Is not true of itself ' is true of 'is not true of itself ' if and only if 'is not true of itself ' is not true of itself; which is to say, 'Is not true of itself ' is true of itself if and only if it is not true of itself. This has the form B if and only if not-B, which is contradictory according to standard logic. This is often called Grelling's paradox, or the heterologicality paradox. (Its inventor Grelling used the term 'heterological' as an abbreviation of 'is not true of itself'.) 2. RUSSELL'S PARADOX FOR PROPERTIES Grelling's paradox concerns linguistic expressions, but it is arguable that underlying it is a more basic paradox concerning properties, one that is in essence due to Russell. The property of being red is instantiated by all and only the red things; the property of having mass greater than 10 kilograms is instantiated by all and only things that have mass greater than 10 kilograms. More generally, for any ¹ See Lavine 1994, pp. 76 ff. ² See the opening sentence of Myhill 1984. ³ The paradox Russell himself had in mind was probably closer to the property paradox than to the paradox for mathematical sets. Russell originally raised the paradox in connection with Frege's theory of extensions. Extensions were supposed to be like mathematical sets in obeying an axiom of extensionality: no two extensions were instantiated by exactly the same things. But they were to be like conceptual properties in that every predicate had one, and that they didn't fall into a hierarchy of ranks. ¹⁵ Actually some such solutions are disguised semi-classical solutions (see previous footnote); in this case, it is the extent to which reasoning by cases needs restricting that depends on the degree to which there's a failure of predicates to express properties. ¹⁶ Moreover, if one thinks that the predicates for which excluded middle is suspect are exactly those that don't express properties, this is likely to prejudice the type of Paracomplete solution one entertains. e.g. one might think that if a predicate doesn't express a property, then no disjunction of that predicate with another could express a property either; but that would force a rather unattractive Paracomplete solution with a very weak logic.

No abstract

Any interesting version of materialism requires not only that there be no irreducibly mental objects, but also that there be no irreducibly mental properties: the idea that although people and certain higher, animals do not contain any immaterial substance, nonetheless they have certain mental properties that are completely unexplainable in physical terms, is an idea that very few people who regard themselves as materialists would find satisfying} The unsatisfactoriness of postulating irreducibly mental properties is the source of the two main problems in the philosophy of mind. The first and more widely discussed problem is the problem raised by experiential properties, for instance the property of feeling pain: a materialist needs to provide a believable account of such properties according to which those properties are not irreducibly mental. The second problem, raised by Brentano, 2 is the problem of intentionality. Many mental properties -believing, desirilag, and so forth -appear to be relational properties: more precisely, they appear to relate people to non-linguistic entities called propositions. So any materialist who takes believing and desiring at face value -any materialist who admits that belief and desire are relations between people and propositions -any such materialist must show that the relations in question are not irreducibly mental. 'Brentano felt that this could not be done; and since he saw no alternative to viewing belief and desire as relations to propositions, he concluded that materialism must be false.The first half of this paper is an examination of Brentano's problemthat is, of the problem (which Brentano believed to be unsolvable) of giving a materialistically adequate account of believing, desiring, and so forth. Unlike Brentano, I take it as unquestionable (given what we know about the world) that materialism is true; I also take it as unquestioned for the purposes of this paper that people do believe and desire. These two assumptions together amount to the assumption that Brentano's problem can be solved; what I shall be interested in is the question of how it can be solved, that is, of what the assumption that it can be solved shows about the nature of belief and desire. I will be particularly concerned to show that Erkenntnis 13 (1978) 9-61.

The paper offers a solution to the semantic paradoxes, one in which (1) we keep the unrestricted truth schema "True( A ) ↔ A", and (2) the object language can include its own metalanguage. Because of the first feature, classical logic must be restricted, but full classical reasoning applies in "ordinary" contexts, including standard set theory. The more general logic that replaces classical logic includes a principle of substitutivity of equivalents, which with the truth schema leads to the general intersubstitutivity of True( A ) with A within the language.The logic is also shown to have the resources required to represent the way in which sentences (like the Liar sentence and the Curry sentence) that lead to paradox in classical logic are "defective". We can in fact define a hierarchy of "defectiveness" predicates within the language; contrary to claims that any solution to the paradoxes just breeds further paradoxes ("revenge problems") involving defectiveness predicates, there is a general consistency/conservativeness proof that shows that talk of truth and the various "levels of defectiveness" can all be made coherent together within a single object language.