Abstract
Leitgeb (2005) proposes a new approach to semantic paradoxes, based upon a direct definition of the set of grounded sentences in terms of dependence upon non-semantic state of affairs. In the present paper, we account for the extensional disagreement between this dependence approach and more familiar alethic approaches. In order to do so, we study the behavior of dependence jumps and alethic jumps, and provide an equivalence result for the two approaches.
Both authors contributed equally to this work and are listed in alphabetical order.
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- 1.
There is no way to do without a principled criterion that decides which T–equivalences hold. Looking just at maximal coherent classes will not do. McGee has shown that taking simply this restriction, the kinds of candidate sets is virtually unrestricted (McGee 1992)
- 2.
Interestingly, the same is not true of \({\text{Tr}} \lceil 2+2=4 \rceil\vee{\text{Tr}}{ \lceil \lnot 2+2=4 \rceil}\) since a priori both \(2+2=4\) and \(2+2\neq 4\) may at the same time be absent from the truth predicate.
- 3.
Leitgeb shows that this definition confirms our intuition that groundedness means referring (in)directly to the world, or, in his terminology, non–semantic states of affairs: φ is ungrounded if, but not only if, there exists a sequence \((\psi_n)_{n\in{\mathbb N}^{*}}\) with \(\psi_n\in{\cal L}_{\text{Tr}}\); \(\psi_0=\phi\) and for every \(n\in{\mathbb N}\), there is a set \(\Uppsi_{n+1}\) such that ψ n depends on \(\Uppsi_{n+1}\) essentially and \(\psi_{n+1} \in \Uppsi_{n+1}\) (Leitgeb 2005, p. 169).
- 4.
Note that using a weak Kleene scheme would not make things really better, since it would not ground \(2+2=4 \vee \lambda\), which is L–grounded. More on this below.
- 5.
We are particularly indebted to Øystein Linnebo for pointing out to us that this simple fact was the crucial feature involved in the comparison between Leitgeb’s approach and familiar alethic jumps.
- 6.
MEADOWe came to know (Meadows 2013) only after finishing a first version of the present paper. The main difference with the present work is that Meadows (2013) does not arrive at the equivalence result by an analysis of conditions (V) and (L) in alethic jumps. As a consequence, the lesson he draws from the equivalence is different: because of the asymmetry problem, Meadows only concludes that properly spelling out the dependence approach makes it equivalent to the supervaluational approach. Our conclusions are less pessimistic. We agree with Meadows that \({\Upphi_{\text{lf}}}\) is not satisfactory. But as discussed in sect. 19.7, we consider that the question whether there exists a conceptually well-motivated dependence fixed point different from supervaluational fixed point is still open. The upshot of the present paper consists in elucidating what is the specific difference between alethic jumps and dependence jumps.
References
Cantini, A. (1990). A theory of formal truth arithmetically equivalent to id 1. Journal of Symbolic Logic, 55, 244–259.
Kripke, S. (1975). Outline of a theory of truth. The Journal of Philosophy, 72 (19), 690–716.
Leitgeb, H. (2005). What truth depends on. Journal of Philosophical Logic, 34, 155–192.
Leitgeb, H. (2008). Towards a logic of type-free modality and truth. In D. Costas et al. Logic colloquium 2005 (pp. 68–85) Dimitracopoulos: Cambridge University Press.
McGee, V. (1992). Maximal consistent sets of instances of Tarski’s schema (T). Journal of Philosophical Logic, 21, 235–241.
Meadows, T. (2013). Truth, dependence and supervaluation: Living with the ghost. Journal of Philosophical Logic, 42 (2), 221–240.
Tarski, A. (1955). A lattice–theoretical fixpoint theorem and its applications. Pacific journal of Mathematics,5(2), 285–309. http://projecteuclid.org/euclid.pjm/1103044538.
van Vugt, F. (2009). What makes a sentence be about the world? Master’s thesis, Cogmaster, Ecole Normale Supérieure, Paris (France).
Yablo, S. (1993). Paradox without self-reference. Analysis, 53(4), 251–252.
Acknowledgements
We would like to thank Serge Bozon, Paul Égré, Hannes Leitgeb, Øystein Linnebo, Philippe de Rouilhan, and Jönne Speck for their insightful comments on earlier versions of this research. We are particularly grateful to Øystein Linnebo who greatly helped us clarify the rationale behind results proven in (van Vugt 2009). We also need to thank various audiences in Gothenburg, London and Paris for their helpful feedback. The present work originates in the second author’s master thesis under the supervision of the first author (van Vugt 2009). Results in the present work were obtained independently of (Meadows 2013), which proves Proposition 8 supra as its main result (see Footnote 6 for a more detailed discussion). The work of the first author was partly supported by the ESF-funded project ‘Logic for Interaction’, a Collaborative Research Project under the Eurocores program LogICCC.
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Bonnay, D., Vugt, F. (2015). Groundedness, Truth and Dependence. In: Achourioti, T., Galinon, H., Martínez Fernández, J., Fujimoto, K. (eds) Unifying the Philosophy of Truth. Logic, Epistemology, and the Unity of Science, vol 36. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9673-6_18
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