A logic for propositions with indefinite truth values.

*(English)*Zbl 0564.03020The purpose of this paper is to report results connecting the system of a logic for propositions with indefinite probabilities as truth values and based on game theory. In this logic, propositions are asserted by some speaker called the proponent and are directed against another speaker called the opponent. A simple position consists of a collection (with repetitions allowed) of propositions asserted by the opponent and a collection of propositions asserted by the proponent. The debates between the two speakers may be viewed as games (with perfect information) on the graph whose vertices are positions and whose edges are moves. It is shown that the Lindenbaum algebra of positions is a lattice and, moreover, is isomorphic to a completely free lattice. The results concerning propositions are special cases of the results for positions, so we have a logic PR which is non-distributive and Post’s axiom is false. The author establishes that PR is a lattice under conjunction and disjunction but it is not modular or even orthomodular and a free lattice can be embedded in PR; to date, no axiom system is known for PR or for the logical identities in propositional variables.

Reviewer: A.S.Karpenko

##### Keywords:

propositions with indefinite probabilities as truth values; game theory; Lindenbaum algebra of positions
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##### References:

[1] | G. Birkhoff, Lattice Theory, 3rd ed., Am. Math. Soc. Coll. Publ., Vol. 25 (1967) |

[2] | R. A. Dean, Completely free lattices generated by partially ordered sets, Transaction of the American Mathematical Society, Vol. 83 (1956), pp. 238-249. · Zbl 0072.02101 |

[3] | R. P. Dilworth, Lattices with unique complements, Transactions of the American Mathematical Society, Vol. 57 (1945), pp. 123-154. · Zbl 0060.06103 |

[4] | R. Giles, Formal languages and the foundations of physics, in: Proceedings of the International Research Seminar on Abstract Representation in Mathematical Physics, London, Ontario, 1974, Reidel, Dordrecht. |

[5] | R. Giles, A pragmatic approach to the formalization of empirical theories, in: Proceedings of the Conference on Formal Methods in the Methodology of Empirical Sciences, Warsaw, June 1974, Ossolineum, Wroc?aw, and Reidel, Dordrecht, 1976. · Zbl 0361.02021 |

[6] | R. Giles, A logic for subjective belief, in: W. L. Harper and C. A. Hooker (eds.), Foundations of Probability Theory, Statistical Inference and Statistical Theories of Science, Vol. 1, Reidel, Dordrecht, 1975. · Zbl 0323.02043 |

[7] | R. Giles, Postscript to ?A pragmatic approach to the formalization of empirical theories?, (unpublished notes), April 1975. |

[8] | R. Giles, A nonclassical logic for physics, in: R. Wójcicki (ed.), Selected Papers on ?ukasiewicz Sentential Calculi, Ossolineum, Wroc?aw, 1976. |

[9] | G. F. Liddell, A logic based on game theory (Ph. D. thesis), Queen’s University, Kingston, Canada, 1975. |

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