Skip to main content
SpringerLink
Log in

Naive Structure, Contraction and Paradox

  • Published:
Topoi Aims and scope Submit manuscript

Abstract

Rejecting structural contraction has been proposed as a strategy for escaping semantic paradoxes. The challenge for its advocates has been to make intuitive sense of how contraction might fail. I offer a way of doing so, based on a “naive” interpretation of the relation between structure and logical vocabulary in a sequent proof system. The naive interpretation of structure motivates the most common way of blaming Curry-style paradoxes on illicit contraction. By contrast, the naive interpretation will not as easily motivate one recent noncontractive approach to the Liar paradox.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. In Shapiro (2011), I used the kind of noncontractive logic discussed by Slaney and Restall to address a version of Curry’s paradox. In reply, Zardini (2013) defends the different noncontractive response to paradox proposed in Zardini (2011). While both approaches block the Curry derivation in Sect. 2.1 the same way, they differ in how they block the Liar derivation presented there.

  2. If logical equivalence is too much to demand, and we can have only equivalence in virtue of arithmetic or syntax, a suitably modified proof will show that arithmetic or syntax implies absurdity.

  3. I use Gentzen-style rules of left and right introduction for the conditional and for the truth predicate, but much of what follows could be reformulated in terms of introduction and elimination rules.

  4. This is because the truth rules are not used prior to cut. In accepting the subderivation’s conclusion, Ripley differs from an earlier advocate of restricting transitivity (Weir 2005, 2014).

  5. In the presence of \(\rightarrow\)R and identity, both \(\rightarrow\)L and cut are derivable using the usual modus ponens rule \(\rightarrow\)E. Hence Beall and Ripley must reject \(\rightarrow\)E as well. Unlike Beall, Ripley keeps the sequent \(\alpha \to \beta, \alpha \vdash \beta\).

  6. It has two main sources, both dating to the 1970s and pursued mainly in the context of naive set theory: Grishin’s work on BCK logic (see e.g. Došen 1993) and work in weak relevant logic (see e.g. Slaney 1990).

  7. These include additive linear logic and many weak relevant logics.

  8. Zardini notes that another route to paradox proceeds by reductio from \(T\langle\lambda\rangle \vdash \lnot T\langle\lambda\rangle\) to \(\vdash \lnot T\langle\lambda\rangle\), and from there to \(\vdash T\langle\lambda\rangle\) and contradiction (2011, p. 502). Instead of blaming acceptance of reductio on excluded middle (cf. Field 2008, p. 8), he blames it on contraction. There is also the option of accepting the route to contradiction via reductio, despite taking the above Liar subderivation to fail on account of illicit contraction: see Weber (2014).

  9. Slaney uses this approach to motivate a noncontractive reply to our Curry subderivation, but not to our Liar subderivation: inferring \(\bot\) on the assumption of \(T\langle\kappa\rangle \to \bot\) and \(T\langle\kappa\rangle\) involves applying one body of information to another, whereas inferring \(T\langle\lambda\rangle \wedge \lnot T\langle\lambda\rangle\) on the twofold assumption of \(T\langle\lambda\rangle\) does not.

  10. On this view, why do we obtain a contradiction from a twofold supposition of \(T\langle\lambda\rangle\)? One answer might be that states of affairs allow a type/token distinction, and that the atemporal dynamics won’t permit multiple tokens of the state of affairs expressed by \(T\langle\lambda\rangle\) to suffer “self-removal” together.

  11. However, it isn’t clear to me to what extent Weber’s fascinating explanation of “why contraction fails” counts as a metaphysical one. Though I won’t be able to pursue the issue here, I believe that his explanation is largely compatible with the “naive” answer to the Intelligibility Question I will propose.

  12. For a defense of this perspective on the turnstile, see Shapiro (2011). There, I explore drawing “deflationary” conclusions about the nature of logical consequence. The present proposal is independent of such motivations.

  13. If ‘true in the object language’ is supposed to express the notion expressed by the object language’s naive truth predicate T, as I’ll assume, our metatheory will have to be non-classical.

  14. Gentzen’s own suggestion for how to understand his comma structure is naive in both these senses. He writes that the sequent \(\alpha_1, \ldots, \alpha_m \vdash \beta_1, \ldots, \beta_n\) “has, with regard to its content, exactly the same significance as the formula” \((\alpha_1 \wedge \cdots \wedge \alpha_m) \supset (\beta_1 \lor \cdots \lor \beta_n), \) from which it differs only in its “formal structure” (Gentzen 1935, pp. 180/290, 191/295, my translation). A closer parallel, which came to my attention after this paper was complete, is Schütte (1950, pp. 62–63). Instead of giving an interpretation of sequent structure, Schütte views himself as replacing sequent proof systems with proof systems involving only object-language sentences. But that characterization is misleading. Provided we substitute ⇒ for the intuitionistic conditional in Schütte’s system K 3, the expression \(\Upgamma \Rightarrow \alpha \Rightarrow \beta \Rightarrow \gamma\) appearing in his rules looks like a notational variant of our metalinguistic \(\Upgamma / \alpha \Rightarrow \beta \vdash \gamma\) (on the naive construal of the slash and turnstile).

  15. Gentzen contrasts “structure inference figures” with “logical-symbol inference figures” (“operational inference figures” in Szabo’s translation). From this paper’s perspective, the widespread tendency to refer to the operator-involving rules as the “logical rules” is unfortunate.

  16. See likewise Sambin et al. (2000) and Rivenc (2005, ch. 7). As explained in note 14, my inverted perspective has precedent in Gentzen. There are similar suggestions in Anderson and Belnap (1975, \(\S \S\) 7.2-3) and Brady (2006, pp. 93–94). To the extent that it gives priority to connectives, the present perspective is also akin to the “Scottish Plan” of Read (1988) and to Priest (2014).

  17. The horizontal line may be understood as a metalanguage conditional, so that the rule says that if \(\alpha \vdash \beta\) and \(\gamma \vdash \alpha\), then \(\gamma \vdash \beta\). The ‘if \(\ldots\) then’ here shouldn’t be understood as a consequence connective, since (as explained in the next note) the logics I’m motivating won’t in general yield the truth of ((α ⇒ β) ∧ (γ ⇒ α)) ⇒ (γ ⇒ β).

  18. Routley et al. (1982, pp. 268–278) and Brady (2006, p. 29) argue, against Anderson and Belnap, that prefixing and suffixing should be rejected for a consequence connective, whose transitivity should instead be expressed by conjunctive syllogism: \((\alpha \Rightarrow \beta) \wedge (\gamma \Rightarrow \alpha) \vdash \gamma \Rightarrow \beta\). The difference is important in the present context. Sequent proof systems that avoid prefixing/suffixing use non-associative structure, distinguishing between \((\alpha, \beta), \gamma \vdash \delta\) and \(\alpha, (\beta, \gamma) \vdash \delta\). No such distinction is expressible with my sequence-forming slash. This isn’t the place to examine the issue, but I’m unconvinced by the considerations behind objections to prefixing/suffixing, e.g. a view of consequence as “content containment” or intuitions about what counts as a “sufficient ground.” Incidentally, conjunctive syllogism can be added using a restricted contraction principle. Only in the presence of structural exchange is doing so equivalent to adding full contraction (Restall 1994, pp. 43–44).

  19. Here I’m presupposing that \(\Updelta\) in Cut⇒ is nonempty. Indeed, we haven’t introduced an interpretation for empty antecedents. For discussion, see Sect. 4.2

  20. The consequents of (1) and (2) are cited as “fallacies of modality” by Anderson and Belnap (1975, pp. 21, 23). Restall (2006, p. 296) notes the latter’s connection with exchange.

  21. Provided derivations establishing that \(\alpha \vdash \beta\) are construed as proofs of the theorem α ⇒ β, the rules of Table 2 derive the Hilbert-system axioms and rules for TW⇒ in Martin and Meyer (1982): prefixing/suffixing in axiom and rule forms, rule transitivity, and identity. Conversely, these axioms/rules suffice to prove the ⇒-statement substituted for by each instance of Trans⇒. Alternatively, we can introduce a constant t governed by Left Push and Left Pop from Sect. 4.2 (with slash in place of semicolon), and regard derivations of \(t \vdash \alpha\) as proofs of the theorem α.

  22. This form of Curry’s paradox is discussed by Whittle (2004) and Shapiro (2011); for the history of related paradoxes, see Read (1979) and Beall and Murzi (2013, pp. 153–156). For reasons explained by Ketland (2012) and Cook (2014), the claim that there is a Curry paradox involving ⇒ is problematic (cf. Murzi and Shapiro, forthcoming). One objection is that the truth rules may not count as preserving logical consequence. A second objection is that \(\zeta\) can’t be logically equivalent to \(T\langle\zeta\rangle \Rightarrow \bot\). The first objection’s weight depends on whether Cook is right that logical consequence must be preserved under uniform substitution of non-logical expressions. The second may be sidestepped in various ways. Cook shows how to modify \(\zeta\) so that logical equivalence isn’t required for deriving paradox. Alternatively, Ripley (2012) shows how, by taking \(\langle\alpha\rangle\) as our notation for whatever individual constant is the language’s “distinguished name” for α, we can stipulate that \(\zeta\) is the very sentence \(T\langle\zeta\rangle \Rightarrow \bot\).

  23. For one proposed explanation, see Weber (2014).

  24. This version of the paradox is the focus of Beall and Murzi (2013); it is also pointed out by Whittle (2004, p. 319n3) and Shapiro (2011, p. 336n29).

  25. Priest (2014) stresses that Curry paradoxes involving a conditional can be derived using the absorption sequent \(\alpha \rightarrow (\alpha \rightarrow \beta) \vdash \alpha \rightarrow \beta\) instead of structural contraction. But it doesn’t follow that we can block such a paradox by rejecting absorption while retaining structural contraction. Moreover, validity Curry too can be derived without structural contraction provided we assume the sequent \(Val(\langle\alpha\rangle, \langle Val(\langle\alpha\rangle, \langle\beta\rangle)\rangle) \vdash Val(\langle\alpha\rangle, \langle\beta\rangle)\).

  26. I’ll only gesture at how we might let sequents generalize over higher-degree consequence conditionals. One approach would have \(\Upgamma\) be a sequence of mark/sentence pairs, writing \(\langle\langle\oslash, \alpha_1\rangle, \langle/, \alpha_2 \rangle, \langle;, \alpha_3,\rangle\rangle\) as α 1/α 2; α 3. Some rules below would need to be rewritten to allow for contexts involving the slash. We might also want a rule letting us derive \(\Upgamma; \alpha \vdash \beta\) from \(\Upgamma/\alpha \vdash \beta\) (cf. Asmus 2009, p. 409), in which case rejecting contraction on the slash would require rejecting it on the semicolon. Alternatively, we could formulate rules ⇒L and ⇒R using only the semicolon. Here ⇒R would let us derive \(\Upgamma \vdash \alpha \Rightarrow \beta\) from \(\Upgamma; \alpha \vdash \beta\) subject to the restriction that \(\Upgamma\) contains only compounds of sentences of form γ ⇒ δ. Cf. Zardini (2013, p. 586) and Anderson and Belnap (1975, \(\S13.1\)).

  27. The noncontractive logics we have been considering are all subsystems of Łukasiewicz’s infinitely-valued logic \(\L_\infty\), which is known to support a non-trivial naive truth theory. However, it remains an open problem which of them, unlike \(\L_\infty\), can support arithmetic ω-consistently. See Hájek et al. (2000) and Bacon (2013).

  28. Without Left \(\hbox{Push}_{\rightarrow}\), we won’t get identity for our conditional: \(t \vdash \alpha \rightarrow \alpha\). It could be added separately.

  29. We could also add Zardini’s rules for ⇒, as sketched at the end of note 26.

  30. This is one reason I haven’t proposed a naive construal of Zardini’s proof system according to which the comma in his rules ∧L, ∧R, \(\rightarrow\)L, \(\rightarrow\)R, and weakening substitutes for the conditional rather than conjunction. Another reason is that the logical rules for the conditional involve the comma. If the comma were in turn introduced to substitute for \(\rightarrow\), that would violate the desideratum from Sect. 3.3 that logical considerations be maximally disentangled from structure-defining ones.

  31. The assumptions are (4) and the transitivity of ⇒. From (7), we get the truth of \((T\langle \kappa \rangle \wedge T\langle \kappa \rangle) \Rightarrow \bot\), whence (6) would give us the truth of \(T\langle \kappa \rangle \Rightarrow \bot\). Assuming (4), we could then derive the truth of \(t \Rightarrow (T\langle \kappa \rangle \rightarrow \bot)\) and thus also of \(t \Rightarrow T\langle \kappa \rangle\). Transitivity then lands us in paradox.

  32. That option would also allow us to block the route to Curry paradox described in note 31 by disallowing the inference from \((T\langle \kappa \rangle \rightarrow \bot) \wedge T\langle \kappa \rangle \vdash\, \bot\) to \(T\langle \kappa \rangle \vdash \bot\). By contrast, in the proof system described in this section \((T\langle \kappa \rangle \rightarrow \bot) \wedge T\langle \kappa \rangle \vdash \bot\) is not derivable in the first place.

  33. See Belnap (1993). Zardini’s structure-involving rules at least yield \(\alpha \wedge (\beta \lor \gamma) \vdash (\alpha \wedge \beta) \lor \gamma\).

  34. Here, again, I should note that how strong a logic we can obtain, while keeping naive truth theory and an ω-consistent arithmetic, remains an open question. Cf. note 27.

  35. For an explanation of “intensional” antecedent structure as “representing fusion,” see Priest (2014).

  36. It might be objected that my own proposal involves the extraneous connective ⇒. But I have argued that our ordinary talk of logical consequence can be seen as generalizing over claims that could be expressed with a consequence connective.

References

  • Anderson A, Belnap N (1975) Entailment: the logic of relevance and necessity, vol 1. Princeton University Press, Princeton

    Google Scholar 

  • Asmus C (2009) Restricted arrow. J Philos Log 38:405–431

    Article  Google Scholar 

  • Bacon A (2013) Curry’s paradox and ω-inconsistency. Stud Logica 101:1–9

    Article  Google Scholar 

  • Beall J (2013) Free of detachment: logic, rationality, and gluts. Nous. doi:10.1111/nous.12029

  • Beall J, Murzi J (2013) Two flavors of Curry’s paradox. J Philos 110:143–165

    Google Scholar 

  • Belnap N (1993) Life in the undistributed middle. In: Schroeder-Heister P, Došen K (eds) Substructural logics. Oxford University Press, Oxford, pp 31–41

  • Brady R (2006) Universal Logic. CSLI Publications, Stanford

    Google Scholar 

  • Cook R (2014) There is no paradox of logical validity! Logica Universalis. doi:10.1007/s11787-014-0094-4

  • Deutsch H (2010) Diagonalization and truth functional operators. Analysis 70:215–217

    Article  Google Scholar 

  • Došen K (1989) Logical constants as punctuation marks. Notre Dame J Formal Log 30:362–381

    Article  Google Scholar 

  • Došen K (1993) A historical introduction to substructural logics. In: Schroeder-Heister P, Došen K (eds) Substructural logics. Oxford University Press, Oxford, pp 1–30

  • Dunn JM (1993) Partial gaggles applied to logics with restricted structural rules. In: Schroeder-Heister P, Došen K (eds) Substructural logics. Oxford University Press, Oxford, pp 63–108

  • Field H (2008) Saving truth from paradox. Oxford University Press, Oxford

    Book  Google Scholar 

  • Geach PT (1955) On Insolubilia. Analysis 15:71–72

    Article  Google Scholar 

  • Gentzen G (1935) Untersuchungen über das logische Schliessen. Mathematische Zeitschrift 39:176–210, 405–431. Translated by M. E. Szabo in Gentzen G (1969) Collected Papers. North-Holland, Amsterdam, pp. 68–131

  • Girard JY (1987) Linear logic. Theoret Comput Sci 50:1–102

    Article  Google Scholar 

  • Hájek P, Paris J, Shepherdson J (2000) The liar paradox and fuzzy logic. J Symb Log 65:339–346

    Article  Google Scholar 

  • Ketland J (2012) Validity as a primitive. Analysis 72:421–430

    Article  Google Scholar 

  • Mares E, Paoli F (2013) Logical consequence and the paradoxes. J Philos Log. doi:10.1007/s10992-013-9268-4

  • Martin E, Meyer R (1982) Solution to the P-W problem. J Symb Log 47:869–887

    Article  Google Scholar 

  • Meyer RK, Routley R, Dunn JM (1979) Curry’s paradox. Analysis 39:124–128

    Article  Google Scholar 

  • Murzi J, Shapiro L (forthcoming) Validity and truth-preservation. In: Achourioti D, Fujimoto K, Galinon H, Martínez-Fernández J (eds) Unifying the philosophy of truth. Springer, Dordrecht

  • Paoli F (2002) Substructural logics: a primer. Kluwer, Dordrecht

    Book  Google Scholar 

  • Priest G (1980) Sense, entailment, and modus ponens. J Philos Log 9:415–435

    Article  Google Scholar 

  • Priest G (2006) In contradiction. Oxford University Press, Oxford. Expanded edition (first published 1987)

  • Priest G (2014) Fusion and confusion. This issue of Topoi

  • Read S (1979) Self-reference and validity. Synthese 42:265–274

    Article  Google Scholar 

  • Read S (1988) Relevant Logic: A Philosophical Examination of Inference. Basil Blackwell, Oxford

    Google Scholar 

  • Restall G (1994) On logics without contraction, PhD thesis, The University of Queensland

  • Restall G (2000) An introduction to substructural logics. Routledge, London

  • Restall G (2005) Multiple conclusions. In: Hájek P, Valdés-Villanueva L, Westerståhl D (eds) Logic, methodology and the philosophy of science: proceedings of the Twelfth International Congress. King’s College Publications, London, pp 189–205

  • Restall G (2006) Relevant and substructural logics. In: Gabbay D, Woods J (eds) Handbook of the history of logic, volume 7: Logic and the modalities in the twentieth century. Elsevier, North Holland, pp 289–398

  • Ripley D (2012) Conservatively extending classical logic with transparent truth. Rev Symb Log 5:354–378

    Article  Google Scholar 

  • Ripley D (2013) Paradoxes and failures of cut. Australas J Philos 91:139–164

    Article  Google Scholar 

  • Rivenc F (2005) Introduction à la logique pertinente. Presses Universitaires de France

  • Routley R, Meyer RK, Plumwood V, Brady RT (eds) (1982) Relevant logics and their rivals I. Ridgeview, Atascadero

  • Sambin G, Battilotti G, Faggian C (2000) Basic logic: reflection, symmetry, visibility. J Symb Log 65:1284–1300

    Article  Google Scholar 

  • Schütte K (1950) Schlußweisen-Kalküle der Prädikatenlogik. Math Ann 122:47–65

    Article  Google Scholar 

  • Shapiro L (2011) Deflating logical consequence. Philos Q 61:320–42

    Article  Google Scholar 

  • Shapiro L (2013) Validity curry strengthened. Thought 2:100–107

    Google Scholar 

  • Slaney J (1990) A general logic. Australas J Philos 68:74–88

    Article  Google Scholar 

  • Weber Z (2014) Naïve validity. Philos Q 64:99–114

    Google Scholar 

  • Weir A (2005) Naïve truth and sophisticated logic. In: Beall J, Armour-Garb B (eds) Deflationism and paradox. Oxford University Press, Oxford, pp 218–249

  • Weir A (2014) A robust non-transitive logic. This issue of Topoi

  • Whittle B (2004) Dialetheism, logical consequence and hierarchy. Analysis 64:318–326

    Article  Google Scholar 

  • Zardini E (2011) Truth without contra(di)ction. Rev Symb Log 4:498–535

    Article  Google Scholar 

  • Zardini E (2013) Naive Modus Ponens. J Philos Log 42:575–593

    Article  Google Scholar 

Download references

Acknowledgments

Ancestors of portions of this paper were presented at the Arché Centre (University of St Andrews), the Logica Symposium (Academy of Sciences of the Czech Republic), the UConn Logic Group, the Munich Center for Mathematical Philosophy (LMU Munich), and the Logos Research Group (University of Barcelona). I thank all the audiences for much helpful discussion, in particular Jc Beall, Colin Caret, Øle Hjortland, Julien Murzi, Francesco Paoli, Stephen Read, Dave Ripley, Marcus Rossberg, and Elia Zardini. I’m especially grateful for written comments from Dave Ripley, Zach Weber, and two referees.

Author information

Authors and Affiliations

  1. University of Connecticut, Storrs, CT, USA

    Lionel Shapiro

Authors
  1. Lionel Shapiro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lionel Shapiro.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shapiro, L. Naive Structure, Contraction and Paradox. Topoi 34, 75–87 (2015). https://doi.org/10.1007/s11245-014-9235-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11245-014-9235-x

Keywords

Search

Navigation