Abstract
This paper gives a personal overview of the current situation concerning the logical theory of intensional predicates. It is shown how several intensional notions, when logically treated as predicates, yield liar-like paradoxes. Some consistent semantic and axiomatic theories of intensional predicates are presented and discussed. To conclude, an inquiry is made into the conceptual relation between truth and intensional notions.
I am grateful to Benedikt Löwe for inviting me to present a paper at the Foundations of the Formal Sciences II conference. Thanks to Volker Halbach, Hannes Leitgeb and Philip Welch for helpful comments and for information about their joint work in progress.
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References
On the operator approach, the desired forms of quantification could be realised by introducing propositional quantifiers into the formal language. But then, given sufficient expressive power, liar-like paradoxes can be generated, as in the predicate approach. For details, see [Gri93].
Due to Kaplan and Montague, 1960. The paradox was also, seemingly independently, discovered by Myhill. Cf. [Myh60].
Cf. [Mon63].
A stands for the Gödel number of A.
See [Tho0 80].
Cf. [Hor1 Lei01].
Cf. [FriShe87, p. 14].
Cf. [McG85].
Cf. [Tar56a].
Cf. [Kri175].
Cf. [Fef91].
Cf. [Can90].
Cf. [Hal0 94].
Cf., e.g., [Mor186].
The approach sketched here is worked out in more detail in [Hor198].
A good introduction to possible worlds semantics for modal logic is [Hug1Cre96].
For details, consult [Hal0LeiWel∞].
See the articles in [Sha185a]. Reinhardt has also contributed significantly to this research program. See, e.g., [Rei086a].
This is clearly brought out in [And83].
This way of formulating a formal theory of Predicate Epistemic Arithmetic is due to Friedman and Sheard in [FriShe87].
History is our witness here. [Göd33] contains the first attempt to formulate an axiomatic theory of absolute provability (treated as an operator). He lists the S4 principles.
See [Ger170, p. 36–37]; [FriShe87, p. 7(chart 1)]; [Nie91, p. 36]. PEA is considerably stronger than the system that was proposed in [Myh60, p. 469–470], which appears to be one of the earliest attempts to consistently formalise informal provability as a primitive predicate.
This observation is due to Fitch, cf. [Fit063]. In the philosophical community, Fitch’s argument has generated an extensive discussion. For an overview, see, e.g., [Wil100, Chapter 12].
For proofs and a more detailed discussion of these results, see [Hor102].
Cf. [Rei086a].
Something similar holds for Cantini’s system VF.
In this respect, we have now argued ourselves into a position that is in line with the HalbachLeitgeb-Welch approach.
Note that the same question can be asked about many popular axiomatic theories of truth. In my opinion, logical and philosophical questions regarding the relation between inner logic and outer logic have until now not received the attention they deserve.
For proofs of these results, the reader is again referred to [Hor102].
This observation is due to Halbach.
Cf. [Göd33].
An elegant proof of this theorem is given in [FriFla86].
If one takes a Tarskian point of view, then one will insist on one more parameter: one will insist that truth is also always relative to a language.
Note, in this context, that the notion of actually having an informal proof is not, as far as we know, subject to liar-like paradoxes.
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© 2003 Springer Science+Business Media Dordrecht
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Horsten, L. (2003). The Logic of Intensional Predicates. In: Löwe, B., Malzkom, W., Räsch, T. (eds) Foundations of the Formal Sciences II. Trends in Logic, vol 17. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0395-6_7
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DOI: https://doi.org/10.1007/978-94-017-0395-6_7
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