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m-valued sub-system of (m + n)-valued propositional calculus1

Published online by Cambridge University Press:  12 March 2014

Tzu-Hua Hoo
Tzu-Hua Hoo*
Affiliation:
National Peking University
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An (m + n)-valued propositional calculus2 may happen to be a subsystem of an m-valued propositional calculus , though the converse is never true. This fact may give us the impression that, as m grows, the content of becomes meagre. The present treatment is intended to remove this impression by constructing a complete, m-valued sub-system of any (m + n)-valued propositional calculus.

In the following we adopt the customary, autonymous mode of speech according to which symbols belonging to the object calculi or languages are used in the syntactic language as names for themselves, and juxtaposition serves to denote juxtaposition.

2.1 = df stands for definational identity in the syntactic language.

2.11 ≡ stands for definational identity in the object calculi.

2.2 ∊, ⊂, ∩, {x1, …, xn} are used in their meanings as customarily employed in the theory of sets—∊ for class membership, ⊂ for proper inclusion, ∩ for the product operation of classes, {x1, …, xn} for the class with x1, …, xn as its only elements.

2.3 x, y, z are used as unspecified natural numbers including 0. m, n, i, j are used as unspecified natural numbers other than 0.

2.401 Definition. δ = df as the function of two variables defined for any x, y such that

2.41 Definition. For m ≧ 2, ιm = df the function of two variables denned on the set {0, …, m − 1} such that ιm (x, y) = y − x for x ≦ y and ιn(x, y) = 0 for x > y.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 14 , Issue 3 , September 1949 , pp. 177 - 181
Copyright
Copyright © Association for Symbolic Logic 1949

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Footnotes

1

I wish to express my gratitude to Prof. Alonzo Church for his helpful suggestions in the preparation of the manuscript, and to Prof. Rudolf Carnap for his reading an early draft of the paper.

References

2 Łukasiewicz, J. and Tarski, A., Untersuchungen über den Aussagenkalkül, Comptes rendus des séances de l'Académie des Sciences at des Letters de Varsovie, Classe III, vol. 23 (1930), pp. 30–50Google Scholar.

3 An m-valued propositional calculus is treated here in a way somewhat different from the usual. For example, 0, instead of 1, is used as the designated value. It may appear inconvenient, but its use here is intended to facilitate comparative study with one of the present writer's forthcoming papers, An ℵ0-valued propositional calculus.

4 See Rosser, J. B. and Turquette, A. R., Axiom schemes for m-valued propositional calculi, this Journal, vol. 10 (1945), pp. 61–82Google Scholar.

5 A matrix is to mean here an ordered quadruple of a set and three functions f, g1, and g2. It is not necessary for the present purpose to split the class into two disjoint sets as Łukasiewicz and Tarski did in their paper, cited in footnote 2 of this paper.

6 By introducing a definition (or more exactly an explicit definition) into a formalism, we mean here an extension of the syntactical formalism in three aspects: (1) the extension of the list of primitive symbols to contain the new symbols introduced in the definiendum; (2) the extension of the formation rules by allowing the definiendum as well-formed; (3) the extension of the transformation rules by introducing a new rule of replacement such that when the definiens occurs in a formula, it may be replaced by the definiendum of the very definition in question. The last of these three is most essential.

7 Or more exactly: , if and only if there is a finite sequence q1, …, qn, such that qn = p and for any i, 1 ≦ i ≦ n, qi, either belongs to or arises out of qi−1 by replacing a partial formula that has the form of the definiens of a definition by the definiendum of d.

8 See 2.1.11.

9 For example, contains the axiom schemata (a2), (a3), …, (a14) as used by Rosser and Turquette in their paper, cited in footnote 4 of this paper.

10 For example, Łukasiewicz's system containing three postulates: (p ⊃ q) ⊃ ((q ⊃ r) ⊃ (p ⊃ r)), (∼p ⊃ p)⊃, p p ⊃ (∼ p ⊃ q) with rules modus ponens and substitution.

11 We may speak of the Wajsberg-Słupecki system with (d1) and (d2) as a three- and two-valued mixed system. For example, the following formulas are among its theorems, CCpq(p ⊃ q), CNp ∼ p, CCNpNq(q ⊃ p), CN ∼p∼p, N∼p⊃ ∼p, Cp ∼ Np, N ∼ p ⊃ N ∼ Np, N ∼ Np ⊃ N ∼ p, CN ∼ Np ∼ Np, ∼N ∼ p, ∼N ∼ Np, etc.

12 In introducing into , however, the following order is to be observed: if i i > i 2, precedes in introduction.