ABSTRACT
Bertrand Russell distinguishes material implication from formal implication in Principles, Principia, and Introduction to Mathematical Philosophy. For propositional functions, he distinguishes among the necessary, the possible, and the impossible. He also distinguishes both material implication and formal implication from deductively valid inference. Russell’s construction of the physical world is avowedly Leibnizian; it is constructed from both actual and theoretically ideal private worlds of monadic consciousnesses, that is, from the perspectives of both actual and theoretically possible observers. Nicholas Rescher says that “in mathematics it is altogether otiose to differentiate between the actual and the necessary, and there is no room at all for the contingently possible”. To say that Russell rejects modality when he makes each modality a property of propositional functions is just as absurd as saying that he rejects existence when he makes existence a property of propositional functions.
