Abstract
The present paper introduces the notion of the probabilistic truth degree of a formula by means of Borel probability measures on the set of all valuations, endowed with the usual product topology, in classical two-valued propositional logic. This approach not only overcomes the limitations of quantitative logic, which require the probability measures on the set of all valuations to be the countably infinite product of uniform probability measures, but also remedies the drawback that probability logic behaves only locally. It is proved that the notions of truth degree, random truth degree in quantitative logic and the probability of formulas in probability logic can all be brought as special cases into the unified framework of the probabilistic truth degree. Thus quantitative logic and probability logic are unified. It also proves a one-to-one correspondence between deductively closed theories and topologically closed subsets of the space of all valuations, and a one-to-one correspondence between probabilistic truth degree functions and Borel probability measures on the space of all valuations. The second part of the present paper proposes an axiomatic definition of the probabilistic truth degree, and it is finally proved that each probabilistic truth degree function is represented by a unique Borel probability measure on the space of all valuations in the way given in the first part. Thus a theory which we call probabilistic and quantitative logic in the framework of classical propositional logic is established.
Similar content being viewed by others
References
Hailperin T. Probability logic. Notre Dame J Form Logic, 1984, 25: 198–212
Nilsson N J. Probabilistic logic. Artif Intell, 1986, 28: 71–87
Adam E W. A Primer of Probability Logic. Stanford: CSLI Publications, 1998. 11–34
Hailperin T. Sentential Probability Logic. London: Associated University Press, 1996. 187–212
Wang G J, Fu L, Song J S. Theory of truth degrees of propositions in two-valued logic. Sci China Ser A-Math, 2002, 45: 1106–1116
Wang G J, Leung Y. Integrated semantics and logic metric spaces. Fuzzy Set Syst, 2003, 136: 71–91
Li B J, Wang G J. Theory of truth degrees of formulas in Łukasiewicz n-valued propositional logic and a limit theorem. Sci China Ser F-Inf Sci, 2005, 48: 727–736
Li J, Wang G J. Theory of truth degrees of propositions in the logic system L *n . Sci China Ser F-Inf Sci, 2006, 49: 471–483
Wang G J, Duan Q L. Theory of (n) truth degrees of formulas in modal logic and a consistency theorem. Sci China Ser F-Inf Sci, 2009, 52: 70–83
Zhou H J, Wang G J. A new theory consistency index based on deduction theorems in several logic systems. Fuzzy Set Syst, 2006, 157: 427–443
Zhou H J, Wang G J, Zhou W. Consistency degrees of theories and methods of graded reasoning in n-valued R 0-logic (NM-logic). Int J Approx Reason, 2006, 43: 117–132
Zhou H J, Wang G J. Generalized consistency degrees of theories w.r.t. formulas in several standard complete logic systems. Fuzzy Set Syst, 2006, 157: 2058–2073
Wang G J, Zhou H J. Quantitative logic. Inform Sciences, 2009, 179: 226–247
Wang G J, Zhou H J. Introduction to Mathematical Logic and Resolution Principle. Beijing/Oxford: Science Press and Alpha Science International Limited, 2009. 257–324
Wang G J, Hui X J. Randomization of classical inference patterns and its application. Sci China Ser F-Inf Sci, 2007, 50: 867–877
Kolmogorov A N. Foundations of Probability. New York: Chelsea Publishing Co, 1950. 2–12
Hamilton A G. Logic for Mathematicians. London: Cambridge University Press, 1978. 27–36
Halmos P R. Measure Theory. New York: Springer, 1974. 154–160, 183
Cohn D L. Measure Theory. Boston: Birkhuser, 1980. 196–296
Munkres J R. Topology. 2nd ed. Beijing: China Machine Press, 2004. 119–125
Zhou H J, Wang G J. Characterizations of maximal consistent theories in the formal deductive system ℒ* (NM-logic) and Cantor space. Fuzzy Set Syst, 2007, 158: 2591–2604
Mill J V. Infinite-Dimensional Topology. Amsterdam: North-Holland, 1988. 17–136
Wang G J, Wang W, Song J S. Topology on the set of maximal consistent propositional theories and the Cantor ternary set (in Chinese). J Shaanxi Normal Univ(Nat Sci Ed), 2007, 35: 1–5
Zhang D X, Li L F. Syntactic graded method of two-valued propositional logic formulas (in Chinese). Acta Electron Sinica, 2008, 36: 325–330
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhou, H., Wang, G. Borel probabilistic and quantitative logic. Sci. China Inf. Sci. 54, 1843–1854 (2011). https://doi.org/10.1007/s11432-011-4268-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11432-011-4268-x