Substructural Logics

ABSTRACT: This paper contends that Stoic logic (i.e. Stoic analysis) deserves more attention from contemporary logicians. It sets out how, compared with contemporary propositional calculi, Stoic analysis is closest to methods of backward proof search for Gentzen-inspired substructural sequent logics, as they have been developed in logic programming and structural proof theory, and produces its proof search calculus in tree form. It shows how multiple similarities to Gentzen sequent systems combine with intriguing dissimilarities that may enrich contemporary discussion. Much of Stoic logic appears surprisingly modern: a recursively formulated syntax with some truth-functional propositional operators; analogues to cut rules, axiom schemata and Gentzen's negation-introduction rules; an implicit variable-sharing principle and deliberate rejection of Thinning and avoidance of paradoxes of implication. These latter features mark the system out as a relevance logic, where the absence of duals for its left and right introduction rules puts it in the vicinity of McCall's connexive logic. Methodologically, the choice of meticulously formulated meta-logical rules in lieu of axiom and inference schemata absorbs some structural rules and results in an economical, precise and elegant system that values decidability over completeness.

According to the logical inferentialist, the meaning of a logical connective is determined by the inference rules that govern its use. Proof theoretic semantics attempts to make this idea precise in a proof theoretic framework, using for example natural deduction or sequent calculus rules. Since Prior's infamous connective tonk much of proof theoretic semantics have been occupied with formal anti-tonk conditions which rule out ill-behaved connectives (e.g. conservativeness, harmony). Common between them is that inference rules only succeed in determining the meaning of a connective if the proof theoretic conditions are fulfilled. On the traditional account, however, such conditions are insensitive to substructural dimensions of proof theory, e.g. the distinction between additive and multiplicative connectives. We argue that proof theoretic semantics ought to have the resources to attribute different meanings to substructurally distinct connectives. Subsequently we show how to develop a notion of proof theoretic harmony that preserves substructural distinctions from introduction to elimination rules. The substructural account of harmony can rule out cases of nonconservativeness that previous accounts have not dealt with.

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 (MMM).

This paper argues: first, that a presentist, powers based, diachronic account of modality can provide a satisfactory account of our intuitions about modality as well as a compelling rebuttal to alternative accounts; second that taking this account seriously requires a more radically attenuated modal logic than even partisans of such an account realize. Attempting to navigate this problem, I provide a formal, non-normal semantics for a tensed modal logic, along with a philosophical interpretation of it.

It is common to find among substructural theorists the idea that to consider the logical consequence as a relation between sets (of premises and conclusions) is enough to guarantee the validity or admissibility of the structural rule of contraction. In this article, we intend to argue against such a thesis. Firstly, it's not clear that the very rule of contraction can be formulated with sets, since they don't distinguish between multiple occurrences of a formula. Secondly, even if it were possible to express the rule of contraction in this way, the thesis would be false since we show one counterexample to it-the logic K3/LP-appealing to the notion of mixed metainferences.

This paper aims to provide a logical background for Combinatory Categorial Grammar (CCG) and its typological discussions. Based on the Curry-Howard correspondence between Gentzen-style proof systems and Lambek Lamda Calculi, and those between Hilbert-style proof systems and substructural BCWK-logic, I define a new class of logic which provides subclasses for each of the substructural combinatory logics, called Subdirectional Combinatory Logic, and propose that CCG is a subdirectional logic of a relevance logic (Combinatory Hypothesis). This hypothesis allows us to discuss typological parameters in universal grammar in terms of the presence/absence of a certain directional combinators.

We overview the logic of Bunched Implications (BI) and Separation Logic (SL) from a perspective inspired by Hiroakira Ono's algebraic approach to substructural logics. We propose generalized BI algebras (GBI-algebras) as a common framework for algebras arising via " declarative resource reading " , intuitionistic generalizations of relation algebras and arrow logics and the distributive Lambek calculus with intuitionistic implication. Apart from existing models of BI (in particular, heap models and effect algebras), we also cover models arising from weakening relations, formal languages or more fine-grained treatment of labelled trees and semistructured data. After briefly discussing the lattice of subvarieties of GBI, we present a suitable duality for GBI along the lines of Esakia and Priestley and an algebraic proof of cut elimination in the setting of residuated frames of Galatos and Jipsen. We also show how the algebraic approach allows generic results on decidability, both positive and negative ones. In the final part of the paper, we gently introduce the substructural audience to some theory behind state-of-art tools, culminating with an algebraic and proof-theoretic presentation of (bi-) abduction.

This article introduces substructural epistemic logics by combining relational models for modal distributive substructural logics with Kripke models for normal modal logics. The main result is a general completeness theorem. It is also noted that the substructural epistemic framework is suitable for modelling non-monotonic information update and non-classical relational belief revision.

The article adds a negative modal operator ~ to normal modal epistemic logics. ~A is seen as representing the set of propositions taken as sufficient grounds to refute A within a specific epistemic context. An operator of contextual belief is defined in terms of ruling out ~A. A number of epistemic logics with different variants of modal negation are explored. It is shown that, in most of the logics, contextual belief avoids the logical omniscience problem. It is also shown that using modal negation in public announcement logic yields a specific form of non-monotonic information update, related to changes of epistemic context. Two variants of modal negation are linked to the Relevant Alternatives theory of knowledge and to abstract argumentation frameworks. The technical results include complete axiomatizations of a number of epistemic logics with modal negation and a proof or a reduction schema for ~ within public announcement logic.

We introduce a number of logics to reason about collective propositional attitudes that are defined by means of the majority rule. It is well known that majoritarian aggregation is subject to irrationality, as the results in social choice theory and judgment aggregation show. The proposed logics for modelling collective attitudes are based on a substructural propositional logic that allows for circumventing inconsistent outcomes. Individual and collective propositional attitudes, such as beliefs, desires, obligations, are then modelled by means of minimal modalities to ensure a number of basic principles. In this way, a viable consistent modelling of collective attitudes is obtained.

This work contributes to the theory of judgment aggregation by discussing a number of significant non-classical logics. After adapting the standard framework of judgment aggregation to cope with non-classical logics, we discuss in particular results for the case of Intuitionistic Logic, the Lambek calculus, Linear Logic and Relevant Logics. The motivation for studying judgment aggregation in non-classical logics is that they offer a number of modelling choices to represent agents' reasoning in aggregation problems. By studying judgment aggregation in logics that are weaker than classical logic, we investigate whether some well-known impossibility results, that were tailored for classical logic, still apply to those weak systems.

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 v-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.

We will present a three-valued consequence relation for metainferences, called MI. MI no valid metainference, and thus no valid inference either. MI's consequence relation for metainferences is based on TS' consequence relation for inferences, in the sense that the standard for the conclusion is more demanding that the one for premises. A metainference is valid in MI iff for every valuation v, if v satisfies every premise according to ST, v satisfies the conclusion according to TS. Nevertheless, MI is informative about the meta-metainferences, or metainferences of level 2. We will afterwards develop a hierarchy of consequence relations MIn for metainferences of level n (for 1 ≤ n ≤ ω). Each MIn invalidates every metainference of level n or less, but is informative about metainferences of a higher level. Finally, we will present a logic L empty based on the hierarchy of MIns that is genuinely empty, in the sense that no metainference of any level is valid in it.

Under a proper translation, the languages of propositional (and quantified relevant logic) with an absurdity constant are characterized as the fragments of first order logic preserved under (world-object) relevant directed bisimulation. Furthermore,  the properties of pointed models axiomatizable by sets of propositional relevant formulas have a purely algebraic characterization. Finally, a form of the interpolation  property holds for the relevant fragment of first order logic.

Monoidal logics were introduced as a foundational framework to analyse the proof theory of deontic logic. Building on Lambek’s work in categorical logic, logical systems are defined as deductive systems, that is, as collections of equivalence classes of proofs satisfying specific rules and axiom schemata. This approach enables the classification of deductive systems with respect to their categorical structure. When looking at their proof theory, however, one can see that there are similarities between monoidal and substructural logics. The purpose of the present paper is to address this issue and highlight the differences between these two approaches. We argue that monoidal logics provide a more flexible foundational framework that enables a finer analysis of the relationship between negation(s) and other logical connectives. We show that the elimination of double negation(s) is independent from the de Morgan dualities, that monoidal deductive systems are not necessarily weakly distributive and that deductive systems satisfying the elimination of double negation(s) and the law of excluded middle are not necessarily classical.

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 ...

In this paper, consistency is understood as the absence of the negation of a theorem, and not, in general, as the absence of any contradiction. We define the basic constructive logic BKc1 adequate to this sense of consistency in the ternary relational semantics without a set of designated points. Then we show how to define a series of logics extending BKc1 within the spectrum delimited by contractionless minimal intuitionistic logic. All logics defined in the paper are paraconsistent logics.

Adding a transparent truth predicate to a language completely governed by classical logic is not possible. The trouble, as is well-known, comes from paradoxes such as the liar and curry. Because of these paradoxes, theories of truth typically use a non- transparent truth predicate preserving classical logic. Revisionary approaches to logic take a different path. Semantic paradoxes are a central motivation for standard non-classical-logic-based truth theories: usually a logic weaker than classical logic. But curry paradox makes this task really complicated. It is well known that any transitive logic with transparent truth is in trouble because the Curry paradox, if it features a conditional connective validating identity, modus ponens, and contraction. Recently, Cobreros, Egre, Ripley and van Rooij have proposed an alternative approach based on a non-transitive notion of logical consequence. Adopting 3- valued models, they develop a permissive consequence in which the validity of an argument goes from strictly true premises to a tolerant true conclusion. In the internal level, this framework allows to deal with semantic paradoxes adopting transparent truth to mesh with classical logic. Nevertheless, in the external level where we are interested on the closure properties on the set of valid arguments, some results are disturbing. In this paper, I show that transitivity is not the only metainference that is missed. In particular, I prove that the strict-tolerant external notion of validity is just LP validity. This result shows that permissive consequence is too weak to express a suitable notion of external entailment.

This paper provides an inferentialist motivation for a logic belonging in the connexive family, by borrowing elements from the bilateralist interpretation for Classical Logic without the Cut rule, proposed by David Ripley. The paper focuses on the relation between inferentialism and relevance , through the exploration of what we call relevant assertion and denial, showing that a connexive system emerges as a symptom of this interesting link. With the present attempt we hope to broaden the available interpretations for connexive logics, showing they can be rightfully motivated in terms of certain relevantist constraints imposed on assertion and denial.

This paper defines a Sahlqvist fragment for relevant logic and establishes that each class of frames in the Routley-Meyer semantics which is definable by a Sahlqvist formula is also elementary, that is, it coincides with the class of structures satisfying a given first order property calculable by a Sahlqvist-van Benthem algorithm. Furthermore, we show that some classes of Routley-Meyer frames definable by a relevant formula are not elementary.

The hierarchy of metainferential logics defined in Barrio et al ([3]) and Pailos ([22]) recovers classical logic, either in the sense that every classical (meta)inferential validity is valid at some point in the hierarchy (as is stressed in [3]), or because a logic of a transfinite level defined in terms of the hierarchy shares its validities with classical logic (as in [22]). Scambler ([31]) presents a major criticism to this approach. He argues that this hierarchy cannot be identified with classical logic in any way, because it recovers no classical antivalidities. And if it can do that, so is the case with a parallel hierarchy based on TS, that recovers every classical antivalidity, but none of its validities. We embrace Scambler criticism and develop a new hierarchy based on the previous two. This new hieararchy recovers both every classical validity and every classical antivalidity. Moreover, we will show that contingencies need to be taken into account, and that none of the logics so far presented are enough to capture classical contingencies. But we will redefine our new hierarchy in such a way that it captures not only every classical validity, but also every classical antivalidity and contingency.

We will present all the mixed and impure disjoint three-valued logics based on the Strong Kleene schema. Some, but not all of them, are (inferentially) empty logics. We will also provide a recipe to build philosophical interpretations for each of these logics, and show why the kind of permeability that characterized them is not such a bad feature. Finally, we will assess the kind of pluralism this position is compromised with.

ABSTRACT: This paper shows that, for the Hertz–Gentzen Systems of 1933 (without Thinning), extended by a classical rule T1 (from the Stoics) and using certain axioms (also from the Stoics), all derivations are analytic: every cut formula occurs as a subformula in the cut’s conclusion. Since the Stoic cut rules are instances of Gentzen’s Cut rule of 1933, from this we infer the decidability of the propositional logic of the Stoics. We infer the correctness for this logic of a “relevance criterion” and of two “balance criteria”, and hence (in contrast to one of Gentzen’s 1933 results) that a particular derivable sequent has no derivation that is “normal” in the sense that the first premiss of each cut is cut-free. We also infer that Cut is not admissible in the Stoic system, based on the standard Stoic axioms, the T1 rule and the instances of Cut with just two antecedent formulae in the first premiss. (OPEN ACCESS)

In this paper, we discuss the approach based on Social Choice Theory and Judgment Aggregation to the definition of collective reasoning. We shall make explicit the aggregative nature of the notion of collective reasoning that is defined in the Judgment Aggregation account and we shall stress that the notion of logical coherence plays a fundamental role in defining collective attitudes. Unfortunately, as several results in Judgment Aggregation show, coherence is not compatible with fair aggregation procedures. On closer inspection, the notion
of coherence that is jeopardized by Judgment Aggregation is based on classical logic. In this work, we propose to revise the standard view of rationality of Judgment Aggregation by exploring the realm of non-classical logics. In particular, we will present possibility results for substructural logics. Those logics, we
argue, provide a viable notion of collective reasoning.

University Of Aberdeen This paper explores how a semantics for Prior’s infamous connective tonk should be, a connective defined by inference rules that trivialize the logic of a deductive system if that logic is supposed to be transitive. To avoid triviality, one must reject transitivity and in a relatively recent paper, Roy Cook develops a semantics for tonk with non-transitive entailment. However, I show in this paper that a cut-free sequent calculus for tonk - the arguably most natural and simplest deductive system for a non-transitive logic - can neither be complete with respect to Cook’s semantics nor with respect to a semantics with non-transitive entailment based on a semantics for vagueness and transparent truth developed by Cobreros et al. It is argued that the failure to adequately represent tonk is connected with the fact that tonk is not uniquely defined in a cut-free sequent calculus system unless the logic is in addition non-reflexive. To remedy this, the paper develops a semantics with non-transitive and non-reflexive entailment based on the idea that complex formulae are true or false relative to them being assessed as premise or as conclusion.(Received August 05 2014)

The paper presents an exhaustive menu of nonmonotonic logics. The options are individuated in terms of the principles they reject. I locate, e.g., cumulative logics and relevance logics on this menu. I highlight some frequently neglected options, and I argue that these neglected options are particularly attractive for inferentialists.

In previous work, I (albeit implicitly) generalized the results of Barrio, Pailos and Szmuc on meta-inferential validity beyond all the finite stages of meta-inferences to stage omega. This short note shows that there is no reason to stop there, since the results naturally generalize up to all transfinite orders once these have been suitably defined.

We show how to embed a framework for multilateral negotiation, in which a group of agents implement a sequence of deals concerning the exchange of a number of resources, into linear logic. In this model, multisets of goods, allocations of resources, preferences of agents, and deals are all modelled as formulas of linear logic. Whether or not a proposed deal is rational, given the preferences of the agents concerned, reduces to a question of provability, as does the question of whether there exists a sequence of deals leading to an allocation with certain desirable properties, such as maximising social welfare. Thus, linear logic provides a formal basis for modelling convergence properties in distributed resource allocation.