Oretic

1. Introduction

In three recent papers, Everett (1993, 1994 and 1996), Anthony Everett has produced a number of interesting and interrelated arguments against a dialetheic solution to the semantic paradoxes. The purpose of this paper is to assess the effectiveness of these arguments. The relationship between the papers is roughly this. Everett (1994) argues that a dialetheic solution to the paradoxes falls foul of certain kinds of Curry paradox (left pincer); Everett (1993) argues that it falls foul of extended paradoxes (right pincer); Everett (1996) argues that any way of avoiding one pincer drives one firmly into the other (the fork). In the next section I will set up the background for the discussion. I will then take the papers in turn. I will argue that Everett's strategic onslaught fails, on all fronts. There are, however, interesting lessons to be learned from his campaign.

Background

The paradigm semantic paradoxes are set up with a truth predicate, T, and some sort of fixed-point construction. If we use angle brackets as notation for a naming device, then a liar sentence is one of the form [lambda], where [Mathematical Expression Omitted] Applying an instance of the T-schema, [Mathematical Expression Omitted], and transitivity gives [Mathematical Expression Omitted], whence the law of excluded middle (or various other principles) gives [Mathematical Expression Omitted]. If we let [perpendicular to] be the absurdity constant, which entails everything,(1) then the Curry paradox is a sentence of the form K, where [Mathematical Expression Omitted]). Using the T-schema for K and transitivity gives us [Mathematical Expression Omitted]. The principle of absorption (or contraction) ([Mathematical Expression Omitted]) then gives us [Mathematical Expression Omitted], and a couple of applications of modus ponens give [perpendicular to].

In Priest (1987) and elsewhere, I have given an account of how I think these paradoxes should be handled. The liar paradox is sound: the contradiction at its conclusion is true. What is challenged is the view that this is unacceptable. In particular, the principle of inference ex contradictione quodlibet ([Mathematical Expression Omitted]) is invalid. This is demonstrated using a semantics according to which sentences may take some nonempty subset of the usual truth values, {t,f}, but otherwise life is as normal in classical logic.

The situation concerning the Curry paradox is rather different. We are not at liberty to accept its conclusion. It delivers us triviality without the detour through ECQ. The solution advocated in Priest (1987) is simply that the conditional operator, [Mathematical Expression Omitted], of the T-schema, does not satisfy absorption.(2) Now there are many semantics for the conditional that invalidate absorption, e.g., algebraic or (ternary relation) world semantics for relevant logics, various semantics for linear logic, continuum-valued semantics for fuzzy logics, etc. And the philosophical foundations of these semantics is a matter of active research.(3) Pretty much any account of the conditional that invalidates absorption would suffice (Priest, 1987, p. 102). However, for the sake of definiteness, I there contented myself with a simple (binary relation) possible-worlds account. The conditional of the T-schema, [Mathematical Expression Omitted], is taken to be a strict implication. (The worlds are also 3-valued, with truth values {t}, {t,f} and {f} - though this is not pertinent to the discussion of conditionality.) The accessibility relation on worlds is not unrestrictedly reflexive, and a counter-model to absorption is then easy to construct. If (and only if) reflexivity fails at world w of some interpretation, then it is quite possible to have [alpha] and [Mathemarical Expression Omitted] true at w, without having [beta] true there. The rest is routine.(4)

Let me finish these background considerations with one further comment. There is an important conceptual distinction between truth and truth-in-an-interpretation, or in the case of languages with intensional connectives, truth-at-a-world-in-an-interpretation. And this is often forgotten in discussions of paradox. Truth is a property (or at least a monadic predicate); it may be invoked for many different purposes, so far in this paper, it has appeared only in the object language in which the paradoxical reasoning is performed. Truth (at a world) in an interpretation is a set-the-oretic relation between a sentence and the value t; unlike truth, it is a model-theoretic notion, and has one primary use: to provide a metatheoretic definition of validity. Maybe it is a constraint on notions of interpretation that there be (a world in) an interpretation, truth at which is extensionally equivalent to truth simpliciter. But that is another matter.

3. Absorption regained

Let us now turn to Everett's (1994) left pincer which concerns Curry paradoxes. The aim here is to define some new conditional, ] , such that: (a) ] satisfies contraction, (b) we have the version of the T-schema with ] (or rather, its corresponding biconditional) as the main connective. The Curry paradox then goes through as before.

Everett gives two such conditionals. Let us take the second first. (In Everett (1994) this is written as a pair of stacked `[right arrow]'s.) Take any interpretation, if the accessibility relation is reflexive at a world, let us call the world itself "reflexive". The truth conditions for ] are simply that [alpha] ] [beta] is true at world w iff for all reflexive worlds accessible to w, if [alpha] is true as w, so is [beta]. If [Mathematical Expression Omitted] is true at world w then at all worlds accessible to w where [alpha] is true, [beta] is true. A fortiori, then, [alpha] ] [beta] is true at w. Thus we have (b) since it follows from the T-schema couched in terms of [Mathematical Expression Omitted].

Turning to (a), Everett claims that ] satisfies absorption.(5) The reason given for this is that even models that invalidate the rule validate it (p. 415). For justification of this we are referred back to a passage where Everett explains why a counter-model for absorption requires reflexivity to fail. The exact nature of this argument is less than clear, since it goes by way of discussing one particular counter-model. However, there are, as far as I can see, only two distinct ways of spelling out the general argument, and both fail, as I shall now explain.

The first way is as a direct argument. Suppose that w is a world of some interpretation, and that [alpha] ] (alpha] ] [beta]), is true at it. We need to show that [alpha] ] [beta] B is also true at it, i.e., that for all accessible reflexive w', if [alpha] is true at w' so [beta]. So suppose that w' is reflexive and that [alpha] is true there. Then by applications of modus ponens - crucially, more than one - we infer that [beta] holds at w'. We now apply the rule of Conditional Proof (CP, if-introduction) to infer: if [alpha] is true at w', then [beta] is true at w'.

This reasoning is classically unimpeachable. However, it applies CP in the metalanguage. Moreover, it does this where an antecedent has been used multiple times to deduce the consequent. This is exactly a contraction. If one were to spell out the reasoning without using natural deduction techniques one would see that at the crucial point absorption is used. (I leave this as an exercise.) Alternatively, one might just note that if one is allowed CP in this form in a natural deduction system (together with modus ponens) a proof of absorption is quickly forthcoming. So an unrestricted form of CP is at least as strong as absorption.

We see, then, that this metatheoretic reasoning for the validity of absorption uses absorption (or something at least as strong). As I have insisted in many places,(6) the logic of meta-theoretic reasoning must be the same as that of the object-language. Hence, reasoning in such a way is not dialetheically acceptable, and to insist on it would simply beg the question. The argument therefore fails,(7) There are, in fact, general reasons why any attempt to define a conditional that satisfies absorption must suffer the same fate, though I will not go in to this here. (A discussion can be found in Priest 1990 [sections] 7.)

4. A semantic limbo

The second way of interpreting Everett's argument is not as a direct, but as a reductio, argument and goes as follows. Suppose that we had a countermodel to absorption for ]. Then at some reflexive world, w, of the interpretation, we would have to have [alpha] and [alpha] ] [beta] true, and [beta] untrue. But since w is reflexive and [alpha] ] [beta] is true at w, so is [beta]. Contradiction.

The problems with this argument are twofold. First, reductio is not generally valid in dialetheic logic. This argument therefore fails since it begs the question, just as the first one did. Secondly, even if the reductio were correct, it would show only that there is no interpretation, I, and world w, in I, such that [alpha] ] ([alpha] ] [beta]) is true at w, and [alpha] ] [beta] is not true at w. The logical form of this is: [Mathematical Expression Omitted]. The statement that the inference is valid is, in the same notation, [Mathematical Expression Omitted]. And the inference from the first of these to the second is not valid for an intensional conditional, [Mathematical Expression Omitted]. Again, therefore, the argument fails.(8)

This argument does raise an interesting question, however. The connective ], as defined, appears to make perfectly good dialetheic sense, though it might not have the properties that a classical logician takes it to have. What properties do the semantics show it to have? It is not difficult to check through the reasoning that shows that, e.g., identity, [alpha] ] [alpha], is valid. It is also simple to come up with counter-models for principles such as [alpha] ] ([alpha] ^ [beta]). Now consider absorption itself We cannot show that this is valid - provided we stick to valid reasoning. But if we try to give a counter-model, we run into contradiction. There would therefore appear to be no counter-models to it either.

It would, seem, then, that there must be a class of inferences, including absorption, that can neither be shown to be valid nor be shown to be invalid by these semantics. In particular, the dialetheist is committed to saying that absorption for ] is invalid, and hence to claiming that there are invalid inferences that the semantics cannot show to be so - at least, as these are interpreted classically. Situations of this kind are not unknown in non-classical logics. For example, Kripke semantics for intuitionist predicate logic are known to be complete classically, but incomplete intuitionistically, provided that Church's Thesis is intuitionistically correct.(9) Hence, for the intuitionist, the dual phenomenon arises: inferences that are valid, but a classical interpretation of the semantics cannot demonstrate them to be so.

In this situation, there are various possibilities. One is that we give up the claim that semantics are constitutive of validity. Validity is to be defined proof-theoretically, semantics are a heuristic tool only, useful in many contexts, but with limitations of which we are aware. The second is that we retain attachment to the primacy of semantics, but change the semantics to other, classically intelligible, semantics. This is always an option.(10)

The third option is the most interesting, but also the most problematic. Non-classical logics make possible mathematical structures that are classically unintelligible. It may be possible to use this space. For example, the Law of Excluded Middle, [Mathematical Expression Omitted], is intuitionistically false, but no classical counter-example can be given, or this would provide an argument for [Mathematical Expression Omitted]. However, using versions of the theory of choice sequences, inconsistent with the classical theory of reals, it is possible to give a counterexample to the Law.(11)

In the dialetheic case, we might try to produce counter-examples to principles such as absorption for ] using inconsistent structures. In the light of the preceding discussion, it is not difficult to see how this might go. Consider the model with one reflexive world, w, such that p is true at w, and q both is and is not true at w. (Note that this specification is inconsistent.) Since p and q are true at the only accessible world, p ] q is true there, as, therefore, is p ] (p ] q). But since p, is true at w and q is not, p ] q, is not true there. Hence, the inference is invalid.

The status of this model is problematic, even from a dialetheic point of view. It requires more than that some sentence is both true and false at some world of a model, that is, that it takes the value {t,f} there (which is quite consistent given the semanticndly, even if the reductio were correct, it would show only that there is no interpretation, I, and world w, in I, such that [alpha] ] ([alpha] ] [beta]) is true at w, and [alpha] ] [beta] is not true at w. The logical form of this is: [Mathematical Expression Omitted]. The statement that the inference is valid is, in the same notation, [Mathematical Expression Omitted]. And the inference from the first of these to the second is not valid for an intensional conditional, [Mathematical Expression Omitted]. Again, therefore, the argument fails.(8)

This argument does raise an interesting question, however. The connective ], as defined, appears to make perfectly good dialetheic sense, though it might not have the properties that a classical logician takes it to have. What properties do the semantics show it to have? It is not difficult to check through the reasoning that shows that, e.g., identity, [alpha] ] [alpha], is valid. It is also simple to come up with counter-models for principles such as [alpha] ] ([alpha] ^ [beta]). Now consider absorption itself We cannot show that this is valid - provided we stick to valid reasoning. But if we try to give a counter-model, we run into contradiction. There would therefore appear to be no counter-models to it either.

It would, seem, then, that there must be a class of inferences, including absorption, that can neither be shown to be valid nor be shown to be invalid by these semantics. In particular, the dialetheist is committed to saying that absorption for ] is invalid, and hence to claiming that there are invalid inferences that the semantics cannot show to be so - at least, as these are interpreted classically. Situations of this kind are not unknown in non-classical logics. For example, Kripke semantics for intuitionist predicate logic are known to be complete classically, but incomplete intuitionistically, provided that Church's Thesis is intuitionistically correct.(9) Hence, for the intuitionist, the dual phenomenon arises: inferences that are valid, but a classical interpretation of the semantics cannot demonstrate them to be so.

In this situation, there are various possibilities. One is that we give up the claim that semantics are constitutive of validity. Validity is to be defined proof-theoretically, semantics are a heuristic tool only, useful in many contexts, but with limitations of which we are aware. The second is that we retain attachment to the primacy of semantics, but change the semantics to other, classically intelligible, semantics. This is always an option.(10)

The third option is the most interesting, but also the most problematic. Non-classical logics make possible mathematical structures that are classically unintelligible. It may be possible to use this space. For example, the Law of Excluded Middle, [Mathematical Expression Omitted], is intuitionistically false, but no classical counter-example can be given, or this would provide an argument for [Mathematical Expression Omitted]. Howevera counterexample to the Law.(11)

In the dialetheic case, we might try to produce counter-examples to principles such as absorption for ] using inconsistent structures. In the light of the preceding discussion, it is not difficult to see how this might go. Consider the model with one reflexive world, w, such that p is true at w, and q both is and is not true at w. (Note that this specification is inconsistent.) Since p and q are true at the only accessible world, p ] q is true there, as, therefore, is p ] (p ] q). But since p, is true at w and q is not, p ] q, is not true there. Hence, the inference is invalid.

The status of this model is problematic, even from a dialetheic point of view. It requires more than that some sentence is both true and false at some world of a model, that is, that it takes the value {t,f} there (which is quite consistent given the semantics); it requires the evaluation function itself to be inconsistent, so that a sentence both is and is not true at a world, i.e., t both is and is not in its value at that world. It is not clear that one can prove the existence of this kind of interpretation. Moreover, if models of this kind are legitimate, then a lot more than absorption is going to turn out to be invalid. For example the conditional [Mathematical Expression Omitted] is invalid (though valid as well), since in the above model p is true at w but p isn't. (This may not be as damaging as it sounds. We reason correctly with formally invalid inferences all the time: every inference is of the form [Mathematical Expression Omitted], which is invalid. A good argument is one which instantiates some valid form of inference.) The viability of this line of thought obviously requires much further consideration, what turns out to be invalid may depend on sensitive questions about how, exactly, the semantics are set up. Moreover, to consider the issue here would take us a long way out of our way. So let us drop the subject, and return to Everett's arguments.

h as [alpha] ] ([alpha] ^ [beta]). Now consider absorption itself We cannot show that this is valid - provided we stick to valid reasoning. But if we try to give a counter-model, we run into contradiction. There would therefore appear to be no counter-models to it either.

It would, seem, then, that there must be a class of inferences, including absorption, that can neither be shown to be valid nor be shown to be invalid by these semantics. In particular, the dialetheist is committed to saying that absorption for ] is invalid, and hence to claiming that there are invalid inferences that the semantics cannot show to be so - at least, as these are interpreted classically. Situations of this kind are not unknown in non-classical logics. For example, Kripke semantics for intuitionist predicate logic are known to be complete classically, but incomplete intuitionistically, provided that Church's Thesis is intuitionistically correct.(9) Hence, for the intuitionist, the dual phenomenon arises: inferences that are valid, but a classical interpretation of the semantics cannot demonstrate them to be so.

In this situation, there are various possibilities. One is that we give up the claim that semantics are constitutive of validity. Validity is to be defined proof-theoretically, semantics are a heuristic tool only, useful in many contexts, but with limitations of which we are aware. The second is that we retain attachment to the primacy of semantics, but change the semantics to other, classically intelligible, semantics. This is always an option.(10)

The third option is the most interesting, but also the most problematic. Non-classical logics make possible mathematical structures that are classically unintelligible. It may be possible to use this space. For example, the Law of Excluded Middle, [Mathematical Expression Omitted], is intuitionistically false, but no classical counter-example can be given, or this would provide an argument for [Mathematical Expression Omitted]. Howeverrption for ] using inconsistent structures. In the light of the preceding discussion, it is not difficult to see how this might go. Consider the model with one reflexive world, w, such that p is true at w, and q both is and is not true at w. (Note that this specification is inconsistent.) Since p and q are true at the only accessible world, p ] q is true there, as, therefore, is p ] (p ] q). But since p, is true at w and q is not, p ] q, is not true there. Hence, the inference is invalid.

The status of this model is problematic, even from a dialetheic point of view. It requires more than that some sentence is both true and false at some world of a model, that is, that it takes the value {t,f} there (which is quite consistent given the semantics); it requires the evaluation function itself to be inconsistent, so that a sentence both is and is not true at a world, i.e., t both is and is not in its value at that world. It is not clear that one can prove the existence of this kind of interpretation. Moreover, if models of this kind are legitimate, then a lot more than absorption is going to turn out to be invalid. For example the conditional [Mathematical Expression Omitted] is invalid (though valid as well), since in the above model p is true at w but p isn't. (This may not be as damaging as it sounds. We reason correctly with formally invalid inferences all the time: every inference is of the form [Mathematical Expression Omitted], which is invalid. A good argument is one which instantiates some valid form of inference.) The viability of this line of thought obviously requires much further consideration, what turns out to be invalid may depend on sensitive questions about how, exactly, the semantics are set up. Moreover, to consider the issue here would take us a long way out of our way. So let us drop the subject, and return to Everett's arguments.

5. Enthymematic conditionals

The second way that Everett (1994) suggests of obtaining a conditional that will allow the Curry argument is with the use of a logical constant, R, which holds in just reflexive worlds. In effect, [alpha] ] [beta] is now defined, enthymematically, as ([alpha] [conjunction] R) [Mathematical Expression Omitted].(12) Condition (a) is satisfied since the enthymematic biconditional now follows simply from the straight one; and, Everett (1994) claims, so is condition (b).

Before we consider this, it is worth noting that it is not at all obvious that there is a logical constant of the required kind. Why should we suppose there to be one? We are told (Everett 1994, p. 416) that we can define one explicitly, e.g., by the phrase "This world is reflexive". Now, the vocabulary of this definition is metatheoretic, and not normally taken to be a part of the object language. But as I have already argued, there can be no objection to this. It also contains demonstratives, which may be more problematic. Can we be sure that in an arbitrary world denotations of demonstratives get fixed correctly? But leaving this issue aside, this definition will not do. Assuming that demonstratives do function as required, R, so defined, is true at w iff"wRw" is true at w. It does not follow from this that wRw, since we do not have: [phi] is true at w iff [phi], truth-at-w and truth are not the same thing.(13)

If language contains no expression that will already do the job, maybe we can just add a new one with the appropriate truth conditions: R is true at world w iff w is reflexive. It is not clear, form a dialetheic point of view, that truth conditions of this kind succeed in giving the constant a determinate sense. If they did, we could just as well introduce a constant, [Mathematical Expression Omitted], true in just those worlds that access @, the actual world. Now consider a world, w, where everything is in the extension of the truth predicate. At such a word, for any [alpha], we have T <[Mathematical Expression Omitted]>, and so, by the T-schema, [Mathematical Expression Omitted]. By assumption w accesses @; hence @ is trivial. there can therefore be no such constant.(14)

Anyway, setting these worries aside, the main problem with the argument again concerns condition (b). Everett (1994) claims that ], defined enthymematically using R, satisfies absorption. The situation is, however, exactly the same as for the first conditional we considered. In fact, this case is essentially the same as the first. The constant R simply allows us to pack the truth conditions of the first conditional into the object-language. But moving from the metalanguage to the object language, like transposing the key of a piece of music, changes nothing structural.

For example, consider the direct argument fo the validity of absoption, which goes as follows. Suppose that ([Mathematical Expression Omitted]) is true at worl w of some intepretation. Let w' be any accessible world where [alpha] [conjunction] R is true. By modus ponens, ([Mathematical Expression Omitted] is true at w'. Since [alpha] [conjunction] R is true at w', w' is reflexive. Hence by modus ponenes again. [beta], is true at w'. thus if [alpha] [conjunction] R is true at w', so is [beta], i.e., ([Mathematical Expression Omitted] is true at w. Note that this argument appeals tot the supposition that [alpha] [conjunction] R is true at w' multiple times before applying Conditional Proof. Hence it uses absorption.

6. Hypercontradictions

Let us move on to Everett's (1993) right pincer. This concerns extended paradoxes. A number of people have thought that extended paradoxes sink a dialetheic account of the semantic paradoxes, just as they sink all consistent accounts: we need only consider the sentence that says of itself that it is false and not also true. This thought is incorrect, however. If we consider a fixed point, [phi], of the form [Mathematical Expression Omitted], then, reasoning in the usual way we can establish that [Mathematical Expression Omitted] - and, for good measure, F<[phi]>. This is not a problem: the aim was never to get rid of contradictions, but to accommodate them.

Timothy Smiley (Priest and Smiley 1993) has given an apparently more virulent extended paradox employing, not the notion of truth, but the notion of truth in an interpretation. He invites us to consider a fixed point, [psi], of the form [v.sub.@] ([psi]) = {f}, where [v.sub.@] is that evaluation whose assignments are in accord with the truth (simpliciter). Reasoning employing this, we conclude that t = f, and triviality rapidly ensues (Priest and Smiley, 1993, pp. 30f.).

There are a couple of possible responses to this argument (Priest and Smiley, 1993, p. 50). One is to modify one's semantics so that the set of non-empty subsets of {t,f} does not exhaust the semantic values. Any member of the set of non-empty subsets of this set may also be a value, as may non-empty subsets of this, and so on. This allows the semantic value of [psi] to be one of these hypercontradictions, and the argument to triviality is broken. A second response we will come to in a moment.

Everett (1993) argues that even given this semantic machinery, there is an extended paradox.(15) Given any set, s, let [eta](s) be the set of its urelements. We consider the fixed point, [theta], of the form [eta](v.sub.@])[theta])) = {f}. Simple reasoning then yields t = f as before. The argument, as does Smiley's original, requires us to suppose that we have the equivalence between t [element of] [v.sub.@]([theta]) and [theta] in a detachable form, which there is ground to doubt (Priest and Smiley, 1993, p. 52). However, with this reservation, it seems to me that the argument of Everett (1993) is correct here.

7. Functionality

A second response (given in Priest and Smiley 1993, p. 50f.) is to retain the original semantics, but to formulate semantic evaluations as relations, not as functions. As is shown there, this allows for the expression of everything that needs to be said, whilst invalidating Smiley's argument. There is an obvious repair of Smiley's argument, formulated with the help of additional set-theoretic machinery. This is also shown to fail. Everett (1993, fn. 6), gives a very similar argument, but using truth instead of truth-in-an-interpretation. We define a fixed point, [theta], of the form [X.sub.[theta]] = {<F>}, where [X.sub.[theta]] = {P;(P = <T> [disjunstion] P = <F>) [conjunction] T<P<[theta]>>}. (Angle brackets again are used to form names, this time of predicates.) From this it is supposed to follow that <T> = <F>. Triviality follows in a few steps.

The argument, as provided by Everett (in correspondence), depends on the premise that [X.sub.[theta]] = {<F>} [disjunction] <T> [element of] [X.sub.[theta]].(16) Now, I see no hope of proving this premise without using dialetheically invalid moves. Even given that we can establish that <F> [element of] [X.sub.[theta]] , we need to show that when the first disjunct, but not the second, obtains, [Mathematical Expression Omitted] <F>. And there are real problems about showing this. From right to left, for example, we may have that [Mathematical Expression Omitted]; but to infer that [Mathematical Expression Omitted], and so [Mathematical Expression Omitted], requires the invalid [Mathematical Expression Omitted]. This fails in the semantics of Priest (1987, p. 112) even when [alpha] is a necessary truth.(17)

This objection depends on formulating set theory using intensional connectives; and it might be thought that the argument will work if we formulate it using extensional connectives, as usual. Now there may, indeed, be good reasons for supposing that set-theory ought to be formulated with extensional connectives (see Goodship, 1996). However, if we do this, then the abstraction schema of set theory takes the form:

[Mathematical Expression Omitted]

(where [equivalent] is the material biconditional, defined as usual: [alpha] [equivalent] [beta] is ([alpha] [contains] [beta]) [conjunction] ([beta] [contains] [alpha]), where [alpha] [conjunction] [beta] is [Mathematical Expression Omitted] [alpha] [conjunction] [beta]) and we can no longer detach from left to right, or vice versa, without using the invalid disjunctive syllogism. Inferences of this form are employed at several places in the above proof, and so it still fails.

The second line of response to the original problem is therefore untouched by Everett's arguments. Before considering the rest of Everett (1993), let me digress for a moment. As Priest and Smiley (1993, p.49f.) set up the situation, a contrast is drawn between extended paradoxes employing the notion of truth, simpliciter, which are anodyne, and extended paradoxes employing the notion of truth-in-an-interpretation, which need not beone may come to realise that their beliefs, say about religion or politics, are inconsistent, by being made to assert an explicit contradiction. The assertion of [Mathematical Expression Omitted] [alpha] in this context does not express a rejection of [alpha]. Ex hypothesi, they do accept [alpha] - at least until they revise their beliefs.(25)

The question of when an utterance expresses a denial is an interesting one, which I shall not pursue here. Certainly, an utterance of [Mathematical Expression Omitted] [alpha] may express a denial of [alpha], as, nearly always, does an utterance of [alpha] [Mathematical Expression Omitted]. Even this need not do so, however. In the mouth of someone who believed everything, [alpha] [Mathematical Expression Omitted] would not express a denial - nothing would.(26) As a category, denial is sui generis, and cannot be reduced to a kind of assertion. And it is one whose employment is open to the dialetheist just as much as the non-dialetheist.

To summarise the discussion concerning (4), what we have seen is that the case given for the claim that we must have a way of expressing things that rules other things out does not stand up. Yet the dialetheist does have ways of ruling things out anyway, in whichever of the above senses this is supposed to be taken. So this part of the argument fails.

Let us turn finally, to (5). The claim here is that if there is a way of ruling something out, this will give rise to triviality-inducing extended paradoxes. Is this true? We noted that [alpha] [Mathematical Expression Omitted] rules [alpha] out, in one sense. If we formulate an extended paradox with this notion, we obtain a fixed point, [kappa], of the form T ([kappa]) [Mathematical Expression Omitted]. This is, of course, exactly Curry's paradox; and it poses no problem if absorption fails, which, according to me, it does.

The other way of ruling something out that we considered was denial, but denial is an illocutionary act, not a connective, and does not give rise to extended paradoxes in the way one might expect. About the closest one can get is a sentence like "I deny this sentence" (call this [beta]) uttered as a performative. This would be the denial of something true, and therefore, presumably, something not to be uttered. If, on the other hand, one asserts [beta] then it is false, and so not to be uttered either. The rational person, it would seem, neither asserts nor denies [beta].(27) Once one throws in a normative element, however, new paradoxes concerning assertion and denial are forthcoming. Consider, for example, the sentence "It is irrational to assert this sentence". Call this [gamma]. Someone who asserts this is asserting something, and at the same time asserting that it is irrational to assert it. This is irrational. Hence, it is irrational to assert [gamma]. But we have just established this (i.e., [gamma]). Hence it is rational to assert it.(28) This is certainly not a triviality-generating paradox, however, and so not a problem for the dialetheist. It is a problem for those who espouse consistency. But I will leave others to worry about that. The important point is that the final step of the argument we have been considering fails.

9. Conclusion

I have now finished discussing the arguments of Everett (1996), and with it, all three of Everett's papers.(29) As I have argued, the central contentions of each fail. In particular, there is no connection of the kind envisaged between extended paradoxes and Curry paradoxes (though there is a connection of a different kind, as we saw in the last section). The discussion has raised some issues that require further consideration, but that may be left for another occasion.(30)

(1) This makes perfectly good sense dialetheically. See Priest 1987, [sections] 8.5. (2) There is another solution, recently advocated by Goodship (1996), that is in some ways more appealing. This is that the conditional of the T-schema is simply a material conditional, [implication]. Absorption for the material conditional holds, but detachment fails. Hence, the Curry paradox is blocked. In fact, T<[kappa]> [implication] [Mathematical Expression Omitted] is logically equivalent to [Mathematical Expression Omitted] and so the Curry paradox collapses into the Liar paradox. (3) Everett (1996, fn. 9) notes the work on contraction-free logics, but claims that this does not undermine the intuitive plausibility of absorption, since the various semantics are not plausible candidates for those of natural language conditionals. Now many of the semantics in question operate with highly intuitive notions such as sense and information-flow, whose relationships with conditionality are intimate ones. Though I do not claim that this is yet the case, I think it very likely that developments in these semantics may come to undermine the apparent plausibility of absorption for some kinds of conditionals, in just the way that Seventeenth Century developments in dynamics undermined the apparent plausibility of various Aristotelian claims about motion, such as that things in motion naturally come to rest. (4) Only a little is said in Priest 1987 about why the accessibility relation is not generally reflexive. I now think that the best story to tell is one according to which it fails at non-normal worlds, these being worlds where the laws of logic are different (making it possible to consider counter-logical conditionals); in particular, in this case, where modus ponens fails. See Priest 1992. The fact that it fails only at worlds where logic is different shows why such a failure does not count against [Mathematical Expression Omitted] being a conditional operator. (This line of thought indicates a reply to Everett 1994, fn.3.) (5) Or strictly speaking, the corresponding conditional, but the difference is inessential here. Note, though, that Everett's discussion starts off with the semantics being three-valued, but slides without warning into using two-valued semantics. (6) E.g., Priest 1989, 1990, and forthcoming. (7) Others have given trivialisation arguments which commit exactly this fallacy. For example, Denyer (1989) attempts such an argument which begs the question over the disjunctive syllogism. See Priest 1989. (8) One can infer the material conditional, [Mathematical Expression Omitted], but this will not do as a paraconsistent statement of validity; for since [implication] does not support detachment, neither would valid arguments. See, further, Priest 1990, [sections] 5. (9) See Dummett 1977, [sections] 5.6, esp. p. 259. (10) Boolean negation gives rise to exactly the same situation as Everett's ]. In this case, appropriate counter-models can be defined by changing from world semantics to an algebraic semantics. (See the appendix of Priest (forthcoming).) The same trick will not work for ]. Its definition makes sense only given the resources of world semantics, and so it has no obvious algebraic counterpart. Maybe so much the worse for it. (11) See Fraenkel Bar-Hillel and Levy 1973, p. 260, or Dummett 1977, p. 84. (12) This is not quite the way that Everett (1994) sets it up. The fixed point is defined as [kappa], where [Mathematical Expression Omitted], but since T distributes over conjunction T<R> effectively becomes the suppressed premise of the enthymeme, and this holds at all and only the worlds that R does, by the instance of the T-schema: T<R> [Mathematical Expression Omitted]. I set things up in the way I have in order to make the discussion more economical. (13) Everett (1994, fn. 5) suggests using [Mathematical Expression Omitted] as an alternative suppressed premise. This is clearly a lot stronger, but if it is true at w, so is wRw, and the rest of the argument proceeds as before. The problem here is even more transparent. From the fact that [Mathematica Expression Omitted] is true at a world, we certainly cannot infer that [Mathematical Expression Omitted] (is true). (14) I suggested (Priest 1990), following Belnap, that the fact that the addition of a piece of logical machinery to a theory gives a non-conservative extension may show that it lacks determinate sense. The constant [Mathematical Expression Omitted] - and, if the rest of Everett's argument is right, R - obviously fails this test. As Everett (1994, p. 418) notes, this argument presupposes that the theory contains the T-schema; but that's exactly what I am presupposing. (15) Priest and Smiley (1993, p. 50) envisage the possibility of this. Everett's claim (1993, p. 6) that the paper says that it will be impossible, given the extended semantics, to construct a triviality-inducing extended paradox is just incorrect. (16) For the record, it is an argument by dilemma, and goes as follows. Suppose that [X.sub.[theta]] {<F>}, i.e., [theta]; then by the T-schema and a couple of simple moves, <T> satisfies the defining condition of [X.sub.[theta]], i.e., <T> [epsilon] [X.sub.[theta]] = {<F>}, and so <T> = <F>. Alternatively, suppose that <T> [epsilon] [X.sub.[theta]]; then by the definition of [X.sub.[theta]] and a couple of applications of the T-schema, [theta], i.e., [X.sub.[theta]] = {<F>}; so again <T> = <F>. (17) With a slightly different account of the conditional, it can be shown that any argument of this kind must fail. Brady (1989) demonstrates that naive set theory with an underlying relevant logic is non-trivial, even when the abstraction schema holds in the very strong form: [Mathematical Expression Omitted] where [alpha] is any condition, even one that contains [gamma]. For any closed [alpha], we may define <[alpha]> as {z;[alpha]}, for some fixed variable, z, and T[chi] as [phi] [epsilon] [chi]. It follows straight away that for any closed [alpha], [Mathematical Expression Omitted]. (See Priest 1990, fn. 9.). The theory therefore includes a theory of truth. Moreover, the strong abstraction schema gives us fixed points of the kind required for self-reference. Let [alpha]([chi] be any formula of one free variable, [chi]. By the schema, there is a set, s, such that: [Mathematical Expression Omitted] It follows that: [Mathematical Expression Omitted] whence there is a fixed point [beta] such that [beta] [Mathematical Expression Omitted] [alpha] (<[beta>). Hence, there is a non-trivial theory of sethood, truth and self-reference. (18) This was pointed out to me by Uwe Peterson. (19) When the argument is rehearsed more briefly in Everett 1993, it is, in fact, given in this more general form. (20) See Priest 1987, [sections] 8.4 for a defence of the claim that consistency is a default assumption. (21) Similar claims have been made by Parsons (1990) and Batens (1990). In particular, Parsons also claims that a dialetheic solution to the paradoxes is in exactly the same situation concerning the inexpressibility of certain notions as are consistent solutions. These claims are discussed and rejected in Priest 1995. (22) Everett (1996) tends to put the point in terms of truth, simpliciter, but since it is model-theoretic validity we are talking about, it should be truth in an interpretation. (23) It might, of course, be argued that a sentence cannot express any claim unless it rules out something, and in particular, its negation. Such an argument would just be fallacious. See Priest forthcoming, [sections] 10. (24) This is what Everett means by "ruling out", as may be inferred from Everett (1996, p. 14). He says there that a semantics with a simple truth predicate cannot express the claim that a sentence is not true (or takes an undesignated value) in such a way as to rule out its negation, whilst one employing model-theoretic apparatus can. There is a running together here of truth and truth-in-an-interpretation. (The language of designated values belongs to model theory.) But setting this aside, why does Everett think that this is so? Given a paraconsistent logic for the model-theoretic metatheory, v([alpha]) = {f} no more makes it logically impossible for t [epsilon] v ([alpha]) to hold than F([alpha]) [Mathematical Expression Omitted] makes it logically impossible for T <[alpha]> to hold. But from v ([alpha]) = {f} and t [epsilon] v([alpha]) it does follow that t = f, and triviality ensues. Hence, the first expression does rule out the possibility that [alpha] is true (in v) in the sense in question. (25) For further discussion of the issue, see Priest and Smiley 1993, p. 36ff. and, especially, Priest forthcoming, [subsections] 9, 10. (26) A different counter-example was provided by Everett himself in correspondence. Consider a Curry paradox, [kappa], of the form T<[kaoppa]> [Mathematical Expression Omitted]. This sentence ought to be rejected, but one cannot do so by uttering [Mathematical Expression Omitted], since this commits the utterer to [kappa] itself. (27) Various other near-paradoxes of this kind are discussed in Parsons 1984. (28) See Priest 1995, p. 61. (29) The final part Everett (1996) generalises his argument to other paradoxes, such as the heterological paradox and its currification. There seem to me to be some more moves in this which are rather too fast. But this part raises nothing essentially novel, so I will not discuss it. (30) I would like to thank Anthony Everett for helpful discussions on earlier drafts of this essay, which clarified a number of issues. I would also like to thank the editor of Mind for a very generous and helpful set of written comments.

REFERENCEC

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