A CLOSURE SYSTEM FOR ELEMENTARY SITUATIONS

Transcription

1 Bulletin of the Section of Logic Volume 11:3/4 (1982), pp reedition 2009 [original edition, pp ] Bogus law Wolniewicz A CLOSURE SYSTEM FOR ELEMENTARY SITUATIONS 1. Preliminaries In [4] a lattice SE of elementary situations has been described. Its dual atoms are possible worlds, and their totality is the logical space SP. The join is x; y, and A B is the corresponding product of subsets. The zero of SE is the empty elementary situation, and the impossible on λ is its unit. Following Loś maximal proper ideals of SE are called realizations. They are all principal here. In [5] a family V of complete sets of verifiers, or V -sets for short, has been defined on SE. These are non-empty SE -sets (i.e., subsets of SE ) such that for any V V : A V V, and A Q V implies A V, for any Q-space Q. (A Q-space is an antichain in SE intersecting every realization R at exactly one of its elements. In particular, SP and Q o = {o} are Q-spaces). Being closed under arbitrary intersection (cf. T1 in [5]), V is a closure system on SE. Put V (A) = {V V : A V }, for any SE -set A. Then < SE, V > is the corresponding closure space. I.e., A V (A), A B implies V (A) V (B), and V (V (A)) = V (A). SE -sets are V -equivalent (written A V B ) iff they always intersect the same realizations. In particular (cf. T7 in [5]) we have: V 1 V V 2 iff V 1 = V 2, for any V 1, V 2 V.

2 A Closure System for Elementary Situations Orthogonality in SE As in [4], x is incompatible with y iff x; y = λ. Write this x y. Then is an orthogonality relation in MacLaren s sense (cf. [2]). I.e., (1) x y implies y x; (2) x x implies x y, for all y SE ; (3) (x z iff y z) implies x = z. (1) is obvious. For (2): x x iff x = λ. And (3) is equivalent to the separation axiom 1.10 in [4], with SE being dually atomic by axioms 1.8 there. To see the last point transpose axiom I.e., we have: (x w iff y w) x = y. w SP Now x w iff x; w λ, for any w SP, by 1.8. Hence 1.10 may be put as (x w iff y w) x = y, which clearly implies (3). For the converse transpose (3): for some z SE, x y = ((x z and y z) or ( x z and y z)). Suppose the former: x; z = λ and y; z λ. Thus z w x w, for any w SP. But y; z w, for some w SP, by 1.8. Hence y w and x w, for some w SP. So (3) implies Closure and V equivalence Set = {λ}, and let A be the set of all y SE incompatible with every x A. Then A A =, provided A. (Note that = SE, but SE = ). As in [2] call A closed iff A = ((A ) ) = A. We claim (T1) V (A) = A, but to make this good we need some lemmas. For any SE -sets A, B we have clearly: (T2) B A iff either A B =, or A =, or B =. Moreover, with Q being any Q-space, we have: (T3) A (B Q) = A B =.

3 136 Bogus law Wolniewicz Indeed, suppose A B. If either is empty, so is A (B Q). Otherwise x; y λ, for some x A, y B. Thus x; y belongs to some realization R. But Q is a Q-space, and R is an ideal. So there is an z Q R such that (x; y); z λ. Hence A (B Q). Thus (T4) A is a V -set. Indeed, λ A, so A is never empty. In view of T2 we have also B A A, with A, B implying A (B A ) =. Finally B Q A implies B A, in view of T2 and T3. Note that for any realization R: A R = iff A R. Hence by transposition, substitution A /A, and transitivity: (T5) A R = iff A R =, for any realization R. I.e., (T6) A v A. Now A A by T2. So V (A) A by T4. Hence A R = implies V (A) R =. So in view of T5 A R = implies V (A) R =, for any realization R. The converse follows from A V (A). Thus (T7) A v V (A). From T6 and T7 we have V (A) v A, and since are V -sets we get T1. As A = A (cf. [2]), T1 yields corollary: (T8) And from T7 we have (T9) A = B iff V (A) = V (B). A v B iff V (A) = V (B). So SE -sets are V -equivalent iff they have equal closures. 4. Dual pseudocomplementation for minimal SE -sets In SE all chains are assumed finite (cf. 3.3 in [4], and section 2 in [5]). Setting MinA = {x A : for no x A, x < x} we have shown in [3] that the family Min = {A SE : A = MinA} of minimal SE -sets is a distributive lattice under the join A B = Min(A B), and the meet A + B = Min(A B). Min is also bounded, with as its unit, and Q 0 as its zero. The partial ordering is involvement: A B (read A involves B ) iff

4 A Closure System for Elementary Situations 137 x A y B y x. Clearly A Q 0. Moreover, A implies A. Hence the family Min = Min is a sublattice of Min, with as its unit. The round-trip map A c = Min(A ), sending A into P (SE ) and back into Min, gives the dual pseudocomplement for Min. I.e., we have for any A, X Min : (T10) A A c =, and A X = X A c. Indeed, note firstly that A B = iff A B =. Now for the former: since A, we have A A =. So A A =, and by M10 of [3] A A = A A c. For the latter assume the antecedent. Thus A X =, so X A by T2. Hence X A, which in turn (cf. T2 in [5]) implies X A c. For any A SE we have also: (T11) A = (MinA). Indeed, MinA A, so A (MinA) as in [2]. Conversely, take any y (MinA). Thus x ; y = λ, for all x MinA. All chains in SE being finite, there is for any x A some x MinA such that x x. Hence x y for all x A, i.e. y A. 5. Situations The family S = {A c : A Min } is the skeleton (cf. [1]) of the dually pseudocomplemented distributive lattice < Min,, +, c, Q c, >. Hence < S,, +, c, Q 0, > is a Boolean algebra with the same ordering of involvement, and the meet defined as A B = (A + B) cc. A simple computation starting at the middle and using only the corollary (MinA) c of T11 show that (T12) (A + B) cc = MinV (A B). The elements of S are situations. These are the objectives of propositions (cf. [5]). As we have seen, any situation S S may be presented as S = S(X) = MinV (X), for some X SE. So (T13) Situations are the minima of closed SE -sets.

5 138 Bogus law Wolniewicz References [1] G. Grätzer, General Lattice Theory, Berlin [2] M. D. MacLaren, Atomic Orthocomplemented Lattices, Pacific Journal of Mathematics, vol. 14 (1964), pp [3] B. Wolniewicz, Some Formal Properties of Objectives, this Bulletin, vol. 8 (1979), no. 1, pp [4] B. Wolniewicz, On the Lattice of Elementary Situations, this Bulletin, vol. 9 (1980), no. 3, pp [5] B. Wolniewicz, The Boolean Algebra of Objectives, this Bulletin, vol. 10 (1981), no. 1, pp Institute of Philosophy Warsaw University