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Naive Modus Ponens and Failure of Transitivity

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Abstract

In the recent paper “Naive modus ponens”, Zardini presents some brief considerations against an approach to semantic paradoxes that rejects the transitivity of entailment. The problem with the approach is, according to Zardini, that the failure of a meta-inference closely resembling modus ponens clashes both with the logical idea of modus ponens as a valid inference and the semantic idea of the conditional as requiring that a true conditional cannot have true antecedent and false consequent. I respond on behalf of the non-transitive approach. I argue that the meta-inference in question is independent from the logical idea of modus ponens, and that the semantic idea of the conditional as formulated by Zardini is inadequate for his purposes because it is spelled out in a vocabulary not suitable for evaluating the adequacy of the conditional in semantics for non-transitive entailment. I proceed to generalize the semantic idea of the conditional and show that the most popular semantics for non-transitive entailment satisfies the new formulation.

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Notes

  1. (MP) is modus ponens as inference and (CPr) is restricted conditional proof as meta-inference. Notice that (CPr) lacks side-context as compared to principle (CP) if Γ, A ⊢ B then Γ ⊢ A ⇒ B. It is important that a conditional expressing entailment-facts satisfies (CPr) but not (CP) for the same reason that a □ representing validity satisfies necessitation but not A ⊢ □A: to avoid that validity collapses into truth.

  2. It is worth mentioning that Zardini himself developed a non-transitive logic for vagueness to deal with sorites paradoxes in [12]. While one might think that Zardini is thus criticizing his own approach, Zardini explains in a footnote that “I dont think that any of the problems [...] applies in the case of the use of non-transitive logics for dealing with vagueness made in my work”.[13, p.580n11]

  3. For a proof, see [13, p.577].

  4. The formulae in Γ∗ and Δ∗ are l o g i c a l ⇒, i.e. of the form A ⇒ B or of the form ¬A′, A′∧B′, A′∨B′ or A′→B′ with A′ and B′ being l o g i c a l ⇒ .

  5. In this paper, I assume the following relationship between entailment and validity: an inference from Γ to Δ is valid if and only if Γ entails Δ.

  6. The distinction between validity and meta-validity corresponds to the distinction between inference rules and rules of proof. For the latter distinction, see [11] and [5].

  7. The official terminology in [7] and [1] is in terms of assertion and not truth to avoid revenge paradoxes associated with the two additional truth-predicates “is strictly true” and “is tolerantly true”. See for example footnote 12 in [3]. However, for presentational purposes and simplicity, I stick to talk of strict and tolerant truth in this paper.

  8. For this reading of a trivalent valuation, see [6].

  9. Note that the conditional in the semantics in [7] is actually too strong to express entailment-facts. As mentioned above in footnote 1, we would not want a ⇒ representing entailment to satisfy unrestricted conditional proof (CP). For more considerations against the unrestricted variant, see [4, p.6]. To obtain a suitable conditional, the semantics can be extended to a frame semantics where the points of evaluation are three-valued in accordance with Strong Kleene and the new conditional is defined via an (at least reflexive) accessibility-relation on the points of evaluation. We can however safely ignore this issue in this section because the section is concerned only with the feature of the conditional that validates modus ponens, i.e. its behaviour as premise, and not the feature that validates conditional proof, i.e. its behaviour as conclusion.

  10. It is also necessary to validate (MP) to satisfy the generalized semantic idea of the conditional. Consider for example the paraconsistent logic LP that fails to validate (MP). To obtain LP, one defines validity as backwards falsity-preserving, meaning that there is no model such that all of the premisses are assigned either 1 or \(\frac {1}{2}\) and all of the conclusions are assigned 0. The semantic idea of the conditional requires now that a conditional that has the value 1 or \(\frac {1}{2}\) cannot have an antecedent that is assigned 1 or \(\frac {1}{2}\) but a consequent that is assigned 0. Since it can be the case that a conditional has the value \(\frac {1}{2}\) even if the antecedent is assigned \(\frac {1}{2}\) and the consequent 0 on the Strong Kleene valuation scheme, the semantic idea fails for LP. I thank an anonymous referee for pressing me on this issue.

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Acknowledgments

I am grateful to Franz Berto, Thomas Brouwer and Toby Meadows for discussion of the material and comments on various drafts of the paper. I also thank two anonymous referees for their valuable and encouraging comments.

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  1. Department of Philosophy, University of Aberdeen, Old Brewery, High Street, Aberdeen, AB25 3UB, UK

    Andreas Fjellstad

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Fjellstad, A. Naive Modus Ponens and Failure of Transitivity. J Philos Logic 45, 65–72 (2016). https://doi.org/10.1007/s10992-015-9351-0

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