Lattice BCK logic is the expansion of the well known Meredith implicational logic BCK expanded with lattice conjunction and disjunction. Although its natural axiomatization has three rules named modus ponens, ∨‐rule and ∧‐rule, we show that we can give an equivalent presentation with just modus ponens and ∧‐rule, however it is impossible to obtain an equivalent presentation with modus ponens as unique rule. In this paper we study and characterize all axiomatic extensions of lattice BCK logic with modus ponens as (...) unique rule. We obtain an infinite chain of proper axiomatic extensions with this property. Moreover, we prove that there is no weakest axiomatic extension of Lattice BCK‐logic admitting modus ponens as unique rule. (shrink)

In the literature on vagueness one finds two very different kinds of degree theory. The dominant kind of account of gradable adjectives in formal semantics and linguistics is built on an underlying framework involving bivalence and classical logic: its degrees are not degrees of truth. On the other hand, fuzzy logic based theories of vagueness—largely absent from the formal semantics literature but playing a significant role in both the philosophical literature on vagueness and in the contemporary logic literature—are logically nonclassical (...) and give a central role to the idea of degrees of truth. Each kind of degree theory has a strength: the classical kind allows for rich and subtle analyses of the comparative form of gradable adjectives and of various types of gradable precise adjectives, while the fuzzy kind yields a compelling solution to the sorites paradox. This paper argues that the fuzzy kind of theory can match the benefits of the classical kind and hence that the burden is on the latter to match the advantages of the former. In particular, we develop a new version of the fuzzy logic approach that—unlike existing fuzzy theories—yields a compelling analysis of the comparative as well as an adequate account of gradable precise predicates, while still retaining the advantage of genuinely solving the sorites paradox. (shrink)