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Minimal Logicism
Full text linkPLV (Plural Basic Law V) is a consistent second-order system which is aimed to derive second-order Peano arithmetic. It employs the notion of plural quantification and a first-order formulation of Frege's 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. 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.PLV (Plural Basic Law V) est un système de second ordre cohérent qui vise à dériver l'arithmétique de Peano du second ordre. Il emploie la notion de quantification plurielle et une formulation du premier ordre de la tristement célèbre Loi Fondamentale V de Frege. La sémantique plurielle de George Boolos est remplacée par la Acts of Choice Semantics (ACS) de Enrico Martino, qui est développée à partir de la notion de référence arbitraire en raisonnement mathématique. ACS fournit une forme de logicisme qui est radicalement alternative à celle de Frege et qui est fondée sur l'existence des individus plutôt que sur l'existence des concepts
Logicism, Neo-Logicism, and the Logics of Abstraction
No full textAbstraction principles play a pivotal role in the foundations of mathematics. In the foundational programs in which they have been traditionally adopted, i.e. Frege’s logicism and Scottish neo-logicism, abstraction principles, conceived as definitions, augment a system of full higher-order logic. Not only, however, does such a logical system raise concerns – both philosophical and technical – for the deployment of abstractive definitions; but, also, abstraction principles can subserve alternative logico-philosophical projects depending on variations of the background logic. We first rehearse the role and significance of definitions by abstraction in logicist and neo-logicist accounts. We then discuss the most compelling technical and conceptual issues related to the logics of abstraction: the debate on the status of higher-order logic; the distinction between predicative and impredicative frameworks, and their relation to paradoxes; the adoption of non-classical logics in contemporary conceptions of abstraction. We conclude by addressing the relations between the logical, the epistemological, and the metaphysical significance of abstraction principles
The Logicality of Second-Order Logic
No full textThis chapter argues against predicative analyses of plurality, which force plurals into the familiar mould of singular logic by turning an apparently plural term standing for several objects into a singular predicate standing for a concept or property. Michael Dummett enlists support from Fregean semantics in favour of a predicative analysis, but his arguments do not stand up, either as exegesis of Frege or on their own merits. As well as facing difficulties in eliminating plural content, predicative analyses are sunk by the equivocity objection: they misrepresent single English predicates as equivocal, by treating the predicate differently according as it combines with singular or plural arguments. Although George Boolos does not offer a predicative analysis, it is argued that his second-order treatment of plurals is also sunk by the equivocity objection. And Ian Rumfitt fails in his attempt to avoid the objection by modifying Boolos's scheme
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