Logical Measure Of The Intrinsic Structure Of The Space

Professor Wang Guojun proposed the basic theory of Quantitative Logics and its Randomization based on the infinite countable product of measure space with even probability and the random series of (0,1) at different time, thus establishing a relatively complete theory of quantitative logics in many propositional logics includ-ing Lukasiewicz, (?), Godel, Goguen and their corresponding fuzzy logic systems. However, the study of the structure of logic metric space still seems to be in a be-ginning stage. Although Professor Wang Guojun have worked out the topological characterization of a number of logical properties and the topological characteriza-tions of maximal consistent theories, further studies haven't been conducted even on the classical logic metric space so far. So, in a sense, the characteristic property of logic metric space is still largely unknown. With this view, this paper attempts to probe into this area, and we have obtained a number of theorems that can reflect the intrinsic structure of classical logic metric spaces. The major achievements of this paper are listed as following:First, the paper proved that there exists a subtle reflexive transformation, which keeps an invariant logic equivalence relation and the corresponding algebraic structure, in the classical logic metric space. And this reflexive transformation can naturally deduce an automorphic reflexive transformation on Lindenbaum algebra. Besides, the fixed point theorem of the above-mentioned transformation is also proved.Second, inspired by Professor Wang's theory, this paper probed the structure of logic metric space from another aspect and worded out the 2-modular sub-normed Z-linear space structure of classical logic metric space, making it possible a carrier that has logic structure, topologic structure and linear structure at the same time, thus providing more possible approaches for further study on classical logic metric space.Third, this paper applied the concept of symmetric Boolean function and equilib-rium Boolean function in cryptography to theories of 2-value quantitative logics, and defined the symmetric logic formula, quasi-symmetric logic formula and equilibrium logic formula, etc. On this basis, two apparently opposite properties of symmetric formula are proved:namely, symmetric formulas with n variables only constitute a small portion of n-variable logic formulas and the proportion tends to zero with the increase of n; while at the same time, from another view, the number of symmetric formulas is huge as it can be proved that the truth degree set of symmetric formula is dense in the interval [0,1] as in the case of all logic formulas. Then it is proved in a contrastive way that the equilibrium formulas with n variables only constitute a small portion of n-variable logic formulas and the proportion tends to zero with the increase of n. Meanwhile, the truth degrees of equilibrium formula always equal 1/2, and the arbitrarily small neighborhood of any equilibrium logic formula has a non-equilibrium logic formula, but the truth degrees of the non-equilibrium logic formula tends to 1/2 with the increase of n. And, the representation theorem of symmetric logic formula and equilibrium logic formula is also given in the paper. Finally, by comparing the set of symmetric logic formula and the set of equilibrium logic formula, it's concluded that they intersect, not empty but don't include each other.Fourth, the paper proved that there exist certain special graphics, such as equilateral polygon and right angled triangle in logic metric space. Then it was proved that there do not exist equilateral triangles with side length of more than or equal to 2/3 in logic metric space, but equilateral triangles with side length arbitrarily close to 2/3 do exist. This shows that the distance of the classical logic metric space has their unique properties and is different from common distances.Fifth, the topological properties of the logic pseudo-metric space (F(S),ρ3) are studied. It was prove that logic pseudo-metric space (F(S).ρ3) is a non-complete, non-compact and zero-dimensional space, and it has a nature of so-called " limited connectivity " similar to the Key Fan's property.Sixth, the paper also discussed fuzzy propositional logic with modal logic con-junction, and it was found that there are very different between the fuzzy modal logic and the basic modal logic. It was proved that the tautology of fuzzy modal logic is a valid formula in basic modal logic. And examples showing that the converse is not true were also provided. In addition, the paper applied the theories of quasi-tautology to fuzzy modal propositional logic and studied the property of quasi-tautology, and constructed a category of tautologies and quasi-tautologies on the basis of logic system (?)*...