To Be is to Be the Object of a Possible Act of Choice

Abstract

Aim of the paper is to revise Boolos' reinterpretation of second-order monadic logic in terms of plural quantification ([4], [5]) and expand it to full second order logic. Introducing the idealization of plural acts of choice, performed by a suitable team of agents, we will develop a notion of plural reference. Plural quantification will be then explained in terms of plural reference. As an application, we will sketch a structuralist reconstruction of second-order arithmetic based on the axiom of infiniteà la Dedekind, as the unique non-logical axiom. We will also sketch a virtual interpretation of the classical continuum involving no other infinite than a countable plurality of individuals.

Arbitrary reference, numbers, and propositions

“…More specifically, semantic primitivism rejects the widely held view that semantic facts supervene upon use facts, viz. 10,11 Second answer: Boccuni (2013), Carrara and Martino (2010) and Martino (2001) adopt what Martino (2001) has called Acts of Choice Semantics. As a consequence, the epistemic question is answered as follows.…”

“…I will henceforth omit "natural," for the paper focuses solely on the natural numbers.5 For instance,Clarke-Doane (2008) andSteinhart (2002) have argued that there is more to establishing which set-theoretic progression is the sequence of the numbers than adequacy considerations.6 The label isEklund's (Eklund forthcoming).7 See alsoPickel (forthcoming) for another deflationary explanation using structured propositions.8 Of course, different reductionist views will claim to have more or different specific theoretical benefits. The complete generality of existential instantiation is achieved by letting the rule be valid whatever a is Boccuni (2013),Breckenridge and Magidor (2012),Carrara and Martino (2010),. To illustrate this briefly, take existential instantiation: if we have derived ∃xF(x), we can introduce a new parameter "a" that does not occur in the general formula ∃xF(x) or in any undischarged assumption supporting the formula and assume Fa.…”

On the Consistency of a Plural Theory of Frege’s Grundgesetze

“…Thus, L is provided with polyadic expressive capacity. 5 In PG, natural numbers may be inductively defined as extension-terms, as…”

“…As for bound second-order variables, in [2] a justification for disallowing them in PRC is provided, on the grounds of [5] and [12]. According to [5] and [12], the possibility of directly referring, at least ideally, to any object of a Universe of Discourse is presupposed both by logical and mathematical reasoning, also when non-denumerable domains are concerned. As a consequence, quantification itself logically presupposes the ideal possibility of directly referring to each and every element of a domain, before we consider those elements through generalisation.…”

“…As a consequence, quantification itself logically presupposes the ideal possibility of directly referring to each and every element of a domain, before we consider those elements through generalisation. [5] and [12] call this claim the Thesis of Ideal Reference (TIR). From this perspective, reference to an entity exclusively in terms of a quantification on the domain it belongs to cannot be allowed, because it is required that we are able to directly refer to that entity, even if just in an ideal way, on pain of violating TIR.…”