Quantitative Logic Randomized Study

The most fundamental inference pattern in propositional logic is {A₁,…,A_n}|-A^*. It means,from syntactic viewpoint,A₁→(A₂→…→(A_n→A^*)…) is a theorem, while it means,from semantic viewpoint,A^* is true under any valuation v whenever A₁,…,A_n are true under v.Notice that if the premises are sound or not is not taken into account,and hence this formalized effective inference seems not to be suitable for practical reasoning.In view of this situation,the probability logic emerged from the 70's of the 20^th century,where uncertainty of premises were considered,and uncertainty degree of the conclusion had been deducted by using the Kolmogorov axioms.It is remarkable that the theory is developed by means of individual cases while the probability of one and the same formula varies in different effective inferences,and only a few(mostly two or three) formulas are involved in premises of effective inferences,therefore the theory seems to be locally but not globally.On the other hand,a global quantified logic theory is proposed,where logic concepts are graded into different levels so as to try to establish a bridge between artificial intelligence and numerical computation.In quantified logic every atomic formula has the same truth degree.This is not consistent with corresponding problems in the real world.In fact if a simple proposition in the real world is true or not,or in what extent it is true is uncertain,hence to follow the way of probabilistic AI and develop a probabilistic style quantified logic is certainly a beneficial task.In view of the above analysis the present paper focus on the alliance of quantified logic and probability logic so that quantified logic and most of its results can be considered special cases of the new setting.In the first chapter,a simple and direct proof of the fundamental theorem of probability logic is proposed by introducing the concepts of generating states set and generating probability.Next,by introducing the concept of naturally merged probability the fundamental theorem has been generalized.Moreover,the present paper extends the concept ofδ-consistency degree of finite logic theories to be the concept of(?)-consistency degree and therefore certain relationship between probability logic and quantitative logic has been obtained.In the second chapter,the relationship between the probability distribution of finite formulas and the probability distribution of atomic formulas generating them is discussed, and the concepts of probability truth degree and probability logic pseudo-metric space are proposed by combining quantitative logic and probability logic.It is proved that when the probability distribution is even,the value of probability truth degree is equal to the value of truth degree and the value of probability logic pseudo-metric is equal to the value of pseudo-metric in quantitative logic.Thus a more general pseudo-metric is established in finite theories.In the third chapter,the concept of D- randomized truth degree of formulas in two-valued propositional logic is introduced by means of randomization,and it is proved that the set of values of D- randomized truth degree of formulas has no isolated point in[0,1]. The concepts of D- logic pseudo-metric and D- logic metric space are also introduced and it is proved that there is no isolated point in the space.The new built D- randomized concepts are extensions of the corresponding concepts in quantified logic.Moreover,it is proved that the basic logic connectives are continuous operators in D-logic metric space.Three different types of approximate reasoning pattems are proposed and it is proved that the three different types of approximate reasoning pattems are equivalent to each other and the set of atomic formulas is not totally divergent in D-logic metric space.After D- opening degree is introduced in D- logic metric space,it is proved that the D- opening degree of a theory is equivalent to it's D- divergence degree.Then the D- consistency degree is defined and it can maintain the basic properties in logic metric space.By means of randomization,the concept of randomized truth degree and randomized logic pseudo-metric of formulas in R₀ three-valued propositional logic, Lukasiewicz three-valued propositional logic,Goguen three-valued propositional logic and G(o|¨)del three-valued propositional logic are introduced.The concept of randomized logic metric space is also introduced and it is proved that the new built randomized concepts are extensions of the corresponding concepts in quantified logic.