Abstract
Closure is the idea that what is true about a theory of truth should be true (and therefore expressible) in it. Commitment to closure under truth motivates non-classical logic; commitment to closure under validity leads to substructural logic (nontransitive or noncontractive). These moves can be thought of as responses to revenge problems. With a focus on truth in mathematics, I will consider whether a noncontractive approach faces a similar revenge problem with respect to closure under provability, and argue that if a noncontractive theory is to be genuinely closed, then it must be free of all contraction, even in the metatheory.
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Notes
Routley: “The liberating effect of giving up the classical faith...is immense: ...one is free to return to something like the grand simplicity of naive set theory, to semantically closed natural languages (having abandoned the towering but ill-constructed and mostly unfinished hierarchies of formal languages), and to intuitive accounts of truth, of proof, and of many other intensional notions” (Routley 1979, p. 302).
There is plenty of subtlety to argue over here (Cook 2014). Some version of this problem has long been known in relevant logic, where one can endorse rule modus ponens, \(\varphi , \varphi \rightarrow \psi \vdash \psi \), but must deny the axiom of ‘pseudo modus ponens’, \(\vdash \varphi \wedge (\varphi \rightarrow \psi ) \rightarrow \psi \) (Robert 1978; Restall 1993).
The main claims of this paper harmonize nicely with the recent results in Wansing and Priest (2015).
Notation: \(\varphi , \psi , ...\) are formulas and \(\Gamma , \Delta , ...\) are multisets of formulas. The left side of a sequent is a multiset; for the right, throughout I will be discussing a single conclusion consequence relation.
Lakatos’ name for this story about mathematics (Lakatos 1978).
Here I’ve written the properties horizontally, following Humberstone (2011, p. 55), but you could display them Gentzen style taking ‘if...then’ as a vertical prooftree. The names I, B, K, and C are from combinatory logic.
Without both the law of excluded middle and ex falso quodlibet, the already questionable material conditional \(\varphi \supset \psi := \lnot \varphi \vee \psi \) becomes obsolete: either \(\varphi \supset \varphi \) fails, or \(\varphi , \varphi \supset \psi \vdash \psi \) does. Decades of work have been expended on re-founding a theory of the conditional. See Beall (2013). Perhaps a lot of what is to follow could be avoided by rejecting an operator that obeys \(\Rightarrow \)-introduction, or a similar device like a validity predicate (Shapiro 2011; Chvalovský and Cintula 2012). We will work on the assumption that closure urges otherwise, as in Weber (2014), Caret and Weber (2015).
By transitivity, contraposition holds in the form \(\varphi \Rightarrow \psi \vdash (\psi \Rightarrow \bot ) \Rightarrow (\varphi \Rightarrow \bot )\), but not in the form \(\varphi \Rightarrow \psi \vdash \lnot \psi \Rightarrow \lnot \varphi \). So even if we added all the de Morgan laws, argument by cases \(\varphi \Rightarrow \chi , \psi \Rightarrow \chi \vdash \varphi \vee \psi \Rightarrow \chi \) can hold even though \( \chi \Rightarrow \varphi , \chi \Rightarrow \psi \vdash \chi \Rightarrow \varphi \& \psi \) does not.
It may be that, for logics that don’t validate contraction, it is equally (though perhaps less obviously) implausible that they could support Euclideanism. We take this up in Sect. 5 below.
Lots of mathematics in practice involves reasoning from suppositions that are not proven, and repeating these suppositions, as a referee points out. But at the end of the day, either the supposition will end up being proven, in which case it is a theorem and contracted away; or else the proven fact is a conditional (‘assuming \(\varphi \) and ... and \(\varphi \), then \(\psi \)) and in these cases the number of repetitions will remain uncontracted.
Here is proof of reductio (with \(\Gamma \)s omitted):
One hypothesis to explain the differences between nontransitive and noncontractive approaches is that they have different targets. A look at the literature suggests that one target—truth as a metaphysical property? a concept? a word?—is not shared by all; for a breakdown of the divergence just within deflationist debates, see Wyatt (2016); cf. Scharp (2013, Ch 1).
The reader may compare this section with Wansing and Priest (2015).
Genuine provability is a predicate, not just an operator, but it will save us quine-quotes to ignore that here.
On the other hand, the ‘contract-on-theorems’ idea is to delete only theorems that really are theorems, not just theorems under assumption, as in \(\Box W_1\). We return to this point in Sect. 5.
The nicknames follow linear logic usage, e.g. Terui (2004). The monoidalness axiom, called K, is not to be confused with the monotonicity or weakening rule, called K.
Assuming, as we have been, that having arbitrarily many copies of axioms is not problematic.
One way to follow this out: invoke Grice’s maxims of manner and quantity, to explain why we are or are not explicitly endorsing everything a logic tells us. (Gricean moves are equally available to the nontransitivist, too.) But this is not to be conflated with saying that we can contract on theorems, only using extra-logical resources (cf. Beall 2013). If that kind of response worked here, I suspect it would have worked in Tarskian apologia for non-semantically closed truth theories to begin with.
Thanks to a referee here.
From conversation with Dave Ripley, in Melbourne 2011.
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Acknowledgments
Thanks to audiences at the University of Kyoto, the University of Victoria (Wellington), Université catholique de Louvain, and Ruhr-universität Bochum. Thanks to Guillermo Badia, Petr Cintula, Erik Istre, Hitoshi Omori, Jeremy Seligman, Peter Verdée, and Heinrich Wansing. Thanks especially to the editor of this special issue, and two anonymous referees for great feedback. Funding was provided by the Marsden Fund, Royal Society of New Zealand.
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Weber, Z. On closure and truth in substructural theories of truth. Synthese 199 (Suppl 3), 725–739 (2021). https://doi.org/10.1007/s11229-016-1226-6
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DOI: https://doi.org/10.1007/s11229-016-1226-6