An Extension of Computational Logic

An Extension of Computational LogicAuthor(s): Alan RoseSource: The Journal of Symbolic Logic, Vol. 17, No. 1 (Mar., 1952), pp. 32-34Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2267455 .

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THE JOURNAL OF SYMBOLIC LOGIC

Volume 17, Number 1, March 1952

AN EXTENSION OF COMPUTATIONAL LOGIC

ALAN ROSE

There has recently been developed a system of computational logic1 to which was given an interpretation in terms of the 2-valued propositional calculus. The object of the present paper is to give the corresponding theory for 3-valued logic.2 We use the same notation as Levin, except that instead of using the numeral "2" as a constant, we use "3".

Rules. Interchangeability is as before a primitive notion. For any x, y, z * not excluding 3: (1) If x is interchangeable with y, then y is interchangeable with x. (2) If x is interchangeable with y, and y is interchangeable with z, then x is

interchangeable with z. (3) If x is interchangeable with u, and y is interchangeable with v, then xy

is interchangeable with uv. (4) x is interchangeable with x33. (5) x is interchangeable with xx(x3). (6) x3x is interchangeable with 3. (7) xy3z is interchangeable with zy3x. (8) x(y3z)3z3p13p23 3pk33(xy) is interchangeable with

x(y33)3z3p13p23 * 3pk3(xy),

provided that k is a non-negative integer. (9) 33x is interchangeable with 3. DEFINITION. For any finite n, if x is interchangeable with 33xl3x23 ... 3x.3,

then x is said to possess the v-property and is called a v-sign.

Theorems. The proofs of Theorems (10), (12), (13), (14), and (15) are omitted as these proofs are exactly analogous to proofs of Levin.

(10) x is interchangeable with x. (11) LEMMA. If y' is a subsign of x and y is any sign interchangeable with y',

then the result of substituting y for y' in x, is interchangeable with x. PROOF: A trivial consequence of (3) and (10).

Received April 11, 1950. 1 NATHAN P. LEVIN, Computational logic, this JOURNAL, VOL. 14 (1949), pp. 167-172. 2This theory was developed by Lukasiewicz and later generalized by him in conjunction

with Tarski to m-valued systems. The m-valued systems were discovered independently by Post. The original papers include:

EMIL L. POST, Introduction to a general theory of elementary propositions, American journal of mathematics, vol. 43 (1921), pp. 163-185.

JAN LUKASIEWICZ, 0 logice tr6jwarto~ciowej, Ruch filozoficzny, vol. 5 (1920), pp. 169-171. JAN LUKASIEwIcz and ALFRED TARSKI, Untersuchungen uiber den AussagenkalkWl, Comp-

tes rendus des seances de la Societe des Sciences et des Lettres de Varsovie, Classe III, vol. 23 (1930), pp. 30-50.

J. B. ROSSER and A. R. TURQUETTE, Axiom schemes for m-valued propositional calculi, this JOURNAL, vol. 10 (1945), pp. 61-82.

32



EXTENSION OF COMPUTATIONAL LOGIC 33

(12) xy is interchangeable with yx. (13) X1X23x3 is interchangeable with XiXj3Xk, where i, j, k is any permutation

of 1,2,3. (14) Every v-sign is interchangeable with 33. (15) If x3y3 is a v-sign and x is a v-sign, so is y. (16) x(y3z)3z3pl3p23 ... 3pk3(xy)3 is interchangeable with x(y33)3z3p13p23 ...

3pk3(xy)3, provided that k is a non-negative integer. PROOF: x(y3z)3z3pl3p23 ... 3pk3(xy)3 (8)(11) x(y33)3z3p13p23 ... 3Pk3 (xy)3

Logical interpretation. We now give an interpretation of the system in terms of the 3-valued propositional calculus. In this way we justify the system as a consistent set of rules.

3 denotes any sentence for which the lowest truth-value may be substituted. v-sign is interpreted as a tautologically true sentence. xy is interpreted as the function x!y with the truth-table

xy 12 3 y

1 3 3 3

2 3 3 2

3 3 2 1

x

where 1 is the designated truth-value. This function can be defined in the Lukasiewicz system by

x!y =df. Y -

"x is interchangeable with y" is interpreted as "x and y agree in truth-value." So translated, our postulates. (1) to (9) become theorems of the 3-valued

propositional calculus, verifiable by the truth-table method.

The logical constants. The notation supplies signs for the logical constants. Thus comparing the truth-table for x -+ y with the one for x3y3 we see that for corresponding substitutions we obtain corresponding truth-values. Similarly x3 is intertranslatable with x.

Derivation of Wajsberg's system. To show that all the theorems of the Lukasiewicz propositional calculus are obtainable as v-signs, we select for proof the axiom schemes corresponding to Wajsberg's3 axioms for the system. This is sufficient as we have already proved the modus ponens rule.

8 M. WAJSBERG, Aksjomatyzacja tr6jwartokciowego rachunku zdani, Comptes rendus des

seances de la Societe des Sciences et des Lettres de Varsovie, Classe III, vol. 24 (1931),

pp. 126-148; or J. B. ROSSER and A. R. TURQUETTE, Op. cit., pp. 73-74.



34 ALAN ROSE

Axiom 1 ((X L) X) X

Transl. X3(X3)33X33X3 (4) X3(X3)X33X3 (4) X3(X3)XX3 (4) X3(X3)(X33)X3 (5) X3X3 (6) 33

Axiom 2. X -(Y -X)

Transl. X3(Y3X3)3 (12) Y3X3(X3)3 (7) X3X3(Y3)3 (6) 33(Y3)3

Axiom3. (Y-*X> (X --Y)

Transl. Y33(X3)33(X3Y3)3 (4) Y(X3)33(X3Y3)3 (4) Y(X3)(X3Y3)3 (12) X3Y(X3Y3)3 (12) X3Y3(X3Y)3 (6) 33

Axiom 4. (X Y) ((y Z) (X- Transl. X3Y33(Y3Z33(X3Z3)3)3

(4) X3Y(Y3Z33(X3Z3)3)3 (4) X3Y(Y3Z(X3Z3)3)3 (12) Y3Z(X3Z3)3(X3Y)3 (12) X3Z3(Y3Z)3(X3Y)3 (13) X3(Y3Z)3Z3(X3Y)3 (16) X3(Y33)3Z3(X3Y)3 (4) X3Y3Z3(X3Y)3 (13) X3Y3(X3Y)3Z3 (6) 33Z3.

COLLEGE OF- TECHNOLOGY, MANCHESTER, ENGLAND



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An Extension of Computational Logic