Lukasiewicz's Many-valued Logic and Neoplatonic Scalar Modality
Titel:
Lukasiewicz's Many-valued Logic and Neoplatonic Scalar Modality
Auteur:
Martin, John N.
Verschenen in:
History and philosophy of logic
Paginering:
Jaargang 23 (2002) nr. 2 pagina's 95-120
Jaar:
2002-06-01
Inhoud:
This paper explores the modal interpretation of Lukasiewicz'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 θ, an n-valued Lukasiewicz algebra <C, a¯ , u¯, `, θ, e > 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. Lukasiewicz'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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