Abstract
In this paper I urge friends of truth-value gaps and truth-value gluts—proponents of paracomplete and paraconsistent logics—to consider theories not merely as sets of sentences, but as pairs of sets of sentences, or what I call ‘bitheories,’ which keep track not only of what holds according to the theory, but also what fails to hold according to the theory. I explain the connection between bitheories, sequents, and the speech acts of assertion and denial. I illustrate the usefulness of bitheories by showing how they make available a technique for characterising different theories while abstracting away from logical vocabulary such as connectives or quantifiers, thereby making theoretical commitments independent of the choice of this or that particular non-classical logic. Examples discussed include theories of numbers, classes and truth. In the latter two cases, the bitheoretical perspective brings to light some heretofore unconsidered puzzles for friends of naïve theories of classes and truth.
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Notes
- 1.
A nice example of confusion is David Lewis’ discussion of the orientation of Nassau Street in Lewis (1982).
- 2.
I have argued for them in some detail elsewhere, see Restall (2005). I do not take these considerations to be conclusive, but on the other hand, I have not seen any rival account of the norms of assertion and denial that is in any way a plausible alternative to this picture. I urge defenders of non-classical logic who take the assumptions I have made to be mistaken to develop an alternative account, explaining when the rules (Id) or (Cut) might fail, when governing assertion and denial, and what should be put in their place.
- 3.
Since p ⊢ p, q, we have by ( ¬L), p, ¬p ⊢ q.
- 4.
Since p ⊢ p, we have by ( ¬R), ⊢ p, ¬p, and by the disjunction rule, ⊢ p ∨ ∼ p.
- 5.
Since T is closed under consequence, that is not saying much of course. We could strengthen things by noting that the statements of the theory are undeniable, given commitment to the axioms of the theory, if the axioms X form a set from which all members of T follow.
- 6.
It might be thought that commitment to something like \(\ulcorner A \rightarrow \perp \urcorner \) would do it, where \(\ulcorner \perp \urcorner \) is to be rejected always. Perhaps that will express a feature appropriate for denial, but now the trouble is that it is too strong. In non-classical logics used for the paradoxes, \(\ulcorner A \vee (A \rightarrow \perp )\urcorner \) is rejected (and it must be, lest the liar paradox arise for the ‘negation’ of implying \(\ulcorner \perp \urcorner \)), so \(\ulcorner A\urcorner \) and \(\ulcorner A \rightarrow \perp \urcorner \) are to be rejected. But this means we must have some way of rejecting \(\ulcorner A\urcorner \) which does not involve accepting \(\ulcorner A \rightarrow \perp \urcorner \). So, \(\ulcorner A \rightarrow \perp \urcorner \) may express one kind of rejection, but it is not enough to express the entirety of the notion.
- 7.
Note, to reject the cotheory is not to reject the conjunction of its members, since the cotheory marks what is to be rejected. To reject it is to reject each member, just as to accept a theory is to accept each member of the theory.
- 8.
In sequent presentations of logic, that would be a direct consequence of \(\ulcorner x + 0 = x\urcorner \) by (\(\forall \) R), since x occurs nowhere else in the sequent: it is arbitrary.
- 9.
To accept that \(\vdash x + 0 = x\) is to take \(\ulcorner x + 0 = x\urcorner \) to be undeniable, so it tells us about what is not to be denied. For positive advice on what is to be denied, however, we need to look elsewhere.
- 10.
- 11.
In the absence of contraction or weakening as structural rules governing the conditional, it makes a difference as to whether the induction scheme is formulated as above, or as \(\phi (0) \rightarrow ((\forall x)(\phi (x) \rightarrow \phi (x{^\prime})) \rightarrow (\forall x)\phi (x))\) or as \((\forall x)(\phi (x) \rightarrow \phi (x{^\prime})) \rightarrow (\phi (0) \rightarrow (\forall x)\phi (x))\) or as a myriad of other formulations, each subtly different.
- 12.
This is not an idle worry: Curry’s paradox wreaks havoc with axiom (V), so in the presence of a conditional, the inference from \(\ulcorner p \rightarrow (p \rightarrow q)\urcorner \) to \(\ulcorner p \rightarrow q\urcorner \) is to be rejected (see Meyer et al. 1979). But then, \(\ulcorner p \rightarrow (p \rightarrow q)\urcorner \) and \(\ulcorner p \rightarrow q\urcorner \) express different conditional connections between \(\ulcorner p\urcorner \) and \(\ulcorner q\urcorner \). For the first, two instances of modus ponens are required to get from \(\ulcorner p\urcorner \) to \(\ulcorner q\urcorner \), for the second, one suffices. Which of these conditional notions is to be used in the statement of axiom (V)? This is a genuinely hard problem. Suppose I write \(\ulcorner p \rightarrow (p \rightarrow q)\urcorner \) as \(\ulcorner p {\rightarrow }_{2} q\urcorner \), and replace my theory expressed in terms of \(\ulcorner \rightarrow \urcorner \) with one expressed in terms of \(\ulcorner {\rightarrow }_{2}\urcorner \). Modus ponens holds for \(\ulcorner {\rightarrow }_{2}\urcorner \) as much as it does for \(\ulcorner \rightarrow \urcorner \). What changes? Which is the real conditional? In the absence of a wider semantic story, the difference is vacuous. Yet for the proponent of a non-classical logic, the difference is important, for if there was no difference, the theory is trivial. So, a wider semantic story of some kind must be told.
- 13.
Not only are there debates concerning contraction: there are also debates over relevance. Should the main conditional be taken to express a relevant connection, or should we weaken the condition to involve a prophylactic \(\ulcorner t\urcorner \)?
$$(\forall x)(\forall y)((\forall z)(z \in x \leftrightarrow z \in y) \wedge t \rightarrow x = y)$$Furthermore, does the identity \(\ulcorner x = y\urcorner \) entail \(\ulcorner (\forall z)(x \in z \leftrightarrow y \in z)\urcorner \) or is the connection here not relevance preserving? Options abound.
- 14.
In the rule (Ext ∈ ) we have the side condition that x is absent from Γ and Δ.
- 15.
The terms \(\ulcorner a\urcorner \) and \(\ulcorner b\urcorner \) denote classes, nothing else here. There is no implicit commitment to the effect that different numbers, electrons or tables must have different \(\ulcorner \in \urcorner \)-members.
- 16.
I have myself explored theories and models in which a kind of ‘naïve comprehension’ holds but in which ( ∈ L), ( ∈ R) fail. The simple LP-models of naïve comprehension (see Restall 1992) validate ‘extensionality’ in the weak form
$$a \in \{ x : \phi (x)\} \equiv \phi (a)$$where \(\ulcorner \equiv \urcorner \) is a material conditional. Here, models do not truly validate ( ∈ L) and ( ∈ R), for a class B in which everything both is and isn’t a member validates that material biconditional, doing the job for {x : ϕ(x)} for any predicate \(\ulcorner \phi \urcorner \). A material biconditional with one side ‘both’ true and false is, at least, true. In models in which B does the job of the empty set {x : ⊥ }, we have \(\ulcorner a \in B \equiv \perp \urcorner \) materially true, but we are prepared to assert \(\ulcorner a \in \{ x : \perp \}\urcorner \) but at the same time deny ⊥ . Here ( ∈ { L}) fails. The case for B standing in for the universal set is dual.
- 17.
This is clearly articulated by Read (2004).
- 18.
Yes, we must be careful of the nature of the context \(\ulcorner \phi (\cdot )\urcorner \) and the terms \(\ulcorner a\urcorner \) and \(\ulcorner b\urcorner \). Here there will be no such opaque contexts or non-rigid designators.
- 19.
Yes, we have kept track of assertion and denial. We have not committed ourselves to any particular theory of negation, or even the claim that our language has a single concept of negation. Just as we may be able to express a range of conditional notions, why not a range of negative notions? To think that there is one Russell set is to think that there is one negation.
- 20.
Actually, they use \(\ulcorner \perp \urcorner \), not an arbitrary \(\ulcorner p\urcorner \) used here. Nothing hangs on this, except the formulation here is slightly more general, designed to apply even in the case where we have no special statement taken to entail all others.
- 21.
Diagonalisation, demonstratives, or other devices give you ‘self-reference’ enough for this.
- 22.
How many truth values are there? Using (Ext \({}_{T\{\!\vert \vert \!\}}\)) it seems there are only two, since we can derive \(\vdash \{\!\vert A\vert \!\} =\{\! \vert B\vert \!\},\{\!\vert B\vert \!\} =\{\! \vert C\vert \!\},\{\!\vert C\vert \!\} =\{\! \vert A\vert \!\}\). Using the form of (Ext \({}_{T\{\!\vert \vert \!\}}\)) with weakening built in:
$$\Gamma ,T\{\vert A\vert \}\vdash T\{\vert B\vert \}, \Delta \quad \Gamma {^\prime},T\{\vert B\vert \}\vdash T\{\vert A\vert\}, \Delta {^\prime} \textrm{[\textit{Ext}]} \Gamma ,\Gamma {^\prime} \vdash \{\vert A\vert \} =\{ \vert B\vert \},\Delta ,\Delta {^\prime}$$simply to make the proofs narrow enough to fit on the page, we have
$$T\{\vert B\vert \},T\{\vert C\vert \}\vdash T\{\vert C\vert \},T\{\vert A\vert \} \quad T\{\vert A\vert \}\vdash T\{\vert C\vert \},T\{\vert A\vert \} {[\textit{Ext}]} T\{\vert B\vert \}\vdash T\{\vert C\vert \},T\{\vert A\vert \}, \{\vert C\vert \} =\{ \vert A\vert \} T\{\vert B\vert \},T\{\vert C\vert \}\vdash T \{\vert B\vert \} \rm{ [\textit{Ext}]} T\{\vert B\vert \}\vdash T\{\vert A\vert \}, \{\vert B\vert \} =\{ \vert C\vert \},\{\vert C\vert \} =\{ \vert A\vert \}$$and similarly, we can prove \(T\{\!\vert A\vert \!\}\vdash T\{\!\vert B\vert \!\},\{\!\vert B\vert \!\} =\{\! \vert C\vert \!\},\{\!\vert C\vert \!\} =\{\! \vert A\vert \!\}\), which together give us
$$T\{\vert A\vert \}\vdash T\{\vert B\vert \},\{\vert B\vert \} =\{ \vert C\vert \}, \{\vert C\vert \} =\{ \vert A\vert \}\qquad T\{\vert B\vert \}\vdash T\{\vert A\vert \}, \{\vert B\vert \} =\{ \vert C\vert \},\{\vert C\vert \} =\{ \vert A\vert \} {[\textit{Ext}]} \vdash \{\vert A\vert \} =\{ \vert B\vert \},\{\vert B\vert \} = \{ \vert C\vert \},\{\vert C\vert \} =\{ \vert A\vert \}$$In other words, of any three truth values, two are equal. To prove that there are at least two truth values, more must be done. I suggest finding sentences ⊤ and ⊥ such that \(\{\!\vert \top \vert \!\} =\{\! \vert \perp \vert \!\}\vdash \).
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Acknowledgements
Comments on this paper are very welcome. Please check the webpage http://consequently.org/writing/adnct to post comments and to read comments left by others. Thanks to the Logic Seminar at the University of Melbourne and the audience at wcp4 (including Jc Beall, Patrick Girard and Jerry Seligman), and to Michael De, for comments on this material, and to an anonymous referee, who helpfully pressed on certain points, and asked me to write up the little proof (in footnote 22) that there are at most two truth values. This research is supported by the Australian Research Council, through grant dp0343388, and Max Richter’s 24 Postcards in Full Colour.
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Restall, G. (2013). Assertion, Denial and Non-classical Theories. In: Tanaka, K., Berto, F., Mares, E., Paoli, F. (eds) Paraconsistency: Logic and Applications. Logic, Epistemology, and the Unity of Science, vol 26. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4438-7_6
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