A Simple Treatment of Truth Functions by Alan Ross Anderson; Nuel D. Belnap,Review by: William T. ParryThe Journal of Symbolic Logic, Vol. 28, No. 4 (Dec., 1963), p. 291Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2271312 .

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REVIEWS 291

ALAN Ross ANDERSON and NUEL D. BELNAP, Jr. A simple treatment of truth functions. The journal of symbolic logic, vol. 24 no. 4 (for 1959, pub. 1961), pp. 301-302.

The authors give an axiomatization of the disjunction-negation calculus in which the axioms are all primitive disjunctions (disjunctions of literals) containing a variable and its negate. Where +(A) is a wff with A as disjunctive part, the rules are: From

+(A) to infer +(A); From i(A) and +(B) to infer #(A v B). Decision procedure and proofs of consistency, completeness, and independence follow immediately. The relevance of this system to systems of entailment is indicated.

The authors correct page 302, line 16 to read: "detachment is not known to be derivable for E." They also point out that this paper, and an abstract (this JOURNAL,

vol. 24, p. 320) which develops first-order predicate calculus in a similar way, are anticipated by Kurt Schutte, XVI 155 and XXII 297. WILLIAM T. PARRY

R. L. GoODSTEIN. Models of propositional calculi in recursive arithmetic. Mathe- matica Scandinavica, vol. 6 (1958), pp. 293-296.

The author defines in recursive arithmetic (R.A.) a model for the (N+ 1)-valued propositional calculus of Post. In this model the connectives v, A, -_, and -* are interpreted as p - (p - q), P + (q - p), {1 - [1 - (N p)]}.(p + 1), (N * p) - (N - q) and the designated value is 0. Under this interpretation the propositional identities of the Post calculus go over into arithmetical identities which are provable in R.A. The rules of modus ponens and of mathematical induction (from /(O) = 0 and /(x) = 0 -/(x + 1) = 0 follows /(x) = 0) go over into schemata of inference valid in R.A. This is no longer true for the substitution rule x = y -f /(x) = f(y) if N > 1 and if x = y is interpreted as (x y) + (y - x).

A similar interpretation is given for a system LC which is obtained from the intu- itionistic propositional calculus by adding to its axioms the formula (p -* q) v (q -* p).

The author states that he was unable to find an interpretation in R.A. of the intuitionistic propositional logic. ANDRZEJ MOSTOWSKI

VLADETA VUCKOVIC. Rekurzivni modeli nekih neklasitnih izkaznih rauluna (Rekur- sive Modelle einiger nichtklassischen Aussagenkalkiile). Filozofija (Belgrade), vol. 4 no. 4 (1960), pp. 69-84.

This paper deals with a propositional calculus A which arises from the Heyting calculus H by dropping the axiom (p q) A (p -+ --q) -p -,p and adjoining six new axioms: (i) p v (p -* q), (ii) -I(p A q) (-4p v --q), (iii) --,(p v q) - (--p v q), (iv) (-_-p q) A (-_'p -_ -q) -_ -,-,p, (v) (_-,-p -+ q) A (-_--*p _q) -_ -4p, (vi) --(p A q) -+(-_-_P A -i--,q).

Formulas (iv)-(vi) are intuitionistically provable but (i)-(iii) are not. Hence A and H overlap and neither is contained in the other.

The author carefully compares both systems A and H. Formulas p v -_p and

_-p p are provable in neither of them but the weak form of the law of excluded middle -ip v -__--p is provable in A.

The author sketches further a formal system which he calls the recursive arithmetic of words (R.A.W.). Let Q be the free semigroup with two generators So, Si and with the "void" element 0. The functions Z, So, S1 defined by the equations: Z(X) = 0,

SO(X) = SoX, S1(X) = S1X as well as the identity functions are called the "starting functions." Other functions can be defined from them by means of substitutions and primitive recursions, the typical example of which is: /(X, 0) = a(X), f(X, SoY) = bo(X, Y, /(X, Y)), /(X, S1Y) = b1(X, Y, /(X, Y)), a, bo, and b, being functions previously defined.

The functions thus obtained are called recursive functions of R.A.W. (abbreviated as r.f.).

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