Search

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. Dans [Boccuni 2010], un fragment prédicatif du blv de Frege augmenté de la quantification plurielle illimitée de Boolos interprète pa2. Le principal inconvénient de cette axiomatisation est qu’elle ne récupère pas Frege Arithmetic (fa), en raison des restrictions imposées aux axiomes. Le but du présent article est de montrer comment [Boccuni 2010] peut être étendu de manière cohérente afin d’interpréter fa et par conséquent pa2 d’une manière qui soit parallèle à celle de Frege. Ce faisant, le système présenté sera mis en comparaison avec le système pe dans [Ferreira 2018] et quelques différences pertinentes entre les deux seront mises en évidence.

We utilize and extend the method of shallow semantic embeddings (SSEs) in classical higher-order logic (HOL) to construct a custom theorem proving environment for abstract objects theory (AOT) on the basis of Isabelle/HOL. SSEs are a means for universal logical reasoning by translating a target logic to HOL using a representation of its semantics. AOT is a foundational metaphysical theory, developed by Edward Zalta, that explains the objects presupposed by the sciences as abstract objects that reify property patterns. In particular, AOT aspires to provide a philosphically grounded basis for the construction and analysis of the objects of mathematics. We can support this claim by verifying Uri Nodelman's and Edward Zalta's reconstruction of Frege's theorem: we can confirm that the Dedekind-Peano postulates for natural numbers are consistently derivable in AOT using Frege's method. Furthermore, we can suggest and discuss generalizations and variants of the construction and can thereby provide theoretical insights into, and contribute to the philosophical justification of, the construction. In the process, we can demonstrate that our method allows for a nearly transparent exchange of results between traditional pen-and-paper-based reasoning and the computerized implementation, which in turn can retain the automation mechanisms available for Isabelle/HOL. During our work, we could significantly contribute to the evolution of our target theory itself, while simultaneously solving the technical challenge of using an SSE to implement a theory that is based on logical foundations that significantly differ from the meta-logic HOL. In general, our results demonstrate the fruitfulness of the practice of Computational Metaphysics, i.e. the application of computational methods to metaphysical questions and theories.

Is a logicist bound to the claim that as a matter of analytic truth there is an actual infinity of objects? If Hume’s Principle is analytic then in the standard setting the answer appears to be yes. Hodes’s work pointed to a way out by offering a modal picture in which only a potential infinity was posited. However, this project was abandoned due to apparent failures of cross-world predication. We re-explore this idea and discover that in the setting of the potential infinite one can interpret first-order Peano arithmetic, but not second-order Peano arithmetic. We conclude that in order for the logicist to weaken the metaphysically loaded claim of necessary actual infinities, they must also weaken the mathematics they recover.

Frege’s theorem states that Peano Arithmetic can be interpreted in Frege Arithmetic, a second-order system with one extra-logical axiom, Hume’s Principle. The epistemological implications of this result have been controversial. Starting from Heck’s epistemological analysis of the conceptual relations between cardinality, counting, and equinumerosity and expanding on his discussion of empirical results on numerical cognition, I assess the relative importance of equinumerosity and the successor relation in empirical results on numerical cognition in developmental psychology and linguistic anthropology. I conclude that both concepts independently play a role in the development of number concepts. However, the cognitive explanations of the acquisition of numerical concepts arguably do not lead to a full understanding of numerical concepts and are in various ways problematic from a methodological point of view. I conclude that for the analysis of numerical concepts in mathematical practice, a modest provisional apsychologism is commendable.

The rather unrestrained use of second-order logic in the neo-logicist program is critically examined. It is argued in some detail that it brings with it genuine set-theoretical existence assumption...

Frege's theorem says that second-order Peano arithmetic is interpretable in Hume's Principle and full impredicative comprehension. Hume's Principle is one example of an abstraction principle, while another paradigmatic example is Basic Law V from Frege's Grundgesetze. In this paper we study the strength of abstraction principles in the presence of predicative restrictions on the comprehension schema, and in particular we study a predicative Fregean theory which contains all the abstraction principles whose underlying equivalence relations can be proven to be equivalence relations in a weak background second-order logic. We show that this predicative Fregean theory interprets second-order Peano arithmetic., Comment: Forthcoming in Bulletin of Symbolic Logic. Slight change in title from previous version, at request of referees

As is well known, Frege gave an explicit definition of number (belonging to some concept) in §68 of his Die Grundlagen der Arithmetik

Неологицизм — современное направление в философии математики, связанное с попытками устранить противоречия «Основоположений арифметики» Фреге или разработать иные способы сведения математики к логике. В настоящее время неологицизм неофрегеанского толка признается наиболее близким к целям Фреге, а философская эффективность иных теорий связывается с необходимостью допущения онтологического плюрализма в математике. Мы рассмотрим некоторые неологицистские теории (К. Райт, Э. Залта) и соответствующие дискуссии в аналитической философии с целью выявления сильных и слабых сторон некоторых вариантов неологицизма, попытаемся конкретизировать понятие «неологицизм». The article focuses on neologicism, which is the trend of modern philosophy of mathematics associated with attempts to resolve the contradictions of Frege’s “Foundations of Mathematics” and to develop alternative ways of deriving mathematics from logic. The author examines neologicist theories of C. Wright, B. Linsky and E. Zalta in detail and reviews broader discussions concerning Fregean neologicism. The author highlights comparative advantages of certain neologicist theories and seeks to define the concept of neologicism in a more precise way.

Two thoughts about the concept of number are incompatible: that any zero or more things have a (cardinal) number, and that any zero or more things have a number (if and) only if they are the members of some one set. It is Russell's paradox that shows the thoughts incompatible: the sets that are not members of themselves cannot be the members of any one set. The thought that any (zero or more) things have a number is Frege's; the thought that things have a number only if they are the members of a set may be Cantor's and is in any case a commonplace of the usual contemporary presentations of the set theory that originated with Cantor and has become ZFC.In recent years a number of authors have examined Frege's accounts of arithmetic with a view to extracting an interesting subtheory from Frege's formal system, whose inconsistency, as is well known, was demonstrated by Russell. These accounts are contained in Frege's formal treatise Grundgesetze der Arithmetik and his earlier exoteric book Die Grundlagen der Arithmetik. We may describe the two central results of the recent re-evaluation of his work in the following way: Let Frege arithmetic be the result of adjoining to full axiomatic second-order logic a suitable formalization of the statement that the Fs and the Gs have the same number if and only if the F sand the Gs are equinumerous.

By Frege's Theorem is meant the result, implicit in Frege's Grundlagen, that, for any set E, if there exists a map υ from the power set of E to E satisfying the conditionthen E has a subset which is the domain of a model of Peano's axioms for the natural numbers. (This result is proved explicitly, using classical reasoning, in Section 3 of [1].) My purpose in this note is to strengthen this result in two directions: first, the premise will be weakened so as to require only that the map υ be defined on the family of (Kuratowski) finite subsets of the set E, and secondly, the argument will be constructive, i.e., will involve no use of the law of excluded middle. To be precise, we will prove, in constructive (or intuitionistic) set theory, the followingTheorem. Let υ be a map with domain a family of subsets of a set E to E satisfying the following conditions:(i) ø ϵdom(υ)(ii)∀U ϵdom(υ)∀x ϵ E − UU ∪ x ϵdom(υ)(iii)∀UV ϵdom(5) υ(U) = υ(V) ⇔ U ≈ V.Then we can define a subset N of E which is the domain of a model of Peano's axioms.