Zeitschr. f. math. Logik und Orundlagen d . Math. Ed. 6 , S. 217-218 (1960)

E Q AND THE FIRST ORDER FUNCTIONAL CALCULUS1)

by NUEL D. BELNAP, Jr., in New Haven (Conn.)

1. It is not known w-hether the rule, from and v '$3 t o infer 23, is admissible in EQ.2) We are not therefore in a position to use the usual methods of showing that EQ contains the first order functional calculus. However, we may use the fact that the following system K, (with -, v, and 3 x as primitive) is a complete formulation of the first order functional calculus 3

Axioms: % v a, where 21 has the form p , 17, f ( x l , . . ., x%), or /(xl,. . ., xn). Rules :

I. S t r u c t u r a l rules .

(a) from '2Jl v 81 v 23 v % to infer '2Jl v 23 v % v % (b) from ?I v 8 v % to infer (11 v %

11. I n t r o d u c t i o n rules .

(a) from '8 to infer 91 v '$2 (b) from 5 v % and 5 v % to infer (c) from I?( v 8 to infer ii v ER (d) from ? [ X V % to infer 3 y % y v P (el from G v % to infer g y % y v %

v 8

I n (d) and (e) it is required that x be free in % x just where y is free in %y, and in (e) it is required that x not occur free in %. Association of parentheses is to the right, and the rules remain valid in the absence of 9'2 or Yt. K, is such that in no formula does the same variable occur both free and bound; to admit such formulas, we should need to add a rule for alphabetic change of bound variable in order to retain completeness.

It is now easy to show that EQ contains the first order functional calculus, for in EQ we have as theorems

' 2 J l v % v % v % - + m v v v % v % , % v 2 I v % + % v % , %-+%v%, -

( % V % ) A ( % V % ) + % V ! 8 V % , % V % + z V % , %XV% -+gY%yV%!, --

and a derived rule, to the effect that if v % is a theorem, then so is 3 y '2l y v %. (Conditions on '2Jl, 92, and variables are as above.) Hence, if ?I is a theorem of K, (and hence of the first order functional calculus), it follows by (a) and induction on the length of proof of % in K, that 91 is provable in EQ.

l) This research was supported in part by the Office of Naval Research, Group Psychology Branch, Contract No. SAR/Nonr-609 (16). Permission is granted for reproduction, translation, publication and disposal in whole or in part by or for the U. S. Government.

2, See ALAN Ross A ~ ~ ~ ~ ~ ~ ~ , C o m p l e t e n e s s theorems for the systems E of entailment and EQ of entailment with quantification, this number of this Zeitschrift.

KURT SCHUTTE, SohluBweisen - Krtlkiile der Priidikatenlogik, Math. Ann. 122 (1950/51), 47-65.

15 Ztschr. f . math. Logik

2 18 NUEL D. BELNAP, JR.

2. By introducing certain auxiliary notions and enlarging the set of axioms, it is possible to present a formulation F1* of the first order functional calculus without rules Ia, Ib, and IIe, which is such that each of its rules corresponds to an entailment in the sense of EQ. (The rule IIe of K, does not have this property.)

No ta t ion a n d definit ions. 1. (x) replaces 3 % as primitive. 2. lir is a sequence of zero or more universal quantifiers. 3. 21 is a disjunctive part of 3 , and if 58 v & is a disjunctive part of 91, then so

4. @(%) is a wff of which 21 is a disjunctive part, and @(a) is the result of

5. If 3 is p , fi , f ( x l , . . ., xn), or j(xl , . . ., xn), then 21 is an atom of @(%).

Axioms: I l g is an axiom of F1* provided every molecule of a is an atom

are 2j and &.

replacing 91 in @(a) by 23.

If % is not of the form 93 v 6, then % is a molecule o/ @(a).

and provided atoms 58 and a occur as disjunctive parts of a. Rules:

1. from IT@(%) to infer n@(E) 2. from I?@@) and l7@(23) to infer IT@(% v '23) 3. from n@(% v a) to infer n@(%) 4. from ITQ(GX) t o infer n ~ ( ( y / ) % y ) 5. from n ( z ) @(%z) to infer n @ ( ( y ) %y) 6. from (. . . ( z ) % I z . . .) to infer (. . . ( y )%y . . .)

In rules 4, 5, and 6, it is required that x be free in (21 x just where y is free in (21 y and in rule 5 it is required that the molecules of (21x be the only molecules of Q, (8 x) in which x occurs free.

The soundness of the axioms and rules of F1* is evident. And the completeness of F1* follows from the fact that it contains K,: inductive proofs establish that if the premiss of an application of rule Ia {IIa, IIe} is a theorem of K, , then so is the conclusion; and the axioms and other rules of K, are special cases of the axioms and rules of F1*.l)

Finally, it is apparent that every axiom of F1* is a theorem of EQ, and that every rule of F1* of the form, from 21 {and B) to infer G, corresponds to a theorem of EQ of the form ?( { A % } +G.

- - - l) 3 ~ % 1 z = ~ ~ ( z ) % z , and in the next paragraph, %r\%=, ,@V@.

(Eingegangen am 22. Milrz 1960)