Intensionality and Two-steps Interpretations

In this paper we considered the extension of the First-order Logic (FOL) by Bealer’s intensional abstraction operator. Contemporary use of the term ‘intension’ derives from the traditional logical Frege-Russell’s doctrine that an idea (logic formula) has both an extension and an intension. Although there is divergence in formulation, it is accepted that the extension of an idea consists of the subjects to which the idea applies, and the intension consists of the attributes implied by the idea. From the Montague’s point of view, the meaning of an idea can be considered as particular extensions in different possible worlds. In the case of the pure FOL we obtain commutative homomorphic diagram that holds in each given possible world of the intensional FOL, from the free algebra of the FOL syntax, toward its intensional algebra of concepts, and, successively, to the new extensional relational algebra (different from Cylindric algebras). Then we show that it corresponds to the Tarski’s interpretation of the standard extensional FOL in this possible world.

💡 Research Summary

The paper proposes an extension of classical first‑order logic (FOL) by incorporating Bealer’s intensional abstraction operator ⟪ · ⟫. This operator turns any well‑formed formula into a term that denotes an intensional entity (a “concept”). The authors construct a three‑layered semantic architecture: (1) a syntactic free algebra of FOL expressions, (2) an intensional algebra of concepts, and (3) a new extensional relational algebra distinct from the traditional cylindric algebras.

In the intensional layer, the domain D is partitioned into D⁻¹ (particular objects), D⁰ (propositions, i.e., 0‑ary concepts), and Dⁿ for n‑ary concepts (n ≥ 1). An interpretation function I maps each syntactic object to an element of the appropriate Dⁿ. For example, a virtual predicate φ(x₁,…,xₖ) is mapped to a k‑ary concept I(φ) ∈ Dᵏ, while a closed sentence is mapped to a proposition I(ψ) ∈ D⁰.

Extensionalization is handled by a family of functions h: D → R, where R is the set of all relational extensions (including the truth values {f, t} for propositions). Each h assigns to a concept its extension in a particular possible world: propositions become true/false, n‑ary concepts become subsets of the n‑fold Cartesian product of the object domain, and particulars are identified with themselves.

Possible worlds are represented by a set W together with a bijection is: W ↔ E, where E is the collection of admissible extensionalization functions satisfying Tarski’s constraint (T): h(I(φ/g)) = t iff the tuple of values assigned to the free variables of φ belongs to h(I(φ)). For each world w, the extensionalization h_w = is(w) yields the world‑specific extension of every intensional entity.

The authors call this a two‑step intensional semantics: first, I provides the intensional meaning of each formula; second, the world‑indexed h_w supplies the extensional value. This separates meaning from extension, allowing two formulas that are co‑extensional in every world to retain distinct intensional identities. The paper illustrates this with the predicates “bought” and “sold”: although they may have identical extensions across all worlds, they are represented by different concepts u = I(bought) and v = I(sold) in D¹, preserving their semantic distinction.

A novel relational algebra over R is introduced, featuring natural join, projection, and selection operators. The authors prove a commutative homomorphic diagram: the syntactic free algebra maps homomorphically to the intensional algebra via I, and further to the extensional relational algebra via each h_w. Consequently, the extensional algebra obtained for any world coincides with the standard Tarski interpretation of FOL in that world.

The identity predicate “=” is treated specially: it is mapped to a binary concept Id = I(.=) in D², and for every world w the extensionalization satisfies h_w(Id) = the usual equality relation. An auxiliary function f⟨⟩ ensures that empty relations are represented consistently.

The paper concludes that the two‑step framework resolves a known weakness of Montague semantics—its tendency to collapse distinct intensional meanings when extensions coincide—while preserving the expressive power and soundness of classical FOL. It also opens avenues for further research on computational aspects, infinite domains, and applications to databases and natural‑language semantics.