Logic of Ground

Origin. Kit Fine, "Guide to ground" and "The pure logic of ground" (2012); Correia, "Grounding and truth-functions" (2010) and later work; Schnieder, deRosset. The formal side of the metaphysical grounding literature: a consequence relation for "because" in its explanatory rather than causal sense, where the disjunction is grounded in the disjunct and not the reverse.

Models. Grounding is explanatory determination between facts: the conjunction holds because both conjuncts do. It is not entailment — entailment is reflexive and grounding is not, and every necessary truth is entailed by everything while nothing grounds a necessary truth indiscriminately. It is not causation — causes precede, grounds do not. The logic asks what structural principles this relation obeys once it is not reduced to either.

Formalism.

The relations: Γ < φ full strict ground: the Γ together fully account for φ Γ ≼ φ full weak ground: allows φ ∈ Γ γ ≺ φ partial strict ground: γ is one of several Strict grounds are irreflexive; weak grounds are reflexive.

Structural principles (Fine's pure logic of ground): Non-circularity: not φ < φ Transitivity (cut): if Γ < φ and Δ, φ < ψ then Γ, Δ < ψ Amalgamation: if Γ < φ and Δ < φ then Γ, Δ ≼ φ Subsumption: Γ < φ implies Γ ≺ φ for each γ ∈ Γ

Introduction rules — grounds by connective: φ, ψ < φ ∧ ψ φ < φ ∨ ψ and separately ψ < φ ∨ ψ (disjunction has two grounds, not one; the pair is not a ground) φ(a) < ∃x φ(x) {φ(a) : a ∈ D} < ∀x φ(x)

The asymmetry: φ ∧ ψ entails φ and φ entails φ ∨ ψ. Grounding runs the other way for conjunction and the same way for disjunction — so grounding is not entailment reversed, and neither direction reduces to the other.

Truthmaker semantics for ground (Fine 2012, 2017): State space (S, ⊑) with fusion. Γ < φ iff every fusion of exact verifiers of Γ exactly verifies φ, and no proper part does. The semantics is the same apparatus as exact entailment; the difference is the constraint.

Correia's variant: Grounding as a relation between propositions closed under a factual equivalence, which validates different principles — notably about negation, where Fine and Correia part.

Symbols.

Metatheory. The pure logic of ground is the fragment with no connectives — just the structural principles — and Fine proves it complete for a class of models built from the amalgamation and subsumption conditions. Adding the introduction rules gives the impure logic, whose completeness depends on the truthmaker semantics chosen. Non-circularity is the axiom under attack: putative counterexamples (a fact grounding itself via an infinite descent, or Fine's own "zero-grounded" facts) have generated a literature on whether the relation is well-founded, and the logic assumes rather than argues that it is.

Applies to. Metaphysical explanation and the reduction debates it replaced. Truth-grounding and the semantic paradoxes, where the Liar's ground is its own ungroundedness. Essence, on Fine's account, where essence and ground are the two hyperintensional primitives. Fundamentality and the structure of levels.

Limitations. Whether grounding is one relation or many is unsettled, and the logic presupposes an answer — Fine's strict/weak/full/partial quadrant is a stipulation the metaphysics does not force. The principles are defended by intuitions about cases, and the cases are contested at every point where the systems differ, so the disagreement between Fine and Correia over negation has no formal adjudicator. And the semantics inherits truthmaker semantics' burdens: the state space is posited, not derived, and nothing fixes which states there are.

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