Proof theory for lattice-ordered groups

Proof theory can provide useful tools for tackling problems in algebra. In particular, Gentzen systems admitting cut-eliminationhave been used to establish decidability, complexity, amalgamation, admissibility, and generation results for varieties of residuated lattices corresponding to substructural logics. However, for classes of algebras bearing some family resemblance to groups, such as lattice-ordered groups, MV-algebras, BL-algebras, and cancellative residuated lattices, the proof-theoretic approach has met so far only with limited success.The main aim of this talk will be to introduce proof-theoretic methods for the class of lattice-ordered groups and to explain how these methods can be used to obtain new syntactic proofs of two core theorems: namely, Holland's result that this class is generated as a variety by the lattice-ordered group of order-preserving automorphisms of the real numbers, and the decidability of the word problem for free lattice-ordered groups.

Substructural fuzzy logics

Substructural fuzzy logics are substructural logics that are complete with respect to algebras whose lattice reduct is the real unit interval [0, 1]. In this paper, we introduce Uninorm logic UL as Multiplicative additive intuitionistic linear logic MAILL extended with the prelinearity axiom ((A → B) ∧ t) V ((B → A)∧ t). Axiomatic extensions of UL include known fuzzy logics such as Monoidal t-norm logic MIX and Gödel logic G, and new weakening-free logics. Algebraic semantics for these logics are provided by subvarieties of (representable) pointed bounded commutative residuated lattices. Gentzen systems admitting cut-elimination are given in the framework of hypersequents. Completeness with respect to algebras with lattice reduct [0, 1] is established for UL and several extensions using a two-part strategy. First, completeness is proved for the logic extended with Takeuti and Titani's density rule. A syntactic elimination of the rule is then given using a hypersequent calculus. As an algebraic corollary, it follows that certain varieties of residuated lattices are generated by their members with lattice reduct [0, 1].

Correspondences between gentzen and hilbert systems

Most Gentzen systems arising in logic contain few axiom schemata and many rule schemata. Hilbert systems, on the other hand, usually contain few proper inference rules and possibly many axioms. Because of this, the two notions tend to serve different purposes. It is common for a logic to be specified in the first instance by means of a Gentzen calculus, whereupon a Hilbert-style presentation ‘for’ the logic may be sought—or vice versa. Where this has occurred, the word ‘for’ has taken on several different meanings, partly because the Gentzen separator ⇒ can be interpreted intuitively in a number of ways. Here ⇒ will be denoted less evocatively by ⊲.In this paper we aim to discuss some of the useful ways in which Gentzen and Hilbert systems may correspond to each other. Actually, we shall be concerned with the deducibility relations of the formal systems, as it is these that are susceptible to transformation in useful ways. To avoid potential confusion, we shall speak of Hilbert and Gentzen relations. By a Hilbert relation we mean any substitution-invariant consequence relation on formulas—this comes to the same thing as the deducibility relation of a set of Hilbert-style axioms and rules. By a Gentzen relation we mean the fully fledged generalization of this notion in which sequents take the place of single formulas. In the literature, Hilbert relations are often referred to as sentential logics. Gentzen relations as defined here are their exact sequential counterparts.