https://hal.inria.fr/inria-00076024/document

The calculus of constructions

To explain the idea behind the present paper the following fundamental principle is emphasized. Let X = (X 1,…, X n ) be an n-dimensional vector valued random variable, and let µ(x) =µ(x 1…, x n )be its probability measure (defined on euclidean n-space E n ). Suppose that X has the property that µ(x) =µ(gx) for every element g of a group G of order h of transformations of E n into itself. Let f(x) =f(x 1…, x n ) be a µ-integrable complex valued function on E n Then the expected value of f(x) is 

$$Ef\left( X \right) = \smallint f\left( x \right)d\mu \left( x \right) = \smallint \bar f\left( x \right)d\mu \left( x \right)$$

(1.1)

, where 

$$\bar f\left( x \right) = \frac{1}{h}\sum\limits_{g \in G} f \left( {gx} \right)$$

(1.2)

.

/pdf/a-combinatorial-lemma-and-its-application-to-probability-ckojqvfrt2.pdf

A Combinatorial Lemma and its Application to Probability Theory

Fuzzy Relational Systems

We give a recursion-theoretic characterization of FP which describes polynomial time computation independently of any externally imposed resource bounds. In particular, this syntactic characterization avoids the explicit size bounds on recursion (and the initial function 2|x|·|y|) of Cobham.

/pdf/a-new-recursion-theoretic-characterization-of-the-polytime-3qr62h2f5g.pdf

A new recursion-theoretic characterization of the polytime functions

Implication in intuitionistic fuzzy and interval-valued fuzzy set theory: construction, classification, application

In 1965 Zadeh introduced the concept of fuzzy sets. The characteristic of fuzzy sets is that the range of truth value of the membership relation is the closed interval [0, 1] of real numbers. The logical operations ⊃, ∼ on [0, 1] which are used for Zadeh's fuzzy sets seem to be Łukasiewciz's logic, where p ⊃ q = min(1, 1 − p + q), ∼ p = 1 − p. L. S. Hay extended in [4] Łukasiewicz's logic to a predicate logic and proved its weak completeness theorem: if P is valid then P + Pn is provable for each positive integer n. She also showed that one can without losing consistency obtain completeness of the system by use of additional infinitary rule.Now, from a logical standpoint, each logic has its corresponding set theory in which each logical operation is translated into a basic operation for set theory; namely, the relation ⊆ and = on sets are translation of the logical operations → and ↔. For Łukasiewicz's logic, P Λ (P ⊃ Q). ⊃ Q is not valid. Translating it to the set version, it follows that the axiom of extensionality does not hold. Thus this very basic principle of set theory is not valid in the corresponding set theory.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022481200043000

Intuitionistic fuzzy logic and intuitionistic fuzzy set theory

Bounded arithmetic and the polynomial hierarchy

/pdf/two-applications-of-logic-to-mathematics-1289683j4u.pdf

Two Applications of Logic to Mathematics

In this paper, we study set theory based on quantum logic. By quantum logic, we mean the lattice of all closed linear subspaces of a Hilbert space. Since quantum logic is an intrinsic logic, i.e. the logic of the quantum world, (cf. 1) it is an important problem to develop mathematics based on quantum logic, more specifically set theory based on quantum logic. It is also a challenging problem for logicians since quantum logic is drastically different from the classical logic or the intuitionistic logic and consequently mathematics based on quantum logic is extremely difficult. On the other hand, mathematics based on quantum logic has a very rich mathematical content. This is clearly shown by the fact that there are many complete Boolean algebras inside quantum logic. For each complete Boolean algebra B, mathematics based on B has been shown by our work on Boolean valued analysis 4, 5, 6 to have rich mathematical meaning. Since mathematics based on B can be considered as a sub-theory of mathematics based on quantum logic, there is no doubt about the fact that mathematics based on quantum logic is very rich. The situation seems to be the following. Mathematics based on quantum logic is too gigantic to see through clearly.

Gaisi Takeuti

Papers

Intuitionistic fuzzy logic and intuitionistic fuzzy set theory

Bounded arithmetic and the polynomial hierarchy

Two Applications of Logic to Mathematics

Quantum Set Theory

On a generalized logic calculus