Abstract
This paper aims to explore the relationship between the necessity predicate and the truth predicate by comparing two possible-world interpretations. The first interpretation, proposed by Halbach et al. (J Philos Log 32(2):179–223, 2003), is for the necessity predicate, and the second, proposed by Hsiung (Stud Log 91(2):239–271, 2009), is for the truth predicate. To achieve this goal, we examine the connections and differences between paradoxical sentences that involve either the necessity predicate or the truth predicate. A primary connection is established through two translations that change only one of the predicates to the other while keeping everything else unchanged. We prove that in bijective frames, a set of sentences that contains one of the two semantic predicates has the same paradoxicality as the corresponding set of sentences that contains the other predicate obtained through translation. However, there are substantial differences as well. First, the necessity predicate and the truth predicate, under the two interpretations, cannot be defined by each other. Moreover, for sentences that involve only the truth predicate, their paradoxicality is preserved under the homomorphisms of frames. For sentences containing the necessity predicate, their paradoxicality is preserved under bounded morphisms, but none of these sentences can have their paradoxicality preserved under the extension of frames. Finally, we also show that paradoxical sentences involving the necessity predicate and those involving the truth predicate differ significantly in terms of mirror symmetry, circularity dependence, and frame compactness.
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Montague explicitly associated Tarski’s theorem with the ‘central lemma’ of his paper (Montague, 1963, p. 289), which implies Theorem 3 of his paper (ibid., p. 293), now known as Montague’s theorem. We refer the reader to (McGee, 1991, pp. 25–26) for a lucid exposition of Tarski’s theorem and Montague’s theorem. It is generally believed that Montague’s theorem shows that the necessity modality should be treated as an operator rather than a predicate. See Stern (2013) or (Stern, 2016, pp. 23–68) for a detailed discussion on this issue.
The characterization problem addressed by Halbach et al. (2003) is specifically concerned with the set of all sentences in the formal language containing the necessity predicate, rather than an arbitrary subset of this set. At the same time, it is worth noting that the examples given by Halbach et al. (2003, Section 3, pp. 188–190) show that different frames may require different (paradoxical) sentences to contradict the admissible valuations. Once we realize this point, it naturally leads to the question in what frames a specific set of sentences cannot satisfy the admissible condition for the necessity predicate.
The construction of the liar sentence is a routine diagonalization. Only note that the biconditional given by the diagonalization is provable in elementary arithmetic such as Peano arithmetic. It immediately follows the same biconditional holds in the sense of the equivalence relation \(\equiv \).
For any transfinite ordinal \(\alpha \), Herzberger (1982, pp. 74–75) informally proposes a paradox consisting of sentences \(\lambda ^{\alpha }_{\beta }\) (\(1\le \beta \le \alpha \)), where \(\lambda ^{\alpha }_{1}\) is the statement saying that \(\lambda ^{\alpha }_{\alpha }\) is untrue, \(\lambda ^{\alpha }_{\beta }\) is the statement saying that \(\lambda ^{\alpha }_{\gamma }\) is true if \(\beta =\gamma +1\) and \(\beta >1\), and \(\lambda ^{\alpha }_{\beta }\) is the statement saying that \(\lambda ^{\alpha }_{\gamma }\) is true for all \(\gamma \) with \(1\le \gamma <\beta \) if \(\beta \) is a limit. See also (Yablo, 1985, p. 340). This paradox can be called the \(\alpha \)-cycle liar. The present \(\omega \)-cycle liar is a formulation of Herzberger’s \(\omega \)-cycle liar in the language \({\mathscr {L}}_{T}\). It would be interesting to investigate for which ordinals Herzberger’s \(\alpha \)-cycle liar can be formalized in \({\mathscr {L}}_{T}\). We would like to thank an anonymous referee for bringing this issue to our attention. For the present purpose, we only consider the \(\omega \)-cycle liar.
See, for instance, (Halbach, 2014, p. 32ff) for more details about Feferman’s dot notation.
See (Kunen, 2011, p. 50) for a standard definition of the rank function.
A result from (Halbach et al., 2003, pp. 206–207) states that the entire \({\mathscr {L}}_{\Box }\) is paradoxical in a certain frame but not in the transitive closure of that frame. Proposition 4 is proved in a similar manner to that result. Both proofs involve constructing the transitive closure of a frame obtained by prefixing a frame to the last point of \(\langle \kappa +1, > \rangle \).
See, for instance, (Blackburn et al., 2001, pp. 64–65).
In Kripke (1975, p. 704)’s stage-by stage inductive construction, if a sentence A is true/false at a stage, then the sentence \(T\,\ulcorner {A}\urcorner \) must be true/false at all higher stages. Take the points as the numbers of stages, and define uRv if v is higher than u. Kripke’s construction is based on the idea that if A is true/false at stage u, then \(T\,\ulcorner {A}\urcorner \) must be true/false at stage v with uRv. In this sense, Hsiung’s interpretation of T can be viewed as a generalization of Kripke’s concept of the inductive construction. It explains why the direction of R in the interpretation of T is reversed. We refer the reader to (Hsiung, 2009, pp. 243–248) for more details.
More precisely, we should call it a frame whose binary relation is bijective. In (Lemmon, 1977, p. 60), a frame with a ‘functional’ relation is a frame in which every point can see exactly one point.
See Hsiung (2017) for the definition of Boolean paradoxes and their characterization theorem.
See (Hsiung, 2009, p. 254) for more information about degrees of paradoxicality.
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Acknowledgements
The author has presented earlier versions of this paper at various talks. The author would like to express gratitude to the organizers and audiences, particularly Professors Bo Chen and Jianjun Zhang, for their helpful comments. The author also wishes to thank the anonymous referees of this journal for their constructive suggestions, which have greatly enhanced the quality of the manuscript.
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This research has been supported by Major Program of National Social Science Foundation of China (Grant Number: 18ZDA031) and National Social Science Fund of China (No. 19BZX136).
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Hsiung, M. Necessity predicate versus truth predicate from the perspective of paradox. Synthese 202, 27 (2023). https://doi.org/10.1007/s11229-023-04244-w
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DOI: https://doi.org/10.1007/s11229-023-04244-w