Janusz Czelakowski. B. h : Dom(h ) {T, F } (T -truth, F -falsehood); precisely: 1. If α Dom(h ), then. F otherwise.

Transcription

1 Bulletin of the Section of Logic Volume 3/2 (1974), pp reedition 2012 [original edition, pp ] Janusz Czelakowski LOGICS BASED ON PARTIAL BOOLEAN σ-algebras Let P = P ;, be the language with connectives, (see [2], p. 46) and let P BσA denote the class of all partial Boolean σ-algebras. We introduced in [2] the following consequence operation Cn based on P BσA: Cn(X) (where P, X P ) iff for every P BσAlgebra B and every valuation h : P B it holds: β X (β Dom(h) hβ = 1) ( Dom(h) h = 1) Let us notice that Cn( ) is the set of formulas which are valid in all P BσAlgebras. We shall introduce another consequence operation Cn 1 (Cn 1 : 2 P 2 P ). Cn 1 is defined as follows: Cn 1 (X) iff for every P BσAlgebraB and every valuation h : P B it holds: β X (β Dom(h) hβ = 1) ( Dom(h) h = 1). Let us observe that Cn 1 ( ) is empty (because some variables may not belong to Dom(h)). Let P 0 contain formulas from P and all expressions of the form ( 1, 2,...), where ( 1, 2,...) is any (denumerable) sequence of formulas of P. One can extend every valuation h : P B to the valuation h : P 0 B in a following way: A. The domain of h (Dom(h )) is the least set satisfying the conditions: 1. Dom(h) Dom(h ) 2. ( 1, 2,...) Dom(h ) iff { 1, 2,...} Dom(h). B. h : Dom(h ) {T, F } (T -truth, F -falsehood); precisely: 1. If Dom(h ), then { h T iff h = 1 = F otherwise.

2 32 Janusz Czelakowski 2. If ( 1, 2,...) Dom(h ), then { h ( ( T iff hm 1, 2,...)) = h n for any m, n N F otherwise. Let us notice that h ( ( 1, 2,...)) = T for any ( 1, 2,...) Dom(h ). Instead of the operations Cn and Cn 1 one may consider consequence operations Cn and Cn 1 (Cn, Cn 1 : 2 P 0 2 P 0 ) : γ 0 Cn (Γ) iff for every P BσAlgebraB and every valuation h : P 0 B (defined as above) it holds: γ Γ (γ Dom(h ) h γ = T ) (γ 0 Dom(h ) h γ 0 = T ) and γ 0 Cn 1(Γ) iff for every P BσAlgebraB and every valuations h : P 0 B (defined as above) it holds: γ Γ (γ Dom(h ) h γ = T ) (γ 0 Dom(h ) h γ 0 = T )(γ 0 P, Γ P ). We shall write: Γ = γ when γ Cn 1(Γ). For a given P, let denote the set of all commeasurabilities appearing in ; precisely: 1. If is a variable x ν = { (x ν )}. 2. If is of the form β, then = β { ( β)}. 3. If is of the form { 1, 2,...}, then = n { ( 1, 2,...)} { ( { 1, 2,...})}. If γ is of the form (β 1, β 2,...), then we put γ = βn { (β 1, β 2,...)}. If Γ P 0, then Γ = df γ. γ Γ Let us introduce the following rules of inference: R1 : ( m, n ) (where m, n are any natural numbers) R2 : ( 1, 2 ), ( 1, 2 ),..., ( m, n ),... (the premise of the rule R2 contains all expressions ( m, n ), where m, n are any natural numbers) R3 : ( 1, 2 ), 2 3 ( 1, 3 ) R4 : ( 1, 2 ) ( 1, 2 )

3 Logics Based on Partial Boolean σ-algebras 33 R5 : ( { n : n N}, 1, 2,...) R6 : 1, 2,... S1 : (where β(x β( 1, 2,...) 1, x 2,...) is any formula valid in all Boolean σ-algebras) S2 : 1, S3 : 1, 2,... {n : n N} The last rule S4 says that if we accept a formula, then we accept any commeasurability from. S4 : γ (where γ is any element from ). For instance S4 : x 1 x 2 (x 1, x 2 ). We shall say that γ(γ P 0 ) follows syntactically from Γ (Γ P 0 ), in symbols: Γ γ, iff there exists a sequence {γ ν } ν<µ0 of expressions from P 0 such that for a given ν 0, γ ν0 Γ Γ or there exist indices ν 1, ν 2,... (ν k < ν 0, k = 1, 2,...) such that γ ν0 follows from γ ν1, γ ν2,... by one of the above rules R1 R6, S1 S4. Theorem. For any Γ P 0, γ P 0, Γ = γ iff Γ γ. A formula P is provable, by definition, iff. The following example will show the reason for having chosen just the language with formulas of infinite length. Example. Let σ and S denote the set of all observables and the set of all states of the physical system, respectively. Let us assume that we are given a map p that assigns a real number p(a,, E) in [0, 1] to each triple (A,, E) from σ S B(R). (B(R) denotes the family of all Borel subsets of the set of real numbers R.) We read: p(a,, E) is the probability that a measurement of A for the system in state will lead to a value in E. Axioms for p (see [3], p. 94): A1. p(a,, ) = 0, p(a,, R) = 1 p(a,, E n ) = p(a,, E n ), where E i E j = for i j. A2. If p(a,, E) = p(a,, E) for all and E, then A = A. If p(a,, E) = p(a,, E) for all A and E, then =.

4 34 Janusz Czelakowski A3. If A 1, A 2,... are members of σ and E 1, E 2,... are members of B(R) such that for i j p(a i,, E i ) + p(a j,, E j ) 1 for all in S, then there exist B σ and disjoint Borel sets F 1, F 2,... such that for j = 1, 2,..., p(a j,, E j ) = p(b,, F j ) for all in S. Let ε 0 = df σ B(R). We shall call the members of ε 0 the experimental hypotheses. We read (A, E) as the measurement of A will leas to a value in E. Thus experimental hypotheses are hypotheses about results of measurement of quantity A. Let us define an equivalence relation in ε 0 as follows: (A, E) (B, F ) iff p(a,, E) = p(b,, F ) for all in S. Let L = ε 0 / and (A, E), (B, F ) L. We put (A, E) (B, F ) iff p(a,, E) p(b,, F ) for all. is a partial order in L. 0 = df (A, ) and 1 = df (A, R) are the least element and the greatest on in L, respectively (A is any member of σ). Let (A, E) = df (A, R E). One can prove ([3]) Theorem (M. M aczyński) L,, ; 0, 1 is a orthocomplemented set satisfying the conditions (L1)-(L2) from the definition of a quantum logic ([4], p. 168). It is a postulate of quantum mechanics that L is a isomorphic to the σ-quantum logic of all closed subspaces of a separable, infinite dimensional Hilbert space. So the following axiom seems to be rather natural: A4. L, ; ; 0, 1 is a quantum logic (or, equivalently, L, ; ; 0, 1 is a transitive partial Boolean σ-algebra). Instead of A4 one may take A4. If (A 1, E 1 ), (A 2, E 2 ), (A 3, E 3 ) L are mutually compatible, then (A 1, E 1 ) (A 2, E 3 ) is compatible with (A 3, A 3 ). (The relation of compatibility is defined as in [4], p. 168). It is easy to write the definition of in terms of the function p. We may treat the experimental hypotheses as variables. Of course ε 0 ω 1. Let ε denote the set of formulas built up by means of the sentential variables ε 0 and connectives,. Let h 0 : ε 0 ε 0 /, h 0 (A, E) = df (A, E). h 0 has a unique extension to the valuation h : ε L = ε 0 /. The system M = L, h is then a model (see [2], p. 46).

5 Logics Based on Partial Boolean σ-algebras 35 References [1] S. Kochen and E. P. Specker, Logical structure arising in quantum theory, [in:] The theory of models, ed. by J. W. Addison, L. Henkin and A. Tarski, North-Holland, Amsterdam, [2] J. Czelakowski, Partial Boolean σ-algebras, Bulletin of the Section of Logic of Inst. of Phil. and Soc. Pol. Acad. Sci., vol. 3, no. 1. [3] M. J. M aczyński, Quantum families of Boolean algebras, Bulletin de l Académie Polonaise des Sciences, Série des sciences math., astr. et phys., vol. XVIII, No. 2 (1970). [4] J. Czelakowski, Some remarks on transitive partial Boolean algebras, Bulletin of the Section of Logic of Inst. of Phil. an Soc. Acad. Sci., vol. 2, No. 3. The Section of Logic Institute of Philosophy and Sociology Polish Academy of Sciences