In this paper we treat metasequents—objects which stand to sequents as sequents stand to formulas—as first class logical citizens. To this end we provide a metasequent calculus, a sequent calculus which allows us to directly manipulate metasequents. We show that the various metasequent calculi we consider are sound and complete w.r.t. appropriate classes of tetravaluations where validity is understood locally. Finally we use our metasequent calculus to give direct syntactic proofs of various collapse results, closing a problem left open in (...) French (Ergo, 3(5), 113–131 2016). (shrink)

In this paper we discuss the extent to which the very existence of substructural logics puts the Tarskian conception of logical systems in jeopardy. In order to do this, we highlight the importance of the presence of different levels of entailment in a given logic, looking not only at inferences between collections of formulae but also at inferences between collections of inferences—and more. We discuss appropriate refinements or modifications of the usual Tarskian identity criterion for logical systems, and propose an (...) alternative of our own. After that, we consider a number of objections to our account and evaluate a substantially different approach to the same problem. (shrink)

There is consensus among computer scientists, logicians, and philosophers that good reasoning with qualitative beliefs must have the structural property of cumulative transitivity or, for short, cut. This consensus is typically explicitly argued for partially on the basis of practical and mathematical considerations. But the consensus is also implicit in the approach philosophers take to almost every puzzle about reasoning that involves multiple steps: philosophers typically assume that if each step in reasoning is acceptable considered on its own, the whole (...) chain of reasoning must also acceptable. In this paper I focus on whether there are good philosophical reasons for thinking that the consensus that good reasoning must satisfy cut is true. My central claim is that we should not accept the consensus—good reasoning might not satisfy cut. In particular, I consider four arguments for the consensus and explain why they are unpersuasive. (§2-5). I then show that the issue of whether good reasoning is cut turns on a substantive yet until now unnoticed question in epistemology (§6). (shrink)

Nonclassical theories of truth have in common that they reject principles of classical logic to accommodate an unrestricted truth predicate. However, different nonclassical strategies give up different classical principles. The paper discusses one criterion we might use in theory choice when considering nonclassical rivals: the maxim of minimal mutilation.

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. (shrink)

We will present a three-valued consequence relation for metainferences, called CM, defined through ST and TS, two well known substructural consequence relations for inferences. While ST recovers every classically valid inference, it invalidates some classically valid metainferences. While CM works as ST at the inferential level, it also recovers every classically valid metainference. Moreover, CM can be safely expanded with a transparent truth predicate. Nevertheless, CM cannot recapture every classically valid meta-metainference. We will afterwards develop a hierarchy of consequence relations (...) CMn for metainferences of level n. Each CMn recovers every metainference of level n or less, and can be nontrivially expanded with a transparent truth predicate, but cannot recapture every classically valid metainferences of higher levels. Finally, we will present a logic CMω, based on the hierarchy of logics CMn, that is fully classical, in the sense that every classically valid metainference of any level is valid in it. Moreover, CMω can be nontrivially expanded with a transparent truth predicate. (shrink)

ABSTRACTWe will present 12 different mixed metainferential consequence relations. Each one of them is specified using two different inferential Tarskian or non-Tarskian consequence relations: or. We will show that it is possible to obtain a Tarskian logic with non-Tarskian inferential logics, but also a non-Tarskian logic with Tarskian inferential logics. Moreover, we will show how some of these metainferential logics work better than the corresponding inferential rivals. Finally, we will show how these logics prove that it is not enough to (...) work with inferences as pairs of sets of formulas to obtain a contractive logic. (shrink)

In this paper we treat metasequents—objects which stand to sequents as sequents stand to formulas—as first class logical citizens. To this end we provide a metasequent calculus, a sequent calculus which allows us to directly manipulate metasequents. We show that the various metasequent calculi we consider are sound and complete w.r.t. appropriate classes of tetravaluations where validity is understood locally. Finally we use our metasequent calculus to give direct syntactic proofs of various collapse results, closing a problem left open in (...) French (Ergo, 3(5), 113–131 2016). (shrink)