Naïve set-theoretic dialetheist absolutism

2,641 words

Long ago, I wrote a thesis on paraconsistent approaches to absolute generality. Here’s a relatively self-contained bit, on paraconsistent naïve set theory.

1. Existing practice

Standard model-theoretic semantics regards quantifiers as ranging over set-domains. Thus, in a typical semantics textbook for linguists (Kearns, Semantics: 96) we read that

a quantifier expresses a relation between sets. For example, All ravens are black expresses [that]…the set of ravens is completely included in the set of black things.

To Studd, ‘[t]he obvious place to look for an account for what it is to quantify over a domain is to our best semantic theories’, and so-called mt-semantics (model-theoretic, i.e. ‘cast in set theory’) are the natural starting point (Studd, Everything: 61). mt-semantics has a number of purposes: to account for logical consequence and truth by generalisation over all interpretations; and in linguistics, to ‘shed light on natural language quantification’ (ibid.: 69). Studd defines two toy languages, ${\mathrm{ℒ︀}}_{\text{SU}}$ (the language of set theory) and ${\mathrm{ℒ︀}}_{\text{GQ}}$ (the language of generalised quantifiers). Since both ‘inherit intended interpretations from the corresponding fragments of English’, we can legitimately ask whether mt-interpretations ‘captur[e] the intended interpretations of the first-order language of set theory’.

A similar set-theoretic semantics can be given for ‘some’, ‘most’, and so on (following ibid.: §§ 3.1–2). Most absolutists, however, accept that there is a straightforward difficulty for mt-absolutism, namely that the availability of a universal set leads to Russell’s paradox. (Studd states this more formally but for brevity I omit details (ibid.: § 3.1–2) of what is a fairly uncontroversial conclusion.)

In naïve set theory, there is a universal set. Since the central conceit of dialetheism ‘is its ability simply to absorb certain contradictions’ (Priest, ‘Review of Absolute Generality’), it is only natural to first investigate whether dialetheist mt-absolutism is viable, by investigating the viability of paraconsistent naïve set theory. This conforms to the earliest concrete suggestion of a dialetheist defence of absolutism of which I am aware (ibid.).

The correct solution here is, I think, a dialetheic one. One of the great strengths of dialetheism is its ability simply to absorb certain contradictions; and, in particular, to provide a simple way of allowing a theory to specify its own semantics. …Paraconsistent set theory quantifies over all sets, and provides a set of all sets to play the role of the domain in a semantic interpretation for the language. All this is entirely natural: no new mechanisms and strategies have to be invoked to handle [absolutely unrestricted quantification].

Such a framework allows familiar set-theoretic reasoning. If, for example, more than half of a choir are men, and all men in the choir wear suits, we can reason that more than half of the choir wear suits, by consideration of the cardinalities of the set of members of the choir, men in the choir, and men wearing suits in the choir.

2. The inclosure schema and principle of uniform solution

Priest submits that there is already a well-established dialetheist means by which to respond to the sorts of paradoxes absolute generality appears to generate (ibid.).

Since the contradictions that arise are simple variations of standard paradoxes of self-reference, they all fit the Inclosure Scheme. …Given an interpretation, diagonalisation allows one to construct an object not in its domain (Transcendence); but the object is clearly in the domain of all quantifiable objects, since one quantifies over it (Closure). One should therefore expect inclosure contradictions to arise in this context.

According to the ‘principle of uniform solution’, the ‘same kind of paradox’ should be accorded the ‘same kind of solution’ (Priest, ‘On the Principle of Uniform Solution’: 123). This principle is disputed. For example, Badici has argued that Ramsey was right to distinguish logical and semantic paradoxes, and sought to undermine Priest’s ‘common kind’ claim (Badici, ‘The Liar Paradox and the Inclosure Schema’). Landini has questioned whether Priest accurately reconstructed his first example of an instance of the schema—Russell’s paradox—and suggested that the schema may not apply to ‘paradoxes of “definability”’ (Landini, ‘Russell’s Schema, Not Priest’s Inclosure’). Whether or not Priest is right about the inclosure schema is certainly of interest, but I do not propose to settle that question. What is important is that the principle of uniform solution applied to the inclosure schema is one important motive of Priest’s suggestion.

I shall instead content myself with showing that the Russell and Williamson–Russell reductiones indeed conform to the schema—a fairly routine, if novel, exercise.

Definition (Inclosure). $\mathrm{\Omega }$ is an inclosure just in case there exist properties $𝜑$ and $𝜓$ and a function $𝛿$ such that:

1. $\mathrm{\Omega }=\left\{𝑦:𝜑\left(𝑦\right)\right\}$ exists, and $𝜓\left(\mathrm{\Omega }\right)$; and
2. for any $𝑥\subseteq \mathrm{\Omega }$, if $𝜓\left(𝑥\right)$, then $𝛿\left(𝑥\right)\notin 𝑥$ and $𝛿\left(𝑥\right)\in \mathrm{\Omega }$.

Lemma. A contradiction is derivable from the existence of any inclosure.

Proof. Since $𝜓\left(\mathrm{\Omega }\right)$, $𝛿\left(\mathrm{\Omega }\right)\notin \mathrm{\Omega }$. $\square$

Claim. The Russell reductio conforms to the inclosure schema.

Let $𝜑$ hold just in case $𝑥$ is quantified over absolutely generally, and $\mathrm{\Omega }$ therefore be an absolutely general domain. Let $𝛿$ map a domain $𝐷$ to a collection ${𝑟}_{𝐷}$ comprising non-self-membered collections in $𝐷$. Finally, let $𝜓\left(𝐷\right)$ denote that ${𝑟}_{𝐷}$ (not $𝐷$) is not self-membered.

We now show each requirement. First, by absolute generality, $\mathrm{\Omega }$ exists. Second, ${𝑟}_{𝐷}$’s members are all non-self-membered. If ${𝑟}_{𝐷}$ is self-membered, then it is not a member of ${𝑟}_{𝐷}$, so ${𝑟}_{𝐷}$ isn’t self-membered. Hence $𝜓\left(𝐷\right)$ in general, and in particular $𝜓\left(\mathrm{\Omega }\right)$.

Third, take any domain (which must be a subset of the absolutely general domain). Then ${𝑟}_{𝐷}\notin 𝐷$, otherwise ${𝑟}_{𝐷}\in {𝑟}_{𝐷}$. But at the same time ${𝑟}_{𝐷}$ is a perfectly well-defined collection over which the absolutely general domain quantifies, so ${𝑟}_{𝐷}\in \mathrm{\Omega }$. $\square$

Claim. The Russell–Williamson reductio conforms to the inclosure schema.

Let $𝜑\left(𝑥\right)$ hold just in case $𝑥$ is quantified over absolutely generally, and $\mathrm{\Omega }$ be the absolutely general domain. Then let $𝛿$ map a domain $𝐷\subseteq \mathrm{\Omega }$ to an interpretation ${𝑖}_{𝐷}$ on which $𝑃$ applies to all those interpretations in $𝐷$ that do not self-apply $𝑃$. Finally, let $𝜓\left(𝐷\right)$ denote of a domain that ${𝑖}_{𝐷}$ does not self-apply $𝑃$.

We now show each requirement. First, by absolute generality, $\mathrm{\Omega }$ exists. Second, ${𝑖}_{𝐷}$ interprets $𝑃$ to apply only to interpretations that don’t self-apply $𝑃$. If ${𝑖}_{𝐷}$ self-applies $𝑃$, it does not self-apply $𝑃$; therefore, assuming no truth value gaps, ${𝑖}_{𝐷}$ does not self-apply $𝑃$. Therefore, $𝜓\left(𝐷\right)$. This applies to all $𝐷$; therefore, in general, $𝜓\left(𝐷\right)$ may be assumed; and, in particular, $𝜓\left(\mathrm{\Omega }\right)$.

Third, take any domain (which must be a subdomain of the absolutely general domain). Then ${𝑖}_{𝐷}\notin 𝐷$, otherwise ${𝑖}_{𝐷}$ self-applies $𝑃$. But at the same time it’s a perfectly reasonable interpretation over which to quantify, so ${𝑖}_{𝐷}\in \mathrm{\Omega }$. $\square$

Recapture redux. Both arguments above appear to be proofs by contradiction, which is slightly worrying; indeed, Priest seems to give an exactly parallel proof by contradiction in expounding Russell’s paradox (Priest, Limits: 142). However, by a result of Priest’s (ibid.: n. 7, p. 130 Priest, In contradiction: Theorem 0, § 8.6), for any classical argument $\mathrm{\Gamma }⊢𝜑$, there is a paraconsistent argument $\mathrm{\Gamma }⊩𝜑\vee 𝛽!$ for some $𝛽$. In this case, $𝜑$ is some dialetheia $𝛼!$, and so $\mathrm{\Gamma }⊩𝛼!\vee 𝛽!$, which itself is a contradiction.

One question for any dialetheist is when we should take a contradiction to be true. The schema gives a partial answer: if accepted, it gives a general class of true contradictions.

3. Recapitulation: the attractions of dialetheist absolutism via a universal set

We have at least three motives to consider a defence of absolutism via a universal set. The first is the familiarity of set-theoretic semantic theorising. The second is that the principal difficulty for that set-theoretic approach is alleged paradox, and paraconsistent approaches may alleviate that problem. The third is the principle of uniform solution.

4. Expectations of paraconsistent set theory

Non-negotiables. As stated above, we require a universal set. Non-triviality is non-negotiable. Finally, inferential strength of at least some level is also non-negotiable; a trivial proof theory on which nothing follows from anything else is non-trivial but also insufficient to carry out set theory. (As Incurvati puts it, a set theory should be ‘neither weak nor trivial’ (Incurvati, Conceptions of Set and the Foundations of Mathematics: 103).) Precisely how strong the proof theory and axioms should be is still up for debate.

Recapture? Studd warns against ‘radical reform [of] mathematical practice…set theory, or model-theoretic semantics’ (Studd, Everything: viii). Dialetheists wary of confrontation could attempt to make ‘substantial progress towards recovering classical set theory’.

Priest’s ambitions—logical revisionism but recapture of classical mathematics—are not too far from Studd’s demands (Priest, In contradiction: 221). Most radically, to Routley (and Weber, who approvingly cites him), classical mathematics should be ‘recoverable insofar as it is correct’ (Weber, Paradoxes and Inconsistent Mathematics: 98)—and there is no presupposition that it is, indeed, correct.

Therefore, dialetheism about sets leads to rather thorny further questions about attitudes to orthodox set theory and mathematics. On the other hand, non-set-theoretic articulations of absolutism, perhaps using new expressive resources, may not; and their dialetheist variants may not, therefore, face the same questions.

Naïveté. Naïve set theory, of the sort usually abandoned due to Russell’s paradox, comprises two fairly intuitive axioms: extensionality ($\forall 𝑥\forall 𝑦\left(\forall 𝑧\left(𝑧\in 𝑥\leftrightarrow ︎𝑧\in 𝑦\right)\leftrightarrow ︎𝑥=𝑦\right)$) and naïve comprehension ($\exists 𝑦\forall 𝑥\left(𝑥\in 𝑦\leftrightarrow ︎𝜑\left(𝑥\right)\right)$). (Some require that $𝑦$ should not be free in $𝜑$, whilst others do not; I shall ignore that detail (Incurvati, Conceptions of Set and the Foundations of Mathematics: 103).) Call the result of this nst—naïve set theory.

To follow. Even diehard dialetheists admit that current developments in paraconsistent naïve set theory are not too encouraging (Weber, Paraconsistency in Mathematics: §§ 2.1–2 Priest, ‘Review of The Many and the One’). However, no definitive argument against all ways of developing a paraconsistent naïve set theory has, to my knowledge, been developed or formulated. In order to justify my own pessimism about naïve set-theoretic dialetheist absolutism, I shall rather rapidly follow Incurvati’s fairly recent survey of the literature (Incurvati, Conceptions of Set and the Foundations of Mathematics). The conclusion I reach, however, is quite provisional.

5. The material and relevant strategies

For Russell’s paradox to avoid triviality, we need to avoid ex contradictione quodlibet. (Of course, that is compatible with holding that anything follows from some contradictions.) But that is not all.

For example, suppose that $\to$ is read to validate contraction: $𝜑\to \left(𝜑\to 𝜓\right)⊢𝜑\to 𝜓$, and, in addition, modus ponens. Then triviality follows (ibid.: 103). Indeed, modus ponens and the axiom $⊢𝜑\wedge \left(𝜑\to 𝜓\right)\to 𝜓$ also lead to triviality (ibid.: 104).

The natural question is whether any meaningful set theory at all can be carried out under such constraints. Two strategies naturally present themselves. The material strategy takes $𝛼\to 𝛽$ to be defined as $\neg 𝛼\vee 𝛽$ and weakens the logic sufficiently that $𝛼,\neg 𝛼\vee 𝛽⊬𝛽$. The relevant strategy rejects contraction.

6. The material strategy

ibid.: § 4.3.

nlp combines the axioms of nst with Priest’s logic lp. It is a fairly natural starting point. The essential revision is that disjunctive syllogism fails: $𝜑,\neg 𝜑\vee 𝜓⊬𝜓$. However, not only is nlp’s consequence relation restricted from application of modus ponens on pain of triviality; the transitivity of the material conditional and biconditional are also ruled out. Consider a valuation $𝑣\left(𝜑\right)=𝑇,𝑣\left(𝜓\right)=𝐵,𝑣\left(𝜒\right)=𝐹$; then $𝑣\models 𝜑\to 𝜓,𝜓\to 𝜒$ but $𝑣⊭𝜑\to 𝜒$ (ibid.: 107). This rules out rather a lot of classical reasoning. Moreover we cannot prove such claims as the indiscernibility of identicals: $𝜑\left(𝑥\right),𝑥=𝑦⊬𝜑\left(𝑦\right)$. A dialetheist could claim that, in a contest of intuitions, naïve set-theoretic axioms should take priority over the indiscernibility of identicals, but such a claim seems tenuous at best (ibid.: 109). Priest has little hope for nlp, and I am not one to doubt him (Priest, In contradiction: 250).

Minimal inconsistency (Incurvati, Conceptions of Set and the Foundations of Mathematics: 109–10). Priest has proposed lpm as the background logic. Its consequence relation is stronger, and,

[i]n particular, every classical consequence of a consistent set of premisses is an lpm consequence, since in consistent situations the minimally inconsistent models are just the classical models.

The result is nlpm (naïve set theory with lpm). Unfortunately,

Thomas (Thomas, ‘Expressive limitations of naïve set theory in LP and minimally inconsistent LP’: § 4) has shown that nlpm’s only model is one in which $\forall 𝑥\forall 𝑦\left(𝑥\in 𝑦\wedge 𝑥\notin 𝑦\right)$ holds and identity is standard.…Moreover, nlpm has only one model up to isomorphism, namely a one-element model.

Various intermediate options have also been explored (Incurvati, Conceptions of Set and the Foundations of Mathematics: 111).

In particular, three alternative versions of lpm have been developed by Marcel Crabbé (Crabbé, ‘Reassurance for the logic of paradox’). Using one of them as background logic, the situation is exactly the same as for nlpm… Using the other two alternative versions of lpm as background logic we avoid almost triviality but at the price of not improving upon lp${\begin{array}{}\end{array}}_{=}$ as the background logic: the resulting naïve set theories cannot prove the existence of sets that behave like singleton sets, sets that behave like ordered pairs, and sets that behave like infinitely ascending linear orders.

Accordingly, I provisionally conclude that Incurvati is right: ‘the prospects for…the material strategy look rather dim, and those for developing it on the basis of lp and cognate systems even dimmer’ (Incurvati, Conceptions of Set and the Foundations of Mathematics: 111).

7. The relevant strategy

ibid.: § 4.4.

The relevant strategy takes the conditionals in the axiomatisation of nst to validate modus ponens. There is a candidate non-trivial theory in which a reasonable number of constructions familiar from standard set theory can be carried out: Weber’s ndlq (Weber, ‘Transfinite cardinals in paraconsistent set theory’). The central problem is that the path to the underlying logic, dlq, appears tortured and ad hoc (Incurvati, Conceptions of Set and the Foundations of Mathematics: 117).

Obviously the Counterexample Rule [$𝜑,\neg 𝜓⊢\neg \left(𝜑\to 𝜓\right)$] appears prima facie plausible… But then, disjunctive syllogism also appears prima facie plausible, but almost no paraconsistent logic can include it, on pain of triviality.

Weber argues that ‘we want the strongest [in terms of deductive strength] logic possible that does not explode when given a comprehension principle’ (Weber, ‘Transfinite cardinals in paraconsistent set theory’: 73). As Incurvati points out, however, some dialetheist strategies are both deductively incommensurable (each has deductive consequences absent in the other) and incompatible (on pain of triviality). There is no obvious reason to prefer one to the other. Finally, ‘current attempts to provide it with a genuine principle of extensionality fail, so that it cannot be regarded as a set theory’ (Incurvati, Conceptions of Set and the Foundations of Mathematics: 121).

Moreover, to the extent that a fuller programme of mathematics grounded in nst in ndlq can be worked out (for reasons for optimism, see Weber, Paradoxes and Inconsistent Mathematics), it is unclear what to make of it. For example, indispensability arguments (see e.g. Colyvan, Indispensability) do not appear to apply to abstract objects as conceived of in the resulting revisionist mathematics, since natural science appeals to orthodox, not revisionist, mathematics. Perhaps that is simply an unfair artefact of history—but we can have little faith in that defence of revisionist mathematics unless a substantial programme of natural science is carried out using revisionist mathematics, which prospect, alas, remains distant.

8. The model-theoretic strategy

Incurvati, Conceptions of Set and the Foundations of Mathematics: § 4.5.

A third strategy is to argue that ‘in a context where no sentence is true and false, [dialetheists] may make use of classical logic’; therefore, ‘the dialetheist can also restrict attention to a consistent subdomain of the naïve universe of sets, and use classical logic in deriving things from axioms describing that subdomain’ (ibid.: 121).

One problem is that we must either accept only ‘a proper segment of [a] given model of zf’ or identity that behaves ‘non-standardly’ (ibid.: 124)—neither of which is particularly desirable. Another is that the strategy delivers merely ‘a particular interpretation of nst which contains a model of zfc’, which ‘does not tell us that the true interpretation of set theory enjoys this property’ (Meadows, ‘Unpicking Priest’s Bootstraps’: 182) so far, the only reason to believe that this is so is that it is ‘an appealing picture’ (Priest, In contradiction: 257). There remains much work to do in following this strategy.

9. Conclusion

Whether or not all happy set theories are alike (Hamkins proposes that there is a ‘set-theoretic multiverse’ (Hamkins, ‘The set-theoretic multiverse’)), each paraconsistent set theory we have encountered has been unhappy in its own way. Two particularly common difficulties arise: triviality (or near-triviality); and adhocness. Studd’s verdict, then, seems to be right. However, naïveté has been assumed throughout. That is natural—a defence of absolutism via a universal set can hardly proceed without one. In classical logic, naïveté cannot be wholly accepted—a universal set cannot coëxist with the usual form of comprehension.

References

Badici, Emil, ‘The Liar Paradox and the Inclosure Schema’, Australasian Journal of Philosophy 86.4 (2008), pp 583–596.

Colyvan, Mark, Indispensability: The indispensability of mathematics, An Oxford University paperback, Oxford University Press, 2001, 172.

Crabbé, Marcel, ‘Reassurance for the logic of paradox’, The Review of Symbolic Logic 4.3 (2011), pp 479–485.

Hamkins, Joel David, ‘The set-theoretic multiverse’, The Review of Symbolic Logic 5.3 (2012), pp 416–449.

Incurvati, Luca, Conceptions of Set and the Foundations of Mathematics, Cambridge University Press, 2020, 238.

Kearns, Kate, Semantics, ed , Modern linguistics series, Palgrave Macmillan, 2011, 269.

Landini, Gregory, ‘Russell’s Schema, Not Priest’s Inclosure’, History and Philosophy of Logic 30.2 (2009), pp 105–139.

Meadows, Toby, ‘Unpicking Priest’s Bootstraps’, Thought: A Journal of Philosophy 4.3 (2015), pp 181–188.

Priest, Graham, Limits: Beyond the limits of thought, Cambridge University Press, 1995, 274.

Priest, Graham, ‘On the Principle of Uniform Solution’: ‘On the Principle of Uniform Solution: A Reply to Smith’, Mind 109.433 (2000), pp 123–126, jstor: 2660000.

Priest, Graham, In contradiction: In contradiction: a study of the transconsistent, expanded edition, Oxford University Press, 2006, 327.

Priest, Graham, ‘Review of Absolute Generality’, Notre Dame Philosophical Reviews (2007).

Priest, Graham, ‘Review of The Many and the One’: ‘Review of The Many and the One: a Philosophical Study of Plural Logics’, Notre Dame Philosophical Reviews (2022).

Studd, J.P., Everything: Everything, more or less: A defence of generality relativism, Oxford Philosophical Monographs, Oxford University Press, 2019, 304.

Thomas, Nick, ‘Expressive limitations of naïve set theory in LP and minimally inconsistent LP’, The Review of Symbolic Logic 7.2 (2014), pp 341–350.

Weber, Zach, ‘Transfinite cardinals in paraconsistent set theory’, The Review of Symbolic Logic 5.2 (2012), pp 269–293.

Weber, Zach, Paradoxes and Inconsistent Mathematics, Cambridge University Press, 2021.

Weber, Zach, Paraconsistency in Mathematics, Elements in the Philosophy of Mathematics, Cambridge University Press, 2022.

Related

absolute generality • dialetheism • logic • philo • set theory • TCSP