Zcitachr. f. maih. Logik und Bruadlagm d. Math. Bd. 97, S. 381-383 (1981)

A COMPLETE SEMANTICS FOR IMPLICATIONAL LOGICS

by ROBERT E. KIRK in Seattle, Washington (U.S.A.) As is known from the work of PRIOR [3], there are purely implicational propositional

logics which are incomplete for the KRIPKE semantics [2]. It is the purpose of this paper to provide a semantics which is complete for every implicational propositional logic containing the implicational fragment I of the intuitionistic logic.

By an implicational logic we shall mean a consistent set of purely implicational formulas, in a fixed set V of variables, which contains I and is closed under modus ponens and substitution. A Kripke model !JX = ( M , 5 , G ) consists of a partielly ordered set ( M , 5 ) (the frame of m) together with an assignment G : M + P ( V ) meeting the condition that, G ( x ) E G(y) whenever x 5 y . As usual, a truth value [p], is assigned to each implicational formula p and a E M in accordance with the condi- tions :

(1 ) [ p ] , = 1 if p E G(a), and [p,Ia = 0, otherwise, ( 2 ) [p --+ y], = 1 if for each b 2 a, [yI6 = 1 whenever [via = 1, and [p 3 y], = 0,

otherwise.

Also, as usual, a formula p is said to be true in when [9,Ia = 1 for all a E M ; and, for any implicational logic L, a frame is said to be L-valid when the formulas of L arc true in every Kripke model based on the frame. Finally, L is said to be complete for the Kripke semantics if L is the set of those formulas which are true in every Kripke model based on an L-valid frame.

In [3] PRIOR has shown that the implicational logic

L, = I + (@ -+ a) -+ 4 -+ ( ( (a + 13) + T) + 4

L, = 1 + (((2, -+ a) 3 a) -+ r) -+ (((P

property contains the logic

d + r ) + r),

but that when the given axioms are added to the full intuitionistic propositional logic the resulting logics are identical. This means, of course, that the L,-valid frames are the same as the &valid ones, showing that L, is not complete for the Kripke semantics.

By an expanded Kripke frame we shall mean a triple .F = ( M , 5 S> where ( M , 6 ) is a partially ordered set and S is a non-empty subset of !$(M). For each d define Q ( 9 ) to be { X I there exist W E S and U W such that U* = X } where U* = {m E M I m' E U whenever m' 2 m and m' E W } ; and, for each X , Y E Q ( S ) set X -+ Y = (m E M I m' E Y whenever m' EX and m' 2 m } .

Lemma 1. For each expanded Kripke frame 9 and X , Y E &(St), X -+ Y E Q ( 9 ) .

Proof . Suppose Y = U* where U E W E S. Let T = {m E A' I m' # X - Y for all m' 2 m}. It will be shown that T* = X -+ Y . Suppose first that m E (T* n X) - Y .

382 ROBERT E. KIRK

Then, since m 6 Y , there exists m' 2 m such that m' E W - U . Also, m' 4 Y , and, since X is closed with respect to 5, m' E X . Hence, m' E ( X - Y ) n W , contradicting our assumption that m E T*, and giving T* E X 4 Y. To complete the proof we show that X -+ Y E T*. Suppose that m 4 T*. Then, there must be m' 2 m such that m' E W - T , and m" 2 m' in X - Y. But then, m 4 X --t Y , as desired.

Following DIEGO [l], a Hi2bert algebra B = ( A , +, 1) is a set A together with a binary operation + and a designated element 1 E A satisfying the conditions that for any a, b, c E A ,

(i) a -+ ( b + a ) = 1,

(ii) (a + (b + c ) ) + ((a -+ b ) -+ (a 3 G ) ) = 1, and (iii) a = b whenever a -+ b = 1 = b -+ a.

It is clear that the Lindenbaum algebra of any implicational logic is a Hilbert algebra. For each expanded Kripke frame S = ( M , 5, S> let H ( 9 ) denote the algebra <&(S), -+, M > .

Theorem 1. For each expanded Kripke frame .F, #(S) is a Hilbert algebra.

Proof. Remark first that M E Q(S) since &(F) is non-empty and M = X -+ X for any X E &(S). To show that (i) holds, suppose m 4 X -+ (Y 4 X ) . Then, for some m' 2 m we must have m' E X - (Y -+ X), and there must in turn be m'' m' such that rn" E Y - X. But, since X is closed and m' E X , m" E X , contradicting that m" E Y - X . A similar argument establishes (ii), while (iii) holds trivially.

Our next goal will be to show that every finite Hilbert algebra is 2(F) for some expanded Kripke frame 9, To do this wc shall prove, generalizing slightly a result of URQUHART [4], an analogue of the Stone Representation Theorem for Boolean algebras.

If 8 = ( A , 3, 1) is a Hilbert algebra and T r A , T is said to be a theory of 94 if 1 E T and b E T whenever a, a -+ b E T . A theory is said to be a-maxhzl, for a E A , if a # T and for each b E A , b E T or b + a E T. Let Ma denote the set of a-maximal theories of B and let M denote the set of theories of 8 which are a-maximal for some a E A . Lastly, set $(a) = ( M , g, (Ma I a E A)) . The following well known lemma will be needed in the proof of our next theorem.

Lemma 2 . For any Hilbert algebra 8, a E A and theory T of 8, if a 4 17, then there exists an. extension S of T such that S E Ma.

Proof. (See Theorem 1 of [4].)

Theorem 2. If !J is a finite Hilbert algebra with at least two elements, !J is isomorphic (in a natural way) to Z(.F(%)).

Proof. Let (I be the map given by o(a) = (T E M I a E 27). It will be shown that CT

is the desired isomorphism. For each a E A and X E Ma let cl(X, a) denote {T E M I every extension of T in Ha is also in X } , and note that a(a) = cl(0, a) for each a E A . For, if a E T , T E cl(0, a) trivially; and if T 4 cl(0, a), T has no extension which is a-maximal, and, by Lemma 2, a E T . That CT is 1 - 1 is also immediate from Lemma 2 ; for, if a =+ b, we will have a 3 b + 1 or b -P a $; 1, and there will then exist T E M separating a and b. We show next that a(a -+ b) = a(a) -+ a(b) for all a, b E A . Sup- pose T # a(a) -+ a(b) for some T E M . Then, by Lemma 2 , there will be an extension

A COMPLETE SEMANTICS FOR IMPLICATIONAL LOGICS 383

T' of T which is in a(a) - a@). Thus a E T and b 4 T', implying that a + b 4 T and that a + b # T. Hence T 4 o(a + b) . On the other hand, suppose T 4 o(a 4 b) . Thus, by Lemma 2, there must be a b-maximal extension T' of T such that a b 4 T'. But then a $ T', giving that T $ a(a) 4 a(b). It remains to be shown that 0 is onto. Let X E Q ( 9 ( % ) ) and let U 5 Ma be such that cl(U, a) = X . It will be shown that X = a(f( U ) ) where f ( V ) = {T 4 a 1 T E U} + a and {al, . . . , a,} + a stands for the element a, + (a, + (a3 + . . . (an -, a) . . .)). Suppose f ( U ) E T. If a E T, T bas no extensions which are a-maximal, and T E X . If a 4 T , there exists an extension S of T which is a-maximal. Then, {T + a I T E U} S. Let T E U be such that TI + a 4 S, and note that T' E S. But, since T' and S are both a-maximal, T = S; and T E X . On the other hand, suppose T E X and that S is an extension of T such that {T + a I T E U} E S. If a # S , Lemma 2 guarantees the existence of R 2 S which is a-maximal. Thus, R E U and R 4 a E R, a contradiction. Hence, a E S, and, since S was arbitrary, f ( U ) E T.

By an extended Kripke &el we shall mean a quadruple ( M , 5, S, G ) where ( M , s , S) is an extended Kripke frame and ( M , s , G ) is an ordinary Kripke model such that G - l ( p ) E &((M, r, S)) for each propositional variable p . Paralleling the definition for ordinary frames, we shall call an extended Kripke frame L-valid, where L is an implicational logic, if the theorems of L are true in every extended Kripke model bases on the frame. Finally, an implicational logic L is said to be complete for the extended Kripke semantics if for each 8 4 L there is an extended Kripke model, based on an L-valid extended frame, in which 8 is false.

Theorem 3. Every impl t2 ional logic i s complete for the extended Kripke semantics. Proof. Let L be an implicational logic and suppose 8 4 L. Let the variables of 8

belong to the set V = { p l , . . ., pn) and let 2,(V) be the Lindenbaum algebra for L determined by the formulas whose variables are in V . It is clear that Z L ( V ) is the free L-algebra freely generated by [p,] , . . ., [pn]. Also, since L 2 I , the map [pil l b [ p J L determines a homomorphism of &(V) onto g L ( V ) ; and, since X?,(V) is finite, as DIEGO has shown in [l], ZL(V) must be a finite Hilbert algebra. Because L is closed under substitution Z L ( V ) is L-valid; and, consequently, . F ( Z L ( V ) ) is also L-valid. Finally, since the canonical valuation pi I+ [pi], refutes 8 in ;IL( V ) , the as- signment G given by G(T) = { p i I p L E T} determines an extended Kripke model ( .F(ZL( V ) ) , a) in which 8 is false.

It should be remarked in conclusion that Theorem 3 remains true for the more restrictive class of extended frames F = ( M , s , S) in which each W E S is permitted to contain only 5 -incomparable elements.

References [l] DIEGO, A., Sur les alghbres de Hilbert. Gauthier-Villars, Paris 1966. [2] &EKE, S., Semantical analysis of intuitionistic logic I. In: Formal systems and recursive

functions (J. N. CROSSLEY and M. A. E. DUMIYIETT, Eds.), North-Holland Publ. Comp., Amster- dam 1966, pp. 92 - 130.

[3] PRIOR, A. N., Two additions to positive implication. J. Symb. Logic 29 (1964), 31 -32. [a] URQUHART, A., Implicational formulas in intuitionistic logic. J. Symb. Logic 89 (1974). 661 - 664.

(Eingegangen am 8. Dezember 1979)