Almost (MP)-based substructural logics

This paper is a contribution to the theory of substructural logics. We introduce the notions of (MP)-based and almost (MP)-based logics (w.r.t. a special set of formulae D), which leads to an alternative proof of the well-known forms of the local deduction theorems for prominent substructural logics (FL, FLe, FLew , etc.). Roughly speaking, we decompose the proof of the local deduction theorem into the trivial part, which works almost classically, and the non-trivial part of determining with respect to which set (if any) the logic is almost (MP)-based. We can also show connection of (almost) (MP)-based condition and the proof by cases properties for generalized disjunctions and the description of (deductive) filters generated by some elements of a given algebra. In order to provide as general theory as possible, i.e., to cover more logics than the usual Ono’s definition of substructural logics [6] (i.e. axiomatic extensions of the logic of pointed residuated lattices) we propose a more general notion of substructural logic based on a very weak system lacking not only structural rules, but also associativity of multiplicative conjunction, and consider all its (even non-axiomatic) extensions, expansions (by new connectives), and well-behaved fragments thereof. This defines a wide family of logical systems containing pretty much all prominent substructural logics. Our basic logic will be the non-associative variant for the Full Lambek Calculus [6, 7], here denoted as SL. Its language, LSL, consists of residuated conjunction &, right % and left $ residual implications, lattice conjunction ∧ and disjunction ∨, and truth constants 0, 1. The logic SL is given by the following axiomatic system: ⊢ φ%φ φ,φ%ψ ⊢ ψ φ ⊢ (φ%ψ)%ψ φ%ψ ⊢ (ψ%χ)%(φ%χ) ψ%χ ⊢ (φ%ψ)%(φ%χ) ⊢ φ % ((ψ $ φ) % ψ) φ % (ψ % χ) ⊢ ψ % (χ $ φ) ψ $ φ ⊢ φ % ψ ⊢ φ ∧ ψ % φ ⊢ φ ∧ ψ % ψ φ,ψ ⊢ φ ∧ ψ ⊢ (χ % φ) ∧ (χ % ψ) % (χ % φ ∧ ψ) ⊢ φ%φ∨ψ ⊢ ψ%φ∨ψ ⊢ (φ%χ)∧ (ψ%χ)% (φ∨ψ%χ) ⊢ (χ$φ)∧ (χ$ψ)% (χ$φ∨ψ) ⊢ ψ % (φ % φ& ψ) ψ % (φ % χ) ⊢ φ& ψ % χ ⊢ 1 ⊢ 1 % (φ % φ) ⊢ φ % (1 % φ) Definition 1 A logic L in a language L containing % is a substructural logic if • L is the expansion of the L ∩ LSL-fragment of SL. • for each n, i < n, and each n-ary connective c ∈ L \ LSL holds: φ % ψ, p % φ ⊢L c(χ1, . . . χi, φ, . . . , χn) % c(χ1, . . . χi, ψ, , . . . , χn)