Topological Characterizations Of Some Classical Logic Problems

The characteristic of mathematical logic lies in symbolism, which is quite different from computational mathematics. The former pays attention to formal deduction and strict proof while the latter stresses numerical computation and approximate solution. If mathematical logic is said to be rigid, then computational mathematics is flexible. One may ask naturally: Can we extend the possible application of mathematical logic by implanting the thoughts of numerical computing into it so as to make it flexible? Professor Wang establishes a quantitative logic by grading the basic concept, and thus gives a positive solution to the questions above. The quantitative logic comprises two-valued propositional logic, the Lukasiewicz many valued logics L_n Luk and many valued logics L~* and L_n~*. The concept of truth degree of formulas in two-valued propositional logic system is proposed by grading the concept of tautologies, and the concepts of similarity degree and pseudometric among formulas are introduced therefrom. Then a logic metric space can be obtained. Furthermore, the continuity of the logic connectives with respect to the pseudo-metricρis proved. On the other hand, for any logic theory, what is the connection between its logic property(such as divergency, consistency) and topological property(such as density, interior point, completeness and the like)? The present paper mainly deals with the questions above, as far as I know, the relevant research has not been considered before. we list the main results as follows:(1) The divergent degree of all the logic theories can fill the unit interval [0, 1];(2) Any L-closed theoryΓis consistent if and only ifΓcontains no non-empty circle with positive radius less than 1, and thus the following conclusion is clear: An L-closed theory is consistent if and only ifΓcontains no interior point in logic metric space (F(S),ρ).(3) A theoryΓis fully divergent if and only if D(Γ) is dense in (F(S),ρ).(4) The logic conclusion of any finite theoryΓin F(S) is a closed set in (F(S),ρ), the same is true for any L-closed theory with root.(5) The logic metric space (F(S),ρ) is of zero dimension, furthermore, it has the property of so called finite equal-ball connectedness. that is, any two points can be connected by finite equal balls with radius equal or less than any given positive numberε. Furthermore, the topological characterization of any L-closed theory is given.