Plural Logic

Origin. Boolos argued for plural quantification (1984, 1985). Avoids set-theoretic commitment for second-order-like reasoning. "There are some critics who admire only one another." Plurals in natural language. Linnebo and others developed formal systems.

Models. Quantifying over many at once. First-order: quantify over individuals. Second-order: quantify over sets/properties. Plural: quantify over "some things" without sets. "The Cheerios" denotes many things, not a set of them. Irreducibly plural reference.

Formalism.

Plural terms and predicates:

xx, yy, zz: plural variables
a ≺ xx: a is one of the xx (singular on the left, plural on the right)
xx ≼ yy: the xx are among the yy
P(xx): predicate applying to pluralities

Plural quantifiers:

∃xx.φ(xx): there are some things such that φ
∀xx.φ(xx): for any things, φ

Comprehension: ∃y.φ(y) → ∃xx.∀y.(y ≺ xx ↔ φ(y)) If something is φ, there are the φs.

Non-empty plural terms: Standard: plural terms denote at least one thing. Variant: allow empty pluralities.

Defining sets: Set(xx) iff ∃z.∀y.(y ≺ xx ↔ y ∈ z) Not all plurals form sets (no universal set paradox).

Natural language examples: "Russell and Whitehead wrote Principia" — collective predication. "Some critics admire only each other" — no first-order paraphrase.

Symbols.

Symbol	Unicode	Name	Meaning
xx, yy, zz	—	Plural variables	Range over pluralities; never occupy a singular position
a, b, y, z	—	Singular variables and constants	Range over individuals
≺	U+227A	Precedes / Is one of	a ≺ xx: the individual a is one of the xx
≼	U+227C	Precedes or equals / Among	xx ≼ yy: every one of the xx is one of the yy
∃	U+2203	Existential quantifier	Applied to singular or plural variables
∀	U+2200	Universal quantifier	Applied to singular or plural variables
∈	U+2208	Element of	Set membership; used only to define Set(xx)
↔	U+2194	Biconditional	Form of the comprehension and Set clauses
→	U+2192	Conditional	Antecedent of plural comprehension
φ	U+03C6	Small letter phi	Schematic formula in comprehension
P	—	Predicate	Applied collectively to a plurality
Set	—	Setness predicate	Set(xx) holds when the xx are exactly the members of some set

Metatheory. Boolos (1984, 1985) established the governing fact: plural quantification is equi-expressive with monadic second-order quantification, so plural logic is not intermediate between first and second order but is a rereading of the monadic second-order fragment on which the values of the higher-order variables are many individuals rather than one set. The consequences follow from that identification. Plural arithmetic is categorical, since induction can be stated as "for any things containing zero and closed under successor, everything is one of them," and Dedekind's argument applies; so by incompleteness the validities are not recursively enumerable and no effective complete proof system exists, and compactness fails. Plural comprehension can be taken unrestricted without inconsistency, because a plurality is not an individual: y ≺ yy is well formed and y ≺ y is not, so the Russell instance cannot even be written, and the paradox is blocked by grammar rather than by an axiom. This is what lets plurals do the work of proper classes over ZFC without adding a second sort of object.

Applies to. Foundations of mathematics (avoiding sets). Metaphysics (composition, mereology). Natural language semantics. Collective action (groups without group entities). Set theory alternatives. Philosophy of logic.

Limitations. Expressive reach stops at the monadic: plural variables take many individuals, not many pairs, so relations cannot be quantified and full second-order logic is out of reach without a pairing device — and pairing reintroduces the objects plurals were meant to avoid. Pluralities are not objects, which is what secures consistency but also means they cannot be collected: nothing is one of a plurality of pluralities, so iterating requires an explicit superplural hierarchy, and at that point the claim of ontological innocence has to be argued afresh at each level. The standard systems admit no empty plurality, so plural terms always denote at least one thing and there is no analogue of the empty set; "the things that are not self-identical" is not a legitimate term. Plural comprehension is impredicative exactly as second-order comprehension is, so the strength and the contested part coincide. And the logic is unaxiomatizable: categoricity of plural arithmetic forces the validities to be non-recursively-enumerable, so deduction can never capture the semantics.