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A truth-maker semantics for ST: refusing to climb the strict/tolerant hierarchy

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Abstract

The paper presents a truth-maker semantics for Strict/Tolerant Logic (ST), which is the currently most popular logic among advocates of the non-transitive approach to paradoxes. Besides being interesting in itself, the truth-maker presentation of ST offers a new perspective on the recently discovered hierarchy of meta-inferences that, according to some, generalizes the idea behind ST. While fascinating from a mathematical perspective, there is no agreement on the philosophical significance of this hierarchy. I aim to show that there is no clear philosophical significance of meta-inferences above the first level.

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Notes

  1. Nevertheless, I will stick to the terminology of inferences as abstract entities and not mental or dialogical acts, as this fits better with the talk of meta-inferences, which has come to be dominant in this area of the literature.

  2. In the language I am using here modus ponens corresponds to disjunctive syllogism, as the language has no conditional. The failure of modus ponens in \({{\textsf {L}}}{{\textsf {P}}}\) is what corresponds most closely to the failure of Cut in \({{\textsf {ST}}}\).

  3. Scambler (2020b) has shown how to extend this result into the transfinite.

  4. Scamber uses the material conditional to formulate MP. I move to disjunctive syllogism because my official language here doesn’t have a conditional.

  5. Exact truth-makers are similar to what are sometimes called “minimal truth-makers,” i.e., truth-makers of sentences such that no proper part of them is a truth-maker of the respective sentence (O’Conaill & Tahko 2016; Armstrong, 2004). There are, however, differences. The fusion of truth-makers of each disjunct of a disjunction is an exact truth-maker of the disjunction—by the third disjunct of the clause (or+) below—while having two proper parts that are also truth-makers of the disjunction.

  6. This formulation differs from Fine’s in the quantification over further states u. In the presence of Downward-Closure, the two formulations are equivalent.

  7. Stipulating these constraints for atomic sentences suffices (given the semantic clauses) to enforce them for the whole language.

  8. As is well known, there is a translation between \({{\textsf {L}}}{{\textsf {P}}}\) and \({{\textsf {ST}}}{{\textsf {ST}}}\) (see Barrio et al., 2015; Dicher & Paoli, 2019). Hence, the same translation also maps truth-maker meta-inferential validity, as just defined, into \({{\textsf {L}}}{{\textsf {P}}}\). I don’t think that this shows that \({{\textsf {ST}}}\) is “really” \({{\textsf {L}}}{{\textsf {P}}}\), but I won’t engage this debate here.

  9. I show elsewhere that one way to define \({{\textsf {T}}}{{\textsf {S}}}\) is to reject Exclusivity while holding on to Exhaustivity (Hlobil, 2022). But this won’t help with our current problem because we want to use the same space of models for both logics, just as we do in the strong Kleene semantics.

  10. Barrio and Pailos (2022, p. 94) hint at a similar philosophical interpretation of the hierarchy in terms of weak and strong acceptance and rejection.

  11. I take it that \(\Delta '\) should be \(\Delta _{i}\), and similarly for the other “primed” uppercase Greek letters in this quote.

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Acknowledgements

For invaluable comments and discussion, I would like to thank Lucas Rosenblatt, Robert Brandom, Daniel Kaplan, Ryan Simonelli, Rea Golan, Shuhei Shimamura, Viviane Fairbank, and audiences at the University of Connecticut and at the Tenth Workshop on Philosophical Logic at IIF-SADAF-CONICET. Work on this paper was supported by the EXPRO grant No. 20-05180X of the Czech Science Foundation.

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  1. Department of Philosophy, Concordia University, 2145 MacKay, Montreal, QC, Canada

    Ulf Hlobil

  2. Department of Philosophy, Univerzita Hradec Králové, Hradec Králové, Czechia

    Ulf Hlobil

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Hlobil, U. A truth-maker semantics for ST: refusing to climb the strict/tolerant hierarchy. Synthese 200, 368 (2022). https://doi.org/10.1007/s11229-022-03820-w

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