This chapter constructs an intensional language called Lω. The semantics for Lω requires a new semantic method, one which harks back to the work of Boole, Peirce, and Schroder. This algebraic semantic method does not appeal to possible worlds even as a heuristic. The heuristic that is used is simply that of properties, relations, and propositions, taken at face value, and fundamental logical operations on properties, relations, and propositions. Using this new algebraic method, the chapter defines two notions of validity, one for the first traditional conception of intensional entities and one for the second traditional conception. Then, surprisingly as it might seem, the logics for Lω relative to these two notions of validity are found to be both sound and complete. This way, two complete theories of PRPs are obtained, one ideally suited for modal matters and the other for intentional matters.
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