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Pluralism and Proofs

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Abstract

Beall and Restall’s Logical Pluralism (2006) characterises pluralism about logical consequence in terms of the different ways cases can be selected in the analysis of logical consequence as preservation of truth over a class of cases. This is not the only way to understand or to motivate pluralism about logical consequence. Here, I will examine pluralism about logical consequence in terms of different standards of proof. We will focus on sequent derivations for classical logic, imposing two different restrictions on classical derivations to produce derivations for intuitionistic logic and for dual intuitionistic logic. The result is another way to understand the manner in which we can have different consequence relations in the same language. Furthermore, the proof-theoretic perspective gives us a different explanation of how the one concept of negation can have three different truth conditions, those in classical, intuitionistic and dual-intuitionistic models.

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Notes

  1. For an exposition of this kind of view, see “Negation in Relevant Logics” (Restall 1999).

  2. Bridges and Richman’s Varieties of Constructive Mathematics (1987) contains a helpful outline of the different schools of constructive mathematics. Intuitionism, Russian Constructivism and Recursive Analysis are all theories which can be thought of as either providing different concflicting theories of real numbers, or theories governing distinctive kinds of numbers—such as recursive reals. Bishop-style constructive mathematics is explicitly constructively reasoning about mathematical objects—objects about which we may (if we wish) reason classically, too.

  3. A referee points out that this is something to which all friends of intuitionist logic can agree. Since \(\vdash \lnot (\lnot \lnot p\wedge \lnot p)\), if we have a transparent theory of truth, the intuitionist can endorse \(\lnot (T\langle \lnot \lnot p\rangle \wedge \lnot T\langle p\rangle )\). It is never the case that \(\lnot \lnot p\) is true while \(p\) is not. This is a particularly sharp way of making the point.

  4. There are more motivations than this. One motivation we do not discuss much is the treatment of incompleteness and indeterminacy. We can take it that the tautology \(p\vee \lnot p\) is not settled by each and every thing, but nonetheless take it that \(p\vee \lnot p\) is of necessity made true by something. Is \(p\vee \lnot p\) a logical truth? Depends on the criterion. Not every restricted part of the world makes each instance of \(p\vee \lnot p\) true for it might not involve the feature which makes \(p\) true or rules it out. However, every world involves such a feature (Restall 1996, 2005). (Of course, this phenomenon is not restricted to claims involving negation. We might think, for example, that the tautology \(E\)!\(a \supset a=a\)—if \(a\) exists then it is self identical—is made true, if \(a\) exists, by parts of the world featuring \(a\).)

  5. ‘Situations’ in Logical Pluralism (Beall and Restall 2006) allow for truth value gaps and truth value gluts. Here we will consider only truth value gluts in these situations. They are closer, on this conception, to the inconsistent worlds of “Ways Things Can’t Be” (Restall 1997), which are collections of consistent and complete worlds considered conjunctively.

  6. Of course, there are examples of arguments which are classically valid and neither intuitionistically or dual intuitionistically valid. A simple example is the argument from \(p\wedge \lnot p\) to \(q\vee \lnot q\).

  7. If we allow for two kinds of propositions, one, genuinely hereditary, and another, not heredity, we would have a way for \(-\) to ‘coexist’ after a manner of speaking with \(\mathord {\smile }\), following the intuitionist clause, and \(\mathord {\frown }\), following the dual-intuitionist clause. However, the information expressed by means of the classical clause for ‘\(-\)’ is no longer hereditary, and it is not suitable for use in any context in which this hereditary information is necessary.

  8. And once one is bilingual, we can consider mixed statements. We have, for example, \(\mathord {\smile }A\vdash \mathord {\frown }A\), since if \(x\Vdash \mathord {\smile }A\) then \(x\not \Vdash A\) and hence, \(x\Vdash \mathord {\frown }A\).

  9. I have discussed this construction of points from invalid sequents elsewhere (Restall 2009). In that paper, I show that if we allow multiple conclusion sequents for intuitionist logic and restrict the rules for the conditional and negation, but not conjunction or disjunction, the resulting structure of points is noticeably different to what we have here. As we will see, the structure of points here forms a Beth model for intuitionist logic, in which a disjunction can hold at a point without either disjunct holding at that point. The construction applied to multiple conclusion sequents forms a Kripke model. The distinction between Kripke and Beth models for intuitionist logic will play no role in what follows.

  10. What clause will work in constructions? The bar condition from Beth Semantics (Restall 2009).

  11. We have not provided a uniform truth condition for negation over a frame with points which are constructions, worlds and situations. A referee pointed out that this might be thought to indicate that negation still not truly one thing but three, in this model, because at worlds it is evaluated using the boolean clause, at constructions, a universal forward-looking clause, and at situations, a particular backward-looking clause. To think that is to think that the area of a plane figure is not ‘one thing’ because for circles it can be calculated in one way, ellipses another, squares another, and so on. The fact that the techniques agree where they overlap should help make the point that there are not three separate notions, any more than that the areas of a circle, a square or an ellipse are a different kinds of area.

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Acknowledgments

Thanks to audiences at the “Logical Pluralism” workshop in Tartu, the Logic Seminar at the University of Melbourne, and the Logic or Logics Workshop of the Foundations of Logical Consequence Project at Arché at the University of St Andrews for comments on this paper. I was supported by the ARC Discovery Grants DP0556827 and DP1094962, and Tonio K’s Life in the Foodchain.

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Correspondence to Greg Restall.

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Restall, G. Pluralism and Proofs. Erkenn 79 (Suppl 2), 279–291 (2014). https://doi.org/10.1007/s10670-013-9477-9

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