First-order Nilpotent Minimum Logics: first steps

Details First-order Nilpotent Minimum Logics: first steps First-order Nilpotent Minimum Logics: first steps

2011-03-30

arxiv.org/abs/1103.6025v1

Mathematics

Logic

18 pages

Scientific paper

First-order Nilpotent Minimum Logic was introduced in [EG01]; in [Gis03] it is showed that every finite NM-chain with negation fixpoint is complete w.r.t. the logic NM. In this paper we will show that this last result, in the first-order case, does not hold. We will study the sets of first-order tautologies of some subalgebras of [0,1]_NM: in particular finite NM-chains and other four infinite NM-chains (with and without negation fixpoint). Moreover we will find a connection between the validity, in an NM-chain, of certain first-order formulas and its order type. Finally, we will analyze axiomatization, undecidability and the monadic fragments. We will conclude with some remarks, future directions of research and open problems. This paper has been inspired by the work done, for first-order G\"odel logic, in [BPZ07] and [BCF07]: when possible, we will point out the analogies and the differences with the G\"odel's case.

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