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Deducibility with respect to necessary and impossible propositions

Published online by Cambridge University Press:  12 March 2014

Arnold F. Emch
Arnold F. Emch*
Affiliation:
Chicago
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Extract

Inasmuch as my discussion of the formal properties of “System L,” presented elsewhere, is substantially the same as that presented by Lewis in a previous issue of this Journal, I shall confine my remarks in this brief rejoinder to the problem of deducibility with respect to necessary and impossible propositions—since it is specifically the solution of this problem which will determine the adequacy of the calculus of logical implication as a canon and critique of deductive inference.

The problem, as originally presented, can be stated succinctly in the following inconsistent triad: (1) All necessarily true principles are logically dependent on or deducible from one another, i.e., are equivalent, (2) All principles of logic and mathematics are necessarily true, and (3) Some principles of logic and mathematics are not logically dependent on or deducible from one another, i.e., are independent. These three propositions are logically inconsistent in the sense that the conjunction of any two of them will logically imply the falsity of the third. Now if Lewis should affirm the first and third of these propositions, then he should concede the falsity of the second. This concession, it was believed, he was not likely to make, inasmuch as it would seem to involve a repudiation of a doctrine (the tautological character of every logical and mathematical truth) with which he has particularly identified himself. If, in view of this fact, he should accept, instead, the second and third of these propositions, then he should concede the falsity of the first.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 2 , Issue 2 , June 1937 , pp. 78 - 81
Copyright
Copyright © Association for Symbolic Logic 1937

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References

1 The calculus of logical implication, presented to the Association for Symbolic Logic and the American Mathematical Society, Sept. 1, 1936, and to be published shortly.

2 Vol. 1 no. 3 (1936), pp. 77–86.

3 Consistency and independence in postulational technique, Philosophy of science, vol. 3 no. 2 (1936), pp. 185–196Google Scholar, and Implication and deducibility, this Journal, vol. 1 no. 1 (1936), pp. 27–35Google Scholar.

4 Symbolic Logic, pp. 24, 211–212.

5 This Journal, vol. 1 (1936), p. 85.

6 Langford, C. H.,Singular propositions, Mind, n.s. vol. 37 (1928), see pp. 77–78Google Scholar. Italics mine.

7 Nelson, Everett J., Intensional relations, Mind, n.s. vol. 39 (1930), pp. 440–453CrossRefGoogle Scholar.

7 This Journal, vol. 1 (1936), p. 67.

9 This Journal, vol. 1 (1936), pp. 61-62.

10 The use of “⋄p”, as well as “p ⊰ q” and “p = q” in “System L,” is unfortunate, since these already have a well established interpretation in the calculus of strict implication. In the formal development of “System L” (The calculus of logical implication, loc. cit.), these symbols have been replaced by others.

11 These postulates are ⋄p ⋄q · ∽ Op Oq and (∃p, q)· ⋄[∽⋄∽⋄p∽⋄∽q·∽(p∿q) ∨ ∽ (q ∿p)], respectively. In a more recent work, these two postulates have been replaced by the simpler ⋄p · ∿ · Op and (∃p, q) : ∽⋄∽(pq) · ∽(p=q) respectively.

12 Symbolic Logic, pp. 160–161.