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Introduction to Formal Philosophy pp 147–176Cite as

Neural Network Models of Conditionals

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Abstract

This chapter explains how artificial neural networks may be used as models for reasoning, conditionals, and conditional logic. It starts with the historical overlap between neural network research and logic, it discusses connectionism as a paradigm in cognitive science that opposes the traditional paradigm of symbolic computationalism, it mentions some recent accounts of how logic and neural networks may be combined, and it ends with a couple of open questions concerning the future of this area of research.

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Notes

  1. 1.

    This article is a revised and extended version of: Leitgeb [41]. Some material contained in Leitgeb [37, 38], Ortner and Leitgeb [46], and in the popular and non-technical exposition of logic and neural networks in Leitgeb [39] was used, too.

    We are grateful for generous support received from the Alexander von Humboldt Foundation.

  2. 2.

    Rumelhart et al. [51] is still something like the “bible” of connectionism; Rojas [50] is a nice introduction to neural networks, and at 〈http://plato.stanford.edu/entries/connectionism/〉 the entry on connectionism in the Stanford Encyclopedia of Philosophy can be found – have a look at these for more background information.

  3. 3.

    See Brewka et al. [11] for a very nice overview of nonmonotonic reasoning, Makinson [44] for a comprehensive logical treatment of the subject, and Schurz and Leitgeb [53] for a compendium of articles on cognitive aspects of nonmonotonic reasoning. Ginsberg [24] is an outdated collection of articles but it is still very useful if one wants to see what nonmonotonic reasoning derives from.

  4. 4.

    Brewka et al. [11] includes a very clear and accessible introduction to logic programming.

  5. 5.

    Almost all of the classical approaches to nonmonotonic reasoning from the 1980s, such as default logic, inheritance networks, truth maintenance systems, circumscription, and autoepistemic logic belong to this class of nonmonotonic reasoning mechanisms.

  6. 6.

    For more on the differences between the two sides of nonmonotonic reasoning, see Brewka et al. [11].

  7. 7.

    We should add that there are also results concerning the description of neural network states and processes by means of classical logic, over and above the traditional McCulloch and Pitts approach: see Pinkas [48] and Bechtel [5] for examples.

  8. 8.

    See van Benthem [58] for a nice discussion of this relationship between conditionals and quantifiers; more can be found by consulting the theory of generalized quantifiers – see e.g. Peters and Westerstahl [47].

  9. 9.

    The following famous example is due to Ernest Adams.

  10. 10.

    For a textbook-like overview of the philosophical literature on indicative and subjunctive conditionals, see Bennett [7].

  11. 11.

    Such networks are called ‘inhibition networks’ in Leitgeb [35].

  12. 12.

    A partial order \(\leqslant \) (on S) is a reflexive, antisymmetric, and transitive binary relation, i.e.: for all s ∈ S: \(s \leqslant s\); for all s, s ′∈ S: if \(s \leqslant s^{\prime }\) and \(s^{\prime } \leqslant s\) then s = s ′; for all s 1, s 2, s 3 ∈ S: if \(s_1 \leqslant s_2\) and \(s_2 \leqslant s_3\) then \(s_1 \leqslant s_3\).

  13. 13.

    So 1 here is actually the constant 1-function, i.e., the function that maps each node to the activation value 1.

  14. 14.

    \(\lambda u^{\prime }.\left \langle W(u^{\prime },u),o_{u^{\prime }}(t)\right \rangle \) is the function that maps u ′ to the pair \(\left \langle W(u^{\prime },u),o_{u^{\prime }}(t)\right \rangle \).

  15. 15.

    Gärdenfors [22] is the classic reference, and Hansson [25] is a nice textbook on belief revision.

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  1. Ludwig-Maximilians-University, Munich, Germany

    Hannes Leitgeb

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  1. Hannes Leitgeb
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Correspondence to Hannes Leitgeb .

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  1. Division of Philosophy, Royal Institute of Technology (KTH), Stockholm, Sweden

    Sven Ove Hansson

  2. Center for Information and Bubble Studies, University of Copenhagen, Copenhagen, Denmark

    Vincent F. Hendricks

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Leitgeb, H. (2018). Neural Network Models of Conditionals. In: Hansson, S., Hendricks, V. (eds) Introduction to Formal Philosophy. Springer Undergraduate Texts in Philosophy. Springer, Cham. https://doi.org/10.1007/978-3-319-77434-3_7

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