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We consider the modal μ-calculus due to Kozen, which is a finitary modal logic with least and greatest fixed points of monotone operators. We extend the existing duality between the category of modal algebras with homomorphisms and the category of descriptive modal frames with contractions to the case of having fixed point operators. As a corollary, we obtain completeness results for two proof systems due to Kozen (finitary and infinitary) with respect to certain classes of modal frames. The rules are sound in every model, not only for validity.

In this paper we survey formal techniques for the study of reasoning about knowledge and belief, and consider their application in the areas of distributed computing and nonmonotonic reasoning.

Faced with the question of what matters in survival, we have a choice of seemingly alternative answers. Let us use the term 'person-stage' as in recent articles by David Lewis and Derek Parfit, and let us, like them, call the relation between two person-stages which belong to the same person the 'I-relation'.' Our apparent alternatives are, then, the thesis that the I-relation is what matters (which common sense seems to suggest) and the view that what matters is a psychological relation between person-stages. An important argument for regarding the answers as alternatives is furnished by the case of fission. A person-stage S belongs to a person C. At a later date a person-stage S, belongs to a person C, and a contemporary but non-identical person-stage S2 belongs to C2. Following Lewis, let us call the psychological relation which, arguably, matters in survival (psychological continuity and/or connectedness) the 'R-relation'. Now suppose that S, and S2 are both R-related to S. Are they both I-related to S? It is commonly argued that they cannot be (by Parfit among others, in his i97i article on personal identity).2 Since identity is transitive and symmetrical it is natural to anticipate that the I-relation is transitive and symmetrical. So if S, and S2 are both I-related to S they will have to be I-related to each other. At least one of S, and S2, therefore, is apparently R-related to S without being I-related to S. The moral, allegedly, is that Rand I-relations are not coextensive, and thus cannot both be what matters in survival. Lewis, unlike Parfit, denies that the two answers to the question of what matters are rivals. According to him the I-relation is the same as the R-relation. He claims not only that a person is a maximal Iinterrelated aggregate of person-stages (I shall accept this without discussion, since it seems to be no more than is implicit in the idea of a person-stage), but also that it is a maximal R-interrelated aggregate of person-stages (Lewis, p. 22). He offers an ingenious treatment of the fission case which seeks to make it consistent with his claims. According to Lewis S is a stage of both C, and C2, but C, and C2 are not the same person. So S, and S2 are not I-related, even though they are both I-related to S. Although he believes that the I-relation is symmetrical, by claiming that it is not transitive he tries to undermine the argument summarized in the previous paragraph.3

easily be introduced as a linguistic distinction between two ways of talking about time, although it is arguable that there is a corresponding distinction between two types of temporal facts. The first type of temporal language includes tense proper in the grammatical sense, temporal demonstratives such as 'now' and 'today', and the terms 'past', 'present' and 'future' which at least superficially behave as predicates of events. It is a common opinion that this is not the rag-bag which it might appear to the uninitiated. I shall follow recent practice2 in referring to the whole category as 'tensed language'. One significant feature common to this type of terminology is that any sentence involving an expression of this type has the property that not all tokens of the sentence have the same truth-value. For example a token of the sentence 'it is raining today' uttered on St Swithin's day may have the value true while one uttered on Christmas Day may have the value false. In this respect tensed language is in marked and celebrated contrast to the other type of terminology, to be called here 'untensed'. Untensed language relies on the relational predicates 'earlier than', 'later than' and 'simultaneous with', together with a liberal use of dates. By these means it is possible to produce sentences which are eternal in the sense that every one of their tokens has the same truth-value. If one token of the

In certain awkward cases there is an apparent conflict between competing criteria of personal identity. In my view in many such cases there is no answer to questions of personal identity. There is an indeterminacy in our concept of a person: the identity conditions which we associate with the concept are sufficient to dictate which person is which in a range of familiar cases but not sufficient to determine the answer to the question of identity in the puzzle cases. Derek Parfit suggests at the beginning of his well-known article on personal identity that there need not be a correct answer to questions of identity.1 In the light of his observations the content of the last paragraph may seem to lack novelty. But there is at least one respect in which the matter deserves to be taken further than it is by Parfit. Claims such as the statement that questions of personal identity sometimes lack a correct answer, or the statement that there is indeterminacy in the concept of a person, cry out for semantic elaboration. Indeed they should not be accepted until the necessary detail has been filled in. I shall try to supply some of what is lacking by addressing myself to a semantic problem raised by the thesis that the concept of a person is indeterminate, namely the problem of what we refer to by "a" in "Is a the same person as b?" when the question has no correct answer. It will be helpful to pick an example which we can keep in front of us while considering conflicting criteria of identity. It should be emphasized, however, that my semantic observations do not depend for their interest solely on whether or not there are conflicting criteria of identity in the chosen case. If there are any cases in which there are plausible alternative criteria of identity then my proposal can be applied, mutatis mutandis, in defense of the thesis that questions of identity are unanswerable in those cases. The chosen example is based on the tale of Mr. Bultitude and his schoolboy son, described in Anstey's Vice Versa and made philosophically familiar by Anthony Quinton.2 The bodies of Mr. Paul Bultitude