Containment Logic

Origin. Angell's analytic containment (1977, 1989) and Parry's earlier proscriptive principle are the ancestors. The modern systems: Fine's truthmaker semantics for exact entailment and containment (2016, 2017); Correia's account of factual equivalence and containment (2004, 2010); Yablo's Aboutness (2014), where containment is subject-matter inclusion. Ferguson's Meaning and Proscription in Formal Logic (2017) surveys the family.

Models. A contains B when B's content is part of A's. Not just truth preservation. Intensional structure matters. Relates to grounding and essence.

Formalism.

Containment relation: A ≤ B: A's content contained in B's. Partial order on propositions. Not same as entailment. Structural, not just modal.

Truthmaker containment: What makes A true contains what makes B true. Truthmaker of disjunction ≤ truthmaker of disjuncts. Compositional structure.

Yablo's aboutness: Proposition about certain subject matters. A ⊃ B only if A, B share subject matter. Irrelevance blocked. Content-based implication.

Exact truthmaking: State s exactly makes A true. Not too much, not too little. Precise fit between truthmaker and truth. No extraneous content.

Fine's state space: States (situations) as truthmakers. States can fuse, factor. Bilateral semantics. Part-whole structure.

Subject matter: m(A) = what A is about. A → B requires m(B) ⊆ m(A ∧ B). Connection requirement. Aboutness preservation.

Comparison with relevance: Both require connection. Containment: mereological. Relevance: variable sharing. Different metaphors.

Symbols.

Metatheory. Containment is not entailment, and the divergence is exactly where two classical principles fail. Disjunction introduction fails: A entails A ∨ B but does not contain it, because the verifiers of a disjunction are the verifiers of either disjunct and B contributes content of its own. Absorption fails: A and A ∧ (A ∨ B) are classically equivalent, but the verifiers of the second include fusions a ⊔ b, so the content order is the equational theory of distributive lattices minus absorption. Correia (2004) gave a semantics for Angell's system of analytic containment and proved completeness; Fine (2016) proved completeness of the same system with respect to exact truthmaker content, where containment is a condition on verifier and falsifier sets rather than on truth conditions. Because the relation is not preserved under replacement of classical equivalents, it is hyperintensional in the strict sense: no assignment of sets of possible worlds can represent it, since necessary equivalents are then identified.

Applies to. Philosophy of logic. Metaphysics. Truthmaker theory. Aboutness.

Limitations. Containment is treated only at the first degree: the systems relate truth-functional formulas and say nothing about containment claims nested inside further containment claims, so there is no logic of iterated content. Because absorption fails, the content operations do not form a lattice, and the standard algebraic apparatus for consequence relations does not apply. The relation is not congruential, so classically equivalent formulas are not intersubstitutable and no possible-worlds model represents it. Quantification, modality and predication are open extensions rather than settled parts of any of these systems. And the three main criteria disagree — Parry's vocabulary test, Angell and Fine's verifier test, and Yablo's partition test classify different pairs as containment — while the state space or partition on which each depends is posited rather than derived.

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