Lorenzo Rossi

Belnap (1970, 1973) proposed to formalize the restriction of first-order quantifiers ∀ and ∃ by means of a single sentential connective: a trivalent conditional that takes the semantic value ‘void’ when the antecedent is false. Thus, ‘every A is B’ is represented as ∀x(Ax → Bx) and ‘some A is B’ is represented as ∃x(Ax → Bx)—a notable unification compared to the standard representation in first-order logic that uses distinct connectives. This paper implements Belnap’s program, henceforth called Conditional Reduction, in full generality and applies it to all conservative ⟨1, 1⟩ quantifiers. By combining Conditional Reductions with the Conservativity Theorem, due to Keenan and Stavi, any conservative type ⟨1, 1⟩ quantifier turns out to be equivalent to Boolean combinations of three basic type ⟨1⟩ quantifiers (‘everything’, ‘at least λ many things’ and the possessives) scoping over a trivalent conditional.

Structures are ubiquitous in mathematics. But how should they be understood? Modelists claim they are model-theoretic structures. This thesis can be read in two ways: as a claim about what structures refer to, or about how we conceptualize them. Objects-modelism, developed by Button and Walsh, pursues the first; the second leads to concepts-modelism, which remains underexplored. In this paper we develop and defend a version of concepts-modelism, cognitive modelism, drawing on Carey’s theory of conceptual development, and we show how it addresses the challenges Button and Walsh pose for a conceptual account of mathematical structures.