Continuing to pursue a research direction that we already explored in connection with Gödel-Dummett logic and Ruspini partitions, we show here that Lukasiewicz logic is able to express the notion of pseudo-triangular basis of fuzzy sets, a mild weakening of the standard notion of triangular basis. En route to our main result we obtain an elementary, logic-independent characterisation of triangu...

We present a new method for deciding Gödel-Dummett logic LC. We first characterize the validity of irreducible sequents of LC by the existence of r-cycles in bi-colored graphs and we propose a linear algorithm to detect r-cycles and build counter-models. Then we characterize the validity of formulae by the existence of r-cycles in boolean constrained graphs. We also give a parallel method to de...

A first-order variable (‘x’, ‘y’, . . . ) is a variable that takes the position of singular terms. Accordingly, in a first-order language one can replace the singular term ‘Bucephalus’ in ‘Horse(Bucephalus)’ by a first order variable ‘x’ and get a well-formed formula: 1For further discussion of Frege, see Dummett (1981) (especcially p. 212 ff.), Goldfarb (1997), Wright (1998), Noonan (2006), Ha...

It is proved that the variety of representable idempotent commutative residuated lattices is locally finite. The n-generated subdirectly irreducible algebras in this variety are shown to have at most 3n+1 elements each. A constructive characterization of the subdirectly irreducible algebras is provided, with some applications. The main result implies that every finitely based extension of posit...

Let \(A_{0},A_{1},...,A_{n}\) be (possibly) distintict wffs, \(n\) being an odd number equal to or greater than 1. Intuitionistic Propositional Logic IPC plus the axiom \((A_{0}\rightarrow A_{1})\vee ...\vee (A_{n-1}\rightarrow A_{n})\vee (A_{n}\rightarrow A_{0})\) is equivalent Gödel-Dummett logic LC. However, if even 2, said a sublogic of

Thoughts are essentially structured. That much is agreed by almost everybody who finds it useful to talk about thoughts. Consequently, any account of the nature of thoughts must incorporate an account of their structure. Frege's is the most comprehensive and worked-out account of the nature of thoughts, but recent work has cast doubt on whether he can be credited with a coherent conception of h...

It is well established that various proteins, as gelatin and albumin, can be adsorbed on various adsorbents so that the proteincoated particle of adsorbent behaves electrophoretically as a particle of protein (Loeb, 1923; Freundlich and Abramson, 1928). Dummett and Bowden (1933) have recently reported, however, that the behavior of adsorbed hemoglobin varies with the adsorbent surface. Thus, wh...

Philosophical theories of logical reasoning are intrinsically related to formal models. This holds in particular of Dummett–Prawitz-style proof-theoretic semantics and calculi of natural deduction. Basic philosophical ideas of this semantic approach have a counterpart in the theory of natural deduction. For example, the “fundamental assumption” in Dummett’s theory of meaning (Dummett, 1991, p. ...