Rea Golan

Reason (or material) relations of implication and incompatibility hold in virtue of the meanings of linguistic expressions in a given language. As opposed to formal or logical consequence relations, reason relations are often structurally open, i.e., unclosed under principles such as monotonicity or transitivity. In their recent book, Hlobil and Brandom go even further, allowing for such relations to be radically open, i.e., unclosed under any structural principle. In particular, they reject the restrictive forms of monotonicity and transitivity known as cautious monotonicity (CM) and cumulative transitivity (CT). In the present paper, I argue that such radical openness is objectionable on the grounds that it does not allow us to construct for every set even a weak version of Tarski's notion of a theory, thereby undermining the very notion of consequence. I then lay out minimal principles under which such a construction is possible, and suggest that reason relations should be closed under those principles.

I explore, from a proof-theoretic perspective, the hierarchy of classical and paraconsistent logics introduced by Barrio, Pailos and Szmuc in (Journal o f Philosophical Logic,49, 93-120, 2021). First, I provide sequent rules and axioms for all the logics in the hierarchy, for all inferential levels, and establish soundness and completeness results. Second, I show how to extend those systems with a corresponding hierarchy of validity predicates, each one of which is meant to capture “validity” at a different inferential level. Then, I point out two potential philosophical implications of these results. (i) Since the logics in the hierarchy differ from one another on the rules, I argue that each such logic maintains its own distinct identity (contrary to arguments like the one given by Dicher and Paoli in 2019). (ii) Each validity predicate need not capture “validity” at more than one metainferential level. Hence, there are reasons to deny the thesis (put forward in Barrio, E., Rosenblatt, L. & Tajer, D. (Synthese, 2016)) that the validity predicate introduced in by Beall and Murzi in (Journal o f Philosophy,110(3), 143–165, 2013) has to express facts not only about what follows from what, but also about the metarules, etc.

Natural deduction inference rules seem to reflect the way we actually reason. Hence, many if not most inferentialist theories maintain that meaning is conferred on linguistic expressions by natural deduction rules, rather than the more abstract alternative of sequent rules. In the present paper, I argue, to the contrary, that an inferentialist theory of meaning must take a somewhat metainferential form, whereby the meanings of linguistic expressions—in particular, the logical constants—are conferred by sequent rules, conceived of as licensing inferences between inferences. To that end, I propose that sequent rules are to be understood as inferential semantic clauses, i.e., as playing a role analogous to that of the semantic clauses in truth conditional theories of meaning. I establish this proposal on the basis of an argument to the effect that inferentialists must adopt an extended version of Frege's context principle, according to which only within a complete sentence potentially serving either as a premise or as a conclusion of an inference do words have meaning. The resultant inferentialist theory is not disconnected from the way we actually reason. Moreover, on its basis one can make a meaning-theoretic case for rather unintuitive features of many sequent calculi, such as multiple conclusions and various ways of going substructural.

Proof theorists have long been struggling to provide simple accounts of various modal logics. Drawing on the recent literature on metainferences, I develop in the present paper a novel approach to this challenge: regular sequent systems augmented with natural deduction-like rules for assuming and discharging sequents. Based on this approach, I introduce an elegant calculus for the modal logic S4⁠. Various discharging conditions on assumptions are shown to yield the weaker logics K⁠, D, and ⁠T. The rules in these calculi have attractive proof-theoretic properties: they are explicit, symmetrical, and non-circular, and they enjoy the subformula property.

I introduce a novel hybrid calculus H for the validities and invalidities of classical propositional logic. The calculus H is different in nature from other hybrid calculi that can be found in the literature in that it does not include specific anti-sequent rules. Instead, I add to the sequent rules of classical propositional logic only two structural rules that allow us to introduce and eliminate anti-sequents in our derivations. The resultant system is much simpler than the existing systems in the literature. For one thing, it does not suffer from a proliferation of rules. For another, it is structurally complete: its admissible rules are all derivable.

Bilateralism is the view that the speech act of denial is as primitive as that of assertion. Bilateralism has proved helpful in providing an intuitive interpretation of formalisms that, prima facie, look counterintuitive, namely, multiple-conclusion sequent calculi. Under this interpretation, a sequent of the form $\Gamma \vdash \Delta$ is regarded as the statement that it is incoherent, according to our conversational norms, to occupy the position of asserting all the sentences in $\Gamma$ and denying all the sentences in $\Delta$. Some have argued, based on this interpretation, that the notion of invalidity is as conceptually primitive and important as the notion of validity: whereas the latter is couched in terms of what positions are incoherent and hence untenable, the former is couched in terms of what positions are coherent and hence tenable. My ultimate goal in this paper is to contest this view. Based on a novel technical account of the two notions—one that I find more accurate than the existing accounts in the literature—I shall argue that the notion of incoherence takes conceptual priority over the notion of coherence.

Several authors maintain that we are capable of having inconsistent intentional states, either in cases of illusion, in certain cases of imagination, or because the observable world is (partly) inconsistent and we perceive it as such. These views are all premised on the assumption that inconsistent intentional states—even if acknowledged—are peculiar and have nothing essential to do with our perceptual capacities. In the present article, I would like to present, and argue for, a much stronger thesis: that inconsistent intentional states underlie the possibility of having intentional content in mind. I argue for this thesis based on a Husserlian phenomenological analysis of our grasp of objects, which I formulate in terms of incompatibility semantics.

I introduce a novel and quite intuitive interpretation of the ternary relation that figures in the relational semantics of many relevance logics. Conceptually, my interpretation makes use only of incompatibility and parthood relations, defined over a set of states. In this way, the proposed interpretation—of the ternary relation and the conditional—extends Dunn’s and Restall’s works on negation and the Routley star operator. Therefore, the interpretation is unified, and hence not only intuitive but also parsimonious. Additionally, the interpretation provides us with a cogent argument to the effect that a ternary relation is indispensable, and we cannot merely do with a binary relation.

Graham Priest has formulated the minimally inconsistent logic of paradox (MiLP), which is paraconsistent like Priest’s logic of paradox (LP), while staying closer to classical logic. We present logics that stand to (the propositional fragments of) strong Kleene logic (K3) and the logic of first-degree entailment (FDE) as MiLP stands to LP. That is, our logics share the paracomplete and the paraconsistent-cum-paracomplete nature of K3 and FDE, respectively, while keeping these features to a minimum in order to stay closer to classical logic. We give semantic and sequent-calculus formulations of these logics, and we highlight some reasons why these logics may be interesting in their own right.

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.

Conventionalism about logic is the view that logical principles hold in virtue of some linguistic conventions. According to explicit conventionalism, these conventions have to be stipulated explicitly. Explicit conventionalism is subject to a famous criticism by Quine, who accused it of leading to an infinite regress. In response to the criticism, several authors have suggested reconstructing conventionalism as implicit in our linguistic behaviour. In this paper, drawing on a distinction from proof theory between derivable and admissible rules, I argue that implicit conventionalism has not been stated in a sufficiently precise way, as it leaves open what one needs to say about admissible yet underivable rules. Moreover, it turns out that this challenge cannot be easily met, and that any attempt to meet the challenge makes conventionalism much less attractive a thesis.

Minimally inconsistent LP (MiLP) is a nonmonotonic paraconsistent logic based on Graham Priest's logic of paradox (LP). Unlike LP, MiLP purports to recover, in consistent situations, all of classical reasoning. The present paper conducts a proof-theoretic analysis of MiLP. I highlight certain properties of this logic, introduce a simple sequent system for it, and establish soundness and completeness results. In addition, I show how to use my proof system in response to a criticism of this logic put forward by JC Beall.

The use of models to assign truth values to sentences and to counterexemplify invalid inferences is a basic feature of model theory. Yet sentences and inferences are not the only phenomena that model theory has to take care of. In particular, the development of sequent calculi raises the question of how metainferences are to be accounted for from a model-theoretic perspective. Unfortunately there is no agreement on this matter. Rather, one can find in the literature two competing model-theoretic notions of metainferential validity, known as the ‘global’ notion and the ‘local’ notion. In this article, I argue that, given certain plausible considerations about metainferential validity, the global notion collapses into the local notion.

There is a pervasive view that inference—as opposed, notably, to a grasp of objects—is an intralinguistic process that does not draw on extralinguistic resources. The present paper aims to show that this dichotomy between inferring and grasping objects can be resisted. Specifically, I offer an alternative view: a phenomenological account according to which our most basic inferences draw on our grasp of objects. I motivate this account on the grounds that, although it is restricted to such basic inferences, it has significant implications for the general question of whether inference involves taking one's premises to support one's conclusion. The proposed account implies that basic inferences need not involve corresponding “takings,” (even though such inferences involve conceptual content), but it leaves open the possibility that non-basic inferences do involve corresponding “takings,” while relying on basic inferences. It will turn out that such a disjunctive view, according to which non-basic reasoning is somewhat parasitic on basic reasoning, manages to avoid many of the problems with which the literature on the “taking condition” deals.

Logical pluralists are committed to the idea of a neutral metalanguage, which serves as a framework for debates in logic. Two versions of this neutrality can be found in the literature: an agreed upon collection of inferences, and a metalanguage that is neutral as such. I discuss both versions and show that they are not immune to Quinean criticism, which builds on the notion of meaning. In particular, I show that the first version of neutrality is sub-optimal, and hard to reconcile with the theories of meaning for logical constants, and the second version collapses mathematically, if rival logics, as object languages, are treated with charity in the metalanguage. I substantiate by proving a collapse theorem that generalizes familiar results. Thus, the existence of a neutral metalanguage cannot be taken for granted, and meaning-invariant logical pluralism might turn out to be dubious.