Lax Logic

Origin. Haskell Curry found the modality first: his 1948 Notre Dame lectures, published as A Theory of Formal Deducibility (1957), contain a modality with axiom schemata essentially those below, refined in "The elimination theorem when modality is present" (Journal of Symbolic Logic 17, 1952, pp. 249–265), which proves cut elimination with the modality present; the sequent rules below are the ones Curry suggested. Robert Goldblatt, "Grothendieck topology as geometric modality" (Mathematical Logic Quarterly 27, 1981, pp. 495–529), rediscovered it as "it is locally the case that", with nuclei on Heyting algebras as its algebraic semantics. Michael Mendler's Edinburgh thesis (1993) found it a third time in hardware verification, as correctness up to constraints, and Matt Fairtlough and Mendler, "Propositional Lax Logic" (Information and Computation 137, 1997, pp. 1–33), named the logic and consolidated its proof theory and Kripke semantics. Benton, Bierman and de Paiva, "Computational types from a logical perspective" (Journal of Functional Programming 8(2), 1998, pp. 177–193), proved the fourth reading: under Curry–Howard the logic is Moggi's computational lambda calculus.

Models. Intuitionistic propositional logic with one modality ○ that is possibility-like and necessity-like at once: ○R and ○M are axioms typical of ◇, ○S is typical of □, and no classical modality combines them — adding excluded middle and ¬○false derives both ○M ⊃ M and M ⊃ ○M, so the modality exists only over an intuitionistic base. Its content is one structure found four times: a proof calculus in Curry, the nucleus and the local truth of Grothendieck topologies in Goldblatt, a constraint discipline in Mendler's hardware verification, and the strong monad of Moggi's computational metalanguage. Lax logic is the logic every strong monad satisfies, which makes it the propositional theory of monadic computation (Monads).

Formalism.

Language: M ::= A | M ∧ M | M ∨ M | M ⊃ M | ¬M | ○M (constants true, false; ¬M abbreviates M ⊃ false)

Axioms (Fairtlough–Mendler): ○R : M ⊃ ○M ○M : ○○M ⊃ ○M ○S : (○M ∧ ○N) ⊃ ○(M ∧ N) with modus ponens and the rule: from M ⊃ N infer ○M ⊃ ○N. Over IPC a single scheme suffices: (N ⊃ ○K) ≡ (○N ⊃ ○K).

Sequent rules (at most one formula on the right):

Γ ⊢ M ─────── ○R Γ ⊢ ○M

Γ, M ⊢ ○N ─────────── ○L Γ, ○M ⊢ ○N

Kripke constraint models: C = (W, R_m, R_i, V, F): two preorders with R_m ⊆ R_i; F ⊆ W the fallible worlds, hereditary and validating every formula; V hereditary and full on F. w ⊨ ○N iff for all v with w R_i v there is a u with v R_m u and u ⊨ N. The ∀∃ clause is the two-sided character: necessity over the intuitionistic relation, possibility over the constraint relation. R_m ⊆ R_i is the only frame condition.

Constraint reading: ○M = "M holds under some constraint". Timing analysis of combinational circuits interprets proofs in the delay algebra (ℕ, 0, +, max): ○R is a zero-delay wire, ○M sequential composition (+ of delays), ○S parallel composition (max of delays).

Nucleus reading: ○ is a nucleus j on a Heyting algebra: a ≤ j a, j j a = j a, j(a ∧ b) = j a ∧ j b. On the subobject classifier Ω of a topos this is a Lawvere–Tierney topology, and ○M reads "locally M" (Topos Logic, Sheaf Semantics).

Monadic reading: ○ is a strong monad on propositions; a proof of ○A is a computation returning a proof of A. The term calculus is Moggi's computational lambda calculus λ_c: ○R the unit, ○M the multiplication, ○S the strength (Benton–Bierman–de Paiva 1998, who call the logic CL).

Symbols.

Metatheory. Fairtlough and Mendler (1997) proved cut elimination for the Gentzen system (independently Benton, Bierman and de Paiva), with the standard consequences: the disjunction property, the subformula property, and the admissibility of the converse necessitation rule — from ○M infer M; strong conservativity says more, that erasing every ○ from a PLL theorem leaves an IPC theorem. The logic is decidable with the finite model property, a result Goldblatt (1981) had obtained semantically. The deduction theorem holds, which separates ○ from the necessity of K, T and S4, where the Hilbert presentations refuse it. Completeness holds for the constraint models, with a semantics-preserving translation into Goldblatt's J-space models, and yields a full and faithful Gödel-style embedding of PLL into the classical bimodal logic (S4, S4) — the classical explanation of a modality classical logic cannot host directly. Neither ¬○false nor ○(M ∨ N) ⊃ (○M ∨ ○N) is a theorem, and the fallible worlds exist to falsify the first: a fallible world validates everything, so a constraint leading only to fallible worlds witnesses ○false without inconsistency. On the algebraic side Goldblatt proved completeness for Heyting algebras with a nucleus — his local algebras — tying the logic to the Lawvere–Tierney topologies of Topos Logic and the covers of Sheaf Semantics; on the computational side Benton, Bierman and de Paiva proved strong normalization and confluence for the term calculus, so lax proofs compute.

Applies to. Hardware verification: correctness up to timing constraints, with proof terms carrying delay information. The logic of monadic computation: Moggi's metalanguage, monadic types in functional languages, the logical reading of effect systems (Monads, Algebraic Effects). Topos theory and locale theory: Lawvere–Tierney topologies, nuclei, sheafification, local truth (Topos Logic, Sheaf Semantics). Authorization logics, which read a principal's says as a lax modality. The monadic type theories of Type/Modal, where ○ is a type former with introduction and elimination rules.

Limitations. The modality has no classical life: with excluded middle and ¬○false it collapses to the identity, so there is no classical Kripke semantics for ○ and every classical account goes through the bimodal embedding. It has no dual either: a □ defined over the two relations yields nothing new, because R_m ⊆ R_i already collapses it, so lax logic is irreducibly a one-modality logic. As the logic of monadic computation it is generic — it holds of every strong monad and therefore distinguishes none: no equations between effects, no combination of two monads, no grading, which is where the algebraic-effects and graded-monad refinements take over (Algebraic Effects). And the calibration is propositional: decidability and the finite model property do not simply lift to quantifiers, and first-order lax logic required new cover machinery (Goldblatt, "Cover semantics for quantified lax logic", Journal of Logic and Computation 21(6), 2011).

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