The Cut-Free Approach and the Admissibility-Curry

Abstract

The perhaps most important criticism of the nontransitive approach to semantic paradoxes is that it cannot truthfully express exactly which metarules preserve validity. I argue that this criticism overlooks that the admissibility of metarules cannot be expressed in any logic that allows us to formulate validity-Curry sentences and that is formulated in a classical metalanguage. Hence, the criticism applies to all approaches that do their metatheory in classical logic. If we do the metatheory of nontransitive logics in a nontransitive logic, however, there is no reason to think that the argument behind the criticism goes through. In general, asking a logic to express its own admissible metarules may not be a good idea.Until recently, philosophical common sense had it that, on pain of triviality, you cannot accept all of classical logic, allow for self-reference and let your language contain its own truth predicate (satisfying all Tarski biconditionals). The nontransitive approach to semantic paradoxes has proved common sense wrong. It offers a supraclassical logic, called STT, with a transparent truth predicate (i.e., a truth predicate, T, such that A and T⟨A⟩ are everywhere intersubstitutable) and self-reference (Cobreros et al. 2012(Cobreros et al. , 2013Ripley 2012Ripley , 2013. Transitivity, that is, Cut, fails in STT. But it only fails for paradoxical sentences, like the liar sentence or Curry sentences. That seems like a small price to pay for the full strength of classical logic together with a transparent truth predicate. With that much to recommend itself, I think we should try to hold on to the nontransitive approach as long as we can. My goal in this paper is to show that, contrary to recent criticism (Barrio, Rosenblatt, and Tajer 2016; Rosenblatt 2017), we shouldn't reject the nontransitive approach because it cannot express the admissibility of its own metarules.Along the way, we can observe some general facts about the possibility of expressing, in the object language, which metarules are admissible. In particular, I point out that no one who does her metatheory in classical logic and allows for self-reference can allow her logic to truthfully express which metarules are admissible, according to her own logic.

Expressing logical disagreement from within

“…that a metainference is represented as valid in T just in case that metainference is valid for T . Hlobil (2018) has shown that the obvious way of parsing that principle trivialises any theory defined with a classical meta-theory. Inspired by recent research on metainferences such as Barrio et al (2015), Barrio et al (2020) and Dicher and Paoli (2019), this paper proposes and develops an approach according to which the requirement for metainferences should be restricted from (globally) valid metainferences to locally valid metainferences; that a metainference is represented as valid in T if and only if the metainference is locally valid for T .…”

“…A first stab at a requirement for representing metainferences is presented in the following form by Hlobil (2018) but the idea of a requirement along these lines is also suggested by Barrio et al (2016) and Rosenblatt (2017): 13 if X 0 T Y 0 and . .…”

Metainferences from a Proof-Theoretic Perspective, and a Hierarchy of Validity Predicates

“….. , we note that . 10 is of the form for some , and so, as before, we construct a maximal such that for all finite :…”

“…By "valid," I really mean here "derivable". For, as Hlobil argues in[10], a validity predicate cannot capture the admissible rules on pain of going non-classical in the metalanguage 14. This definition serves as an alternative to the explicit definition of validity in terms of truth preservation.…”

Why a Logic is not only its Set of Valid Inferences

“…Recently, Barrio, Rosenblatt, and Tajer (2015) and Barrio, Pailos, and Szmuc (2018) draw attention to certain paraconsistent aspects of ST (and consequently, of CFOLE). Both papers have shown thatthrough some suitable translation-the set of valid inferences in LP coincides with the set of valid meta-inferences in ST. First among these this objection has been criticized by Hlobil (2018 and2019). In these papers Hlobil argues that (i) the criticism applies to all approaches that do their metatheory in classical logic and (ii) asking a logic to express its own admissible metarules may not be a good idea.…”