False though partly true – an experiment in logic

Journal of Philosophical Logic 32 (6):613-665 (2003)
Abstract
We explore in an experimental spirit the prospects for extending classical propositional logic with a new operator P intended to be interpreted when prefixed to a formula as saying that formula in question is at least partly true. The paradigm case of something which is, in the sense envisaged, false though still "partly" true is a conjunction one of whose conjuncts is false while the other is true. Ideally, we should like such a logic to extend classical logic - or any fragment thereof under consideration - conservatively, to be closed under uniform substitution (of arbitrary formulas for sentence letters or propositional variables), and to allow the substitutivity of provably equivalent formulas salva provabilitate. To varying degrees, we experience some difficulties only with this last ('congruentiality') desideratum in the two four-valued logics we end up giving our most extended consideration to
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