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
aa ≺ xx: a is one of the xx
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	—	Plural variable	Many things
≺	U+227A	One of	Individual among
≼	U+227C	Among	Subplurality
∃xx	—	Plural exists	Some things
∀xx	—	Plural forall	Any things
a, b	—	Singular	Individual

Metatheory. Plural logic is between first and second order in strength. Set theory interpretable via plurals without set-theoretic commitment. Plural comprehension avoids Russell's paradox (no set of all plurals). Decidability: plural monadic logic decidable; full plural logic undecidable. Semantics: plural interpretations or superplurals.

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. Ontological status debated (are plurals really non-set-like?). Superplural reference controversial. Proof theory less developed. Tool support minimal. Interaction with modal operators unclear. Not standard in mathematical practice.