Abstract
The purpose of this chapter is to extend the goal-directed proof methods to strict implication modal logics. We consider this as a first step in order to extend the goal-directed paradigm to the realm of modal logics. Strict implication, denoted by A ⇒ B is read as ‘necessarily A implies B’. The notion of necessity (and the dual notion of possibility) are the subject of modal logics. Strict implication can be regarded as a derived notion: A ⇒ B = □(A → B),where → denotes material implication and □ denotes modal necessity. However, strict implication can also be considered as a primitive notion, and has already been considered as such at the beginning of the century in many discussions about the paradoxes of material implication [Lewis, 1912; Lewis and Langford, 1932].
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