Abstract
We discuss logical difficulties with the naive set theory based on the weak relevant logic DKQ. These are induced by the restrictive nature of the relevant conditional and its interaction with set theory. The paper concludes with some possible ways to mitigate these difficulties.
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Notes
- 1.
Brady (2006) argues for a weak relevant logic without the law of excluded middle (LEM). The loss of LEM in a system already deprived of so many logical inferences makes it very difficult to prove much of anything. Many theorems in Weber’s work explicitly rely on it.
- 2.
MR2 can also be derived in a similar way, using the deduction theorem, A16, and A17.
- 3.
Note the difference between a non-constructive proof and a non-constructed proof. The former is still a visible argument, the latter is not. Even an infinitary proof procedure offers the promise of some kind of visible proof.
- 4.
There’s also the problem that the usual proofs of a deduction theorem for a logic rely on the conditional being capable of contraction.
- 5.
This is a strong reason why the presentation of DKQ is easier in a Hilbert-style axiomatic system: a natural deduction formalization of DKQ would contain a \(\rightarrow _I\) rule that did not intuitively correspond to our usual understanding of conditional proof. Such a natural deduction system can be found in Brady (2006, Chap. 3).
- 6.
It is not clear what the status of \(\lnot (A \rightarrow B) \vdash (A \wedge \lnot B)\) is. A naive notion of relevance implies that \(A \rightarrow B\) can fail to hold because A was not relevant to B rather than it being because A was true and B was false. This rule is not included in DKQ.
Note that in some nonclassical logics (e.g., Nelson’s constructive logics), \(A \wedge \lnot B\) is equivalent to \(\lnot ( A \rightarrow B)\), without forcing the conditional to behave materially. In the context of DKQ, the negation is De Morgan and we have Double Negation Elimination, so materiality of the conditional does follow.
- 7.
We also have evidence in the form of relevant Peano Arithmetic that PA can get away without weakening. This theory has been explored and seems capable of proving most of what we need in doing mathematics. This suggests that the loss of weakening did not inflict too grievous a wound. That said, it was found to not recover all of PA and was thus largely abandoned. See Friedman and Meyer (1992) for more on relevant arithmetic. It is worth noting that the work on this is notoriously difficult to find as a lot of it is unpublished.
- 8.
Note that when we name sets by their extension, we do not know their intension. This greatly reduces what the relevant conditional can do.
- 9.
This also suggests we intuitively think of the conditional that manages set equality as material and truth-functional. For a particular \(z \in B\) that A shares, we have \(z \in A\) and use disjunction introduction to get \(z \notin B \vee z \in A\). On the other hand, for any \(z \notin B\) we can still get \(z \notin B \vee z \in A\). These two observations together gives \(\forall z(z \notin B \vee z \in A)\).
- 10.
And the contraposition fares no better, since that would require a direct proof from \(\lnot (C \rightarrow C)\) to \(x \not = 0\).
- 11.
It also seems difficult in DKQ to produce proofs of propositions of the form \(A \rightarrow (B \rightarrow C)\).
- 12.
That is, by assuming \(z \in x\), we literally have no access to the defining property of the set x. The relevant conditional does not cooperate well in these cases.
- 13.
This appears to be another example where weakening plays a unique role in set theory.
- 14.
Counting defined in the way we normally think of as in set theory, by mappings. The fact that \(R \not = R\) should not suffice to generate injective mappings from arbitrarily sized sets which demonstrate that the singleton set \(\{R\}\) has more than 1 set inside it. For example, there should not exist an injective mapping from a set containing two obviously distinct things like \(\emptyset \) and \(\{\emptyset \}\) to the singleton set \(\{R\}\). The weak negation applied to equality fails to show that two sets are actually “distinct” in a meaningful way with regard to counting.
- 15.
Another conditional will take us beyond what is covered in Brady’s non-triviality proof.
- 16.
Relevant restricted quantification would likely be useful for relevant naive set theory, however the proper framework for it isn’t yet clear and adding it in would take us further from Brady’s non-triviality result. However, we can achieve some of the properties that would be wanted from relevant restricted quantification with a new arrow. More information on restricted relevant quantification can be found in Beall et al. (2006).
- 17.
This is not to say that including it does not expose us to any danger of finding ourselves with a trivial naive set theory. Standard relevant semantics suggest that the addition of the t constant with its usual introduction of rule of \(\vdash t\) may make it possible to prove \(((A \rightarrow B) \wedge A \wedge t) \rightarrow B\). Then one could use a pseudo modus ponens type proof of Curry’s paradox with a slight adjustment to the definition of the Curry set: \(\{x | (x \in x \rightarrow p) \wedge t\}\). Thanks to Edwin Mares for showing me this revenge Curry.
- 18.
The hardest part in adding such a conditional in the present context is that it must be contraction free. See Beall and Murzi (2013) for further exposition on how completely we need to avoid contraction.
- 19.
We cannot have conjunctive syllogism without also validating the proof of Curry’s that uses pseudo modus ponens.
- 20.
These can be formalized in the usual set theory as by tagging each element of a set as distinct. For example the multiset \(\{A, A\}\) would be \(\{ \langle 0, A \rangle , \langle 1, A \rangle \}\) as a normal set.
- 21.
As with the other meta-rules, this meta-rule has an inherently non-constructive nature.
References
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Istre, E., McKubre-Jordens, M. (2019). The Difficulties in Using Weak Relevant Logics for Naive Set Theory. In: Başkent, C., Ferguson, T. (eds) Graham Priest on Dialetheism and Paraconsistency. Outstanding Contributions to Logic, vol 18. Springer, Cham. https://doi.org/10.1007/978-3-030-25365-3_17
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