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Emch's calculus and strict implication

Published online by Cambridge University Press:  12 March 2014

C. I. Lewis
C. I. Lewis*
Affiliation:
Harvard University
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Extract

In a previous number of this Journal, Dr. Arnold F. Emch argues that the mathematical properties of the system of strict implication preclude the interpretation of p⊰q as synonymous with “q is deducible from p” and of ◊(pq), or p o q, as “p and q are consistent.” He proposes a new calculus of propositions in which the primitive ideas of the system of strict implication are retained but an additional idea, “logical consistency,” Op, is introduced, and two new relations, “logical implication,” p∾q, and “logical equivalence,” p = q, are denned. He believes that the properties of this new system are in accord with the facts about deducibility, consistency, and independence of propositions where those of p⊰q and ◊(pq) fail of such accord.

Not all of the objections which Dr. Emch advances against strict implication are given due consideration in what follows. Concerning one especially important point, something will be said at the end of this paper. For the rest, discussion will here be limited to certain formal properties of the two systems, and certain consequences for the interpretation of them.

For brevity, the system of strict implication will be referred to, in what follows, as the “system S,” or merely as S; and Dr. Emch's calculus as the “system L,” or as L.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 1 , Issue 3 , September 1936 , pp. 77 - 86
Copyright
Copyright © Association for Symbolic Logic 1936

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References

1 No. 1, pp. 26–35.

2 Assumptions of S are as given in Lewis and Langford, Symbolic logic. Chapter VI. The alternative notation for ◊(pq), and the corresponding definition (17.01), are omitted here, since they would merely complicate the present discussion. The reprint of Dr. Emch's paper includes an Addendum, announced as forthcoming in No. 2 of this Journal, in which the form of L7, as originally printed, is altered, and three postulates are added. Changes indicated in this Addendum are here made. [This Addendum has since appeared, in No. 2, page 58. Editor.]

3 See Symbolic logic, p. 495, where the reason for including 11.2 is explained.

4 Dr. Emch notes that this theorem is deducibile, in the alternative form (∃p, q):O(pq).O(p∿q) (see his Addendum). He also remarks (p. 33, footnote) that “demonstration of the existence principles of this calculus [L] requires certain other rules of procedure comparable to those specified for the calculus of strict implication.” Such a rule, sufficient for the theorem just cited, and for LT, would be: If (∃p, q, …).ϕ(p, q,…) is asserted, and ϕ(p, q, …)∿Ψ(p, q, …) is asserted, then also (∃p, q, …).Ψ(p, q, …) may be asserted.

5 The name adopted for this correspondence is, of course, arbitrary; it seems desirable to avoid the phrase “mathematical analogues” in view of another comparison, to be mentioned later.

6 One such possible order of the early theorems, up to the deduction of 11.2 would be: 12.15, 12.25, 12.3, 12.1, 12.11, 18.8, 19.74, 12.5, 12.6, 11.2. (Numbers are as in Symbolic logic.) This deduction is due to Dr. W. T. Parry.

7 Proof of (3) and (4) as theorems in S is due to Prof. E. V. Huntington.

8 It is not the whole class of symbolically identical laws in the two systems which is in question. There are such which do not involve any of the symbols ◊, ⊰, =; and hence are independent of the interpretation of ◊ These last coincide with the class of laws which are at once symbolically identical, in S and in L, and logistic analogues. Thus the class whose interpretation is in question may be specified as those laws of L which are symbolically identical with laws of S but not their logistic analogues; or as the laws of L which involve ◊. ⊰, or = , but not O, ∾, or =.

9 This notation, * p, is not, of course, related to the notation a*, used in certain papers of Huntington to represent what is here symbolized by ∼ ◊ p.

10 As has been pointed out, L contains all laws of S in symbolically identical form. Call this class the “subset L◊.” L◊ and L◊ do not coincide. But the remainder of L◊ consists of theorems, like p ∨ ∼ p, which involve none of the symbols ◊ ⊰ =. Thus their interpretation depends only on that of ∼ and the relation pq. And when the interpretation of these is fixed, the interpretation of L◊, depends only on that of L◊.

11 I would not say that there is no wider meaning of “ϕx is consistent with Ψx,” or narrower meaning of “ϕx implies Ψx.”

12 Tautologies of the type p ∨ ∼ p, where p is a proposition, and those of the type ϕx ∨ ∼ ϕx, where ϕx is a propositional function and the whole expression is asserted, are not identical in all their properties. But the above statement holds of both these forms.

13 The postulates and theorems of such a formal system are, in a somewhat special sense, propositional functions: they become true or false when “interpreted.” The real variables, whose occurrence in the formal postulates and formal theorems marks these as propositional functions, are not what are commonly called “the variables”—since these last remain variables when the postulates and theorems are interpreted— but are the class K and the “operations” and “relations.” That is; the formal postulates and theorems become genuine propositions when a determinate meaning is assigned to “clement in K” and determinate meanings to the operations and relations.