A number of authors have objected to the application of non-classical logic to problems in philosophy on the basis that these non-classical logics are usually characterised by a classical metatheory. In many cases the problem amounts to more than just a discrepancy; the very phenomena responsible for non-classicality occur in the field of semantics as much as they do elsewhere. The phenomena of higher order vagueness and the revenge liar are just two such examples. The aim of (...) this paper is to show that a large class of non-classical logics are strong enough to formulate their own model theory in a corresponding non-classical set theory. Specifically I show that adequate definitions of validity can be given for the propositional calculus in such a way that the metatheory proves, in the specified logic, that every theorem of the propositional fragment of that logic is validated. It is shown that in some cases it may fail to be a classical matter whether a given sentence is valid or not. One surprising conclusion for non-classical accounts of vagueness is drawn: there can be no axiomatic, and therefore precise, system which is determinately sound and complete. (shrink)

The subject of Labelled Non-Classical Logics is the development and investigation of a framework for the modular and uniform presentation and implementation of non-classical logics, in particular modal and relevance logics. Logics are presented as labelled deduction systems, which are proved to be sound and complete with respect to the corresponding Kripke-style semantics. We investigate the proof theory of our systems, and show them to possess structural properties such as normalization and the subformula property, which we exploit not (...) only to establish advantages and limitations of our approach with respect to related ones, but also to give, by means of a substructural analysis, a new proof-theoretic method for investigating decidability and complexity of (some of) the logics we consider. All of our deduction systems have been implemented in the generic theorem prover Isabelle, thus providing a simple and natural environment for interactive proof development. Labelled Non-Classical Logics is essential reading for researchers and practitioners interested in the theory and applications of non-classical logics. (shrink)

In this paper we develop what we can describe as a “dual two-sided” cut-free sequent calculus system for the non-classical logics of truth lp, k3, stt and a non-reflexive logic ts which is, arguably, more elegant than the three-sided sequent calculus developed by Ripley for the same logics. Its elegance stems from how it employs more or less the standard sequent calculus rules for the various connectives and truth, and the fact that it offers a rather neat connection between (...) derivable sequents and validity in comparison to the calculus developed by Ripley. (shrink)

The AGM theory of belief revision provides a formal framework to represent the dynamics of epistemic states. In this framework, the beliefs of the agent are usually represented as logical formulas while the change operations are constrained by rationality postulates. In the original proposal, the logic underlying the reasoning was supposed to be supraclassical, among other properties. In this paper, we present some of the existing work in adapting the AGM theory for non-classical logics and discuss their interconnections and (...) what is still missing for each approach. (shrink)

The Knower paradox purports to place surprising a priori limitations on what we can know. According to orthodoxy, it shows that we need to abandon one of three plausible and widely-held ideas: that knowledge is factive, that we can know that knowledge is factive, and that we can use logical/mathematical reasoning to extend our knowledge via very weak single-premise closure principles. I argue that classical logic, not any of these epistemic principles, is the culprit. I develop a consistent theory (...) validating all these principles by combining Hartry Field's theory of truth with a modal enrichment developed for a different purpose by Michael Caie. The only casualty is classical logic: the theory avoids paradox by using a weaker-than-classical K3 logic. I then assess the philosophical merits of this approach. I argue that, unlike the traditional semantic paradoxes involving extensional notions like truth, its plausibility depends on the way in which sentences are referred to--whether in natural languages via direct sentential reference, or in mathematical theories via indirect sentential reference by Gödel coding. In particular, I argue that from the perspective of natural language, my non-classical treatment of knowledge as a predicate is plausible, while from the perspective of mathematical theories, its plausibility depends on unresolved questions about the limits of our idealized deductive capacities. (shrink)

I show that standard dynamic approaches to the semantics of epistemic modals invalidate the classical laws of excluded middle and non-contradiction, as well as the law of epistemic non-contradiction. I argue that these facts pose a serious challenge.

In this paper, we use algebra-valued models to study cardinal numbers in a class of non-classical set theories. The algebra-valued models of these non-classical set theories validate the Axiom of Choice, if the ground model validates it. Though the models are non-classical, the foundations of cardinal numbers in these models are similar to those in classical set theory. For example, we show that mathematical induction, Cantor’s theorem, and the Schröder–Bernstein theorem hold in these models. We also (...) study a few basic properties of cardinal arithmetic. In addition, the generalized continuum hypothesis is proved to be independent of these non-classical set theories. (shrink)

Classical real-valued probabilities come at a philosophical cost: in many infinite situations, they assign the same probability value—namely, zero—to cases that are impossible as well as to cases that are possible. There are three non-classical approaches to probability that can avoid this drawback: full conditional probabilities, qualitative probabilities and hyperreal probabilities. These approaches have been criticized for failing to preserve intuitive symmetries that can be preserved by the classical probability framework, but there has not been a systematic (...) study of the conditions under which these symmetries can and cannot be preserved. This paper fills that gap by giving complete characterizations under which symmetries understood in a certain “strong” way can be preserved by these non-classical probabilities, as well as by offering some results to make it plausible that the strong notion of symmetry here is the right one. Philosophical implications are briefly discussed, but the main purpose of the paper is to offer technical results to help make further philosophical discussion more sophisticated. (shrink)

Dyadic semantics is a sort of non-truth-functional bivalued semantics introduced in Caleiro et al. Logica Universalis, Birkhäuser, Basel, pp 169–189, 2005). Here we introduce an algorithmic procedure for constructing conservative translations of logics characterised by dyadic semantics into classical propositional logic. The procedure uses fresh propositional variables, which we call hidden variables, to represent the indeterminism of dyadic semantics. An alternative algorithmic procedure for constructing conservative translations of any finite-valued logic into classical logic is also introduced. In this (...) alternative procedure hidden variables are also used, but in this case to represent the degree of true or falsehood of propositions. (shrink)

This paper contributes to the generalization of lattice-valued models of set theory to non-classical contexts. First, we show that there are infinitely many complete bounded distributive lattices, which are neither Boolean nor Heyting algebra, but are able to validate the negation-free fragment of \. Then, we build lattice-valued models of full \, whose internal logic is weaker than intuitionistic logic. We conclude by using these models to give an independence proof of the Foundation axiom from \.

The paper is devoted to the contributions of Helena Rasiowa to the theory of non-classical negation. The main results of Rasiowa in this area concerns–constructive logic with strong (Nelson) negation.

n this paper, a non-classical axiomatic system was introduced to classify all moods of Aristotelian syllogisms, in addition to the axiom "Every a is an a" and the bilateral rules of obversion of E and O propositions. This system consists of only 2 definitions, 2 axioms, 1 rule of a premise, and moods of Barbara and Datisi. By adding first-degree propositional negation to this system, we prove that the square of opposition holds without using many of the other rules (...) of classical logic (including double negation elimination). We then show that the Propositional Substructural Logic SLe is the best logic to study Aristotelian Syllogisms. Also, based on the IFLe square of opposition, the rules of conversation and the rules of negation are completely proved in Muzaffar's logic. For this purpose, we used the monadic first-order logic with the same standard deductive apparatus of quantifiers in classical logic, plus the axioms of "some a is an a" and "some not-a is a not-a". Finally, to show that there is no existential commitment to general terms in categorical logic, the Strong Four-Valued Relevant-classical Logic KR4 was used. With the same existential interpretation of the quantifiers and the standard translation of the quarter quantified. (shrink)

A non-classical, non-quantum theory, or NCQ, is any fully consistent theory that differs fundamentally from both the corresponding classical and quantum theories, while exhibiting certain features common to both. Such theories are of interest for two primary reasons. Firstly, NCQs arise prominently in semi-classical approximation schemes. Their formal study may yield improved approximation techniques in the near-classical regime. More importantly for the purposes of this note, it may be possible for NCQs to reproduce quantum results over (...) experimentally tested regimes while having a well defined classical limit, and hence are viable alternative theories. We illustrate an NCQ by considering an explicit class of NCQ mechanics. Here this class will be arrived at via a natural generalization of classical mechanics formulated in terms of a probability density functional. (shrink)

Using the time-dependent wave function we have studied the properties of the atomic transverse motion in an interferometer, and the cause of the non-classical behavior of atoms reported by Kurtsiefer, Pfau, and Mlynek [Nature 386, 150 (1997)]. The transverse wave function is derived from the solution of the two-dimensional Schrödinger's equation, written in the form of the Fresnel–Kirchhoff diffraction integral. It is assumed that the longitudinal motion is classical. Comparing data of the space distribution and of the transverse (...) momentum distribution in interferometers with one and two open slits, it follows that the atomic motion is influenced by the atomic matter wave and violates the laws of classical mechanics. However, the negative values of Wigner's function should not be taken as evidence that the atoms in an interferometer violate the classical statistical law of the addition of positive probabilities. This inference follows from the comparison of properties of Wigner's function and of the de Broglian probability density in phase space. (shrink)

Proofs and countermodels are the two sides of completeness proofs, but, in general, failure to find one does not automatically give the other. The limitation is encountered also for decidable non-classical logics in traditional completeness proofs based on Henkin’s method of maximal consistent sets of formulas. A method is presented that makes it possible to establish completeness in a direct way: For any given sequent either a proof in the given logical system or a countermodel in the corresponding frame (...) class is found. The method is a synthesis of a generation of calculi with internalized relational semantics, a Tait–Schütte–Takeuti style completeness proof, and procedures to finitize the countermodel construction. Finitizations for intuitionistic propositional logic are obtained through the search for a minimal derivation, through pruning of infinite branches in search trees by means of a suitable syntactic counterpart of semantic filtration, or through a proof-theoretic embedding into an appropriate provability logic. A number of examples illustrates the method, its subtleties, challenges, and present scope. (shrink)

This is the first introductory textbook on non-classical propositional logics.

Objects of consideration are various non-classical connectives “hidden” in the classical logic in the form of G˛s with ˛ —a classical connective, and s—a propositional variable. One of them is negation, which is defined as G ⇒ s; another is necessity, which is defined as G ∧ s. The new operations are axiomatized and it is shown that they belong to the 4-valued logic of Lukasiewicz. A 2-point Kripke semantics is built leading directly to the 4-valued logical (...) tables. (shrink)

In order to prove the validity of logical rules, one has to assume these rules in the metalogic. However, rule-circular ‘justifications’ are demonstrably without epistemic value. Is a non-circular justification of a logical system possible? This question attains particular importance in view of lasting controversies about classical versus non-classical logics. In this paper the question is answered positively, based on meaning-preserving translations between logical systems. It is demonstrated that major systems of non-classical logic, including multi-valued, paraconsistent, intuitionistic (...) and quantum logics, can be translated into classical logic by introducing additional intensional operators into the language. Based on this result it is argued that classical logic is representationally optimal. In sec. 6 it is investigated whether non-classical logics can be likewise representationally optimal. The answer is predominantly negative but partially positive. Nevertheless the situation is not symmetric, because classical logic has important ceteris paribus advantages as a unifying metalogic. (shrink)

A violation of Bell’s inequalities is generally considered to be the Holy Grail of experimental proof that a specific natural phenomenon cannot be explained in a classical framework and is based on a non-boolean structure of predications. Generalized quantum theory allows for such non-boolean predications. We formulate temporal Bell’s inequalities for cognitive two-state systems and indicate how these inequalities can be tested. This will introduce the notion of temporally non-local measurements. The Necker-Zeno model for bistable perception predicts a violation (...) of these temporal Bell’s inequalities. (shrink)

Analyzing situations where information is partial, incomplete or contradictory has created a demand for quantitative belief measures that are weaker than classic probability theory. In this paper, we compare two frameworks that have been proposed for this task, Dempster-Shafer theory and non-standard probability theory based on Belnap-Dunn logic. We show the two frameworks to assume orthogonal perspectives on informational shortcomings, but also provide a partial correspondence result. Lastly, we also compare various dynamical rules of the two frameworks, all seen as (...) generalizations of classic Bayes’ conditioning. (shrink)

Circular denitions have primarily been studied in revision theory in the classical scheme. I present systems of circular denitions in the Strong Kleene and supervaluation schemes and provide complete proof systems for them. One class of denitions, the intrinsic denitions, naturally arises in both schemes. I survey some of the features of this class of denitions.

This revised and considerably expanded 2nd edition brings together a wide range of topics, including modal, tense, conditional, intuitionist, many-valued, paraconsistent, relevant, and fuzzy logics. Part 1, on propositional logic, is the old Introduction, but contains much new material. Part 2 is entirely new, and covers quantification and identity for all the logics in Part 1. The material is unified by the underlying theme of world semantics. All of the topics are explained clearly using devices such as tableau proofs, and (...) their relation to current philosophical issues and debates are discussed. Students with a basic understanding of classical logic will find this book an invaluable introduction to an area that has become of central importance in both logic and philosophy. It will also interest people working in mathematics and computer science who wish to know about the area. (shrink)

Recently, it has been proposed to understand a logic as containing not only a validity canon for inferences but also a validity canon for metainferences of any finite level. Then, it has been shown that it is possible to construct infinite hierarchies of "increasingly classical" logics—that is, logics that are classical at the level of inferences and of increasingly higher metainferences—all of which admit a transparent truth predicate. In this paper, we extend this line of investigation by taking (...) a somehow different route. We explore logics that are different from classical logic at the level of inferences, but recover some important aspects of classical logic at every metainferential level. We dub such systems meta-classical non-classical logics. We argue that the systems presented deserve to be regarded as logics in their own right and, moreover, are potentially useful for the non-classical logician. (shrink)

Non‐classical logicians do not typically reject classically valid logical principles across the board. In fact, they sometimes suggest that their preferred logic recovers classical reasoning in most circumstances. This idea has come to be known in the literature as ‘classical recapture’. Recently, classical logicians have raised various doubts about it. The main problem is said to be that no rigorous explanation has been given of how is it exactly that classical logic can be recovered. The (...) goal of the paper is to address this problem. First, I discuss a number of different ways to characterize the idea of classical recapture and I examine how far it can be pushed. Secondly, I argue that a palatable form of classical recapture is given by the thought that classical logic can be retained for statements that are grounded (in Kripke’s sense). To substantiate this view I provide a formal fixed‐point semantics for a language containing a predicate standing for the concept of groundedness and I address a couple of objections that have been deployed against the claim that classical recapture can aid the non‐classical logician’s cause. (shrink)

Any "classical" theory of truth will satisfy tarski's criterion ("p" is true if and only if p), And the principle of bivalence (every proposition is either true or false). A non-Classical theory may be obtained by rejecting these principles: - in fact it is shown that rejection of the second entails rejection of the first. If the resulting non-Classical theory is formalized, A system structurally isomorphic to either s4 or s5 is obtained. An attempt is made to (...) show that the essential insights of intuitionist logic may be preserved if we replace their complex and in many respects "unintuitive" propositional logic by a theory of truth which is non-Classical in the sense described above. (shrink)

In this paper we extend to non-classical set theories the standard strategy of proving independence using Boolean-valued models. This extension is provided by means of a new technique that, combining algebras (by taking their product), is able to provide product-algebra-valued models of set theories. In this paper we also provide applications of this new technique by showing that: (1) we can import the classical independence results to non-classical set theory (as an example we prove the independence of (...) $\mathsf {CH}$ ); and (2) we can provide new independence results. We end by discussing the role of non-classical algebra-valued models for the debate between universists and multiversists and by arguing that non-classical models should be included as legitimate members of the multiverse. (shrink)

Formal ontologies are nowadays widely considered a standard tool for knowledge representation and reasoning in the Semantic Web. In this context, they are expected to play an important role in helping automated processes to access information. Namely: they are expected to provide a formal structure able to explicate the relationships between different concepts/terms, thus allowing intelligent agents to interpret, correctly, the semantics of the web resources improving the performances of the search technologies. Here we take into account a problem regarding (...) Knowledge Representation in general, and ontology based representations in particular; namely: the fact that knowledge modeling seems to be constrained between conflicting requirements, such as compositionality, on the one hand and the need to represent prototypical information on the other. In particular, most common sense concepts seem not to be captured by the stringent semantics expressed by such formalisms as, for example, Description Logics (which are the formalisms on which the ontology languages have been built). The aim of this work is to analyse this problem, suggesting a possible solution suitable for formal ontologies and semantic web representations. The questions guiding this research, in fact, have been: is it possible to provide a formal representational framework which, for the same concept, combines both the classical modelling view (accounting for compositional information) and defeasible, prototypical knowledge ? Is it possible to propose a modelling architecture able to provide different type of reasoning (e.g. classical deductive reasoning for the compositional component and a non monotonic reasoning for the prototypical one)? We suggest a possible answer to these questions proposing a modelling framework able to represent, within the semantic web languages, a multilevel representation of conceptual information, integrating both classical and non classical (typicality based) information. Within this framework we hypothesise, at least in principle, the coexistence of multiple reasoning processes involving the different levels of representation. (shrink)

This paper shows how to conservatively extend theories formulated in non-classical logics such as the Logic of Paradox, the Strong Kleene Logic and relevant logics with Skolem functions. Translations to and from the language extended by Skolem functions into the original one are presented and shown to preserve derivability. It is also shown that one may not always substitute s=f(t) and A(t, s) even though A determines the extension of a function and f is a Skolem function for A.

Provability, Computability and Reflection.

In this paper I urge friends of truth-value gaps and truth-value gluts – proponents of paracomplete and paraconsistent logics – to consider theories not merely as sets of sentences, but as pairs of sets of sentences, or what I call ‘bitheories,’ which keep track not only of what holds according to the theory, but also what fails to hold according to the theory. I explain the connection between bitheories, sequents, and the speech acts of assertion and denial. I illustrate the (...) usefulness of bitheories by showing how they make available a technique for characterising different theories while abstracting away from logical vocabulary such as connectives or quanti- fiers, thereby making theoretical commitments independent of the choice of this or that particular non-classical logic. (shrink)

This work contributes to the theory of judgement aggregation by discussing a number of significant non-classical logics. After adapting the standard framework of judgement 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 judgement aggregation in non-classical logics is that they offer a number of modelling choices to represent agents’ reasoning in aggregation problems. By studying judgement 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. (shrink)

This paper generalises an argument for probabilism due to Lindley [9]. I extend the argument to a number of non-classical logical settings whose truth-values, seen here as ideal aims for belief, are in the set $\{0,1\}$, and where logical consequence $\models $ is given the “no-drop” characterization. First I will show that, in each of these settings, an agent’s credence can only avoid accuracy-domination if its canonical transform is a (possibly non-classical) probability function. In other words, if an (...) agent values accuracy as the fundamental epistemic virtue, it is a necessary requirement for rationality that her credence have some probabilistic structure. Then I show that for a certain class of reasonable measures of inaccuracy, having such a probabilistic structure is sufficient to avoid accuracy-domination in these non-classical settings. (shrink)

In this article, Savage’s theory of decision-making under uncertainty is extended from a classical environment into a non-classical one. The Boolean lattice of events is replaced by an arbitrary ortho-complemented poset. We formulate the corresponding axioms and provide representation theorems for qualitative measures and expected utility. Then, we discuss the issue of beliefs updating and investigate a transition probability model. An application to a simple game context is proposed.

We discuss the interpolation property on some important families of non classical logics, such as intuitionistic, modal, fuzzy, and linear logics. A special paragraph is devoted to a generalization of the interpolation property, uniform interpolation.

In this paper, by suggesting a formal representation of science based on recent advances in logic-based Artificial Intelligence (AI), we show how three serious concerns around the realisation of traditional scientific realism (the theory/observation distinction, over-determination of theories by data, and theory revision) can be overcome such that traditional realism is given a new guise as ‘naturalised’. We contend that such issues can be dealt with (in the context of scientific realism) by developing a formal representation of science based on (...) the application of the following tools from Knowledge Representation: the family of Description Logics, an enrichment of classical logics via defeasible statements, and an application of the preferential interpretation of the approach to Belief Revision. (shrink)

In the literature on self-referential paradoxes one of the hardest and most challenging problems is that of revenge. This problem can take many shapes, but, typically, it besets non-classical accounts of some semantic notion, such as truth, that depend on a set of classically defined meta-theoretic concepts, like validity, consistency, and so on. A particularly troubling form of revenge that has received a lot of attention lately involves the concept of validity. The difficulty lies in that the non-classical (...) logician cannot accept her own definition of validity because it is given in a classical meta-theory. It is often suggested that this mismatch between the consequence relation of the account being espoused and the consequence relation of the meta-theory is a serious embarrassment. The main goal of the paper is to explore whether certain substructural accounts of the paradoxes can avoid this sort of embarrassment. Typically, these accounts are expressively incomplete, since they cannot assert in the object language that certain invalid arguments are in fact invalid. To overcome this difficulty I develop a novel type of hybrid proof-procedure, one that takes invalidities to be just as fundamental as validities. I prove that this proof-procedure enjoys a number of interesting properties and I analyze the prospects of applying it to languages capable of expressing self-referential statements. (shrink)

Justification logics provide frameworks for studying the fine structure of evidence and justification. Traditionally, these logics do not impose any closure requirements on justification. In this paper, we argue that for some applications they should subject justification to closure under some variety of logical consequence. Specifically, we argue, building on ideas from Beall, that the non-classical logic FDE offers a particularly attractive notion of consequence for this purpose and define a justification logic where justification is closed under FDE consequence. (...) We show the resulting logic to be sound and complete. Lastly, we discuss how the closure of justification under FDE ontrasts closure under related non-classical logics and how our approach contrasts with some alternatives. (shrink)