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A system of strict implication

Published online by Cambridge University Press:  12 March 2014

P. G. J. Vredenduin
P. G. J. Vredenduin*
Affiliation:
Arnhem, The Netherlands
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Extract

Lewis has made an attempt to construct a logistic system containing an implication-relation ⊰ in such a way that p⊰q shall have the meaning: q is deducible from p. Lewis and Langford admit that “the one serious doubt which can arise concerning the equivalence of p⊰q to the relation of deducibility … arises from the fact that strict implication has its corresponding paradoxes:

19.74˜⋄ p ⊰ · p ⊰ q,

‘If p is impossible, then p strictly implies any proposition q’; and

19.75 ˜⋄˜p ⊰ · q ⊰ p,

‘If p is necessary, then any proposition q strictly implies p.’”

Indeed, it is not obvious that 19.74 and 19.75 should hold. It is true that in many cases 19.75 (as well as 19.74) is valid; for in the system of Lewis and Langford

p · ˜p · ⊰ q

is provable, and the conformity of this law with real deduction is shown by them. In all cases in which a proposition of the form p·∼p can be derived from an impossible proposition, 19.75 holds. But it is not obvious that there are no other kinds of impossible propositions.

Fortunately it is easy to see what is the origin of the paradoxes. They are introduced into the system by the definition

11.02 p ⊰ q · = ˜ ⋄(p · ˜q).

Assume this definition. Then if q is necessary, p·∼q is impossible, and so, in accordance with 11.02, p⊰q is valid.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 4 , Issue 2 , June 1939 , pp. 73 - 76
Copyright
Copyright © Association for Symbolic Logic 1939

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References

1 C. I. Lewis, A survey of symbolic logic; Lewis and Langford, Symbolic logic (abbreviated S. L.).

2 These numbers are those of the propositions in S. L.

3 S. L., p. 248.

4 S. L., p. 250.

5 S. L., p. 124.

6 S. L., p. 244.

7 S. L., pp. 124–126, 166.

8 These propositions are: 11.02; the third members of 18.1–18.14; the third member of 18.2; the fourth members of 18.3–18.31; 18.61 (if in 18.61 p is substituted for r, then it is possible to deduce the necessary proposition p from an arbitrary proposition q); 18.7; 19.47; the third member of 19.72; 19.73–19.77; 19.83–19.84. Furthermore, it is necessary to replace the definition 17.01 by p∘q. = ◇pq.

17.51 and 17.52 are asserted in another form, namely,

There are no so-called T-principles (S. L., p. 147) in the system N. The T-principles of S. L. are all asserted without T.

9 S. L., p. 127 (12.1) and p. 133 (12.81).

10 S. L., p. 493, group I.

11 S. L., p. 497.

12 S. L., p. 497.