In this paper, I shall sketch a new way to consider a Lindenbaum-Tarski algebra as a 3D logical space in which any one (of the 256 statements) occupies a well-defined position and it is identified by a numerical ID. T...In this paper, I shall sketch a new way to consider a Lindenbaum-Tarski algebra as a 3D logical space in which any one (of the 256 statements) occupies a well-defined position and it is identified by a numerical ID. This allows pure mechanical computation both for generating rules and inferences. It is shown that this abstract formalism can be geometrically represented with logical spaces and subspaces allowing a vectorial representation. Finally, it shows the application to quantum computing through the example of three coupled harmonic oscillators.展开更多
Several nonmonotonic logic systems together with their algebraic semantics are discussed.NM-algebra is defined.An elegant construction of an NM-algebra starting from a Boolean algebra is described which gives rise to ...Several nonmonotonic logic systems together with their algebraic semantics are discussed.NM-algebra is defined.An elegant construction of an NM-algebra starting from a Boolean algebra is described which gives rise to a few interesting algebraic issues.展开更多
The concepts of metric R0-algebra and Hilbert cube of type RO are introduced. A unified approximate reasoning theory in propositional caculus system ? and predicate calculus system (?) is established semantically as w...The concepts of metric R0-algebra and Hilbert cube of type RO are introduced. A unified approximate reasoning theory in propositional caculus system ? and predicate calculus system (?) is established semantically as well as syntactically, and a unified complete theorem is obtained.展开更多
Let F(S)be the free algebra of type(,∨,→)generated by the non_empty set S,it is proved that the logical equivalent relation defined by means of R 0_semantics is a congruence relation on F(S)and the corresponding quo...Let F(S)be the free algebra of type(,∨,→)generated by the non_empty set S,it is proved that the logical equivalent relation defined by means of R 0_semantics is a congruence relation on F(S)and the corresponding quotient algebra is said to be the R 0_semantic Lindenbaum algebra.Taking R 0_semantic Lindenbaum algebra as a prototype,the concepts of implicational lattices and regular implicational lattices which are generalizations of the concept of Boolean algebras are introduced.Besides,the concept of fuzzy implicational spaces is introduced and the representation theorem of regular implicational lattices is obtained by means of fuzzy implicational spaces.In case of Boolean algebras,the corresponding fuzzy implicational spaces are zero_dimensional compact Hausdorff spaces and herefrom it is proved that the famous Stone’s representation theorem of Boolean algebras is a corollary of the representation theorem of regular implicational lattices.展开更多
文摘In this paper, I shall sketch a new way to consider a Lindenbaum-Tarski algebra as a 3D logical space in which any one (of the 256 statements) occupies a well-defined position and it is identified by a numerical ID. This allows pure mechanical computation both for generating rules and inferences. It is shown that this abstract formalism can be geometrically represented with logical spaces and subspaces allowing a vectorial representation. Finally, it shows the application to quantum computing through the example of three coupled harmonic oscillators.
文摘Several nonmonotonic logic systems together with their algebraic semantics are discussed.NM-algebra is defined.An elegant construction of an NM-algebra starting from a Boolean algebra is described which gives rise to a few interesting algebraic issues.
基金supported by the National Natural Science Foundation of China(Grant No.19331010).
文摘The concepts of metric R0-algebra and Hilbert cube of type RO are introduced. A unified approximate reasoning theory in propositional caculus system ? and predicate calculus system (?) is established semantically as well as syntactically, and a unified complete theorem is obtained.
基金supported by the National NaturalScience Foundation(Grant No.19331030).
文摘Let F(S)be the free algebra of type(,∨,→)generated by the non_empty set S,it is proved that the logical equivalent relation defined by means of R 0_semantics is a congruence relation on F(S)and the corresponding quotient algebra is said to be the R 0_semantic Lindenbaum algebra.Taking R 0_semantic Lindenbaum algebra as a prototype,the concepts of implicational lattices and regular implicational lattices which are generalizations of the concept of Boolean algebras are introduced.Besides,the concept of fuzzy implicational spaces is introduced and the representation theorem of regular implicational lattices is obtained by means of fuzzy implicational spaces.In case of Boolean algebras,the corresponding fuzzy implicational spaces are zero_dimensional compact Hausdorff spaces and herefrom it is proved that the famous Stone’s representation theorem of Boolean algebras is a corollary of the representation theorem of regular implicational lattices.
