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A Proximity-Based Understanding of Conditionals

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Transactions on Large-Scale Data- and Knowledge-Centered Systems XXXIV

Part of the book series: Lecture Notes in Computer Science ((TLDKS,volume 10620))

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Abstract

The aim of the present paper is to introduce a new logic, PUC-Logic, which will be used to give a systematic account of well-known counterfactuals conditionals on the basis of a concept of proximity. We will formulate a natural deduction system for PUC-Logic, the system PUC-ND, that will be shown to be sound and complete with respect to the semantics of PUC-Logic. We shall also prove that PUC-Logic is decidable and that the system PUC-ND satisfies the normalization theorem.

R.Q. de Araujo Fernandes—We would like to thank PUC-Rio for the VRac sponsorship and DAAD (Germany) for the Specialist Literature Programme.

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Notes

  1. 1.

    The expression state-of-affairs is here used in an intuititive and very general sense as a kind of “truth-maker”, as that piece of reality that is responsible for the truth of a proposition (as Michael Dummett [16] would put it). There’s a long and important discussion in Philosophy as to the true nature of state-of-affairs, but to get into this discussion is clearly beyond the scope of the presente paper.

  2. 2.

    \(\$_{i}\) gives the neighborhoods around the world i. They are the available strictness to evaluate counterfactuals at i.

  3. 3.

    A \(\phi \)-world is a world in which \(\phi \) holds.

  4. 4.

    The notion of closeness or proximity is based on the work of Lewis; it is a topological notion explained in the end of Sect. 2 and formally defined in Sect. 4.

  5. 5.

    This definition of database which includes a first-order model B and not only the integrity constraints is similar to the definition of a relational database in [17].

  6. 6.

    \(\alpha (b_1,\ldots ,b_n)\) is an abuse of notation; it means that \(b_i\) is assigned to \(x_i\) by means of some assignment function.

  7. 7.

    We are going to use labels in the spirit of labelled deductive systems, as it is used by Gabbay and Negri. Labels help us to push down semantic notions into the syntax (see, for example, [22]).

  8. 8.

    We use the term wff to denote both the singular and the plural form of the expression well-formed formula.

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Authors and Affiliations

  1. Systems Development Center, Brazilian Army, Brasília, DF, Brazil

    Ricardo Queiroz de Araujo Fernandes, Edward Hermann Haeusler & Luiz Carlos Pinheiro Dias Pereira

  2. Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, RJ, Brazil

    Ricardo Queiroz de Araujo Fernandes, Edward Hermann Haeusler & Luiz Carlos Pinheiro Dias Pereira

Authors
  1. Ricardo Queiroz de Araujo Fernandes
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  2. Edward Hermann Haeusler
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  3. Luiz Carlos Pinheiro Dias Pereira
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Correspondence to Ricardo Queiroz de Araujo Fernandes .

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Editors and Affiliations

  1. IRIT, Paul Sabatier University, Toulouse, France

    Abdelkader Hameurlain

  2. FAW, University of Linz, Linz, Austria

    Josef Küng

  3. FAW, University of Linz, Linz, Austria

    Roland Wagner

  4. Polytechnic University of Valencia, Valencia, Spain

    Hendrik Decker

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de Araujo Fernandes, R.Q., Haeusler, E.H., Pereira, L.C.P.D. (2017). A Proximity-Based Understanding of Conditionals. In: Hameurlain, A., Küng, J., Wagner, R., Decker, H. (eds) Transactions on Large-Scale Data- and Knowledge-Centered Systems XXXIV. Lecture Notes in Computer Science(), vol 10620. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55947-5_6

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