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Capturing naive validity in the Cut-free approach

  • S.I. : Substructural Approaches to Paradox
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Abstract

Rejecting the Cut rule has been proposed as a strategy to avoid both the usual semantic paradoxes and the so-called v-Curry paradox. In this paper we consider if a Cut-free theory is capable of accurately representing its own notion of validity. We claim that the standard rules governing the validity predicate are too weak for this purpose and we show that although it is possible to strengthen these rules, the most obvious way of doing so brings with it a serious problem: an internalized version of Cut can be proved for a Curry-like sentence. We also evaluate a number of possible ways of escaping this difficulty.

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Notes

  1. It is more or less standard to make a distinction between laws (e.g. the law of Excluded Middle), rules (e.g. the rule of Explosion) and metarules (e.g. Conditional Proof or Reasoning by Cases). A law establishes that a certain sentence is valid. A rule establishes that the argument that goes from certain sentences to another sentence(s) is valid. Finally, a metarule says that if certain arguments are valid, then another argument is valid. These distinctions are not meant to be exhaustive nor exclusive. They are not exhaustive because there might be perfectly legitimate and intelligible principles which are neither laws, nor rules nor metarules. And they are not exclusive because we can understand a law as a 0-premise rule and, similarly, we can understand a rule as a 0-premise metarule. Later on we’ll have a chance to see various examples of this.

  2. To simplify the discussion below we take validity to be a two-place predicate. Needless to say, nothing important depends on this, since in most logical systems multiple premises can be collected into a single conjunction and multiple conclusions (if our theories allow for such a thing) can be collected into a single disjunction. There are some exceptions to this, like the non-contractive theory supported in Lionel Shapiro (2015). However, we’ll ignore this possibility here.

  3. It is important to point out that by a theory \(\mathcal {T}\) we do not mean a set of formulae closed under some consequence relation. In this context it will be more appropriate to understand a theory as a set of pairs of multisets of formulae closed under certain metarules or, more simply, as a set of arguments closed under certain metarules.

  4. Val is a predicate, so this should be formalized as \(Val(\langle \phi \rangle , \langle \psi \rangle )\), where \(\langle \rangle \) works as a name forming device. However, to ease the notation we will write \(Val(\phi ,\psi )\) instead of \(Val(\langle \phi \rangle , \langle \psi \rangle )\) throughout the paper.

  5. As it stands, VD can be seen as a rule or as a 0-premise metarule. But sometimes VD is presented as

    figure a

    Zardini (2013) suggests that if there are reasons for thinking that the informal idea behind validity detachment is expressed at least partly by this principle, then those are reasons for thinking that naive validity is actually not faithfully captured by the non-transitive approach. However, this version of VD could be said to be nothing more than a variant of Cut. Since we want to consider a Cut-free approach, we need to stick to the first version.

  6. For a natural deduction presentation of the paradox, see Beall and Murzi (2013).

  7. Notice that at two points in the proof we are implicitly relying on the identity between \(\pi \) and \(Val(\pi , \bot )\).

  8. It is worth remarking that the concept of validity we are discussing is not a purely logical concept, for it can be iterated (i.e., sentences about validity can themselves be valid). In fact, the purely logical notion of validity can actually be captured in any first-order arithmetical theory extending Robinson’s arithmetic, as Ketland (2012) and Cook (2014) point out. Also in Field (2008) and Field (forthcoming) it is claimed that under a certain understanding of Val, VD fails, and under another, it is VP that fails. So, the issue is far from being uncontroversial.

  9. The Strict-Tolerant approach has also been used to deal with the truth-theoretic paradoxes, the set-theoretic paradoxes and the paradoxes of vagueness.

  10. ST is often presented as a trivalent system with a Strong Kleene matrix. The valid arguments are those in which, if the premises have value 1 (i.e., are strictly true), the conclusion has value 1 or \(\frac{1}{2}\) (i.e., is tolerantly true). However, since we don’t know if this semantic characterization is supposed to apply also to extensions of ST such as STV, our presentation of STV is proof-theoretic. More specifically, STV will be presented by means of a (multiple conclusioned) sequent calculus. Also, to simplify things we’ll focus on the quantifier-free part of STV.

  11. These are simplified versions of the rules. Strictly speaking, the object language renderings of the rules should mention \(\varGamma \) and \(\Delta \), but we omit them for readability. The same applies to the metarules below.

  12. Whenever necessary, we’ll assume that the language contains a truth constant \(\top \) such that \(\Rightarrow \top \) is an initial sequent of our system. This comes in handy for stating in the object language sentences expressing the validity of laws.

  13. Of course, there are invalid sequents that are not provably invalid in STV, but that is another matter.

  14. Of course, this definition is meant to apply to one-premise metarules as well. So we say that a theory \(\mathcal {T}\) internalizes a metarule \(\mathcal {R}\) of the form

    figure i

    if \(\mathcal {T}\) proves every instance of

    figure j
  15. One thing we should point out is that the definition we’ve given for internalizing a metarule seems to be sensible to the order in which the premises of the metarule occur. So for example the sequent \(\Rightarrow Val(\psi _{1}, \psi _{2})\wedge Val(\phi _{1}, \phi _{2})\rightarrow Val(\chi _{1}, \chi _{2})\) does not strictly count as the internalization of a metarule with left premise \(\phi _{1}\Rightarrow \phi _{2}\), right premise \(\psi _{1}\Rightarrow \psi _{2}\) and conclusion \(\chi _{1}\Rightarrow \chi _{2}\). Of course, this is harmless. The system we are considering deals with multisets and so the exchange metarules are built in. As a consequence \(\wedge \) is a commutative connective and this means that if \(\Rightarrow Val(\psi _{1}, \psi _{2})\wedge Val(\phi _{1}, \phi _{2})\rightarrow Val(\chi _{1}, \chi _{2})\) has a proof, \(\Rightarrow Val(\phi _{1}, \phi _{2})\wedge Val(\psi _{1}, \psi _{2})\rightarrow Val(\chi _{1}, \chi _{2})\) has a proof as well.

  16. We could have instead demanded that for a certain metarule to be internalized the following sequent should have a proof:

    figure l

    And indeed, if we have this in STV we can obviously reach the other sequent by L\(\wedge \) and R\(\rightarrow \).

  17. In Priest and Wansing (2015), the authors also introduce a notion of internalization. But, unlike us, they work with a language that only contains a validity operator and their goal is to show that a variation of the v-Curry paradox that uses external validity (roughly, preservation of theoremhood) is not forthcoming without the use of the appropriate form of Contraction, just like the usual v-Curry paradox. So their purpose is quite different from ours.

  18. Instead of using this multiplicative or non-context sharing version of \(VD^{+}\), we could have used, as one anonymous referee suggests, an additive or context-sharing version of this metarule. This would have the benefits of making the Contraction metarules admissible and of maintaining the validity metarules uniform with the logical metarules of ST, which are additive when they involve two premises and multiplicative when they involve one premise. Of course, given the presence of Weakening and Contraction both metarules are interderivable. In what follows we stick to the non-context sharing version only because it makes some of the proofs below less cumbersome.

  19. In the system like the one we are considering, the standard modal metarule corresponding to K is

    figure r

    where \(\square \varGamma \) stands for the multiset \(\square \gamma \) for each \(\gamma \in \varGamma \).

  20. Thanks to an anonymous referee for suggesting this very interesting alternative.

  21. This metarule should ring bells for anyone acquainted with sequent calculus presentations of the modal logic S4, where the following metarule is given:

    figure t

    As far as we know this way of presenting S4 actually originates with Prawitz’ (1965, p. 74) natural-deduction system for S4 (thanks to Elia Zardini for the reference). A more recent presentation can be found in Negri (2011).

  22. Actually, the metarule we’ve given works properly because metarules only contain one conclusion. But it might also be interesting to consider a version of this metarule with multiple validity predications on the right as well. That is:

    figure v

    We will come back to this metarule below in the discussion of the issue of the admissibility of Cut. But for most of our purposes, we can ignore it.

  23. In a modal sequent calculus, the metarule corresponding to T is as follows:

    figure w
  24. By the way, this was one of the reasons to use \(VD^{+}\) instead of \(VD^{'}\). This sequent cannot be proved from \(VD^{'}\) if Cut is not available.

  25. For an overview of the current situation and of the relevant literature, see Negri (2011).

  26. In modal logic, this principle can be formulated as follows: \(\phi \rightarrow \square \Diamond \phi \).

  27. We are extremely grateful to an anonymous referee for pointing this out to us.

  28. By a primitive metarule we mean a metarule that is an explicit part of the definition of \(STV^{+}\). At this point it seems necessary to recall a very familiar distinction between two ways in which a metarule might be said to hold. It is one thing to say that a metarule holds if there is a proof from the top sequent to the bottom sequent and it is another thing to say that a metarule holds if the bottom sequent is provable whenever the top sequent is provable. A metarule holding in the first sense is sometimes said to be derivable, while a metarule holding in the second sense is usually called admissible. Obviously, a derivable metarule is also admissible, but the converse might fail. Thanks to an anonymous referee and to Dave Ripley for urging us to clarify this matter.

  29. For matters of readability we prefer to show a simplified version of the proof where \(\varGamma \) is \(\phi \) and \(\Delta \) is empty. Strictly speaking, the internalization of R\(\wedge \) should say \(\Rightarrow Val(\bigwedge \varGamma , \phi \vee \bigvee \Delta ) \wedge Val(\bigwedge \varGamma , \psi \vee \bigvee \Delta ) \rightarrow Val(\bigwedge \varGamma , (\phi \wedge \psi )\vee \bigvee \Delta \))). We make a similar simplification for \(VP^{+}\).

  30. Notice that the proof above only shows that derivable metarules can be internalized in \(STV^{+}\) and although we suspect that admissible metarules can be internalized as well, a different kind of proof is needed for that.

  31. A similar problem occurs if a naive truth predicate is available in the language. It has been noted in Ripley (2013) that STV proves that valid arguments preserve truth:

    figure ae

    This seems to be a problem, at least to the extent that ST’s notion of validity is essentially non-truth-preserving.

  32. Curiously, we can also ‘internalize’ the claim that Cut does not hold for the instance involving \(\pi \), in the sense that \(\Rightarrow Val(\top , \pi )\wedge Val(\pi , \bot )\rightarrow \lnot Val(\top , \bot )\) is provable.

  33. Thanks to Elia Zardini for suggesting this.

  34. Of course, the conditional is detachable in the sense that the rule of Modus Ponens (\(\phi , \phi \rightarrow \psi \Rightarrow \psi \)) is provable, but the metarule of Modus Ponens (if \(\Rightarrow \phi \rightarrow \psi \) and \(\Rightarrow \phi \), then \(\Rightarrow \psi \)) does not hold. In fact, it was proved in Barrio et al. (2015) that the metarules that hold in ST are (under a certain translation) the same as the rules that hold in the logic LP (Priest’s Logic of Paradox), so the fact that the rule of Modus Ponens fails to hold in LP is enough to infer that the metarule of Modus Ponens fails to hold in ST. See also Zardini (2013) and Fjellstad (2016) for some discussion of this aspect of ST.

  35. Thanks to Dave Ripley for this clever suggestion.

  36. We need \(VP^{Z}\) because to additively internalize the context-sharing metarules R\(\wedge \), L\(\vee \) and L\(\rightarrow \) without using Contraction, \(VP^{+}\) is not enough.

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Acknowledgments

We are very grateful to Elia Zardini, Dave Ripley and two anonymous referees for extremely helpful comments on previous versions of this paper. Some of this material was presented at conferences in Campinas (CLE) and Buenos Aires (Buenos Aires Logic Group). We also owe thanks to the members of these audiences for their valuable feedback.

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Barrio, E., Rosenblatt, L. & Tajer, D. Capturing naive validity in the Cut-free approach. Synthese 199 (Suppl 3), 707–723 (2021). https://doi.org/10.1007/s11229-016-1199-5

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