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This paper is devoted to show the development of some of the model-theoretic ideas which are clearly present in the main members of the Peano school (Peano himself, Burali-Forti, Pieri and Padoa) as a result of their conception of nominal definitions. Also, their semantic definition of logical consequence (Pieri, Padoa) is viewed as one of the outcomes of that conception. Some examples of their use of the expression “nominal definition” are presented first. Second, the main advantages of this kind of definition, as they saw them, are briefly explained, mainly in a philosophical context. Finally, already in the kernel of the paper, some of the details of the model-theoretic view itself are shown, first in Peano, then in Pieri and Padoa, including in both cases some study of their semantic definitions of logical consequence (Former versions of parts of this paper were read in the symposium “Logics and the foundations of mathematics (1885–1905)”, organized by I. Grattan-Guinness and myself, as a part of the XIXth International Congress of History of Science (Zaragoza, August, 1993), and also in the Department of Mathematics, University of Genova, June, 1995. For the preparation of this paper resources provided by the grant DGICYf PS93-0220 are gratefully acknowledged).

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Nominal Definitions and Logical Consequence in the Peano School

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  • 发表日期：

    2025年

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  • Francisco Rodríguez-Consuegra

摘要

This paper is devoted to show the development of some of the model-theoretic ideas which are clearly present in the main members of the Peano school (Peano himself, Burali-Forti, Pieri and Padoa) as a result of their conception of nominal definitions. Also, their semantic definition of logical consequence (Pieri, Padoa) is viewed as one of the outcomes of that conception. Some examples of their use of the expression “nominal definition” are presented first. Second, the main advantages of this kind of definition, as they saw them, are briefly explained, mainly in a philosophical context. Finally, already in the kernel of the paper, some of the details of the model-theoretic view itself are shown, first in Peano, then in Pieri and Padoa, including in both cases some study of their semantic definitions of logical consequence (Former versions of parts of this paper were read in the symposium “Logics and the foundations of mathematics (1885–1905)”, organized by I. Grattan-Guinness and myself, as a part of the XIXth International Congress of History of Science (Zaragoza, August, 1993), and also in the Department of Mathematics, University of Genova, June, 1995. For the preparation of this paper resources provided by the grant DGICYf PS93-0220 are gratefully acknowledged).