Lattice-Valued Logic: An Alternative Approach to Treat by Professor Dr. Yang Xu, Professor Dr. Keyun Qin, Professor

By Professor Dr. Yang Xu, Professor Dr. Keyun Qin, Professor Dr. Da Ruan, Dr. Jun Liu (auth.)

Lattice-valued good judgment goals at developing the logical origin for doubtful details processing mostly played through people and synthetic intelligence structures. during this textbook for the 1st time a normal creation on lattice-valued good judgment is given. It systematically summarizes study from the fundamental notions as much as contemporary effects on lattice implication algebras, lattice-valued common sense platforms in line with lattice implication algebras, in addition to the corresponding reasoning theories and strategies. The publication offers definitely the right theoretical logical heritage of lattice-valued common sense platforms and helps newly designed clever uncertain-information-processing structures and a large spectrum of clever studying tasks.

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Dubois and Prade [95,100] also investigated the possibilitic logic, and Liau et al. [243,244] introduced a possibilitic residuated implication logics with applications for reasoning, where the semantics of the logics is uniformly based on possibility theory. Each logic in the class is parameterized by at-norm operation on [0,1], and the degree of implication between the possibilities of two formulas explicitly by using residuated implication with respect to the t-norm. , quantum logic, was born of the attempts of Jordan, et al.

X 1\ Y = (x' V y')' = ((y' -+ x') -+ x')' = ((x -+ y) -+ x')'. ((2) =? (1)). x V Y = (x' 1\ y')' = (y' -+ x') -+ y = (x -+ y) -+ y. ((1) =? (3)). If x ~ y, then x V y (x = y and hence -+ y) -+ y = y, it follows that x -+ y = x -+ ((x = (x -+ y) = 1. 1 Lattice Implication Algebras Conversely, suppose that x -4 y = I, then x V Y = (x -4 y) =I-4y ((3) =} ~ -4 Y y, = it follows that x 37 y. (1)). By X -4 ((x -4 y) -4 y) = (x y) -4 -4 (x y) = I, -4 and Y -4 ((x -4 y) -4 y) = (x -4 y) -4 (y =(x-4y)-4I -4 y) = I, it follows that x ~ (x -4 y) -4 Y and y x VY ~ (x On the other hand, for any a E L, if x and ~ (x -4 y) -4 y) -4 y.

3 If L is a lattice H implication algebra, then for any x E L and n E N+, xn = x. 1 Let L be a lattice implication algebra, Xo Or: L - - t L as follows: for any x E L, OI(X) = Xo -+ E L, define 01, x, Or(x) = X -+ xo, then 01 (named a left-mapping) is a lattice homomorphism and Or (named a right-mapping) is a dual lattice homomorphism. Remark 4. 1, 01 and Or are related to Xo. For simplicity, we omit Xo in them. Proof. 2, for any x, y E L, OI(XVy) = = = OI(X 1\ y) = = = Or(X Vy) = = = Or(x 1\ y) = = = Xo -+ (xVy) (xo -+ x) V (xo -+ y) OI(X) V Ol(y), Xo -+ (x 1\ y) (xo -+ x) 1\ (xo -+ y) OI(X) 1\ OI(Y), (x Vy) -+ Xo (x -+ xo) 1\ (y -+ xo) Or(x) 1\ Or(Y), (x 1\ y) -+ Xo (x -+ xo) V (y -+ xo) Or(x) V Or(Y).

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