Semantic paradoxes pose a real threat to logics that attempt to be capable of expressing their own semantic concepts. Particularly, Curry paradoxes seem to show that many solutions must change our intuitive concepts of truth or validity or impose limits on certain inferences that are intuitively valid. In this way, the logic of a universal language would have serious problems. In this paper, we explore a different solution that tries to avoid both limitations as much as possible. Thus, we argue (...) that it is possible to capture the naive concepts of truth and validity without losing any of the valid inferences of classical logic. This approach is called the Buenos Aires plan. We present the logic of truth and validity, $$\mathsf {STTV}_{\omega }$$ based on the hierarchy of logics $$\textsf{ST}_{\omega }$$, whose validity predicate has the same semantic conditions as the material conditional. We argue that $$\mathsf {STTV}_{\omega }$$ is capable of blocking the problematic results while keeping the deductive power of classical logic as much as possible and offering an adequate semantic theory. On the other hand, one could object that it is not possible to reason with $$\mathsf {STTV}_{\omega }$$ because it is not closed under its logical principles. We respond to this objection and argue that the local characterization of validity shows how to make inferences using the logic $$\textsf{ST}_{\omega }$$. (shrink)

ABSTRACT What do we disagree about when we disagree about logic? On the face of it, classical and nonclassical logicians disagree about the laws of logic and the nature of logical properties. Yet, sometimes the parties are accused of talking past each other. The worry is that if the parties to the dispute do not mean the same thing with ‘if’, ‘or’, and ‘not’, they fail to have genuine disagreement about the laws in question. After the work of Quine, this (...) objection against genuine disagreement about logic has been called the meaning-variance thesis. We argue that the meaning-variance thesis can be endorsed without blocking genuine disagreement. In fact, even the type of revisionism and nonapriorism championed by Quine turns out to be compatible with meaning-variance. (shrink)

The historical consensus is that logical evidence is special. Whereas empirical evidence is used to support theories within both the natural and social sciences, logic answers solely to a priori evidence. Further, unlike other areas of research that rely upon a priori evidence, such as mathematics, logical evidence is basic. While we can assume the validity of certain inferences in order to establish truths within mathematics and test scientifi c theories, logicians cannot use results from mathematics or the empirical sciences (...) without seemingly begging the question. Appeals to rational intuition and analyticity in order to account for logical knowledge are symptomatic of these commitments to the apriority and basicness of logical evidence. This chapter argues that these historically prevalent accounts of logical evidence are mistaken, and that if we take logical practice seriously we fi nd that logical evidence is rather unexceptional, sharing many similarities to the types of evidence appealed to within other research areas. (shrink)

One of the most fruitful applications of substructural logics stems from their capacity to deal with self-referential paradoxes, especially truth-theoretic paradoxes. Both the structural rules of contraction and the rule of cut play a crucial role in typical paradoxical arguments. In this paper I address a number of difficulties affecting noncontractive approaches to paradox that have been discussed in the recent literature. The situation was roughly this: if you decide to go substructural, the nontransitive approach to truth offers a lot (...) of benefits that are not available in the noncontractive account. I sketch a noncontractive theory of truth that has these benefits. In particular, it has both a proof- and a model-theoretic presentation, it can be extended to a first-order language, and it retains every classically valid inference. (shrink)

In some recent articles, Cobreros, Egré, Ripley, & van Rooij have defended the idea that abandoning transitivity may lead to a solution to the trouble caused by semantic paradoxes. For that purpose, they develop the Strict-Tolerant approach, which leads them to entertain a nontransitive theory of truth, where the structural rule of Cut is not generally valid. However, that Cut fails in general in the target theory of truth does not mean that there are not certain safe instances of Cut (...) involving semantic notions. In this article we intend to meet the challenge of answering how to regain all the safe instances of Cut, in the language of the theory, making essential use of a unary recovery operator. To fulfill this goal, we will work within the so-called Goodship Project, which suggests that in order to have nontrivial naïve theories it is sufficient to formulate the corresponding self-referential sentences with suitable biconditionals. Nevertheless, a secondary aim of this article is to propose a novel way to carry this project out, showing that the biconditionals in question can be totally classical. In the context of this article, these biconditionals will be essentially used in expressing the self-referential sentences and, thus, as a collateral result of our work we will prove that none of the recoveries expected of the target theory can be nontrivially achieved if self-reference is expressed through identities. (shrink)

It is widely accepted that classical logic is trivialized in the presence of a transparent truth-predicate. In this paper, we will explain why this point of view must be given up. The hierarchy of metainferential logics defined in Barrio et al. and Pailos recovers classical logic, either in the sense that every classical inferential validity is valid at some point in the hierarchy ), or because a logic of a transfinite level defined in terms of the hierarchy shares its validities (...) with classical logic. Each of these logics is consistent with transparent truth—as is shown in Pailos —, and this suggests that, contrary to standard opinions, transparent truth is after all consistent with classical logic. However, Scambler presents a major challenge to this approach. He argues that this hierarchy cannot be identified with classical logic in any way, because it recovers no classical antivalidities. We embrace Scambler’s challenge and develop a new logic based on these hierarchies. This logic recovers both every classical validity and every classical antivalidity. Moreover, we will follow the same strategy and show that contingencies need also be taken into account, and that none of the logics so far presented is enough to capture classical contingencies. Then, we will develop a multi-standard approach to elaborate a new logic that captures not only every classical validity, but also every classical antivalidity and contingency. As a€truth-predicate can be added to this logic, this result can be interpreted as showing that, despite the claims that are extremely widely accepted, classical logic does not trivialize in the context of transparent truth. (shrink)

Against the backdrop of the frequent comparison of theories of truth in the literature on semantic paradoxes with regard to which inferences and metainferences are deemed valid, this paper develops a novel approach to defining a binary predicate for representing the valid inferences and metainferences of a theory within the theory itself under the assumption that the theory is defined with a classical meta-theory. The aim with the approach is to obtain a tool which facilitates the comparison between a theory (...) and its competitors within the theory itself, thereby expressing the disagreement between the theories within the theories. After discussing what we can and should require of an object-linguistic representation of a theory for that purpose, this paper proposes to restrict the representation of valid metainferences to locally valid metainferences, a requirement which turns out to be \-consistent and conservative over classical first-order arithmetic. This approach is then applied to four theories definable on strong Kleene models using a labelled nested sequent calculus. (shrink)

Recently, several authors have pointed out that substructural logics are adequate for developing naive theories that represent semantic concepts such as truth. Among them, three proposals have been explored: dropping cut, dropping contraction and dropping reflexivity. However, nowhere in the substructural literature has anyone proposed rejecting the structural rule of weakening, while accepting the other rules. Some theorists have even argued that this task was not possible, since weakening plays no role in the derivation of semantic paradoxes. In this article, (...) I introduce a theory for naive truth based on the logic resulting from dropping the rule of weakening from classical logic, and maintaining the other structural rules. (shrink)

Logical anti‐exceptionalism is the view that logic is not special, it is continuous with science. This continuity is typically understood in terms of the use of the abductive method in logical theory choice, with logical knowledge resulting from our choice of the theory best accounting for the data. In this paper, we argue for two related claims: (i) that this understanding of the continuity between logic and science faces considerable challenges; and (ii) that such challenges may be avoided by elaborating (...) the continuity of logic with science in terms of the “logic‐as‐models” approach. More specifically, we defend that logical theories should be seen in the lights of the “models‐as‐epistemic‐tools” approach, an approach to the nature of models in empirical science more closely connected with current scientific practice. The result is a version of logical anti‐exceptionalism opening new perspectives for the philosophical understanding of logic. (shrink)

Substructural solutions to the semantic paradoxes have been broadly discussed in recent years. In particular, according to the non-transitive solution, we have to give up the metarule of Cut, whose role is to guarantee that the consequence relation is transitive. This concession—giving up a meta rule—allows us to maintain the entire consequence relation of classical logic. The non-transitive solution has been generalized in recent works into a hierarchy of logics where classicality is maintained at more and more metainferential levels. All (...) the logics in this hierarchy can accommodate a truth predicate, including the logic at the top of the hierarchy—known as CMω—which presumably maintains classicality at all levels. CMω has so far been accounted for exclusively in model-theoretic terms. Therefore, there remains an open question: how do we account for this logic in proof-theoretic terms? Can there be found a proof system that admits each and every classical principle—at all inferential levels—but nevertheless blocks the derivation of the liar? In the present paper, I solve this problem by providing such a proof system and establishing soundness and completeness results. Yet, I also argue that the outcome is philosophically unsatisfactory. In fact, I’m afraid that in light of my results this metainferential solution to the paradoxes can hardly be called a “solution,” let alone a good one. (shrink)

The goal of the paper is to discuss whether substructural non-contractive accounts of the truth-theoretic paradoxes can be philosophically motivated. First, I consider a number of explanations that have been offered to justify the failure of contraction and I argue that they are not entirely compelling. I then present a non-contractive theory of truth that I’ve proposed elsewhere. After looking at some of its formal properties, I suggest an explanation of the failure of structural contraction that is compatible with it.

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)