Title: Suppes–style natural deduction system for probability logic

Abstract:
An elegant way to work with probabilized sentences was proposed by P. Suppes. According to his approach we develop a natural deduction system $\mathbf{NKprob}(\varepsilon)$ for probability logic, inspired by Gentzen’s natural deduction system $\mathbf{NK}$ for classical propositional logic. We use a similar approach as in defining general probability natural deduction system $\mathbf{NKprob}$ (see M. Bori\v ci\’c, Publications de l’Institut Mathematique, Vol. 100(114) (2016), pp. 77–86). Our system will be suitable for manipulating sentences of the form $A^n$, where $A$ is any propositional formula and $n$ a natural number, with the intended meaning ‘the probability of truthfulness of $A$ is greater than or equal to $1-n\varepsilon$’, for some small $\varepsilon >0$.

For instance, the rules dealing with conjunction looks as follows:
$$\frac{A^m\quad B^n}{(A\wedge B)^{m+n}}(I\wedge)\qquad\frac{A^m\quad (A\wedge B)^n}{B^n}(E\wedge)$$
and with implication:
$$\frac{(\neg A)^m\quad B^n}{(A\to B)^{\min\{m,n\}}}(I\to)\qquad\frac{A^m\quad (A\to B)^n}{B^{m+n}}(E\to)$$
The system $\mathbf{NKprob}(\varepsilon)$ will be a natural counterpart of our sequent calculus $\mathbf{LKprob}(\varepsilon)$ (see M. Bori\v ci\’c, Journal of Logic and Computation 27 (4), 2017, pp. 1157–1168).

We prove that our system is sound and complete with respect to the traditional Carnap–Popper type probability semantics.