Abstract
Graham Priest (Analysis 57:236–242, 1997) has argued that Yablo’s paradox (Analysis 53:251–252, 1993) involves a kind of ‘hidden’ circularity, since it involves a predicate whose satisfaction conditions can only be given in terms of that very predicate. Even if we accept Priest’s claim that Yablo’s paradox is self-referential in this sense—that the satisfaction conditions for the sentences making up the paradox involve a circular predicate—it turns out that there are paradoxical variations of Yablo’s paradox that are not circular in this sense, since they involve satisfaction conditions that are not recursively specifiable, and hence not recognizable in the sense required for Priest’s argument. In this paper I provide a general recipe for constructing infinitely many (in fact, continuum-many) such noncircular Yabloesque paradoxes, and conclude by drawing some more general lessons regarding our ability to identify conditions that are necessary and sufficient for paradoxically more generally.
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Notes
Of course, this brief description falls far short of a definition or explication of inferential circularity. Since my purpose here is not to examine inferential circularity directly, however, but merely to identify it so as to set it aside, the description provided suffices.
There are actually two distinct versions of the analysis of circularity presented here that are worth distinguishing. The first, officially adopted above, we might call the weakly epistemic version. This version requires that, if a paradox p is circular, then there must be a recursive function \(f_\mathsf{p}\) codifying the satisfaction conditions of p and witnessing the circularity. The second, which we might call the strongly epistemic version, requires, further, that we know that \(f_\mathsf{p}\) is recursive. Although the weakly epistemic version of the account seems more plausible to the author, officially no stand needs to be taken here. If the arguments to follow do, indeed, show that there are non-circular paradoxes on the weakly epistemic version of the view, then the same arguments will also show this for the strongly epistemic reading (since the stronger epistemic reading is less generous with respect to which functions can witness circularity).
This proof, like similar proofs earlier in the paper, shows that there is no way to extend the standard model of arithmetic so that it coherently interprets all of the sentences in the Yabloeseque construction in question, and the reasoning hence occurs in the metatheory. The reasoning cannot ‘descend’, and be carried out as an explicit deduction of a contradiction from the \(\omega \)-many sentences in the *-variant in question plus, say, standard first-order Peano arithmetic, since any infinitely non*-variant of \(\mathsf{YP}\), like \(\mathsf{YP}\) itself, is consistent (albeit \(\omega \)-inconsistent, which is in effect what the proof shows). The proof-theoretic behavior of *-variants of \(\mathsf{YP}\) (and other variants of Yablo’s construction) is a topic that deserves more attention than I can give it here—see Barrio (2010) and Ketland (2005) for some relevant results, and Picollo (2013) for an extension of the proof-theoretic perspective to higher-order languages.
By “truth of first-order arithmetic” I mean a formula expressible in the language of first-order arithmetic that is true on the standard model of arithmetic. Similar comments apply to “truth of second-order arithmetic” below.
Ketland (2005) also considers a variant of the paradox that involves two principles - a Generalized Yablo Principle:
$$\begin{aligned} (\forall n)(\mathsf{Y}(n) \leftrightarrow (\forall m > n)(\mathsf{sat}(m, \ulcorner \mathsf{Y}(x) \urcorner ))) \end{aligned}$$and a Generalized Satisfaction Scheme:
$$\begin{aligned} (\forall n)(\mathsf{sat}(m, \ulcorner \mathsf{Y}(x) \urcorner ) \leftrightarrow \mathsf{Y}(m)) \end{aligned}$$These two principles are, unlike the infinite list version of \(\mathsf{YP}\), jointly (deductively) inconsistent. Adopting this more restricted (albeit more powerful) formulation of the Yablo paradox gives away too much to Priest, however, since we can only formulate generalizations of the first principle (via diagonalization) of the form:
$$\begin{aligned} (\forall n)(\mathsf{Y}^+(n) \leftrightarrow \Phi (n, \ulcorner \mathsf{Y}^+(x) \urcorner )) \end{aligned}$$if \(\Phi (x, y)\) is expressible within whatever language we are using. If \(\Phi (x, y)\) is expressible, however, then we can easily express the satisfaction conditions of \(\mathsf{Y}^+(n)\) in terms of \(\Phi (x, y)\), and hence the paradox in question is circular in Priest’s sense. Another way of putting this is as follows: The non-circular variants of the Yablo paradox constructed here cannot be formulated in terms of a single principle like the Generalized Yablo Principle.
It is worth noting that I am understanding a theory as a (recursive) collection of principles that (among other things) specify a class of models (those models that satisfy the theory) and hence indirectly specify a collection of truths relative to that theory (those principles that are true on all the specified models). There need not be any effective procedure (e.g. complete deductive system) for determining which principles are and are not true according to the theory. This will become important when we replace first-order Peano arithmetic with its second-order cousin, which is categorical but incomplete, below.
Additionally, no claim is made here that this is at all a plausible understanding of the notion of definability, but only that it is a non-trivial understanding that would result in different functions being definable.
There is, of course, a related proposal whereby \(f_\mathsf{p}\) is definable if and only if there is some \(\Phi (x, y)\) in the language of \({\mathcal {T}}\) such that:
$$\begin{aligned} {\mathcal {T}} \vdash \Phi (\underline{a}, \underline{b}) \end{aligned}$$if and only if:
$$\begin{aligned} f_\mathsf{p}(a) = b \end{aligned}$$Note, however, that the example in the text formulated in terms of second-order PA will fail if we replace \(\vDash \) with \(\vdash \).
A few technical observations: First, it is worth noting that, for any consistent, recursively axiomatizable theory \({\mathcal {T}}\) that (i) is first-order and (ii) contains (in the relevant sense) Robinson Arithmetic, a function is definable relative to that theory (in the sense elucidated above) if and only if it is recursive. The proof of this result depends on the Friedman-Goldfarb-Harrington Theorem, and is beyond the scope of this footnote. Nevertheless, it follows that, in order to obtain an extensionally distinct notion of definability using the schema introduced here, we need to choose a theory \({\mathcal {T}}\) where one of the following hold:
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\({\mathcal {T}}\) is not first-order.
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\({\mathcal {T}}\) does not contain Robinson Arithmetic.
The example in the text is an instance of the former strategy. Thanks are owed to an anonymous referee for pointing out, and emphasizing the importance of, this fact.
Second, I am assuming the standard or full semantics for second-order logic, where the monadic second-order variables range over a domain isomorphic to the powerset of the first-order domain (and similarly for n-ary second-order variables for \(n > 1\)). The argument would fail if we instead use, e.g., Henkin semantics, where the monadic second-order might range over a domain isomorphic to a proper subset of the powerset of the first-order domain.
Third, it turns out that we need not actually add a truth predicate \(\mathsf{T}_0\) to the language of (second-order) PA in order to carry out this construction, since both a truth predicate and a satisfaction predicate adequate on the purely arithmetical first-order expressions of PA are definable (in the ordinary sense) in second-order PA. Simply adding a Tarskian, typed predicate \(\mathsf{T}_0\), however, is simpler.
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Of course, this argument requires that we restrict our attention to recursively axiomatizable theories. If we allowed arbitrary collections of axioms to be theories in the relevant sense, then we could ‘define’ any circularity-witness function \(f_\mathsf{p}\) pointwise in terms of a countably infinite list of axioms.
Another way of putting the point is this: An arbitrary Yabloesque construction is circular only if we can identify a function that expresses its satisfaction conditions and witnesses the circularity. We can only do this, in general, when the satisfaction-codifying function is recursive.
References
Barrio, E. (2010). Theories of truth without standard models and Yablo’s sequences. Studia Logica, 96, 377–393.
Cook, R. (2014). The Yablo paradox: An essay on circularity. Oxford: Oxford University Press.
Ketland, J. (2005). Yablo’s Paradox and \(\omega \)-inconsistency. Synthese, 145, 295–302.
Leitgeb, H. (2002). What is a self-referential sentence? Critical remarks on the alleged (non)-circularity of Yablo’s paradox. Logique et Analyse, 177, 3–14.
Picollo, L. (2013). Yablo’s paradox in second-order languages: Consistency and unsatisfiability. Studia Logica, 101, 601–613.
Priest, G. (1997). Yablo’s paradox. Analysis, 57, 236–242.
Sorensen, R. (1998). Yablo’s paradox and kindred infinite liars. Mind, 107, 137–155.
Urbaniak, R. (2009). ”Leitgeb, “About”, Yablo”, Logique et Analyse, 52(207), 239–254.
Yablo, S. (1993). Paradox without self-reference. Analysis, 53, 251–252.
Acknowledgements
An early version of this paper was presented by Jesse M. Butler at the 2009 APA Central Division Meeting in Chicago, Illinois, where Roy T. Cook was the commentator. Sadly, Jesse passed away in 2013 before publishing this work, and Professor Cook graciously volunteered to oversee the refereeing and revision process for the paper. Jesse’s widow—Ivana Simic—would like to gratefully acknowledge all the work Professor Cook has put in to help prepare the paper for publication, despite the fact that it was written more than 10 years ago and that a lot has happened in the discourse on Yablo’s paradox since. Thanks are also owed to two anonymous referees for exceptionally helpful comments.
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Jesse M. Butler was formerly at University of Florida.
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Butler, J.M. An entirely non-self-referential Yabloesque paradox. Synthese 195, 5007–5019 (2018). https://doi.org/10.1007/s11229-017-1443-7
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DOI: https://doi.org/10.1007/s11229-017-1443-7