Abstract
Many take the lesson of the paradoxes to be that we ought to impose some form of logical revision. It is argued here that this kind of move should not be taken lightly.
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Notes
We will be following the mainstream literature in using \(\ulcorner \phi \urcorner \) to denote (the gödel code of) the name of the formula \(\phi \), where both \(\phi \) and its name belong to the language of the theory. The rest of the paper will also follow tradition insofar as use of gödel corners will for the most part be omitted.
Implicit in the formulation of (Norm) is an assumption which underwrites this entire discussion, which it is therefore worth making explicit here. This is that, while no doubt there are mathematical logicians uninterested in the philosophical debate about truth and philosophers uninterested in the formal study of this concept, there are also those whose interest and work combines both sides of the inquiry. This paper sits—in very good company—in the latter category. For other instances where similar remarks are voiced explicitly, see e.g. (Halbach and Horsten forth.) and Halbach (2001).
Halbach and Horsten (2005) pp. 207–208.
Feferman (1984) p. 95.
As pointed out by an anonymous referee, one might be inclined to think that Formalism is all and only there is to a ‘formal’ theory of truth; and that ‘formal’ theories of truth should be distinguished from (merely) ‘philosophical’ theories, represented instead by Story. As noted earlier, however, we subscribe to the view that a fully satisfactory account of truth must encompass both these components.
It is entirely conceivable that other meta-norms might emerge beyond those countenanced in this paper; far from being cause for concern, this would be a very welcome development, insofar as it would at least partly validate the approach we are proposing. Thanks to an anonymous referee for indirectly pointing this out.
McGee (1989), p. 533 ff.
For an overview, see e.g. Baker (2010).
Tarski (1936), p.164.
Ibid.
Tarski (1944), pp. 348–349.
Or, in an arithmetically-coded language, the numerals of the gödel codes of its sentences.
And indeed this particular controversy is ultimately immaterial to our present concern.
Let \((tr^+,tr^-)\) be a partial interpretation of \(tr\). The kripkean Strong Kleene jump operator is defined by
\(j(tr^+,tr^-)=(j^+(tr^+,tr^-),j^-(tr^+,tr^-))\),
where \(j^+(tr^+,tr^-)=\{\phi :(tr^+,tr^-)\,\vDash\, \phi \}\) and \(j^-(tr^+,tr^-)=\{\phi :(tr^+,tr^-)\,\vDash\, \lnot \phi \}\).
Kripke (1975) pp. 64–65 fn.18.
And it hardly needs mentioning, moreover, that ZFC is itself a thoroughly classical theory.
Kripke (1975) p. 711.
Kripke (1975) pp. 714–715.
As this section is only supposed to serve as a case study, we will not review any of the formal details of Field’s theory and assume some familiarity with the account.
Field (2010) p. 416.
We merely sketch the longer answer here, as it would take us too far afield from the main topic. It is this: because Field is a deflationist, and the story behind [Transparency] is the standard deflationist story according to which asserting \(tr(\phi )\) amounts to no more, no less, than asserting \(\phi \).
And of related classical principles, of course, including in particular DNE.
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Acknowledgments
I am indebted to the audience at the workshop “Paradox and logical revision” for their many valuable questions, criticisms, and suggestions, which helped sharpen the ideas in this paper. I wish to thank in particular Michael Glanzberg, Hannes Leitgeb, Graham Priest, Dave Ripley, and Tim Williamson for their comments. I am equally grateful to the MCMP for hosting the event, and most of all to Julien Murzi and Massimiliano Carrara for giving me the opportunity to contribute to this special issue. Finally, I wish to thank two anonymous referees for their comments.
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Terzian, G. Norms of Truth and Logical Revision. Topoi 34, 15–23 (2015). https://doi.org/10.1007/s11245-014-9259-2
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DOI: https://doi.org/10.1007/s11245-014-9259-2