A pragmatic account of consequence aims to understand logical or semantic consequence in terms of normative pragmatic notions, for instance, commitment or assertibility. There is widely thought to be a crucial problem with pragmatic accounts of consequence: they seem critically susceptible to various sorts of collapse arguments showing that, in the context of a pragmatic account of consequence, the truth of a sentence collapses into its bearing whatever normative status is appealed to in that account, be it commitment, assertibility, or what have you. In this paper we address such arguments, providing an account of semantic consequence in terms of the pragmatic notion of committive consequence and showing that this account is susceptible to none of the problematic collapse arguments that have been proposed in the literature. As a proof of concept, we present a multilateral logic of commitment which validates the key pragmatic inferences involving a commitment operator while avoiding collapses.
As Carnap (1943) first pointed out, some deviant interpretations of logical terms seem consistent with classicists' inferential practice. This has come to be known as Carnap’s Problem.
Bilateralism has been proposed as a principled way to rule out non-standard interpretations (Smiley 1996, Rumfitt 2000). The approach is known to work for propositional logic, but it is unclear how to extend it to first-order languages. In this talk, we will examine what bilateralism can do to pin down the standard interpretation of quantifiers. Along the way, we will sketch different formulations of Carnap's Problem and characterise the classes of (non-normal) interpretations one gets in different bilateralist settings.
Bilateralism is often introduced by replacing a single inferential stance with two opposed but independent ones: assertion and denial, acceptance and rejection, proof and refutation. But when does this two-sided outlook mark a genuine structural difference, rather than dressing ordinary unilateral consequence in bilateral garb?
The talk addresses this question at a deliberately abstract level. Proof systems and semantic structures are not treated as rival foundations, but as concrete implementations of more general patterns involving consequence relations and closure operators: deductive systems implement consequence as derivability, while semantic structures implement it as validity. Rather than beginning with consequence itself, I start from judgment-compatibility between assertion-states and denial-states, mathematically represented by a polarity. This allows us to ask how one-sided closure is transformed when closure is required to act on two qualitatively opposed dimensions at once. In particular, two-sided closure operators give
rise to a distinctive two-sided notion of theory: not a single closed set, but a pair of coordinates registering opposed forms of commitment, whose interaction is part of what closure itself must determine.
A central proposal of the talk is that, once acceptance and rejection are represented independently, familiar non-Tarskian consequence relations can be reconstructed as one-dimensional aspects of a richer
two-dimensional consequence relation. The resulting framework clarifies what is at stake in debates about primitive denial, coordination principles, and bilateral consequence: the decisive issue
is not simply whether a formalism distinguishes two opposed positions, but whether its associated consequence relations and closure operators encode how those positions constrain one another.
The intuitionistic negation has been frequently criticized for its non-constructive flavor. From a bilateralist point of view it seems to make sense to consider not only the notion of proof but also the notion of refutation if you work in a constructive setting. Such a basis is usually agreed upon to give rise to the so-called 'strong negation', introduced by David Nelson for his constructive logics N3 and N4. In this talk I want to share some thoughts on why I do not agree that this negation can be considered constructive and what these considerations entail for the constructive bilateralist.
The notion of metaphysical necessity is controversial, and even if it is conceded that there is such a thing as metaphysical necessity along-side logical necessity, its exact relationship to logical necessity remains a matter of debate. In this paper, I propose to capture the distinction between metaphysical and logical necessity from the perspective of logical bilateralism. Whilst statements of logical and metaphysical necessity share the same (support of) truth conditions, they differ in their (support of) falsity conditions. The distinction is made precise by introducing a certain bi-modal connexive propositional logic.
Bilateralist frameworks mostly come with a toggle negation which internalizes refutations via proofs and vice versa. Yet there are reasons to consider negation-free bilateralism. One reason is that toggle negation is what allows (at least on a formal level) to reduce bilateral systems to unilateral ones, limiting the scope of the framework. Second reason is related to often held stance that refutation should be considered as conceptually prior to negation in a bilateral framework. In this case it follows that the framework in question should be able to hold on its own without being propped by toggle negation.
In this talk I will show how bilateral first-order logics can be formulated in a way that does not rely on toggle negation. On the side of proof-theory multiple-conclusion sequent calculi are used, while semantics is given in the form of both relation sheaves and relational frames. A number of familiar systems or first-order versions of familiar system obtain. Some interesting effects resulting from adding bilateral excluded middle and non-contradiction principles are discussed.
This talk presents a bilateral natural deduction system with general schemas for positive and negative introduction rules, along with corresponding elimination rules. We do not include any coordination principles between the positive and negative rules, and we argue that the usual notion of harmony between introduction and elimination rules is sufficient to be meaning conferring without requiring any additional notion of bilateral harmony. Notably, these rules allow for connectives that introduce provable contradictions and provably underdetermined formulas. We isolate a handful of connectives that are functionally complete with respect to these schemas in the proof-theoretic sense. We also introduce notions of gap- and glut-averse connectives and prove accompanying functional-completeness results.
Bilateralism is a well-known approach to inferentialist semantics for the logical constants of the predicate calculus. What is less well known is that it lends itself to a neat formalisation of modal logic. The purpose of this paper is threefold.
(1) To draw attention to a formalisation of S5 that deserves to be more widely known amongst philosophers with an inferentialist, and particularly bilateralist, bent. It has been provided by Andrzej Indrzejczak and uses a sequent calculus with bilateral features and has, opposed to the most standard formalisation of S5, proof-theoretically desirable properties: it is cut free. Nonetheless, it also stays very close to standard sequent calculus, and this is in part due to a bilateralist feature.
(2) To reformulate Indrzejczak's system to give it a more recognisably bilateralist form.
(3) To transpose the resulting system into natural deduction, thereby extending well-known classical systems of bilateral logic by S5 modal operators.
Graham Priest’s simple three-valued logic LP has many curious properties. It has the same valid formulas as classical logic, but differs from classical logic when it comes to valid sequents. The valid sequents do not uniquely characterise the logic: it is possible to have more than one different LP-“negation”, each of which satisfies all the LP-requirements, without being equivalent. (The situation is not unlike modal operators in your favoured modal logic. A modal logic like S5 does not uniquely determine the meaning of the modal operators. The same goes for LP when characterised in terms of its consequence relation.) One consequence of this fact is that extant proof-first characterisations of LP are unwieldy. In some systems, connectives are given rules featuring negation and rules without; in others, formulas occur positively or negatively signed, and in others, sequents have three positions in which formulas can occur instead of two. This makes relating LP to familiar logics on a proof-first basis difficult.
The simple three-valued logic ST (strict-tolerant logic) also has many curious properties. It has the same valid formulas and valid sequents as classical logic, but differs from classical logic at the level of meta-inferential validity (rules obtaining between sequents). In ST, the Cut rule is not generally valid: from A ⇒ B and B ⇒ C, it need not follow that A ⇒ C. Understanding the distinctive behaviour of ST on an inferential level involves considering not only valid formulas and valid inferences but also valid meta-inferences. However, keeping track of this ever-growing tower of consequence relations is also difficult.
In this talk, I aim to address both of these issues in one go. I will exploit the relationship between LP and ST, and some prior work on bilateral treatments of natural deduction to provide a novel natural deduction proof system for both LP and ST that has the following features:
• Each connective rule is a standard natural deduction rule, familiar from Gentzen.
• Each connective is uniquely characterised by rules governing it.
• The difference between LP and ST on the one hand, and classical logic on the other, is the addition of a purely structural rule.
• The relationship between the valid formulas, the valid sequents, and the valid meta-sequents in each of the logics in question (LP, ST and classical logic) is uniquely and systematically determined by the rules governing the construction of proofs in the underlying calculus.
The aim of this exercise is not is to not only get a better understanding the breadth of the range of options for inferential presentations of logic LP and ST, but to also deepen our understanding of the relationship between natural deduction and the sequent calculus (and meta-inferential relations above the level of the sequent), and the distinctive role of structural rules from each of these perspectives.
Our use of vague predicates in reasoning is often full of uncertainty. In particular, the tolerance principles key to paradox---eg, if this person is tall, then that almost-as-tall person must be tall as well---might be the kind of thing we can have very high confidence in, without being totally certain.
This talk explores a bilateral approach to consequence developed over classical probability functions, and explores applications to reasoning with vague predicates. On the approach we develop, as we raise our standards for "certain enough", we validate more and more classically-valid arguments, until we reach full classical logic by requiring perfect certainty. Relatedly, as we raise our standards for "certain enough", we validate longer and longer sequences of soritical reasoning. But as long as we do *not* require perfect certainty, we can still accommodate sorites sequences.
As many philosophers have noted, we have two takes on the world: "the view from nowhere" and "the view from here". In the latter the cognitive agent occupies a privileged position; in the former they do not. But the two views are contradictory. Reality has two sides, as it were, like a ring made of paper, each side contradicting the other. In fact, the two views are more intimately related than this, since each presupposes the other. Reality is, then, more like what happens when you put a twist in the ring, producing a Möbius strip. There is just one side which is self-contradictory. This paper explores all these matters.