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53149099882
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note
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For instance, in Åqvist and Mullock (1989), we use such a finite framework to develop a detailed theory of causation by agents and the representation of causal issues in Tort and Criminal Law, which is based on a version of Games and Game Theory in Extensive Form. Cases (Anglo-American, Swedish, German) are examined and graphically represented by means of game trees. The main result is that agent causation involves essentially the conjunction of two conditions, a negative (avoidability of harm in the sense of historical possibility of absence of harm) and a positive one (historical necessitation of harm); hence my present concern about the logic of historical necessity and possibility. Again, although the temporal setting of Åqvist and Hoepelman (1981) happens to be infinite, its treatment of the Chisholm Contrary-to-Duty Paradox - a core problem in deontic logic - would be best understood in a finite framework like the one proposed here. Note also the important role played by the historical modalities in that work.
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2
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53149098403
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note
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The object language of our systems is obviously very rich: it contains no less than 11 primitive one-place modal tense operators as well as infinitely many frame constants for each dimension. This raises the question whether our object language can be simplified. As appears from the axiomatics given in Section 6 infra, there are indeed several ways of reducing the number of primitive operators in the systems TWxy. The JPL referee has pertinently suggested one such way, viz. the following: take as primitives the systematic frame constants and additionally the operators and Then define: (Matrix Equation Presented) Another reduction method is intimated in the Appendix, the proof of the Coincidence Lemma 8.3, Case A = B, Remark: take the frame constants still as primitives together with the operators e, w, n and s. Then define the Priorean G, H, G′, H′ and their duals as in Section 6 (preferably by means of explicit categorical definitions as indicated in the next note), the latitudinal and the longitudinal by means of our axiom schema A6(b) in Section 6 infra, the universal by A6(d), (c) and, finally, o as in the list above. In this paper we have chosen not to deal with alternative versions, like those just mentioned, of the axiomatic systems TWxy as presented below. In order for those alternative versions to be adequate they will anyway have to be formulated in such a way that we can prove them to be deductively equivalent to the version given below in Section 6, which seems to be the most perspicuous one (in spite of its large list of axiom schemata). But the matter may well be dealt with on another occasion.
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3
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53149145846
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note
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The referee neatly points out that my 'conditional' definition Def G can, and should, be replaced by the following 'categorical' one: (Matrix Equation Presented) and analogously for Def H, Def G′ and Def H′. This observation is of course perfectly correct, and the reader should establish the equivalence of the two types of definition in the four cases under consideration here.
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