QUANTUM . LOGIC gJJ

Quantum Logic: A logic (血,←, þ), where the proof-theoretic and the semantic-consequence relations violate some characteristic classical principles like the distributivity of conjunction and disjunction. Adjoint: If A is a bounded operator on a Hilbert space , then the adjoint of A is the unique bounded operator A食 that satisfies (A~ φ) = (ψIA* φ) for all vectors ψ, φin the space. Algebraic Model of a Language: A pair (stl.,v) consisting of an algebraic structure stl. and of an interpretation function v that transforms the sentences of the language into elements of stl., preserving the logical form. Algebraic Semantics: The basic idea is that interpreting a formal language means associating to any sentence an element oE an algebraic structure. Attribute: One of a class of properties affiliated with a physical system. At any given moment some of the attributes of the system may be actual. while others are only potential. A."{iomatizable Logic: A logic is a:dümatizable when its concept of proof is d l.'cidable. Boolean Algebra: An orthocomplemented bttice L that satisfìes the distributive bw p ^ (q V r) = (p ^ q) V (p ^ r) for all p , q , r E L. Borel Set: A (real) Borel set is a set that belongs to the smallest collection of subsets of the real numbers IR that contains all open inten;als and is closed under the formation of complements and countable unions. Brouwer-Zadeh Lattice (or Poset): A lattice (or poset) L with smallest element 1 and largest element 0 , equipped with a regular involution ' (a fuzzylike complement) , and an intuitionisticlike complement 丁 sub­ ject to the following conditions for all p ,q ε L: (i) p 三 q 司 q三 P飞(ii) p ^ p= 0, (ii i) p 三 p→, (iv) p-' = p二. Brouwer-Zadeh Logic: A logic that is characterized by the class of all models based on Brouwer-Zadeh lattices. Compatible: A set C of elements in an orthoalgebra L is a compatible set if there is a Boolean subalgebra B of L such that C ç B. Complete Lattice: A lattice in which everγsubset has a least upper bound , or join, and a greatest lower bound , or meet. Completeness: A logic (FL,'t-,乍 is (semantically) complete when all the semantic consequences are proof-theoretic consequences (ifα 乍 ß then α~ ß). Conjunction (in an Orthoalgebra): If L is an orthoalgebra and p , q ε L, then p and q have a conjunction p & q ε L if there is a Boolean subalgebra B of L with p , q ε B , and the meet P ^B q of p and q in B is independent of the choice of B , in which case p & q = P ^B q. Decidable Logic: A logic is decidable when its proof-theoretic consequence relation is decidable. Density Operator: A self-adjoint , nonnegative , trace-class operator W on a Hilbert space , such that tr(W) 1. Direct Sum (or Cartesian Product) of Hilbert Spaces: The direct sum of the Hilbe口 spaces 况 and 页 is the Hilbert space 'iJf. EÐ 'J[ consisting of all ordered pairs (x ,y) with x ε 绽, y ε 页, and with coordinatewise vector operations. The inner product is defined by ((x"y ,)I(xz ,yz)) = (x , L~z) + (y ,lyz). Disjunction (in an Orthoalgebra): If L is an orthoalgebra and P, q ε L , then p and q Quantum Logic 251 ha\'e a disjunction p 十 q if there is a Boolean subalgebra B of L \vith p , q E B and the join p V 8 q of p and q in B is independent of the choice of B, in \\'hich case p + q p\/8 q. Dynamics: The e\'ülution in time of the state of a physical system. Effect: A linear bounded operalor A of a Hilbert space such that for any density operator W , tr(WA)ε[0 ,门. In the unsharp approach to quantum mechanics , effects represent possible experimental propositions. Experimental Proposition: A proposition 飞vhose truth value can be determined by conducting an experimen t. Greatest vs Max缸nal: If L is a partially ordered set (poset) and X ç L , then an element b εX is a greatest element of X if x :s b for all x εX. An element b εX is a maximal element of X if there exists no element xεX with b < x. lnvolution: A mapping p • p' on a poset L satisfying the following conditions for all p , q ε L: (i) p 三 q 司 q' 主三 p' , (ii) p" = p. Involutive Lattice (or Poset): A lattice (or poset) with smallest element 0 and largest element 1, equipped with an involution. Join: If L is a partially ordered set (poset) and p , q ε L, then the join (or least upper bound) of p and q in L , denoted by p V q if it exists , is the unique element of L satisfying the following conditions: (i) p , q 三 p V q and (ii) r ε L with p , q 主三 r 司 p V q :s r. Kripkean Model of a Language: A system (I,R". . . ,Rno ,o". . . ,o",ll,v) consisting of a set 1 of possible worlds , a (possibly empty) sequence of world relations R" . . . , Rm and of world operations 0" . . . , 0月, a family n of subsets of 1 (called the propositions) , and an interpretation function v that transforms the sentences of the language into propositions , preserving the logical form. Kripkean Semantics: The basic idea is that interpreting a formal language means associating to any sentence the set of the possible worlds where the sentence holds. This set is called also the proposition associated to the sentence. Lattice: A partially ordered set (poset) in which everγpair of elements p , q has a least upper bound, or join, p V q and a greatest lower bound , or meet , p ^ q. Least vs Minimal: If L is a partially ordered set (poset) and X ç L , then an element G εX is a least element of X if a 三 x for all