|
Abstract:
|
After a brief introductory survey, this work is divided into two parts. The first
part deals with presence of probability in logic (v. [16], [17], [18], [19], [22], [23] and
[24]), and the second one is devoted to the application of entropy in classification of
many–valued logics (v. [14], [15], [20], [21] and [25]).
The basic idea, dominant in the first part of the work, is to enrich the Gentzen’s
sequent calculus LK for propositional classical logic by a kind of probability operator
defined over the sequents ¡ ` ¢ in order to express the fact that ”the truthfulness
probability of ¡ ` ¢ belongs to the interval [a; b] µ [0; 1]”. We introduce the
following four systems: LKprob, LKprob("), NKprob and LKfuzz. The basic
form of sequents in LKprob is ¡ `b
a ¢ with the above given intended meaning. The
system LKprob(") is focused on the Suppes’ forms ¡ `n ¢ enabling to formalize
the sentence ”the truthfulness probability of ¡ ` ¢ belongs to the interval [1 ¡
n"; 1] µ [0; 1]”, for some n 2 N. The system NKprob presents a natural deduction
counterpart of the sequent calculus LKprob. The models founded on Carnap–
Popper–Leblance probability semantics are defined for each of these calculi and
accompanied by the corresponding soundness and completeness results. Finally, the
calculus LKfuzz is introduced with a more general form of the sequents ¡ `x ¢,
where x is an element of a finite lattice, with the aim to describe a fuzzification
of LK. The meaning of ¡ `x ¢ is that ”x is the fuzziness measure of ¡ ` ¢”.
Models for LKfuzz are given with soundness and completeness results, and a proof–
theoretical treatment of LKfuzz includes the cut–elimination theorem.
The second part of the work explores the fact that each logical system associated
with the partition induced by the corresponding Lindenbaum–Tarski algebra makes
it possible to define its entropy. We define the entropy of a logical system based on
geometric distribution of measures over matching partition of set of formulae. This
iv
definition enables the classification of many–valued propositional logics according to
their entropies. Asymptotic entropy approximations for some infinite–valued logics
are proposed as well. The considered examples include Lukasiewicz’s, Kleene’s and
Priest’s three–valued logics, Belnap’s four–valued logic, G¨odel’s and McKay’s m–
valued logics, and Heyting’s and Dummett’s infinite–valued logics. |