Multiplying logical values

Abstract

The modern history of many-valuedness starts with Lukasiewicz’s construction of three-valued logic. This pioneering, philosophically motivated and matrix based construction, first presented in 1918, was in 1922 extended to n-valued cases, including two infinite ones. Soon several constructions of many-valued logic appeared and the history of the topic became rich and interesting. However, as it is widely known, the problem of interpretation of multiple values is still among vexed questions of contemporary logic. With the paper, which essentially groups my earlier settlements, from [3], [4], [7] and [8], I intend to put a new thread into discussion on the nature of logical many-valuedness. The topics, touched upon, are: matrices, tautological and non-tautological many-valuedness, Tarski’s structural consequence and the Lindenbaum–Wojcicki completeness result, which supports the Suszko’s claim on logical two-valuedness of any structural logic. Consequently, two facets of many-valuedness — referential and inferential — are unravelled. The first, fits the standard approach and it results in multiplication of semantic correlates of sentences, and not logical values in a proper sense. The second many-valuedness is a metalogical property of inference and refers to partition of the matrix universe into more than two disjoint subsets, used in the definition of inference.