Extendability and Paradox

Hellman, Geoffrey; Cook, Roy T.

doi:10.1007/978-3-319-96274-0_5

Geoffrey Hellman⁴ &
Roy T. Cook⁴

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 9))

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2 Citations

Abstract

In this essay we examine the revenge problem as it arises with respect to accounts of both the set-theoretic and the semantic paradoxes. First we review revenge as it arises in the set-theoretic setting – the Burali-Forti paradox – and outline its modal-structural resolution, highlighting the roles played by the logic of plurals, modal principles, and especially the extendability of models of set theory on this account. We then we turn to the semantic paradoxes, especially the Liar, and develop an analogy between the problems of expressive incompleteness and revenge affecting proposals to resolve the semantic paradoxes, on the one hand, and, on the other, the always incomplete and extendable nature of domains of sets on the modal structural approach to set theory. We then argue for a corresponding parallelism in resolutions of the set-theoretic and the semantic paradoxes. Focusing on recent accounts stemming from the work of Martin and Woodruff (Philosophia, 5(3):213–217, 1975) and Kripke (Journal of Philosophy, 72:690–716, 1975), we formulate a modal account of the extendability of languages on the Embracing Revenge account of the semantic paradoxes (see, e.g. Cook, Embracing Revenge: On the Indefinite Extendability of Language, 2007; Schlenker, Review of Symbolic Logic, 3(3):374–414, 2010) analogous to the formulation of extendability principles for set theoretic universes on the modal structural approach. Finally, however, we examine an interesting dis-analogy via a meta-revenge version of the Liar paradox that seems to have no analogue in the set-theoretic context, and we show how the solution to this puzzle also highlights even deeper connections between the modal-structural account of set theory and the Embracing Revenge account of truth and semantics.

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Notes

1.
For some interesting history of the Burali-Forti paradox, see Ferreirós (2007), Menzel (1984), and Moore and Garciadiego (1981).
2.
It is worth noting the oddness of the axiom of foundation in this respect. The presence of foundation, which rules out both the existence of self-membered sets and of infinitely ‘descending’ membership chains, is often invoked in explanations for why ZFC is not susceptible to the Russell, Cantor, and Burali-Forti paradoxes. But, in fact, foundation plays no role in blocking the paradoxes: it is not the fact that ZFC proves that the problematic sets do not exist, but rather the fact that ZFC minus foundation does not prove that these sets exist, that provides the explanation (if any) for the adequacy of ZFC as an alternative to naïve set theory. After all, adding an additional axiom – in this case, foundation – to a theory makes it more, rather than less, likely that the theory is inconsistent.
3.
Talk of ‘pluralities’ is, as usual, strictly a façon de parler, not a reference to any special objects.
4.
Note that a downwards closed plurality xx can ‘contain’ non-ordinals as well.
5.
That $\lambda $ must be the least strict upper bound on the $\beta \beta $ follows similarly.
6.
Note that the proof of Corollary 5.2.2 (as well as the proof of Theorem 5.2.1, on which it depends) does not invoke Desideratum 2. The alert reader will have noticed that Desideratum 2 also leads to problems closely related to Burali-Forti. As usually stated, Hartogs’ Theorem governs arbitrary sets of ordinals, but the generalization in the language of plurals applies to ‘arbitrary collections’, as it were. So again, if it makes sense to refer to ‘absolutely all ordinals’, then this would present an exception to Hartogs’ Theorem, as there could then be no initial ordinal, cardinally – hence ordinally – greater than all ordinals! (‘Initial ordinal’ here just means ‘ordinal cardinally greater than any earlier ordinal’ .)
7.
Here we mean ‘model’ in a broad sense, as any universe of discourse with an interpretation function assigning ordered pairs of elements of the universe to binary relation-symbol $\in $, along with standard interpretations of the plural quantifiers. We explicitly do not restrict ‘model’ to require that the domain be a set.
The use of second-order Replacement is essential in Zermelo’s proofs of the quasi-categoricity of ZFC2 (see the next note).
8.
These conditions are best understood as only ‘up to isomorphism’. A major achievement of Zermelo (1930) was to present a general proof of the quasi-categoricity of the ZFC2 axioms, viz. that given any two models (without urelements) , one is an end-extension of a model isomorphic to the other.
It should also be noted that Hilary Putnam independently formulated an extendability principle very much in the spirit of Zermelo’s in Putnam (1967), although it applied to models of Z rather than those of ZFC . Also, Putnam was the first to formulate it explicitly as a modal principal.
9.
For more on the modal structural approach generally, see Hellman (1989). The application of modal structuralism to the Burali-Forti paradox discussed here builds on and extends the account found in Hellman (2011).
10.
Officially, there is no need to quantify over worlds. Everything needed for pure and applied mathematics can be expressed directly via the modal operators.
11.
To get the effect of assertions of relations of structures across worlds, e.g. that two such are isomorphic, we can get by with additional assumptions of compossibility of models satisfying the relevant conditions. For further details, see chapter 1 of Hellman (1989).
12.
Indeed, these expressions of actualism involving plural constructions sound even more tautologous than those involving classes. In effect, we get instances of plurals comprehension such as:

There are only the things satisfying condition $\Phi $ that exist.

and, in counterfactual circumstances C:

There would only be those things satisfying $\Phi $ that would then exist.
13.
Note, however, that this is not guaranteed for arbitrary choices of $\Phi $ and $\Psi $: For example, if $\Phi $ is “pairs coding a bijective order-preserving map from all ordinals to all accessible ordinals”, and $\Psi $ is “pairs coding a bijective order-preserving map from all ordinals to all ordinals some of which are inaccessible”, clearly there is no way for:
$$\begin{aligned} (\exists xx)(\exists yy)(\Phi (xx) \wedge \Psi (yy)) \end{aligned}$$
to be satisfied in a single (ZFC2) model.
14.
Actually, in some instances where we might wish to formalize informal claims of the form “some ordinal is $\Phi $”, stronger formalizations such as:
$$\begin{aligned} \Diamond (\exists \mathcal {M}_1)\Box (\forall \mathcal {M}_2)(\mathcal {M}_1 \subseteq _\mathsf{end} \mathcal {M}_2 \rightarrow (\exists x)(x \in \mathsf{On}_\mathcal {M}(x) \wedge \Phi (x))) \end{aligned}$$
which express that there is an ordinal that satisfies $\Phi (x)$, and there will continue to be such an ordinal in any extension of the current model, will be more apt.
15.
As typically described, the hierarchy is defined only for languages indexed by finite ordinals, but it can be extended into the transfinite, a refinement inessential for our purposes here. The languages must be indexed by ordinals (or some other well-ordered collection of objects) where ${\mathcal {L}}_\alpha $ is an extension of ${\mathcal {L}}_\beta $ if and only if $\alpha > \beta $ – otherwise a version of the Yablo paradox arises, see Visser (2002).
16.
The use of the somewhat ambiguous term ‘semantic status’ is intentional, since one can read the additional value in Kripke’s construction as a ‘gap’ (sentences assigned this value receive no genuine truth value), a ‘glut’ (sentences assigned this value receive more than one genuine truth value), or in any number of other ways. Similar comments apply to the ‘Embracing Revenge’ semantics sketched below, which adds infinitely many such additional statuses. Different such readings will result in different classes of designated values, and hence different consequence relations. Although we prefer the ‘gappy’ reading, all of the arguments presented below affect only the semantics, and are thus compatible with consequence relations corresponding to ‘glutty’ readings (and other sorts of readings) of that semantics.
17.
The parallelism between this general reasoning and that of Tarski’s proof of his celebrated theorem on the indefinability of arithmetic truth in the language of arithmetic is, of course, not accidental.
18.
Cook’s most recent work on this topic has involved collaboration with Nicholas Tourville – see, e.g. Tourville and Cook (2016).
19.
Note that, in a Kripke/Martin-Woodruff style three-valued semantics, there is no monotonic binary truth-functional connective $*(A, B)$ such that $*(A, A)$ is always true (other than the trivial connective that outputs true for any arguments). Thus there is no truth-functional connective (other than the trivial one) $\rightarrow ^*$ such that, for any expression $\Phi $, $\Phi \rightarrow ^* \Phi $ is a theorem. Hence, for the Tarski T-scheme to even be expressible in a fixed-point language, we require a better, non-truth-functional conditional. For good, general assessments of these sorts of issues within Kripke’s fixed-point approach, see, Gupta (1982) and Hellman (1985).
20.
For a useful overview of these sorts of objections, see Priest (2010). We restrict our attention to objections relevant to the task at hand.
21.
Field goes further, and introduces a hierarchy of successively weaker negations $\lnot _\mathsf{Ch}\mathsf{D}$, $\lnot _{\ Ch}\mathsf{D}\mathsf{D}$, $\lnot _{\ Ch}\mathsf{D}\mathsf{D}\mathsf{D}$, etc. Although each of these is weaker than the next, none is equivalent to $\lnot _\mathsf{Ex}$, on pain of contradiction.
22.
Actually, Field resists the dilemma posed by what we are calling exclusion negation for the given fixed-point language, arguing that, since the preferred logic for handling the paradoxes is non-classical and excluded middle does not hold (on his paracomplete approach), what we have been calling exclusion negation (for a given fixed-point language) is not really a legitimate notion. This is a natural result for a paracomplete theory of the semantic paradoxes, since the acceptance of the inter-substitutivity of $\Phi $ and $\mathsf{T}(\ulcorner \Phi \urcorner )$ forces a renunciation of excluded middle for any notion of negation expressible in the language. And exclusion negation, by definition, obeys excluded middle. As we will see momentarily, however, there is an attractive way of transcending this situation.
23.
The technical details differ among Cook (2007) , Cook (2009), Schlenker (2010), and Tourville and Cook (2016). Here we will follow the most recent, and most powerful, formulation of the view – that found in Tourville and Cook (2016) – but all of the points made here also apply to the expressively weaker formal systems found in the earlier papers.
24.
We are indebted to Kit Fine for this observation.
25.
The alert reader might notice that the notation of this purported meta-liar is potentially problematic: the predicate should attribute truth to the result of concatenating the $\beta \mathrm{{th}}$-level negation (construed as a syntactic object) with the formal truth-predicate (followed by a free variable). As written, however, the negation sign is being used rather than merely mentioned, and, as will soon emerge, this turns out to be the key to ER’s immunity to any purported meta-Liar. Now, in the case where $\beta \le \alpha $ (and hence $\lnot ^\mathsf{W}_\beta \in \mathcal {L}_\alpha $), the predicate:
$$\begin{aligned} \lnot ^\mathsf{W}_\beta (\mathsf{T}(x)) \end{aligned}$$
is equivalent to the explicit formulation just described via an application of the relevant Tarski T-sentence. In such a case ($\beta \le \alpha $), the $\beta $th level exclusion negation is available in $\mathcal {L}_\alpha $ for use, so the disquotation is harmless. But of course, disquotation is entirely illegitimate (in fact, impossible!) when $\beta > \alpha $, as explained below.
26.
Hartry Field makes a similar move in Field (2003), from which we have brazenly stolen the idea (see also Field 2008).
27.
Of course, we could obtain something like a meta-Liar for $\mathcal {L}_\alpha $ by adding an operator $\mathsf{meta}_\alpha (x)$ such that:
- $\mathsf{meta}_\alpha (x)$ is true of x if and only if there is some weak negation $\lnot ^\mathsf{W}_\beta $ that can be modeled in $\mathcal {L}_\alpha $ such that $\lnot ^\mathsf{W}_\beta \mathsf{T}((x))$ is true of x.
- $\mathsf{meta}_\alpha (x)$ applied to x is false otherwise.
(Note that $\beta $ need not be less than $\alpha $). The sentence that results from diagonalizing on such a notion would express a version of the meta-Liar – albeit one relativized to $\mathcal {L}_\alpha $ – and would not be interpretable in terms of the semantics for $\mathcal {L}_\alpha $. The $\mathsf{meta}_\alpha (x)$-Liar would, however, be interpretable on the semantics for some $\mathcal {L}_\beta $ for sufficiently large $\beta $ – large enough such that any negation that can be modeled/mentioned in $\mathcal {L}_\alpha $ can be used on $\mathcal {L}_\beta $. But this still does not get us a genuine meta-Liar, since we can repeat the construction, adding a new ${\mathcal {L}}_\beta $-relative notion ${\mathsf{meta}_\beta }(x)$ such that:
- $\mathsf{meta}_\beta (x)$ is true of x if and only if there is some weak negation $\lnot ^\mathsf{W}_\gamma $ that can be modeled in $\mathcal {L}_\beta $ such that $\lnot ^\mathsf{W}_\gamma \mathsf{T}((x))$ is true of x.
- $\mathsf{meta}_\beta (x)$ applied to x is false otherwise.
But now we can construct a new $\mathcal {L}_\beta $-relativized meta-Liar, which (assuming that the set theory in $\mathcal {L}_\beta $ guarantees the existence of $\aleph _\gamma $ for any $\gamma < \beta $) cannot be interpreted in the semantics for $\mathcal {L}_\beta $, but can be interpreted on the semantics for any language high enough in the hierarchy such that it allows one to use any negation that can be mentioned/modeled in $\mathcal {L}_\beta $. And so on.
This construction is notable in that it allows us to make ‘large’ jumps in the hierarchy. On the standard revenge construction, adding the resources required to describe the semantics of a particular $\mathcal {L}_\alpha $ ‘pushes’ us up to $\mathcal {L}_{\alpha +1}$. Adding $\mathsf{meta}_\alpha (x)$ to $\mathcal {L}_\alpha $, however, will ‘push’ us to an $\mathcal {L}_\beta $ where $\beta $ is significantly larger than $\alpha +1$. In particular, $\beta $ must be greater than any ordinal $\gamma $ whose existence is guaranteed by the set theory contained in $\mathcal {L}_{\alpha }$. This is yet another reflection of the fact that the languages $\mathcal {L}_\beta $ whose semantics can be set-theoretically modeled in a particular $\mathcal {L}_\alpha $ far outstrip $\mathcal {L}_\alpha $ itself.
It is important to note, however, that the difference between the $\mathsf{meta}_\alpha (x)$ construction and more familiar revenge-style constructions is, in a sense, a difference of degree, not of kind – while $\mathsf{meta}_\alpha (x)$-constructions ascend the hierarchy of languages more quickly, the ascent is of the same sort, and in the end this is just one more sequence of increasingly strong revenge Liars.
28.
We assure you, the author of Cook (2007) did not have any explicit links to the MS view in mind when writing these passages, although that seems almost hard to believe in retrospect!
29.
One interesting question is whether the relationship is parasitic or symbiotic. In other words, is there some natural way to understand the modal structuralist account as parasitic on the embracing revenge account via obtaining $\mathsf{EP}_\mathsf{MS}$ as a consequence of $\mathsf{EP}_\mathsf{ER}$ plus some principle governing which set-theoretic structures exist relative to each language $\mathcal {L}_\alpha $? We set aside examination of this question for another time.
30.
We would like to thank Harty Field, Kit Fine, and Nicholas Tourville for helpful conversations on the issues raised in this paper.

References

Beall, J. (Ed.). (2007). Revenge of the liar: New essays on the liar paradox. Oxford: Oxford University Press.
Google Scholar
Cook, R. (2007). Embracing revenge: On the indefinite extendability of language. In Beall (2007), pp. 31–52.
Google Scholar
Cook, R. (2009). What is a truth value, and how many are there? Studia Logica, 92, 183–201.
Article MathSciNet Google Scholar
Ferreirós, J. (2007). The early development of set theory. In The stanford encyclopedia of philosophy. http://plato.stanford.edu/entries/settheory-early/.
Field, H. (2003). A revenge-immune solution to the semantic paradoxes. Journal of Philosophical Logic, 32(2), 139–177.
Article MathSciNet Google Scholar
Field, H. (2008). Saving truth from paradox. Oxford: Oxford University Press.
Book Google Scholar
Gabbay, D., & Guenthner, F. (Eds.). (2002). The handbook of philosophical logic (Vol. 11). Dordrecht: Kluwer Academic.
MATH Google Scholar
Gupta, A. (1982). Truth and paradox. Journal of Philosophical Logic, 11, 1–60.
Article MathSciNet Google Scholar
Hellman, G. (1985). Review of Martin and Woodruff, 1984, Kripke, 1984, Gupta, 1984b, and Herzberger, 1984. Journal of Symbolic Logic, 50(4), 1068–1071.
Article Google Scholar
Hellman, G. (1989). Mathematics without numbers: Towards a modal-structural interpretation. Oxford: Oxford University Press.
MATH Google Scholar
Hellman, G. (2011). On the significance of the Burali-Forti paradox. Analysis, 71(4), 631–637.
Article MathSciNet Google Scholar
Kripke, S. (1975). Outline of a theory of truth. Journal of Philosophy, 72, 690–716.
Article Google Scholar
Martin, R., & Woodruff, P. (1975). On representing true-in-L in L. Philosophia, 5(3), 213–217.
Article Google Scholar
Menzel, C. (1984). Cantor and the Burali-Forti paradox. The Monist, 67, 92–107.
Article Google Scholar
Moore, G., & Garciadiego, A. (1981). Burali-Forti’s paradox: A reappraisal of its origins. Historia Mathematica, 8, 319–350.
Article MathSciNet Google Scholar
Priest, G. (2010). Hopes fade for saving truth. Philosophy, 85(331), 109–140.
Article Google Scholar
Putnam, H. (1967). Mathematics without foundations. In Putnam (1975), pp. 43–59.
Google Scholar
Putnam, H. (1975). Mathematics, matter, and method: Philosophical papers (Vol. I). Cambridge: Cambridge University Press.
MATH Google Scholar
Schlenker, P. (2010). Super liars. Review of Symbolic Logic, 3(3), 374–414.
Article MathSciNet Google Scholar
Tarski, A. (1944). The semantic conception of truth. Philosophy and Phenomenological Research, 4(3), 341–376.
Article MathSciNet Google Scholar
Tourville, N., & Cook, R. (2016). Embracing the technicalities: Expressive completeness and revenge. Review of Symbolic Logic, 9(2), 325–358.
Article MathSciNet Google Scholar
van Heijenoort, J. (Ed.). (2002). A sourcebook in mathematical logic: 1879–1931. Cambridge: Harvard University Press.
MATH Google Scholar
Visser, A. (2002). Semantics and the liar paradox. In Gabbay and Guenthner (2002), pp. 149–240.
Google Scholar
Zermelo, E. (1930). Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre. Fundamenta Mathematicae, 16, 29–47.
Article Google Scholar

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University of Minnesota, Minneapolis, MN, USA
Geoffrey Hellman & Roy T. Cook

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Department of Philosophy, University of Minnesota Twin Cities, Minneapolis, MN, USA
Geoffrey Hellman
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Roy T. Cook

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Hellman, G., Cook, R.T. (2018). Extendability and Paradox. In: Hellman, G., Cook, R. (eds) Hilary Putnam on Logic and Mathematics. Outstanding Contributions to Logic, vol 9. Springer, Cham. https://doi.org/10.1007/978-3-319-96274-0_5

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DOI: https://doi.org/10.1007/978-3-319-96274-0_5
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