PG (Plural Grundgesetze) is a consistent second-order system which is aimed to derive second-order Peano arithmetic. It employs the notion of plural quantification and a few Fregean devices, among which the infamous Basic Law V. George Boolos’ plural semantics is replaced with Enrico Martino’s Acts of Choice Semantics (ACS), which is developed from the notion of arbitrary reference in mathematical reasoning. Also, substitutional quantification is exploited to interpret quantification into predicate position. ACS provides a form of logicism which is radically (...) alternative to Frege’s and which is grounded on the existence of individuals rather than on the existence of concepts. (shrink)

We can introduce singular terms for ordered pairs by means of an abstraction principle. Doing so proves useful for a number of projects in the philosophy of mathematics. However there is a question whether we can appeal to the abstraction principle in good faith, since a version of the Caesar Problem can be generated, posing the worry that abstraction fails to introduce expressions which refer determinately to the requisite sort of object. In this note I will pose the difficulty, and (...) then propose a solution. Since my solution appeals to a plausible constraint on the introduction of new expressions to a language, it is of interest independently of the particular case of terms for pairs. Since these provide the occasion for discussion we should nonetheless review the use of abstraction for pairs before the argumentative business of the paper commences. (shrink)

PG (Plural Grundgesetze) is a predicative monadic second-order system which is aimed to derive second-order Peano arithmetic. It exploits the notion of plural quantification and a few Fregean devices, among which the infamous Basic Law V. In this paper, a model-theoretical consistency proof for the system PG is provided.

I present the traditional debate about the so called explanation of Russell’s paradox and propose a new way to solve the contradiction that arises in Frege’s system. I briefly examine two alternative explanatory proposals—the Predicativist explanation and the Cantorian one—presupposed by almost all the proposed solutions of Russell’s Paradox. From the discussion about these proposals a controversial conclusion emerges. Then, I examine some particular zig zag solutions and I propose a third explanation, presupposed by them, in which I emphasise the (...) role of an implicit premise in the derivation of the paradox. In conclusion, I propose a different zig zag solution obtained by the adoption of a negative free logic. (shrink)

In [Boccuni 2010], a predicative fragment of Frege’s blv augmented with Boolos’ unrestricted plural quantification is shown to interpret pa2. The main disadvantage of that axiomatisation is that it does not recover Frege Arithmetic fa because of the restrictions imposed on the axioms. The aim of the present article is to show how [Boccuni 2010] can be consistently extended so as to interpret fa and consequently pa2 in a way that parallels Frege’s. In that way, the presented system will be (...) compared with the system pe in [Ferreira 2018] and some relevant differences between the two will be highlighted. (shrink)