  Many researchers want to unify probability and logic by defining logical
probability or probabilistic logic reasonably. This paper tries to unify
statistics and logic so that we can use both statistical probability and
logical probability at the same time. For this purpose, this paper proposes the
P-T probability framework, which is assembled with Shannon's statistical
probability framework for communication, Kolmogorov's probability axioms for
logical probability, and Zadeh's membership functions used as truth functions.
Two kinds of probabilities are connected by an extended Bayes' theorem, with
which we can convert a likelihood function and a truth function from one to
another. Hence, we can train truth functions (in logic) by sampling
distributions (in statistics). This probability framework was developed in the
author's long-term studies on semantic information, statistical learning, and
color vision. This paper first proposes the P-T probability framework and
explains different probabilities in it by its applications to semantic
information theory. Then, this framework and the semantic information methods
are applied to statistical learning, statistical mechanics, hypothesis
evaluation (including falsification), confirmation, and Bayesian reasoning.
Theoretical applications illustrate the reasonability and practicability of
this framework. This framework is helpful for interpretable AI. To interpret
neural networks, we need further study.
