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Exhibiting interpretational and representational validity


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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  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.


  • Barwise, J., & Etchemendy, J. (1999). Language, proof and logic. Stanford: CSLI Publications.

    Google Scholar 

  • Baumgartner, M., & Lampert, T. (2008). Adequate formalization. Synthese, 164, 93–115.

    Article  Google Scholar 

  • Beall, J. C., & Restall, G. (2006). Logical pluralism. Oxford: Oxford University Press.

    Google Scholar 

  • Blau, U. (1977). Die dreiwertige Logik der Sprache. Berlin: de Gruyter.

    Google Scholar 

  • Brun, G. (2004). Die richtige Formel. Philosophische Probleme der logischen Formalisierung. Ontos: Frankfurt a.M.

    Google Scholar 

  • Brun, G. (2008). Formalization and the objects of logic. Erkenntnis, 69, 1–30.

    Article  Google Scholar 

  • Brun, G. (2012). Adequate formalization and De Morgan’s argument. Grazer Philosophische Studien, 85, 325–335.

    Google Scholar 

  • Bueno, O., & Shalkowski, S. A. (2009). Modalism and logical pluralism. Mind, 118(470), 295–321.

    Article  Google Scholar 

  • Chiswell, I., & Hodges, W. (2007). Mathematical logic. Oxford: Oxford University Press.

    Google Scholar 

  • Chomsky, N. (1977). Essays on form and interpretation. Amsterdam: North-Holland.

    Google Scholar 

  • Davidson, D. (1967). The logical form of action sentences. In N. Rescher (Ed.), The logic of decision and action (pp. 81–95). Pittsburgh: University of Pittsburgh Press.

    Google Scholar 

  • Davidson, D. (1984). Inquiries into truth and interpretation. Oxford: Oxford University Press.

    Google Scholar 

  • Epstein, R. L. (1990). The semantic foundations of logic: Propositional logics. Dordrecht: Kluwer.

  • Epstein, R. L. (1994). The semantic foundations of logic: Predicate logic. Oxford: Oxford University Press.

    Google Scholar 

  • Etchemendy, J. (1990). The concept of logical consequence. Cambridge: Harvard University Press.

    Google Scholar 

  • Field, H. (2009). Pluralism in logic. Review of Symbolic Logic, 2(2), 342–359.

    Article  Google Scholar 

  • Gómez-Torrente, M. (1996). Tarski on logical consequence. Notre Dame Journal of Formal Logic, 37, 125–151.

    Article  Google Scholar 

  • Hale, B. (1996). Absolute necessities. Philosophical Perspectives, 10, 93–117.

    Google Scholar 

  • Hodes, H. T. (1984). Logicism and the ontological commitments of arithmetic. Journal of Philosophy, 81(3), 123–149.

    Article  Google Scholar 

  • Hodges, W. (2001). Elementary predicate logic. In D. Gabbay and F. Guenthner (Eds.), Handbook of philosophical logic (2 ed.), Volume 1, (pp. 1–130). Dordrecht: Kluwer. Academic Publishers.

  • Jackson, B. (2007). Beyond logical form. Philosophical Studies, 132, 347–380.

    Article  Google Scholar 

  • Lampert, T., & Baumgartner, M. (2010). The problem of validity proofs. Grazer Philosophische Studien, 80, 79–109.

    Google Scholar 

  • Massey, G. J. (1975). Are there any good arguments that bad arguments are bad? Philosophy in Context, 4, 61–77.

    Google Scholar 

  • McFetridge, I. (1990). Logical necessity: Some issues. In J. Haldane & R. Scruton (Eds.), Logical necessity and other essays by Ian G. McFetridge (pp. 135–154). London: Aristotelian Society.

  • Montague, R. (1974). Formal philosophy: Selected papers of Richard Montague. New Haven: Yale University Press.

    Google Scholar 

  • Ray, G. (1996). Logical consequence: A defense of Tarski. Journal of Philosophical Logic, 25(6), 617–677.

    Article  Google Scholar 

  • Read, S. (1994). Formal and material consequence. Journal of Philosophical Logic, 23, 247–265.

    Article  Google Scholar 

  • Russell, G. (2008). One true logic? Journal of Philosophical Logic, 37(6), 593–611.

    Article  Google Scholar 

  • Sainsbury, M. (2001). Logical forms. An introduction to philosophical logic (2nd ed.). Malden, MA: Blackwell Publishing.

    Google Scholar 

  • Shalkowski, S. A. (2004). Logic and absolute necessity. Journal of Philosophy, 101, 55–82.

    Google Scholar 

  • Shapiro, S. (1998). Logical consequence: Models and modality. In M. Schirn (Ed.), The Philosophy of Mathematics Today (pp. 131–156). Oxford: Oxford University Press.

  • Shapiro, S. (2005). Logical consequence, proof theory, and model theory. In S. Shapiro (Ed.), The Oxford Handbook of Philosophy of Mathematics and Logic (pp. 651–671). Oxford: Oxford University Press.

  • Sher, G. (1996). Did Tarski commit ‘Tarski’s fallacy’? Journal of Symbolic Logic, 61, 653–686.

    Article  Google Scholar 

  • Tarski, A. (1956). Logic, semantics, metamathematics. Oxford: Clarendon Press.

  • Vecsey, Z. (2010). On the pluralistic conception of logic. Pragmatics & Cognition, 18, 1–16.

    Article  Google Scholar 

  • Wittgenstein, L. (1995). Tractatus logico-philosophicus. Suhrkamp: Frankfurt a. M.

    Google Scholar 

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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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Correspondence to Michael Baumgartner.

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Baumgartner, M. Exhibiting interpretational and representational validity. Synthese 191, 1349–1373 (2014).

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