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Strengthening the Russellian argument against absolutely unrestricted quantification

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Abstract

The Russellian argument against the possibility of absolutely unrestricted quantification can be answered by the partisan of that quantification in an apparently easy way, namely, arguing that the objects used in the argument do not exist because they are defined in a viciously circular fashion. We show that taking this contention along as a premise and relying on an extremely intuitive Principle of Determinacy, it is possible to devise a reductio of the possibility of absolutely unrestricted quantification. Therefore, there are intuitive reasons to believe that the counter-argument fails to support the possibility of absolutely unrestricted quantification.

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Notes

  1. In a sense, the idea that the paradoxes of set theory suggest a kind of indefinite extensibility of the universe of sets goes back to Zermelo (1930).

  2. For a dissenting voice regarding this point, see Rayo (2021).

  3. E.g. Studd (2019).

  4. This formulation substitutes intensional entailment for quantification in the expression of relativism to the end of avoiding the use of the same generality whose possibility relativism denies. We cannot lay out the details here but in this we follow (Luna, 2008; Luna & Taylor, 2010).

  5. Maybe, strictly speaking, a definition of \(R_{U}\) would be something like \(R_{U} = \left\{ {x|x \notin x} \right\}\) or \(R_{U} = \iota x\left[ {\forall y\left[ {y \in x \leftrightarrow y \notin y} \right]} \right]\), where ‘\(\iota\)’ is Russell’s definite description operator. I beg the reader to allow me the freedom of taking for simplicity instances of comprehension as definitions.

  6. Other Russellian objects can be used to this end, as in the argument devised (though not endorsed) by Williamson (2003), which is laid out and discussed in the “Appendix”.

  7. Though the late Laurence Goldstein did not write this passage in the context of a discussion of absolutism, he later used it (in personal communication with the author) against the Russellian argument for relativism. This piece is for the most part a response to his argument, even if one that comes far too late.

  8. Email to the author on 7/28/2012 19:51 UTC + 02:00.

  9. Note the difference between determinacy and decidability: even if (Con-P) and (EM-P) apply for every \(x\) in \(u\), there may still be some \(x\) in \(u\) for which \(P\left( x \right)\) is not humanly decidable.

  10. It goes without saying that we leave aside all along fuzzy classes. The membership relation of non-fuzzy classes is affected by no vagueness; as a consequence, it is subject not just to the logical law of Contradiction but to the law of Excluded Middle as well. So, we suggest to read (Con-\(\in\)), (EM-\(\in\)), and (B) below in Inference 3 as relativized to non-fuzzy classes. Alternately, we can exclude all fuzzy classes from \(U\) and define it as a universe extendable by no object other than fuzzy classes.

  11. The possible objection that the circularity of (C) need not make (B) circular because \(R_{U}\) could fail to exist, so that (C) may be no instance of (B) is unwarranted, since there are circular definitions attempting at defining objects that for whatever reason do not exist; those definitions are such that if the object purportedly being defined existed, the definition would be circular. No one would argue about the following definition (and leaning on the fact that its first clause ensures the nonexistence of the object purportedly defined) that it is not circular due to its second clause: \(\iota x\left[ {x \ne x \& \forall y\left[ {y \in x \leftrightarrow y \in x} \right]} \right]\). The case is the same with respect to (B): if \(R_{U}\) existed, (C) would render (B) circular; therefore, (B) is circular.

  12. For simplicity, we can assume that if \(x\) is not an interpretation of \(L\), then for all \(y\), \(x{ \nVdash }P\left( y \right)\).

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  1. IES Doctor Francisco Marin, Siles, Jaén, Spain

    Laureano Luna

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Appendix: Williamson’s argument

Appendix: Williamson’s argument

We briefly explain Williamson’s argument, show how the Goldsteinian Circularity Counter-argument would go and how it can be harnessed for a reductio of absolutely unrestricted quantification.

Let \(L\) be a first-order language with at least one predicate letter \(P\); let \(U\) be an absolutely inextensible universe of discourse; as such \(U\) contains all interpretations of \(L\); let \(I^{*}\) be an interpretation of \(L\) over \(U\) such that for each interpretation \(i\) of \(L\) in \(U\),

$$\left( {\text{E}} \right)I^{*} { \Vdash }P\left( i \right)\;{\text{iff}}\;i\,{ \nVdash }\,P\left( i \right),$$

which is the Russellian clause. As \(I^{*}\) is in \(U\), this Russellian contradiction ensues:

$$\left( {\text{F}} \right)I^{*} { \Vdash }P\left( {I^{*} } \right)\;{\text{iff}}\;I^{*} { \nVdash }P\left( {I^{*} } \right).\;$$

So, if we assume an inextensible \(U\), we are led to a contradiction. This is Williamson’s Russellian argument against absolutely unrestricted quantification.

Let us look for the circularity in the definition of \(I^{*}\). Note that we have not really defined \(I^{*}\) but only its Russellian clause; it is all we really need: (F) as part of the definition of \(I^{*}\) is no less circular than (C) as part of the definition of \(R_{U}\). So, it is unable to determine whether \(I^{*} \in I^{*} \left( P \right)\) or not.

On the other hand, it follows from the very definition of interpretation that the relation \(x{ \Vdash }P\left( y \right)\) complies withFootnote 12

$$({\text{Con}} - { \Vdash })\forall xy \sim \left[ {x{ \Vdash }P\left( y \right) \& x{ \nVdash }P\left( y \right)} \right];$$
$$({\text{EM}} - { \Vdash })\;\forall xy\left[ {x{ \Vdash }P\left( y \right) \vee x{ \nVdash }P\left( y \right)} \right].$$

Therefore, our argument in section V applies: assuming its premises, if there is an inextensible universe of discourse \(U\), then both (E) is and is not affected by indeterminacy due to circularity.

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Luna, L. Strengthening the Russellian argument against absolutely unrestricted quantification. Synthese 200, 182 (2022). https://doi.org/10.1007/s11229-022-03672-4

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