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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**Example text**

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).