Strict implication
Strict implication
As early as 1912, C. I. Lewis projected a kind of implication between propositions, to be called strict implication, which should more nearly accord with the usual meaning of “implies” than does material implication (see logic, formal, 1). should make “p implies q” synonymous with “q is deducible from p,” and should avoid such so-called paradoxes of material implication as the theorem [p ? q] ? [q ? p]. The first satisfactory formulition of a calculus of propositions with strict implication appeared in 1920, and this system, and later modified forms of it. have since been extensively investigated. An essential feature is the introduction of modalities through the notation (say) M[p], to mean “p is possible” (Lewis uses a diamond instead of M). The strict implication of q by p is then identified with ~M[p ~q], whereas the material implication p ? q is given by ~[p ~q]. In 1932 Lewis, along with other modifications, added a primitive formula (involving the binding of propositional variables by existential quantifiers) which renders definitively impossible an interpretation of the system which would make Mp the same as p and strict implication the same as material implication. Consistency of the system, including this additional primitive formula, may be established by means of an appropriate four-valued propositional calculus, the theorems of the system being some among the tautologies of the four-valued propositional calculus. — A.C.
Lewis and Langford,
Symbolic Logic, New York and London, 1932.
F. V. Huntington,
Postulates for assertive conjunction, negation, and equality, Proceedings of the American Academy of Arts and Sciences, vol. 72, no. 1, 1937.
W. T. Parry,
Modalities in the Survey system of strict implication, The Journal of Symbolic Logic, vol. 4 (1939). pp 137-154.