Zeitmhr. 1. naath. Lo& und Grundlagen d. Math. Bd. 18, S. 237-210 (1972)

ISOMORPHISM BETWEEN C 1 AND C2

by ALEX BLUM in Ramat-Gan (Israel)

0. Let K be a two-sorted first order predicate calculus containing a denumerable list of individual-variables of a sort we call “object” variables, one individual- variable of a sort we call “world” variables, and a denumerable list of n + 1-adic predicate letters (for all n: n 2 0) the n + l t h place of which is to be filled by the world-variable, and the remaining places, if any, are to be filled by object- variables.

We show that the system C1 consisting of the set of wffs of K and the relation of K-deducibility (from now kK) is isomorphic to the system C2, consisting of the set of wffs of modal predicate logic S5* and the relation of S5*-deducibility (from now ts5.). Hence we have the philosophically curious result that S5* formalizes the notion of entailment iff K does.

As a corollary, we extend SLOMSON’S result [a, theorem 21 of the undecidability of L , to K with one dyadic predicate; where L is like K with one dyadic predicate, except that L contains a denumerable list of individual-variables of both sorts. We also show that any decision precedure 6 for the first order monadic predicate calculus, is a decision precedure for 55 . And finally, we specify a set of wffs of dyadic first order predicate logic which is undecidable.

1. The two-sorted first order predicate-calculus K is built up from the following elements :

(a) A denumerable list of individual-object-variables, xl, x 2 , x 3 , . . . ; (b) An individual-world-variable, w; (c) A denumerable list of n + 1-adic predicate letters (n: n 2 0) whose n +- l th

argument place is to be filled by the individual-world-variable, and the remaining places, if any are to be filled by individual-object-variables; Fil, Fi2, Pi3, . . ., FL1, Fi2, FL3, . . ., Pi1, Fi2, Pi3, . . .;

(4 The symbols, -, v, (4, (z2) , (x3) . . . , (w), ( and 1. The symbols - , 3, E and ( 3 4 , ( 3 x 2 ) , ( 3 x 3 ) , . . ., (3w), are introduced in the

usual manner. The formation rules, axiom schemes and rules of inference of K , are the standard ones for many-sorted first order predicate calculi and may be found in CHURCH [l, pp. 339-3401 and WANG [5, pp. 322-3231.

In our formulation, S5* is built up from: (a) A denumerable list of individual-object-variables, x l , x2, x 3 , . . .; (b) A denumerable list of n-adic predicate letters, F i , Fg, F t , . . . , F:, F:, F;, . . . ,

J;, F; , F:, . . .; and 1 4 The symbols, - , v, (4 , (4 , (z3) , . . - , , ( and ).

238 ALEX BLUM

The symbols, a , 3, =, (3x1), (Ix,), ( 3 x 3 ) , . . .) 0, are introduced in the usual manner. The formation rules are the standard ones; and the axiom schemes and rules of inference are the ones used by KWKE in [3].

2. Lemma 1. There exists a function 97 which is a one-to-one correspondence between the set of wffs of S 6" and K .

This Lemma is an immediate consequence of our definition of 9) given by the following table :

S5*: xl, x2. x 3 , . . .

Y

K : x,, xz, x3,. . . 55*: (XI), k), h),. . ., (3x1), (Ix,), ( 3 x 3 ) , . . ., v, -;, I>, E , O , 0 ( and )

F i , F i , F:, . . ., P:,li':,F!, . . ., F i , F : , F i ) . . ., t -1

t

F;l, Pi2, F:3, . . ., FA1, . . ., Fil,FAz, Fi3, . . .,

+ K: (XI), (zJ, ( ~ 3 1 , . . (3x1)) (gx,), (3x3) s * * * ) V, - 3 * , 3, (w)> ( 3 ~ ) ( and ) .

We now prove our main result, that: Theorem 1.

Proof. Let W be some non-empty set (of worlds) {el,. . ., e i , . . .}, D some non- empty set (of objects) {ul, . . . , u i , . . .} , and let V* and V be value-assignments satisfying the conditions to be stated below. Then ( W , D, V*) is an SB*-model (with respect to which S5* is complete [3]) and ( W , B , V ) is a K-model (with respect to which K is complete [5, pp. 326-329, see also 1, pp. 339 -3401)l).

17": (i) To each individual-object-variable x, V* assigns some elements of D (i,e,,

(ii) To each n-adic predicate letter y , V* assigns a set of ordered (n + 1)-tuples, each of the form (ul, . . . , u,, e,) , where each u is an element of D and ei E W .

1. [Atomic formula]. If y is any n-adic predicate letter, then V* (y (xl, . . . , xn) , ei) = 1 if ( V * ( x , ) , . . ., V*(x,J, ei) E V * ( y ) . Otherwise V*(y(x , , . . ., x,,,), e J = 0 .

2. [V*-1. For any wff 01 and any eiE W , V*(-or, ei) = 1 if V*(a , eJ = 0 . Otherwise V*( -oc, ei) = 0.

3. [ V*v]. For any wffs 01 and p and any w i E W , V* ((01 v ,!I), e i ) = 1 if V* (a , e i ) = 1 or V*(,!I, e i ) = 1. Otherwise V*((a v p ) , ei) = 0 . 4. [V*(a)]. For any wff oc, any individual-object-variable a , and any ei E W ,

V* ( ( a ) a , ei) = 1 if for every S 5* assignment V*' which makes the.same assignment as V* does to all variables other than a , V*' (01 , ei) = 1 . Otherwise V* ( ( a ) 01 , e J = 0.

5. [V*o]. For any wff a and any eiE W , V*(ua , e,) = 1 if for every ej E W , V* (oc, e j ) = 1. Otherwise V*(oa, e i ) = 0 .

is an isomorphism from C 1 to C2.

V * ( x ) = u) .

1) Our formulation of the SS*-model is taken from HUGHES and CRESSWELL [Z, pp. 73-74, pp. 146-1471. Our formulation of the K-model is an adoptation of [Z, pp. 136-1371.

ISOMORPHISM BETWEEN 01 AND 0 2 239

A wff 01 of S5* is SB*-valid if and only if for every S5*-model ( W , D, V*) , V*(oc, e i ) = 1 for every e i E W .

V : (i) V ( w ) = e . (ii) V ( z ) = u . (iii) To each n + 1-adic predicate letter p, V assigns a set of ordered (n + 1)-tuples, each of the form (ulr . . . , u,, e i ) , where each uE D and e i E W .

1. [Atomic formula]. If y is an (n + 1)-adic predicate letter, V(y(xl, . . ., x,, w)) = 1 if (V(xl), . . ., V(z,), V ( w ) ) E V ( y ) . Otherwise V ( y ( x l , . . ., z,, w)) = 0.

2. [V-1. If LX is a wff, then V ( - a ) = 1 if V ( a ) = 0. Otherwise V ( - a ) = 0. 3. [Vv]. If 01 and /3 are wffs, then V(0r v /3) = 1 if V ( a ) = 1 or V(P) = 1. Other-

wise V(01vp) = 0. 4. [V(a)] . If 01 is a wff and a is an individual-variable, then V ( ( a ) a ) = 1 if for

every K assignment V' which makes the same assignment as V does to all variables other than a , v'(01) = 1. Otherwise V'((a) a) = 0 .

A wff 01 is K-valid iff for every K-model ( W , D, V ) , V ( a ) = 1.

From these models it immediately follows that there is no assignment that V* can make to a formula a which V cannot make to its pcounterpart a' and con- versely. In particular, V* can make a falsifying assignment to a iff V can make a falsifying assignment to a'. Hence a formula 01 is valid iff its qJ-counterpart is. But 01 1 /3 iff 01 3 /? is valid. Thus C 1 and C2 are isomorphic.

3. We now prove

Corol lary 1. K with one dyadic predicate letter i s undecidable.

Proof . S6* with one monadic predicate letter is undecidable, [a, Theorem 41. But from the one-to-one correspondence y and theorem 1, it immediately follows that C1' consisting of the set of wffs of K with one dyadic predicate letter and t K is isomorphic to C2' consisting of the set of wffs of S6* containing only one monadic predicate letter and ks5*.

Corollary 1, extends SLOMSON'S result [4, Theorem 21 from a system L contain- ing a denumerable set of (what we called) individual-world-variables, to a system which contains only one.

Corol lary 2 . If 6 is a decision procedure for the first order monadic predicate calculus, i t is a decision precedure for SS.

Proof . Again, from and Theorem 1, it immediately follows, that C1" consist- ing of the set of wffs of K containing only monadic predicate letters and t~ is iso- morphic to C2" consisting of the set of wffs of S B and t S 5 . But monadic K is a subsystem of the first order predicate calculus.

We let 01 be the name of a particular individual variable and we let B be any individual variable, other than a . Now let G be a monadic predicate letter and H a dyadic predicate letter, then

Corol lary 3. The set of wffs of the first order predicate calculus containing G b and H/301, is undecidable.

240 ALEX BLUM

Proof. By corollary 1 , K with one dyadic predicate letter is undecidable. By WANG’S translation-schema [5, p. 3231, the Wma-transforms of the set of wffs of K with one dyadic predicate letter is the set of wffs consisting of Ga and HBa. But by WANa [5, pp. 324, 3291 a set of wffs of K is decidable iff their WANa-trans- forms are.

References [l] CHURCH, A., Introduction to Mathematical Logic, Vol. 1. Princeton, N. J., 1956. [2] HUQHES, G. E., and M. J. CRESSWELL, An Introduction to Modal Logic. Methuen, London

1968. [3] KRIPKE, S., A Completeness theorem in modal logic. J. Symb. Logic 24 (1959), 1-15. [4] SLOMSON, A. R., An undecidable two-sorted predicate calculus. J. Symb. Logic 34 (1969),

[S] WANQ, H., A Survey of Mathematical Logic, Studies in logio and the foundations of mathe- 21-23.

matics. North-Holland, Amsterdam 1963.

(Eingegangen am 28. Juni 1971)