strict implication, deducibility and the deduction theorem

Strict Implication, Deducibility and the Deduction TheoremAuthor(s): Ruth Barcan MarcusSource: The Journal of Symbolic Logic, Vol. 18, No. 3 (Sep., 1953), pp. 234-236Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2267407 .

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THE JOURNAL OF SYMBOLIC LOGIC

Volume 18, Number 3, Sept. 1953

STRICT IMPLICATION, DEDUCIBILITY AND THE DEDUCTION THEOREM

RUTH BARCAN MARCUS

Lewis and Langford' state, "... it appears that the relation of strict implication expresses precisely that relation which holds when valid deduction is possible. It fails to hold when valid deduction is not possible. In that sense, the system of strict implication may be said to provide that canon and critique of deductive inference which is the desideratum of logical investigation." Neglecting for the present other possible criticisms of this assertion, it is plausible to maintain that if strict implication is intended to systematize the familiar concept of deducibility or entailment, then some form of the deduction theorem should hold for it. The purpose of this paper i's to analyze and extend some results previously established2 which bear on the problem.

We will begin with a rough statement of some relevent considerations. Let the system S contain among its connectives an implication connective 'I' and a conjunction connective '&'. Let A1, A2 .I., A ,n F B abbreviate that B is provable on the hypotheses Al, A2, ..., An for a suitable def- inition of "proof on hypotheses", where A1, A2, . ., An B are well-formed expressions of S.

Three deduction theorems for S would be:

I. If Al, A2) . . ., A,, F B then Al. A2) . . *, An-l F An I B. II. If A1, A2, ..., An F B then (A1 & A2& ... & An)I B.

III. If A1 F B then A I B.

It is well known that for the system of material implication, a theorem corresponding to I, and consequently to II and III is provable. For the Lewis systems3 this is not the case. It was established4 that for S2, no theorem is available corresponding to either I, II or III. More explicitly (since the Lewis systems involve two implication connectives), neither of

Received November 5, 1951. 1 C. I. LEWIS and C. H. LANGFORD, Symbolic logic, New York and London, 1932,

p. 247. 2 The deduction theorem in a functional calculus of first order based on strict implication,

this JOURNAL, vol. 11 (1946), pp. 115-118. The results of this paper were obtained for functional extensions of the Lewis systems S2 and S4. The results of the present paper also obtain for the corresponding functional extensions of the Lewis systems under discussion.

3 See LEWIS and LANGFORD, op. cit., particularly Appendix II. 4 For this and subsequent references to what has been established with respect to

the deduction theorem in S2 and S4, see my paper, op. cit., on the deduction theorem.

234



STRICT IMPLICATION AND THE DEDUCTION THEOREM 235

the following is provable:

1. If Al, A2, .. ., An F B then Al, A2,**., An1l F An D B. 2. If Al, AV . . . An F B then Al, A2 **, Anl F An - B.

Since S2 is an extension of SI, the above establishes the non-provablility of a deduction theorem for both of these systems. It appears to the author that in view of Lewis and Langford's proposed analysis, a theorem corresponding to III should (at the least) be available. It would be a curious explication of the concept of deducibility if, although B followed from the premise A, B could not be said to be deducible from A.

For S4 it was established that 1 holds, and 2 holds if the following condition is satisfied:5

3. A_ 0,-H 1, A ---oH2.*** An-~

The above proof may also be paralleled for S5 which is an extension of S4. That 2 does not hold unconditionally for S4 and S5 can be shown by a matrix of Wajsberg6 which satisfies the axioms and rules of S4 and S5. Although A, A D B F B can be shown in S4 and S5, A -3 ((A D B) -q B) does not always have a designated value.

These results as they stand are perhaps misleading. From them Rosen- bloom7 concludes that "The contention that from the standpoint of the interpretation as deducibility, 'strict' implication is a more satisfactory operation than 'material' implication is consequently untenable until a system based on the former is constructed in which the deduction theorem is proved... ". In our previous paper, it was not made clear whether theorems corresponding to II or III were provable for S4 (and consequently S5), which indeed they are.8 This is established by using the following two theorems of S4:

4. If A is provable then N is provable.9 5. (Alh :: . . (An-l :: (An d B) ............. )-((Al. A2 .....An)

v B).

From 1 and 4 we have:

6. If A-B then A- B.

5 As suggested originally by the referee, this condition could have been weakened to

A1 > H1, A2 H2* An-l Hn-v 6 See LEwIs and LANGFORD, op. cit., the Group III matrix on p. 493. 7 PAUL ROSENBLOOM, The elements of mathematical logic, New York, 1950,

p. 60. 8 A theorem corresponding to III for S5 was established by R. CARNAP in Modalities

and quantification, this JOURNAL, vol. 11 (1946), p. 56. 9 See J. C. C. MCKINSEY and ALFRED TARSKI, Some theorems about the sentential

calculi of Lewis and Heyting, this JOURNAL, vol. 13 (1948), p. 5.



236 RUTH BARCAN MARCUS

From 1, 4 and 5 we have:

7. If A1, A2, . . ., An F B then (Al 3A23 ... An) i B.

The weakness of the deduction theorem for S4 and S5 lies in that we cannot prove 2 unconditionally for n> 1. Whether the absence of 2 makes S4 or S5 inadequate for a systematization of the concept of deducibility, as Rosen- bloom contends, would depend on a more detailed pre-systematic analysis of the concept. We hope to discuss this elsewhere at a later time.

NORTHWESTERN UNIVERSITY



strict implication, deducibility and the deduction theorem

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