Borel型概率计量逻辑

Borel probabilistic and quantitative logic

视全体赋值之集为通常乘积拓扑空间,利用该空间上的Borel概率测度在二值命题逻辑中引入了公式的概率真度概念.该方法既克服了计量逻辑学要求赋值集上的概率测度必须为均匀概率测度的无穷可数乘积的局限,又弥补了概率逻辑学只讲局部而缺乏整体性的不足;证明了计量逻辑学中公式的真度、随机真度以及概率逻辑学中公式的概率等概念都可作为本文提出的概率真度的特例而纳入到统一的框架中,从而实现了计量逻辑学与概率逻辑学的融合与统一;证明了逻辑闭理论与赋值空间中的拓扑闭集是一一对应的以及概率真度函数与赋值空间上的Borel概率测度是一样多的等若干结论;本文的第4节给出了公式的概率真度的公理化定义,证明了公式集上满足Kolmogorov公理的任一[0,1]值函数均可由赋值空间上的某Borel概率测度按本文的方法所表出,从而建立了二值命题逻辑框架下的概率计量逻辑的理论体系.

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.