The (Non-)classicality of (Non-)classical Mathematics

Abstract

Purpose

Graham Priest has recently argued that the distinctive trait of classical mathematics is that the conditional of its underlying logic—that is, classical logic—is extensional. In this article, I aim to present an alternate explanation of the specificity of classical mathematics (and actually for L-mathematics, for a significative amount of instances of 'L').

Method

I examine Priest's argument for his claim and show its shortcomings. Then I deploy a model-theoretic presentation of logics that allows comparing them, and the mathematics based on them, more fine-grainedly.

Results

Such a model-theoretic presentation of logics suggests that the specific character of classical logic consists in the structure that it confers to its truth values and in the structure of the evaluation indices of its formulas, and that this trait is useful to explain the specific character of the logics and the mathematics based on them.

Conclusion

The extensionality of the conditional in classical logic is a by-product of other structural features of a logic, which are more likely to be what gives a kind of mathematics based on it its specific character.

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Notes

  1. 1.

    As Priest paper is still forthcoming, the reader might find difficult to assess my rejoinder. In order to mitigate that obstacle, the reader can find the abstract here http://sites.google.com/site/afterconsequencesunilog/events/philosophy-of-mathematics-workshop-mathematical-pluralism and a video recording of a talk based on the paper here: http://www.youtube.com/watch?v=wCNigpWMEdg.

  2. 2.

    I deliberately refuse to follow the quite widespread tradition of assuming that '⊃' denotes a “classical”, “extensional” or “Boolean” conditional. I use it to designate an arbitrary conditional, a task that other authors, Priest included, leave to '→.'

  3. 3.

    The principles are these:

    -\( A \wedge B{ \vdash }A \) (and B)

    -If \( C{ \vdash } A \) and \( C{ \vdash }B \) , then \( C{ \vdash }A \wedge B \)

    -\( A\left( {{\text{or}}\;B} \right){ \vdash }A \vee B \)

    -If \( A \vdash C \) and \( B{ \vdash }C \) , then \( A \vee B{ \vdash }C \)

    \( A \wedge (B \vee C){ \vdash }(A \wedge B) \vee (A \wedge C) \) .

  4. 4.

    Here is a sketch of the proof. Suppose \( \left( {\varepsilon = 0} \right) \vee \neg \left( {\varepsilon = 0} \right) \), an instance of, \( A \vee \neg A \) , holds. Suppose \( \neg \left( {\varepsilon = 0} \right) \) . For any real numbers r and n, if \( r \ne 0 \) , then \( r^{n} \ne 0 \) . Then, \( \varepsilon^{n} \ne 0 \) , but this contradicts Nilpotence. Suppose now (\( \varepsilon = 0 \) ). Substitute then in Microcancellation, putting ‘1’ instead of ‘x’ and ‘0’ instead of ‘y’. One gets then that if 0(1) = 0(0), then 1 = 0. But this contradicts Microcancellation, as the conditional obtained is false (the antecedent is true, but the consequent is not). Therefore, it is not the case that \( \left( {\varepsilon = 0} \right) \vee \neg \left( {\varepsilon = 0} \right) \) . Hence, the axiomatization of the reals is different from the usual one when one thinks that there are reals satisfying Nilpotence and Microcancellation. Again, see Bell (1998) for a detailed account.

  5. 5.

    For examples of mathematics built upon LP, see Priest (2006: chapters 17 and 18) and the references therein. On the other hand, although fuzzy logics have dominated the realm of many-valued mathematics, see Tye (1994) for a set theory built upon K3.

  6. 6.

    A list differs from a set in that both the order and the number of appearances of its elements matter (and in this, although not in the first feature, a list resembles a multiset). For example, the list {a, b, c} is a different list from {a, c, b}, and the list {a, b} is not the same as {a, b, b}.

  7. 7.

    In alethic modal logic, the so-called “actual world” is a typical designated index, but it is not the only one. For several reasons, one might like to pay special attention to initial worlds (worlds that can access every other one) or terminal worlds (worlds that are accessible from every other one). And nowadays, the actual world is very important in some definitions of ‘necessity’ or modal definitions of ‘logical validity’. Pure logic might allow as possible some weird things, and a notion of “real possibility” may be desired. Then, possibility would not be defined simply as truth at some index, but as truth at some index in a certain set of indices, say, those closer to the actual world given some suitable metric. Similarly, is logical validity going to be defined as truth preservation across all worlds? Then, it might be that no argument is valid. If that is much to swallow, validity can be defined as truth preservation across all “normal” (or “really possible”, or what one fancies) worlds. A set of designated indices is implicit in all those constructions. See Beall (2010) for further discussion.

  8. 8.

    That an index i is maximal with respect to an index j means that for every index x, if j ≤ x, then x ≤ i. What the compatibility of j with i exactly means depends on substantial philosophical ideas, but at least the following characterization might be admitted without much problems: j is compatible with i if (1) Rij, (2) there is another relation between i and j, denoted ‘Cij’, such that C is symmetric, non-reflexive and that, for all indexes x and y, if Cij, x ≤ i and y ≤ j, then Cxy.

  9. 9.

    Cases where W is not the singleton of the empty set and the R’s are distinct from the identity relation typically serve to model modal logics, specifically to give admissible truth conditions for the typical modal connectives (necessity, possibility, knowledge, belief, etc.). Nonetheless, those “complex” W’s and R’s are not restricted to logics with typical modal connectives: The case of relational semantics for intuitionistic logic is well known, and certain relevant logics require W with more than one element and R’s distinct from the identity, especially to interpret the conditional with a ternary relation to avoid the so-called fallacies of relevance. See Beall et al. (2012) for an overview.

  10. 10.

    Think, for example, in the usual truth table for conjunction in classical zeroth-order logic. It is usually said that the table exhibits the truth conditions for conjunction. I disagree: The table (or what is usually called ‘truth condition’) mixes a truth condition (or rather generally, a satisfiability condition), namely, that the value of a conjunction is the infimum of the values of the conjuncts, and certain specific inputs for such a satisfiability condition, for example, that there are exactly two values, linearly ordered, etc.

  11. 11.

    The logic FDE (first degree entailment) also belongs to that family of logics under this presentation, as it considers admissible both the relation that relates a formula to both truth values (as in LP) and the one that does not relate it to either (as in K3). See Priest (2008: chapters 7 and 8).

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Acknowledgements

I would like to thank Michèle Friend and Mihir K. Chakraborty for putting this special issue together, and for their help to produce a better paper than the originally submitted. Special thanks are deserved to an anonymous referee for their comments, corrections and suggestions, as well as to Charlie Donahue and the audience at the IV Conference of the Latin American Association for Analytic Philosophy for discussion, and to Elisángela Ramírez-Cámara for discussion and her invaluable help in producing a readable English version. This work was written under the support of the PAPIIT projects IA401015 “Tras las consecuencias. Una visión universalista de la lógica (I)” and IA401117 “Aspectos filosóficos de las lógicas contraclásicas”.

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Estrada-González, L. The (Non-)classicality of (Non-)classical Mathematics. J. Indian Counc. Philos. Res. 34, 365–377 (2017). https://doi.org/10.1007/s40961-017-0097-7

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Keywords

  • Conditional
  • Extensionality
  • Classical mathematics
  • Non-classical mathematics
  • LP
  • K3
  • Satisfiability conditions