Abstract
The semantic paradoxes and the paradoxes of vagueness (‘soritical paradoxes’) display remarkable family resemblances. In particular, the same non-classical logics have been (independently) applied to both kinds of paradoxes. These facts have been taken by some authors to suggest that truth and vagueness require a unified logical framework (see e.g. [3, 5]). Some authors go further, and argue that truth is itself a vague or indeterminate concept (see e.g. [4, 7]). Importantly, however, there currently is no identification of what the common features of semantic and soritical paradoxes exactly consist in. This is what we aim to do in this work: we analyze semantic and soritical paradoxes, and develop our analysis into a theory of paradoxicality. The unification of the paradoxes of truth and vagueness we propose here has a wide scope, but for the sake of concreteness we focus on four three-valued logics.
We would like to thank two anonymous referees for their comments on the previous version of the paper, that greatly helped us to improve it. This is an extended abstract of [1]. Due to space constraints, proofs are omitted. We refer the interested reader to [1] for details.
R. Bruni—This work was supported by the Italian Ministry of Education, University and Research through the PRIN 2017 program “The Manifest Image and the Scientific Image” prot. 2017ZNWW7F_004.
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Notes
- 1.
In the sense of [8, Chap. 5].
- 2.
This meta-inferential formulation of diagonalization is required, because the ‘usual’ form of diagonalization (‘weak diagonalization’) involving a biconditional is not available in some of the theories we consider.
- 3.
Unlike the T-models of [2], soritical models do not impose reflexivity or symmetry on \(\sim _\mathsf{P}\).
- 4.
This of course include vague atomic sentences of the form \(\mathsf{P}(c)\).
- 5.
This entails that terms of \(\mathcal {L}_3\) may end up being possibly infinite strings of symbols.
- 6.
One could argue that SSe, TTe, TSe, and STe are not, strictly speaking, logics or, conversely, that they treat truth-predication and vague atomic sentences as (quasi-)logical expressions. This matter is largely terminological, so we leave it aside here.
- 7.
If M=S, then Definition 14 requires that all formulas in \(\Gamma _\lambda \) be S-true in the first place. In turn (see Definition 13), this requires that the set of equations associated with all of the formulas in \(\Gamma _\lambda \) be solvable - in the sense of Definition 12 - by setting the assignment to 1 (i.e., by putting \(\alpha (v_\lambda )=1\) in this case). However, this cannot happen due to the set of equations associated to \(\lambda \) being \(\{v_\lambda =1-w,w=v_\lambda \}\).
- 8.
Like before, if M=S, then for the argument to be valid it is required that all of the sentences in \(\Gamma _\sigma \) be S-true, i.e. that the set of equations associated to them be solvable by setting the value of the principal variable to 1. In particular, this requires that value 1 is assigned to the variables corresponding to every \(\mathsf{P}(a_i)\), which cannot happen in soritical models.
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Bruni, R., Rossi, L. (2021). A Unified Approach to Semantic and Soritical Paradoxes. In: Ghosh, S., Icard, T. (eds) Logic, Rationality, and Interaction. LORI 2021. Lecture Notes in Computer Science(), vol 13039. Springer, Cham. https://doi.org/10.1007/978-3-030-88708-7_3
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