Por Bruno Jacinto (Universidade de Lisboa) e José Mestre (Stirling (SASP) e LanCog (Lisboa)).
Abstract: Russell argued for the logicist claim that arithmetic is reducible to logic on the basis of the derivability of the Peano axioms in his type-theoretic system. Unlike his logicist ally Frege, who conceived of numbers as individuals, Russell thought of them as attributes of attributes of individuals. As a result, Russell's brand of logicism faced a number of distinctive problems. Chief among these is that his derivation of the Peano axioms relies on the assumption that the domain of individuals is infinite. As critics have pointed out, this assumption does not constitute a logical law. The aim of the present talk is to pave the way for a novel argument for the reducibility of arithmetic to logic. Departing from the Russellian view that numbers are attributes of attributes of individuals, a result will be presented to the effect that the axioms of Peano arithmetic are theorems of a modal and type-theoretic system committed solely to a potential infinity of individuals. In addition, we will examine the prospects of formulating and defending a novel form of logicism about arithmetic on the basis of this result.
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