Lukasiewicz's Many-valued Logic and Neoplatonic Scalar Modality

History and Philosophy of Logic 23 (2):95-120 (2002)
This paper explores the modal interpretation of ?ukasiewicz's n -truth-values, his conditional and the puzzles they generate by exploring his suggestion that by ?necessity? he intends the concept used in traditional philosophy. Scalar adjectives form families with nested extensions over the left and right fields of an ordering relation described by an associated comparative adjective. Associated is a privative negation that reverses the ?rank? of a predicate within the field. If the scalar semantics is interpreted over a totally ordered domain of cardinality n and metric ?, an n-valued Lukasiewicz algebra is definable. Privation is analysed in terms of non-scalar adjectives. Any Boolean algebra of 2 n ?properties? determines an n + 1 valued Lukasiewicz algebra. The Neoplatonic ?hierarchy of Being? is essentially the order presupposed by natural language modal scalars. ?ukasiewicz's ≈ is privative negation, and ? proves to stand for the extensional (antitonic) dual if ? then for scalar adjectives, especially modals. Relations to product logics and frequency interpretations of probability are sketched
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DOI 10.1080/01445340210154330
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John N. Martin (2001). Proclus and the Neoplatonic Syllogistic. Journal of Philosophical Logic 30 (3):187-240.

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