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Switch to: References
Citations of:
Semantic analysis of orthologic
Journal of Philosophical Logic 3 (1/2):19 - 35 (1974)
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We present and defend the Australian Plan semantics for negation. This is a comprehensive account, suitable for a variety of different logics. It is based on two ideas. The first is that negation is an exclusion-expressing device: we utter negations to express incompatibilities. The second is that, because incompatibility is modal, negation is a modal operator as well. It can, then, be modelled as a quantifier over points in frames, restricted by accessibility relations representing compatibilities and incompatibilities between such points. (...) |
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We introduce game-theoretic semantics for systems without the conveniences of either a De Morgan negation, or of distribution of conjunction over disjunction and conversely. Much of game playing rests on challenges issued by one player to the other to satisfy, or refute, a sentence, while forcing him/her to move to some other place in the game’s chessboard-like configuration. Correctness of the game-theoretic semantics is proven for both a training game, corresponding to Positive Lattice Logic and for more advanced games for (...) |
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ABSTRACTCorrespondence and Shalqvist theories for Modal Logics rely on the simple observation that a relational structure is at the same time the basis for a model of modal logic and for a model of first-order logic with a binary predicate for the accessibility relation. If the underlying set of the frame is split into two components,, and, then frames are at the same time the basis for models of non-distributive lattice logic and of two-sorted, residuated modal logic. This suggests that (...) |
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This article introduces, studies, and applies a new system of logic which is called ‘HYPE’. In HYPE, formulas are evaluated at states that may exhibit truth value gaps and truth value gluts. Simple and natural semantic rules for negation and the conditional operator are formulated based on an incompatibility relation and a partial fusion operation on states. The semantics is worked out in formal and philosophical detail, and a sound and complete axiomatization is provided both for the propositional and the (...) |
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We investigate complex algebras of the form arising from a frame where, and exhibit their abstract algebraic and logical counterparts. |
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This article addresses and resolves some issues of relational, Kripke-style, semantics for the logics of bounded lattice expansions with operators of well-defined distribution types, focusing on the case where the underlying lattice is not assumed to be distributive. It therefore falls within the scope of the theory of Generalized Galois Logics, introduced by Dunn, and it contributes to its extension. We introduce order-dual relational semantics and present a semantic analysis and completeness theorems for non-distributive lattice logic with n -ary additive (...) |
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The contribution of this paper lies with providing a systematically specified and intuitive interpretation pattern and delineating a class of relational structures and models providing a natural interpretation of logical operators on an underlying propositional calculus of Positive Lattice Logic and subsequently proving a generic completeness theorem for the related class of logics, sometimes collectively referred to as Generalized Galois Logics. |
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In this paper, we give a possible characterization of the distributivity on bi-approximation semantics. To this end, we introduce new notions of special elements on polarities and show that the distributivity is first-order definable on bi-approximation semantics. In addition, we investigate the dual representation of those structures and compare them with bi-approximation semantics for intuitionistic logic. We also discuss that two different methods to validate the distributivity—by the splitters and by the adjointness—can be explicated with the help of the axiom (...) |
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I present a new approach to the logic and semantics of vagueness. |
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I propose a comprehensive account of negation as a modal operator, vindicating a moderate logical pluralism. Negation is taken as a quantifier on worlds, restricted by an accessibility relation encoding the basic concept of compatibility. This latter captures the core meaning of the operator. While some candidate negations are then ruled out as violating plausible constraints on compatibility, different specifications of the notion of world support different logical conducts for negations. The approach unifies in a philosophically motivated picture the following (...) |
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Is there a notion of contradiction—let us call it, for dramatic effect, “absolute”—making all contradictions, so understood, unacceptable also for dialetheists? It is argued in this paper that there is, and that spelling it out brings some theoretical benefits. First it gives us a foothold on undisputed ground in the methodologically difficult debate on dialetheism. Second, we can use it to express, without begging questions, the disagreement between dialetheists and their rivals on the nature of truth. Third, dialetheism has an (...) |
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This paper presents an axiomatization of a class of set-theoretic conditional operators using minimization of the geodesic distance defined as the shortest path generated by the accessibility relation on a frame. The objective of this modeling is to define conditioning based on a notion of similarity generated by degrees of indistinguishability. |
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We address the old question whether a logical understanding of Quantum Mechanics requires abandoning some of the principles of classical logic. Against Putnam and others (Among whom we may count or not E. W. Beth, depending on how we interpret some of his statements), our answer is a clear "no". Philosophically, our argument is based on combining a formal semantic approach, in the spirit of E. W. Beth's proposal of applying Tarski's semantical methods to the analysis of physical theories, with (...) |
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We group the existing variants of the familiar set-theoretical and truth-theoretical paradoxes into two classes: connective paradoxes, which can in principle be ascribed to the presence of a contracting connective of some sort, and structural paradoxes, where at most the faulty use of a structural inference rule can possibly be blamed. We impute the former to an equivocation over the meaning of logical constants, and the latter to an equivocation over the notion of consequence. Both equivocation sources are tightly related, (...) |
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In this paper we show how ideas coming from two areas of research in logic can reinforce each other. The first such line of inquiry concerns the "dynamic turn" in logic and especially the formalisms inspired by Propositional Dynamic Logic (PDL); while the second line concerns research into the logical foundations of Quantum Physics, and in particular the area known as Operational Quantum Logic, as developed by Jauch and Piron (Helve Phys Acta 42: 842-848, 1969), Pirón (Foundations of Quantum Physics, (...) |
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A logic is called 'paraconsistent' if it rejects the rule called 'ex contradictione quodlibet', according to which any conclusion follows from inconsistent premises. While logicians have proposed many technically developed paraconsistent logical systems and contemporary philosophers like Graham Priest have advanced the view that some contradictions can be true, and advocated a paraconsistent logic to deal with them, until recent times these systems have been little understood by philosophers. This book presents a comprehensive overview on paraconsistent logical systems to change (...) |
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The paper deals with polymodal languages combined with standard semantics defined by means of some conditions on the frames. So, a notion of "polymodal base" arises which provides various enrichments of the classical modal language. One of these enrichments, viz. the base £(R,-R), with modalities over a relation and over its complement, is the paper's main paradigm. The modal definability (in the spirit of van Benthem's correspondence theory) of arbitrary and ~-elementary classes of frames in this base and in some (...) |
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It is known that propositional relevant logics can be conservatively extended by the addition of a Heyting (intuitionistic) implication connective. We show that this same conservativity holds for a range of first-order relevant logics with strong identity axioms, using an adaptation of Fine’s stratified model theory. For systems without identity, the question of conservatively adding Heyting implication is thereby reduced to the question of conservatively adding the axioms for identity. Some results in this direction are also obtained. The conservative presence (...) |
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“Paraconsistent” means “beyond the consistent” [3, 15]. Paraconsistent logics tolerate inconsistencies in a way that traditional logics do not. In a paraconsistent logic, the inference of explosion A, ∼AB is rejected. This may be for any of a number of reasons [16]. For proponents of relevance [1, 2] the argument has gone awry when we infer an irrelevant B from the inconsistent premises. Those who argue that inconsistent theories may have some logical content but do not commit us to everything, (...) |
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I wish to present a proof that vagueness is impossible. Of course, vagueness is possible; and so there must be something wrong with the proof. But it is far from clear where the error lies and, indeed, all of the assumptions upon which the proof depends are ones that have commonly been accepted. This suggests that we may have to radically alter our current conception of vagueness if we are to make proper sense of what it is. |
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This paper will take into account the Lindenbaum property in Orthomodular Quantum Logic (OQL) and Partial Classical Logic (PCL). The Lindenbaum property has an interest both from a logical and a physical point of view since it has to do with the problem of the completeness of quantum theory and with the possibility of extending any semantically non-contradictory set of formulas to a semantically non-contradictory complete set of formulas. The main purpose of this paper is to show that both OQL (...) |
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The paper describes in detail the procedure of identification of the inner language and an inner logico of a physical theory. The procedure is a generalization of the original ideas of J. von Neuman and G. Birkhoff about quantum logic. |
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Epistemic modals have peculiar logical features that are challenging to account for in a broadly classical framework. For instance, while a sentence of the form $$p\wedge \Diamond \lnot p$$ (‘p, but it might be that not p’) appears to be a contradiction, $$\Diamond \lnot p$$ does not entail $$\lnot p$$, which would follow in classical logic. Likewise, the classical laws of distributivity and disjunctive syllogism fail for epistemic modals. Existing attempts to account for these facts generally either under- or over-correct. (...) |
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We introduce and study a class ofbetweenness algebras—Boolean algebras with binary operators, closely related to ternary frames with a betweenness relation. From various axioms for betweenness, we chose those that are most common, which makes our work applicable to a wide range of betweenness structures studied in the literature. On the algebraic side, we work with two operators ofpossibilityand ofsufficiency. |
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In the literature there has been evidence that a kind of relational structure called a quantum Kripke frame captures the essential characteristics of the orthogonality relation between pure states of quantum systems, and thus is a good qualitative mathematical model of quantum systems. This paper adds another piece of evidence by providing a tensor-product construction of two finite-dimensional quantum Kripke frames. We prove that this construction is exactly the qualitative counterpart of the tensor-product construction of two finite-dimensional Hilbert spaces over (...) |
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A class of ordered relational topological spaces is described, which we call orthomodular spaces. Our construction of these spaces involves adding a topology to the class of orthomodular frames introduced by Hartonas, along the lines of Bimbó's topologization of the class of orthoframes employed by Goldblatt in his representation of ortholattices. We then prove that the category of orthomodular lattices and homomorphisms is dually equivalent to the category of orthomodular spaces and certain continuous frame morphisms, which we call continuous weak (...) |
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We investigate a hitherto under-considered avenue of response for the logical pluralist to collapse worries. In particular, we note that standard forms of the collapse arguments seem to require significant order-theoretic assumptions, namely that the collection of admissible logics for the pluralist should be closed under meets and joins. We consider some reasons for rejecting this assumption, noting some prima facie plausible constraints on the class of admissible logics which would lead a pluralist admitting those logics to resist such closure (...) |
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Orthomodular logic is a weakening of quantum logic in the sense of Birkhoff and von Neumann. Orthomodular logic is shown to be a nonlinear noncommutative logic. Sequents are given a physically motivated semantics that is consistent with exactly one semantics for propositional formulas that use negation, conjunction, and implication. In particular, implication must be interpreted as the Sasaki arrow, which satisfies the deduction theorem in this logic. As an application, this deductive system is extended to two systems of predicate logic: (...) |
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In this paper we present two families of probability logics (denoted _QLP_ and \(QLP^{ORT}\) ) suitable for reasoning about quantum observations. Assume that \(\alpha \) means “O = a”. The notion of measuring of an observable _O_ can be expressed using formulas of the form \(\square \lozenge \alpha \) which intuitively means “if we measure _O_ we obtain \(\alpha \) ”. In that way, instead of non-distributive structures (i.e., non-distributive lattices), it is possible to relay on classical logic extended with (...) |
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In Sect. 1 it is argued that systems of logic are exceptional, but not a priori necessary. Logics are exceptional because they can neither be demonstrated as valid nor be confirmed by observation without entering a circle, and their motivation based on intuition is unreliable. On the other hand, logics do not express a priori necessities of thinking because alternative non-classical logics have been developed. Section 2 reflects the controversies about four major kinds of non-classical logics—multi-valued, intuitionistic, paraconsistent and quantum (...) |
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Per il dialeteismo ci sono contraddizioni vere. Questa concezione filosofica ha assunto una forma chiara e definita a partire dal lavoro del filosofo e logico Graham Priest – uno dei suoi padri fondatori, nonché uno dei suoi più strenui difensori. Questo libro intende portare il dialeteismo all’attenzione di un ampio pubblico, che non sia solo quello degli addetti ai lavori. Il volume è suddiviso in due parti. La prima include le cinque lezioni su "Dialeteismo e storia della filosofia" tenute da (...) |
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One of the main problems of the orthoframe approach to quantum logic was that orthomodularity could not be captured by any first-order condition. This paper studies an elementary and natural class of orthomodular frames that can work around this limitation. Set-theoretically, the frames we propose form a natural subclass of the orthoframes, where is an irreflexive and symmetric relation on X. More specifically, they are partially-ordered orthoframes with a designated subset. Our frame class contains the canonical orthomodular frame of the (...) |
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Paraconsistent quantum logics are weak forms of quantum logic, where the noncontradiction and the excluded-middle laws are violated. These logics find interesting applications in the operational approach to quantum mechanics. In this paper, we present an axiomatization, a Kripke-style, and an algebraic semantical characterization for two forms of paraconsistent quantum logic. Further developments are contained in Giuntini and Greuling's paper in this issue. |
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According to Quine, in any disagreement over basic logical laws the contesting parties must mean different things by the connectives or quantifiers implicated in those laws; when a deviant logician ‘tries to deny the doctrine he only changes the subject’. The standard semantics for intuitionism offers some confirmation for this thesis, for it represents an intuitionist as attaching quite different senses to the connectives than does a classical logician. All the same, I think Quine was wrong, even about the dispute (...) |
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. Relational semantics for nonclassical logics lead straightforwardly to topological representation theorems of their algebras. Ortholattices and De Morgan lattices are reducts of the algebras of various nonclassical logics. We define three new classes of topological spaces so that the lattice categories and the corresponding categories of topological spaces turn out to be dually isomorphic. A key feature of all these topological spaces is that they are ordered relational or ordered product topologies. |
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After observing that the truth conditions of connectives of non–classical logics are generally defined in terms of formulas of first–order logic, we introduce ‘protologics’, a class of logics whose connectives are defined by arbitrary first–order formulas. Then, we introduce atomic and molecular logics, which are two subclasses of protologics that generalize our gaggle logics and which behave particularly well from a theoretical point of view. We also study and introduce a notion of equi-expressivity between two logics based on different classes (...) |
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We study the computational complexity of the universal and quasi-equational theories of classes of bounded distributive lattices with a negation operation, i.e., a unary operation satisfying a subset of the properties of the Boolean negation. The upper bounds are obtained through the use of partial algebras. The lower bounds are either inherited from the equational theory of bounded distributive lattices or obtained through a reduction of a global satisfiability problem for a suitable system of propositional modal logic. |
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This paper considers some issues to do with valuational presentations of consequence relations, and the Galois connections between spaces of valuations and spaces of consequence relations. Some of what we present is known, and some even well-known; but much is new. The aim is a systematic overview of a range of results applicable to nonreflexive and nontransitive logics, as well as more familiar logics. We conclude by considering some connectives suggested by this approach. |
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In his 1964 note, ‘Two Additions to Positive Implication’, A. N. Prior showed that standard axioms governing conjunction yield a nonconservative extension of the pure implicational intermediate logic OIC of R. A. Bull. Here, after reviewing the situation with the aid of an adapted form of the Kripke semantics for intuitionistic and intermediate logics, we proceed to illuminate this example by transposing it to the setting of modal logic, and then relate it to the propositional logic of what have been (...) |
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In this paper, we establish the first-order definability of sequents with consistent variable occurrence on bi-approximation semantics by means of the Sahlqvist–van Benthem algorithm. Then together with the canonicity results in Suzuki (2011), this allows us to establish a Sahlqvist theorem for substructural logic. Our result is not limited to substructural logic but is also easily applicable to other lattice-based logics. |
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This article proposes a basic logic for application in physics dispensing with the Principle of Excluded Middle. It is based on the article “Matrix Based Logics for Application in Physics (RMQ) which appeared 2009. In his article with Stachow on the Principle of Excluded Middle in Quantum Logic (QL), Peter Mittelstaedt showed that for some suitable QLs, including their own, the Principle of Excluded Middle can be added without any harm for QL; where ‘without any harm for QL’ means that (...) |
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