Ruth Barcan Marcus. Strict implication, deducibility, and the deduction theorem. The journal of symbolic logic, vol. 18 (1953), pp. 234–236.

1954 ◽  
Vol 19 (4) ◽  
pp. 294-294
Author(s):  
P. G. J. Vredenduin
Keyword(s):  
Symbolic Logic ◽  
Deduction Theorem ◽  
Strict Implication ◽  
Barcan Marcus

The deduction theorem in a functional calculus of first order based on strict implication

1946 ◽  
Vol 11 (4) ◽  
pp. 115-118 ◽  
Author(s):  
Ruth C. Barcan
Keyword(s):  
Functional Calculus ◽  
Symbolic Logic ◽  
Deduction Theorem ◽  
Strict Implication ◽  
First Order ◽  
Fundamental Theorems

In a previous paper, a functional calculus based on strict implication was developed. That system will be referred to as S2. The system resulting from the addition of Becker's axiom will be referred to as S4. In the present paper we will shw that a restricted deduction theorem is provable in S4 or more precisely in a system equivalent to S4. We will also show that such a deduction theorem is not provable in S2.The following theorems not derived in Symbolic logic will be required for the fundamental theorems XXVIII* and XXIX* of this paper. We will state most of them without proofs.

Strict implication, deducibility and the deduction theorem

1953 ◽  
Vol 18 (3) ◽  
pp. 234-236 ◽  
Author(s):  
Ruth Barcan Marcus
Keyword(s):  
Deductive Inference ◽  
Logical Investigation ◽  
Deduction Theorem ◽  
Strict Implication ◽  
Suitable Definition ◽  
Definition Of

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 is to analyze and extend some results previously established 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, …, An ⊦ B abbreviate that B is provable on the hypotheses A1, A2, …, An for a suitable definition of “proof on hypotheses”, where A1, A2, …, An, B are well-formed expressions of S.

Modalities in the Survey system of strict implication

1939 ◽  
Vol 4 (4) ◽  
pp. 137-154 ◽  
Author(s):  
William Tuthill Parry
Keyword(s):  
Finite Number ◽  
Symbolic Logic ◽  
Strict Implication ◽  
Survey System ◽  
Image Position

Professor C. I. Lewis, in Lewis and Langford's Symbolic logic, designates the system (S2) determined by the postulates used in Chapter VI—namely, 11.1–7 (B1–7) andas the system of strict implication. For certain reasons, he prefers it to either the earlier system (S3) determined by the stronger set of postulates of his Survey of symbolic logic, as emended, namely, A1–7 andor the system (S1) determined by the weaker set of postulates B1–7 or A1–7.But Lewis and others, following O. Becker, have also given consideration to systems which contain some additional principle effecting the reduction of complex modalities to simpler ones. Notable are the system (S4) determined by B1–7 pluswhich includes (is stronger than) S3; and the system (S5) determined by B1–7 pluswhich includes S4.It seems worth while to investigate further the system of the Survey (emended), S3, which is intermediate between S2 and S4. This is the purpose of the present paper. We first prove some additional theorems in S2 and S3. These enable us to reduce all the complex modalities in S3 to a finite number, viz. 42; and it is shown that no further reduction is possible. Finally, several systems which include S3 are considered.

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