avoid paradox to insist on the formal constraints and definitions to which Ketland refers. We get the same problem with a slight modification of his definition (7) together with a reasonable stipulation concerning the extension of predicate “P,” where both the definition and the assumption satisfy Ketland's formal constraints, as we write instead:
| (1) ∀x∀y(P(x + y) ↔ x < y)||Definition|
| (2) ∀x∀y(P(x + y) ↔ P(y + x))||Assumption|
| (3) ∀y(P(0 + y) ↔ 0 < y)||1|
| (4) P(0 + 1) ↔ 0 < 1||3|
| (5) 0 < 1||Theorem of arithmetic|
| (6) P(0 + 1)||4,5|
| (7) ∀y(P(1 + y) ↔ 1 < y)||1|
| (8) P(1 + 0) ↔ 1 < 0||7|
| (9) ¬(1 < 0)||Theorem of arithmetic|
|(10) ¬P(1 + 0)||7,8|
|(11) P(1 + 0)||2,6|
|(12) P(0 + 1) ∧ ¬P(0 + 1)||10,11|
Ketland's formal constraint, requiring that whatever higher-order predicate variables appear in the definiens also occur in the definiendum, is therefore neither a necessary nor a sufficient impediment to logical paradox. Reckless predicates get us into logical trouble regardless of whether or not we observe Ketland's restriction. This is evident in the paradoxical constructions presented above, ∀Fn+1∀Fn(Fn+1(Fn) ↔ ∀Gn(Fn+1(Gn) ↔ ¬Fn+1(Gn)) and ∀Fn+1∀Fn(Fn+1(Fn) ↔ ∀Gn+1(Gn+1(Fn) ↔ ¬Gn+1(Fn)), in which the knotty predicate “Fn+1” is introduced in two different ways. The only thing such paradoxes prove is that the predicates themselves harbor hidden logical inconsistencies, that they are syntactically and semantically problematic, and hence are to be avoided in rigorously defining concepts and developing a logically consistent formalism. Logic itself is never to blame for the mischief caused when intrinsically paradoxical applied concepts are brought into logical structures.
We are not required to accept any and every logically regimented predicate into applied logic. Nor are we automatically committed to the existence of a corresponding property whenever we can formalize such a predicate. The fact that a predicate “P” can be defined in a language does not by itself guarantee that even such logical constructions as Pa, ¬Pa, or ∃x(Px ∨ ¬Px), and so on, must have truth value. Whether they do or not can depend on such extrasyntactical factors as the preferred predicate-quantificational semantics, on the content of “P,” if we are intensionalists, on the contingent state of the world and the existent entities it happens to include and exclude, if we are extensionalists, or, if we are intuitionists, of all extant sound proofs or disproofs of all logical and mathematical theorems and antitheorems, among other kinds of potentially semantically relevant background.
Moreover, if that were the issue, which emphatically it is not, we could always trivially satisfy Ketland's formal constraint on definitions in such cases as my typed definition of heterologicality without affecting its function in the derivation of a typed version of Grelling's paradox. It can be done in a variety of ways, for example, as follows:
∀Fn+1∀Fn((Hn+1(Fn) ∧ Fn+1 = Fn+1) ↔ ¬Fn+1(Fn))
Performing all the same step-by-step uniform substitutions as in my argument for Grelling's revenge produces logical inconsistency and a harmless identity statement within the purely formal constraints on definitions that Ketland recommends. Since the constraints are supposed to be purely formal anyway, it should make no difference that the addition is trivial. The typed Grelling's paradox goes through, and Ketland should have no legitimate complaint to raise about the syntactical probity of the definition by which the argument proceeds.
Any of these considerations taken by itself, and with greater force when considered collectively, implies that the syntactical constraint on definitions Ketland invokes should at most apply only and exclusively to variables in object place rather than predicate place.