Location: KELLER 313


Title: SEQUENT CALCULUS FOR CLASSICAL LOGIC PROBABILIZED

Abstract: Gentzen’s approach to deductive systems, and Carnap’s and Popper’s treatment of probability in logic were two fruitful ideas that appeared in logic of the mid-twentieth century. By combining these two concepts, the notion of sentence probability, and the deduction relation formalized in the sequent calculus, we introduce the notion of ‘probabilized sequent’ $\Gamma\vdash_a^b\Delta$ with the intended meaning that “the probability of truthfulness of $\Gamma\vdash\Delta$ belongs to the interval $[a,b]$”. This method makes it possible to define a system of derivations based on ‘axioms’ of the form $\Gamma_i\vdash_{a_i}^{b_i}\Delta_i$, obtained as a result of empirical research, and then infer conclusions of the form $\Gamma\vdash_a^b\Delta$. We discuss the consistency, define the models, and prove the soundness and completeness for the defined
probabilized sequent calculus.

<a href=”https://math.hawaii.edu/home/depart/theses/PhD_2023_Kunwar.pdf“>Dissertation draft</a>

Date: Monday, April 17
Time: 2:30 – 3:20
Room: Keller 313

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.