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Exhibiting interpretational and representational validity

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Abstract

A natural language argument may be valid in at least two nonequivalent senses: it may be interpretationally or representationally valid (Etchemendy in The concept of logical consequence. Harvard University Press, Cambridge, 1990). Interpretational and representational validity can both be formally exhibited by classical first-order logic. However, as these two notions of informal validity differ extensionally and first-order logic fixes one determinate extension for the notion of formal validity (or consequence), some arguments must be formalized by unrelated nonequivalent formalizations in order to formally account for their interpretational or representational validity, respectively. As a consequence, arguments must be formalized subject to different criteria of adequate formalization depending on which variant of informal validity is to be revealed. This paper develops different criteria that formalizations of an argument have to satisfy in order to exhibit the latter’s interpretational or representational validity.

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Notes

  1. The problem of systematizing logical formalization has also been addressed from a different angle in the literature. Instead of developing criteria of adequate formalization, authors as Davidson (1984), Chomsky (1977), Massey (1975), or Montague (1974), implicitly or explicitly subscribe to the ambitious project to define effective formalization procedures. This second thread in the formalization literature will be sidestepped in this paper.

  2. Two FOL formulas \(\phi \) and \(\psi \) are said to be unrelated iff neither \(\phi \) is a specification of \(\psi \) nor vice versa and there does not exist a formula \(\chi \) that is a specification of both \(\phi \) and \(\psi \). For the relevant notion of specification cf. (Brun (2004), p. 320) or (Lampert and Baumgartner (2010), pp. 89–90).

  3. For details on this model-theoretic background cf. any textbook, e.g. (Chiswell and Hodges (2007), Sect. 7.3); Barwise and Etchemendy (1999, Sect. 18.2).

  4. Similarly, (Shapiro (1998), p. 137).

  5. Essentially, this is what (Shapiro (2005), p. 663) dubs the blended view of models. I prefer to stick to Etchemendy’s original label “interpretational view”, because it emphasizes the core difference from the representational view.

  6. Cf. e.g. (Brun (2004), p. 210), (Baumgartner and Lampert (2008), p. 108). Correctness can also be defined syntactically. In this paper, I am only going to use the semantic variant of correctness. For details on the syntactic one cf. Baumgartner and Lampert (2008).

  7. Correspondence schemes have a very different function in the representational framework. In that framework they fix the interpretation of formal languages. Cf. Sect. 4 below.

  8. As we shall see in Sect. 4, (\(r\)-COR) faces an analogous termination problem. Cf. also (Baumgartner and Lampert (2008), p. 97).

  9. In this respect the version of \(r\)-correctness presented here differs considerably from the versions developed by Blau (1977) and Brun (2004), who both only require correspondence of truth conditions relative to the subset of models \(\mathfrak{M }^{\scriptscriptstyle \phi }_k\) that provide what Blau and Brun call suitable interpretations of \(\phi \). For a detailed criticism of this latter approach cf. Baumgartner and Lampert (2008).

  10. According to the convention adopted in Sect. 3, I say that arguments and their formalizations are true if they are truth-preserving in corresponding possible worlds and models, respectively, and false otherwise.

  11. In Baumgartner and Lampert (2008), we argue that, from the representational perspective, there is no clear distinction between logical formalization and semantic analysis.

  12. In this regard, \(r\)-correctness differs significantly from \(i\)-correctness. While formalizations of a statement \(\varPhi \) that introduce arbitrary tautologous elements not contained in \(\varPhi \) cannot be seen to model\(^*\) permissible reinterpretations of \(\varPhi \), formalizations of \(\varPhi \) model\(^*\) the truth conditions of \(\varPhi \) even if they feature an excessive tautologous surplus, provided that (IN) is respected.

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Acknowledgments

I am particularly indebted to Timm Lampert for countless discussions about the topics of this paper and for our common work on logical formalization. Moreover, I thank Wolfgang Spohn, Jochen Briesen, Alexandra Zinke, Brian Leahy and the anonymous referees of this journal for helpful comments on earlier drafts. Finally, the paper has profited from discussions with audiences at two research colloquia at the Humboldt University Berlin and the University of Konstanz.

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  1. Institute of Cognitive Science, University of Osnabrück, Albrechtstrasse 28, 49076 , Osnabrück, Germany

    Michael Baumgartner

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Baumgartner, M. Exhibiting interpretational and representational validity. Synthese 191, 1349–1373 (2014). https://doi.org/10.1007/s11229-013-0331-z

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