Our aim is to show that translating the modal graphs of Moretti’s “n-opposition theory” (2004) into set theory by a suited device, through identifying logical modal formulas with appropriate subsets of a characteristic set, one can, in a constructive and exhaustive way, by means of a simple recurring combinatory, exhibit all so-called “logical bi-simplexes of dimension n” (or n-oppositional figures, that is the logical squares, logical hexagons, logical cubes, etc.) contained in the logic produced by any given (...) modal graph (an exhaustiveness which was not possible before). In this paper we shall handle explicitly the classical case of the so-called 3(3)-modal graph (which is, among others, the one of S5), getting to a very elegant tetraicosahedronal geometrisation of this logic. (shrink)

There are several three-valued logical systems that form a scattered landscape, even if all reasonable connectives in three-valued logics can be derived from a few of them. Most papers on this subject neglect the issue of the relevance of such logics in relation with the intended meaning of the third truth-value. Here, we focus on the case where the third truth-value means unknown, as suggested by Kleene. Under such an understanding, we show that any truth-qualified formula in a large range (...) of three-valued logics can be translated into KD as a modal formula of depth 1, with modalities in front of literals only, while preserving all tautologies and inference rules of the original three-valued logic. This simple information logic is a two-tiered classical propositional logic with simple semantics in terms of epistemic states understood as subsets of classical interpretations. We study in particular the translations of Kleene, Gödel, ᴌukasiewicz and Nelson logics. We show that Priest’s logic of paradox, closely connected to Kleene’s, can also be translated into our modal setting, simply by exchanging the modalities possible and necessary. Our work enables the precise expressive power of three-valued logics to be laid bare for the purpose of uncertainty management. (shrink)

Since the bilateral commerce between the USA and China changes greatly and Chinese government continues to reform and delegate power in the economic field, it is of importance to study the Uniform Commercial Code of the USA. The study specifically studies the features of modal verbs in the UCC through the comparison with the United States Code and Frown. With the help of Antconc, the distribution of modal verbs, the modal value of the three sets of corpora (...) are examined and analyzed based on the theory of Halliday’s systemic-functional grammar. The results show that there are significant differences between the three sets of corpora and the three main features of modal verbs in the UCC are: the high frequency use of may, the relatively lower frequency use of legal modal verb shall, and the much higher frequency of must and may not. The seemingly contrary usages within the UCC are discussed and explained with the background of the legislation and the legal thoughts of the major legislator in a sociosemiotic approach. With both the quantitative and qualitative analysis combining the examples, the study hopes to shed light on the related legal translation and further enhance the understanding of the modal verbs in the UCC as well as legal discourse in a more general context. (shrink)

Graph-based frames have been introduced as a logical framework which internalises an inherent boundary to knowability (referred to as ‘informational entropy’), due, e.g. to perceptual, evidential or linguistic limits. They also support the interpretation of lattice-based (modal) logics as hyper-constructive logics of evidential reasoning. Conceptually, the present paper proposes graph-based frames as a formal framework suitable for generalising Pawlak's rough set theory to a setting in which inherent limits to knowability exist and need to be considered. Technically, the present (...) paper establishes systematic connections between the first-order correspondents of Sahlqvist modal reduction principles on Kripke frames, and on the more general relational environments of graph-based and polarity-based frames. This work is part of a research line aiming at: (a) comparing and inter-relating the various (first-order) conditions corresponding to a given (modal) axiom in different relational semantics; (b) recognising when first-order sentences in the frame-correspondence languages of different relational structures encode the same ‘modal content’; (c) meaningfully transferring relational properties across different semantic contexts. The present paper develops these results for the graph-based semantics, polarity-based semantics, and all Sahlqvist modal reduction principles. As an application, we study well known modal axioms in rough set theory (such as those corresponding to seriality, reflexivity, and transitivity) on graph-based frames and show that, although these axioms correspond to different first-order conditions on graph-based frames, their intuitive meaning is retained. This allows us to introduce the notion of hyperconstructivist approximation spaces as the subclass of graph-based frames defined by the first-order conditions corresponding to the same modal axioms defining classical generalised approximation spaces, and to transfer the properties and the intuitive understanding of different approximation spaces to the more general framework of graph-based frames. The approach presented in this paper provides a base for systematically comparing and connecting various formal frameworks in rough set theory, and for the transfer of insights across different frameworks. (shrink)

We articulate a relational understanding of modality, and show it at work in a survey of pre-order models, set-lifting, and modal languages.

In 1958 Arnould Bayart, 1911-1998, produced a semantics for first and second-order S5 modal logic, and in 1959 a completeness proof for first-order S5, and what he calls a 'quasi-completeness' proof for second-order S5. The 1959 paper is the first completeness proof for modal predicate logic based on the Henkin construction of maximal consistent sets, and indeed may be the easier application of the Henkin method even to propositional modal logic. The semantics is in terms of possible (...) worlds, which, Bayart notes, can be anything at all. The present paper provides an English translation of both these papers together with an historical introduction and a logical commentary. (shrink)

Although modern modal logic came about largely after Peirce’s death, he anticipated some of its key aspects, including strict implication and possible worlds semantics. He developed the Gamma part of Existential Graphs with broken cuts signifying possible falsity, but later identified the need for a Delta part without ever spelling out exactly what he had in mind. An entry in his personal Logic Notebook is a plausible candidate, with heavy lines representing possible states of things where propositions denoted (...) by attached letters would be true, rather than individual subjects to which predicates denoted by attached names are attributed as in the Beta part. New transformation rules implement various commonly employed formal systems of modal logic, which are readily interpreted by defining a possible world as one in which all the relevant laws for the actual world are facts, each world being partially but accurately and adequately described by a closed and consistent model set of propositions. In accordance with pragmaticism, the relevant laws for the actual world are represented as strict implications with real possibilities as their antecedents and conditional necessities as their consequents, corresponding to material implications in every possible world. (shrink)

Viewing the language of modal logic as a language for describing directed graphs, a natural type of directed graph to study modally is one where the nodes are sets and the edge relation is the subset or superset relation. A well-known example from the literature on intuitionistic logic is the class of Medvedev frames $\langle W,R\rangle$ where $W$ is the set of nonempty subsets of some nonempty finite set $S$, and $xRy$ iff $x\supseteq y$, or more liberally, where (...) $\langle W,R\rangle$ is isomorphic as a directed graph to $\langle \wp(S)\setminus\{\emptyset\},\supseteq\rangle$. Prucnal [32] proved that the modal logic of Medvedev frames is not finitely axiomatizable. Here we continue the study of Medvedev frames with extended modal languages. Our results concern definability. We show that the class of Medvedev frames is definable by a formula in the language of tense logic, i.e., with a converse modality for quantifying over supersets in Medvedev frames, extended with any one of the following standard devices: nominals (for naming nodes), a difference modality (for quantifying over those $y$ such that $x\not= y$), or a complement modality (for quantifying over those $y$ such that $x\not\supseteq y$). It follows that either the logic of Medvedev frames in one of these tense languages is finitely axiomatizable---which would answer the open question of whether Medvedev's [31] "logic of finite problems'' is decidable---or else the minimal logics in these languages extended with our defining formulas are the beginnings of infinite sequences of frame-incomplete logics. (shrink)

In this paper, we shall give another proof of the faithfulness of Blass translation of the propositional fragment \ of Leśniewski’s ontology in the modal logic \ by means of Hintikka formula. And we extend the result to von Wright-type deontic logics, i.e., ten Smiley-Hanson systems of monadic deontic logic. As a result of observing the proofs we shall give general theorems on the faithfulness of B-translation with respect to normal modal logics complete to certain sets (...) of well-known accessibility relations with a restriction that transitivity and symmetry are not set at the same time. As an application of the theorems, for example, B-translation is faithful for the provability logic \ ), that is, \ \ \ \supset \Box \phi \). The faithfulness also holds for normal modal logics, e.g., \, \, \, \. We shall conclude this paper with the section of some open problems and conjectures. (shrink)

ABSTRACT This paper reports on an empirical performance analysis of four modal theorem provers on benchmark suites of randomly generated formulae. The theorem provers tested are the Davis-Putnam-based procedure KSAT, the tableaux-based system KRZIS, the sequent-based Logics Workbench, and a translation approach combined with the first-order theorem prover SPASS. Our benchmark suites are sets of multi-modal formulae in a certain normal form randomly generated according to the scheme of Giunchiglia and Sebastiani [GS 96a, GS 96b]. We investigate (...) the quality of the random modal formulae and show that the scheme has some shortcomings, which may lead to mistaken conclusions. We propose improvements to the evaluation method and show that the translation approach provides a viable alternative to the other approaches. (shrink)

We introduce the subject of modal model theory, where one studies a mathematical structure within a class of similar structures under an extension concept that gives rise to mathematically natural notions of possibility and necessity. A statement φ is possible in a structure (written φ) if φ is true in some extension of that structure, and φ is necessary (written φ) if it is true in all extensions of the structure. A principal case for us will be the class (...) Mod(T) of all models of a given theory T—all graphs, all groups, all fields, or what have you—considered under the substructure relation. In this article, we aim to develop the resulting modal model theory. The class of all graphs is a particularly insightful case illustrating the remarkable power of the modal vocabulary, for the modal language of graph theory can express connectedness, k-colorability, finiteness, countability, size continuum, size ℵ1, ℵ2, ℵω, ℶω, first ℶ-fixed point, first ℶ-hyper-fixed point, and much more. A graph obeys the maximality principle φ(a)→φ(a) with parameters if and only if it satisfies the theory of the countable random graph, and it satisfies the maximality principle for sentences if and only if it is universal for finite graphs. (shrink)

Modal logic provides an elegant way to understand the notion of potential infinity. This raises the question of what the right modal logic is for reasoning about potential infinity. In this article I identify a choice point in determining the right modal logic: Can a potentially infinite collection ever be expanded in two mutually incompatible ways? If not, then the possible expansions are convergent; if so, then the possible expansions are branching. When possible expansions are convergent, the (...) right modal logic is S4.2, and a mirroring theorem due to Linnebo allows for a natural potentialist interpretation of mathematical discourse. When the possible expansions are branching, the right modal logic is S4. However, the usual box and diamond do not suffice to express everything the potentialist wants to express. I argue that the potentialist also needs an operator expressing that something will eventually happen in every possible expansion. I prove that the result of adding this operator to S4 makes the set of validities Pi-1-1 hard. This result makes it unlikely that there is any natural translation of ordinary mathematical discourse into the potentialist framework in the context of branching possibilities. (shrink)

This contribution deals with a number of challenges faced by translation historians of the French Revolution. First, the challenge of extending the translation category by including a range of transfer modalities that gravitate around translation. Second, the challenge of finding a balance between two historians’ viewpoints: those who want to know what the Revolution has meant for translation and those who want to know what translation has meant for the Revolution. Third, the challenge of relating (...) the different scales of translators’ activities taking simultaneously or successively place in local, national and transnational settings. (shrink)

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 a reduction of the first to the latter may be possible, encoding Positive Lattice Logic as a fragment of Two-Sorted, Residuated Modal Logic. The reduction is analogous to the well-known Gödel-McKinsey-Tarski translation of Intuitionistic Logic into the S4 system of normal modal logic. In this article, we carry out this reduction in detail and we derive some properties of PLL from corresponding properties of First-Order Logic. The reduction we present is extendible to the case of lattices with operators, making use of recent results by this author on the relational representation of normal lattice expansions. (shrink)

In this paper, we study three representations of lattices by means of a set with a binary relation of compatibility in the tradition of Ploščica. The standard representations of complete ortholattices and complete perfect Heyting algebras drop out as special cases of the first representation, while the second covers arbitrary complete lattices, as well as complete lattices equipped with a negation we call a protocomplementation. The third topological representation is a variant of that of Craig, Haviar, and Priestley. We then (...) extend each of the three representations to lattices with a multiplicative unary modality; the representing structures, like so-called graph-based frames, add a second relation of accessibility interacting with compatibility. The three representations generalize possibility semantics for classical modal logics to non-classical modal logics, motivated by a recent application of modal orthologic to natural language semantics. (shrink)

We present optimizations for the modal logic programming system MProlog, including the standard form for resolution cycles, optimized sets of rules used as meta-clauses, optimizations for the version of MProlog without existential modal operators, as well as iterative deepening search and tabulation. Our SLD-resolution calculi for MProlog in a number of modal logics are still strongly complete when resolution cycles are in the standard form and optimized sets of rules are used. We also show that the labelling (...) technique used in our direct approach is relatively better than the Skolemization technique used in the translation approaches for modal logic programming. (shrink)

This work is divided in two papers (Part I and Part II). In Part I, we introduced the class of Rare-logics for which the set of terms indexing the modal operators are hierarchized in two levels: the set of Boolean terms and the set of terms built upon the set of Boolean terms. By investigating different algebraic properties satisfied by the models of the Rare-logics, reductions for decidability were established by faithfully translating the Rare-logics into more standard modal (...) logics (some of them contain the universal modal operator).In Part II, we push forward the results from Part I. For Rare-logics with nominals (present at the level of formulae and at the level of modal expressions), we show that the constructions from Part I can be extended although it is technically more involved. We also characterize a class of standard modal logics for which the universal modal operator can be eliminated as far as satifiability is concerned. Although the previous results have a semantic flavour, we are also able to define proof systems for Rare-logics from existing proof systems for the corresponding standard modal logics. Last, but not least, decidability results for Rare-logics are established uniformly, in particular for information logics derived from rough set theory. (shrink)

Jaśkowski’s discussive logic D2 was formulated with the help of the modal logic S5 as follows (see [7, 8]): \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${A \in {D_{2}}}$$\end{document} iff \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\ulcorner\diamond{{A}^{\bullet}}\urcorner \in {\rm S}5}$$\end{document}, where (–)• is a translation of discussive formulae from Ford into the modal language. We say that a modal logic L defines D2 iff \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} (...) \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm D}_{2} = \{A \in {\rm For^{\rm d}} : \ulcorner\diamond{{A}^{\bullet}}\urcorner \in {\it L}\}}$$\end{document}. In [14] and [10] were respectively presented the weakest normal and the weakest regular logic which (†): have the same theses beginning with ‘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\diamond}$$\end{document}’ as S5. Of course, all logics fulfilling the above condition, define D2. In [10] it was prowed that in the cases of logics closed under congruence the following holds: defining D2 is equivalent to having the property (†). In this paper we show that this equivalence holds also for all modal logics which are closed under replacement of tautological equivalents (rte-logics).We give a general method which, for any class of modal logics determined by a set of joint axioms and rules, generates in the given class the weakest logic having the property (†). Thus, for the class of all modal logics we obtain the weakest modal logic which owns this property. On the other hand, applying the method to various classes of modal logics: rte-logics, congruential, monotonic, regular and normal, we obtain the weakest in a given class logic defining D2. (shrink)

In this paper, we generalize the set-theoretic translation method for polymodal logic introduced in [11] to extended modal logics. Instead of devising an ad-hoc translation for each logic, we develop a general framework within which a number of extended modal logics can be dealt with. We first extend the basic set-theoretic translation method to weak monadic second-order logic through a suitable change in the underlying set theory that connects up in interesting ways with constructibility; then, (...) we show how to tailor such a translation to work with specific cases of extended modal logics. (shrink)

The main objective o f this descriptive paper is to present the general notion of translation between logical systems as studied by the GTAL research group, as well as its main results, questions, problems and indagations. Logical systems here are defined in the most general sense, as sets endowed with consequence relations; translations between logical systems are characterized as maps which preserve consequence relations (that is, as continuous functions between those sets). In this sense, logics together with translations form (...) a bicomplete category of which topological spaces with topological continuous functions constitute a full subcategory. We also describe other uses of translations in providing new semantics for non-classical logics and in investigating duality between them. An important subclass of translations, the conservative translations, which strongly preserve consequence relations, is introduced and studied. Some specific new examples of translations involving modal logics, many-valued logics, para- consistent logics, intuitionistic and classical logics are also described. (shrink)

We propose a translation into Modal Logic of the ideas that formalise belief change in the Situation Calculus. This translation is extended to the case of revision. In the conclusion is presented a set of open issues.

Charles Peirce incorporates modality into his Existential Graphs by introducing the broken cut for possible falsity. Although it can be adapted to various modern modal logics, Zeman demonstrates that making no other changes results in a version that he calls Gamma-MR, an implementation of Jan Łukasiewicz's four-valued Ł-modal system. It disallows the assertion of necessity, reflecting a denial of determinism, and has theorems involving possibility that seem counterintuitive at first glance. However, the latter is a misconception that (...) arises from overlooking the distinction between the intermediate truth values that are assigned to possibly true propositions as either X-contingent or Y-contingent. Any two propositions having the same ITV are possible together, while any two propositions having different ITVs, including those that are each other's negation, are possible individually yet not possible together. Porte shows that Ł-modal can be translated into classical logic by defining a constant for each ITV such that its implication of another proposition asserts the latter's possibility, while its conjunction with another proposition asserts the latter's necessity. These are expressed in the Alpha part of EG without broken cuts, simplifying derivations and shedding further light on Łukasiewicz's system, as long as graphs including either of the constants are properly interpreted. Ł-modal and Gamma-MR thus capture the two-sided nature of possibility as the limit between truth and falsity in Peirce's triadic conception. (shrink)

We introduce an effective translation from proofs in the display calculus to proofs in the labelled calculus in the context of tense logics. We identify the labelled calculus proofs in the image of this translation as those built from labelled sequents whose underlying directed graph possesses certain properties. For the basic normal tense logic Kt, the image is shown to be the set of all proofs in the labelled calculus G3Kt.

ABSTRACT This paper follows on previous papers which present and evaluate various decision procedures for modal logics. We consider new test sets and systems that have been recently proposed in the literature. This new experimental analysis confirm previous experimental results in showing that SAT based decision procedures, i.e., the procedures built on top of decision procedures for propositional satisfiability, are more efficient than tableau based decision procedures. They also confirm previous evidence of an easy-hard-easy pattern in the satisfiability curve (...) for modal K. Finally, on these tests, SAT based decision procedures are also more efficient than the recently proposed decision procedures based on translation methods. These results contradict some of the claims presented in previous papers by other authors. (shrink)

This paper presents the topological arrangements in four geometrical figures of modal propositions and their derivative relations by means of Peirce's gamma graphs and their rules of transformation. The idea of arraying the gamma graphs in a geometric and symmetrical order comes from Peirce himself who in a manuscript drew two cubes in which he presented the derivative relations of some gamma graphs. Therefore, Peirce's insights of a topological order of gamma graphs are extended here (...) backwards from the cube to the line and the square; and then forwards from the cube to the four-dimensional polyhedron. (shrink)

A semantics for quantified modal logic is presented that is based on Kleene's notion of realizability. This semantics generalizes Flagg's 1985 construction of a model of a modal version of Church's Thesis and first-order arithmetic. While the bulk of the paper is devoted to developing the details of the semantics, to illustrate the scope of this approach, we show that the construction produces (i) a model of a modal version of Church's Thesis and a variant of a (...) modal set theory due to Goodman and Scedrov, (ii) a model of a modal version of Troelstra's generalized continuity principle together with a fragment of second-order arithmetic, and (iii) a model based on Scott's graph model (for the untyped lambda calculus) which witnesses the failure of the stability of non-identity. (shrink)

Peirce introduced Existential Graphs in late 1896, and they were systematically investigated in his 1903 Lowell Lectures. Alpha graphs for classical propositional logic constitute the first part of EGs. The second and the third parts are the beta graphs for first-order logic and the gamma graphs for modal and higher-order logics, among others. As a logical syntax, EGs are two-dimensional graphs, or diagrams, in contrast to the linear algebraic notations. Peirce's theory of EGs is (...) not only a theory of logical syntax but also a proof theory for many logics. A set of transformation rules of graphs defines a proof system for a graphical logic. In this... (shrink)

In an article dating back in 1992, Kosta Došen initiated a project of modal translations in substructural logics, aiming at generalising the well-known Gödel–McKinsey–Tarski translation of intuitionistic logic into S4. Došen's translation worked well for (variants of) BCI and stronger systems (BCW, BCK), but not for systems below BCI. Dropping structural rules results in logic systems without distribution. In this article, we show, via translation, that every substructural (indeed, every non-distributive) logic is a fragment of a (...) corresponding sorted, residuated (multi) modal logic. At the conceptual and philosophical level, the translation provides a classical interpretation of the meaning of the logical operators of various non-distributive propositional calculi. Technically, it allows for an effortless transfer of results, such as compactness and the Löwenheim-Skolem property and it opens up new directions for deducing properties of substructural logic systems by establishing appropriate transfer theorems. (shrink)

This paper develops the model theory of normal modal logics based on partial “possibilities” instead of total “worlds,” following Humberstone [1981] instead of Kripke [1963]. Possibility semantics can be seen as extending to modal logic the semantics for classical logic used in weak forcing in set theory, or as semanticizing a negative translation of classical modal logic into intuitionistic modal logic. Thus, possibility frames are based on posets with accessibility relations, like intuitionistic modal frames, (...) but with the constraint that the interpretation of every formula is a regular open set in the Alexandrov topology on the poset. The standard world frames for modal logic are the special case of possibility frames wherein the poset is discrete. The analogues of classical Kripke frames, i.e., full world frames, are full possibility frames, in which propositional variables may be interpreted as any regular open sets. We develop the beginnings of duality theory, definability/correspondence theory, and completeness theory for possibility frames. The duality theory, relating possibility frames to Boolean algebras with operators (BAOs), shows the way in which full possibility frames are a generalization of Kripke frames. Whereas Thomason [1975a] established a duality between the category of Kripke frames with p-morphisms and the category of complete (C), atomic (A), and completely additive (V) BAOs with complete BAO-homomorphisms (these categories being dually equivalent), we establish a duality between the category of full possibility frames with possibility morphisms and the category of CV-BAOs, i.e., allowing non-atomic BAOs, with complete BAO-homomorphisms (the latter category being dually equivalent to a reflective subcategory of the former). This parallels the connection between forcing posets and Boolean-valued models in set theory, but now with accessibility relations and modal operators involved. Generalizing further, we introduce a class of principal possibility frames that capture the generality of V-BAOs. If we do not require a full or principal frame, then every BAO has an equivalent possibility frame, whose possibilities are proper filters in the BAO. With this filter representation, which does not require the ultrafilter axiom, we are lead to a notion of filter-descriptive possibility frames. Whereas Goldblatt [1974] showed that the category of BAOs with BAO-homomorphisms is dually equivalent to the category of descriptive world frames with p-morphisms, we show that the category of BAOs with BAO-homomorphisms is dually equivalent to the category of filter-descriptive possibility frames with p-morphisms. Applying our duality theory to definability theory, we prove analogues for possibility semantics of theorems of Goldblatt [1974] and Goldblatt and Thomason [1975] characterizing modally definable classes of frames. In addition, we discuss analogues for possibility semantics of first-order correspondence results in the style of Lemmon and Scott [1977], Sahlqvist [1975], and van Benthem [1976a]. Finally, applying our duality theory to completeness theory, we show that there are continuum many normal modal logics that can be characterized by full possibility frames but not by Kripke frames, that all Sahlqvist logics can be characterized by full possibility frames that contain no worlds, and that all normal modal logics can be characterized by filter-descriptive possibility frames. (shrink)

The task this essay set for itself is a reconsideration of the status of the “object” in contemporary forms of philosophical realism that postulate “flat ontologies.” I argue that the theoretical construction of the “object” often comes about in these ontologies through a fetishistic disavowal that effectively makes these objects speak. As a result, the construction of the generalized field of objectivity passes through a double articulation. On the one hand, since contemporary realism defines itself as a rejection of all (...) forms of linguistic idealism, it also tries to shift the focus away from human language as the primary medium of the construction of objectivity. On the other hand, however, this demotion of language proceeds in these works simultaneously with the elevation of the concept of “translation” to an ontological principle: these non-linguistic objects exist through their perpetual translations of each other. The fetishistic disavowal at work in realism introduces the modality of fiction into the very heart of objectivity. This fictional dimension constitutive of objectivity can be described through an engagement of the Kantian notion of “purposiveness.” I argue that these translations that supposedly constitute objectivity rest on the fundamental presupposition that guides the entire Kantian system: we must presuppose purposiveness even where we can detect no evidence of it at all. Hence, today, the theory of the “democracy of objects” must be supplemented by its necessary correlate, a theory of the “conspiracy of objects.”. (shrink)

Full proofs of the Gödel incompleteness theorems are highly intricate affairs. Much of the intricacy lies in the details of setting up and checking the properties of a coding system representing the syntax of an object language (typically, that of arithmetic) within that same language. These details are seldom illuminating and tend to obscure the core of the argument. For this reason a number of efforts have been made to present the essentials of the proofs of Gödel’s theorems without getting (...) mired in syntactic or computational details. One of the most important of these efforts was made by Löb [8] in connection with his analysis of sentences asserting their own provability. Löb formulated three conditions (now known as the Hilbert-Bernays-Löb derivability conditions), on the provability predicate in a formal system which are jointly sufficient to yield the Gödel’s second incompleteness theorem for it. A key role in Löb’s analysis is played by (a special case of) what later became known as the diagonalization or fixed point property of formal systems, a property which had already, in essence, been exploited by Gödel in his original proofs of the incompleteness theorems. The fixed point property plays a central role in Lawvere’s [7] category-theoretic account of incompleteness phenomena (see also [10]). Incompleteness theorems have also been subjected to intensive investigation within the framework of modal logic (see, e.g.[4], [5]). In this formulation the modal operator takes up the role previously played by the provability predicate, and the derivability conditions on the latter are translated into algebraic conditions (the so-called GL, i.e., Gödel–Löb, conditions) on the former. My purpose here is to present a framework for incompleteness phenomena, fully compatible with intuitionistic or constructive principles, in which the idea of a coding system is retained, only in a 2 simple, but very general form, a form wholly free of syntactical notions. As codes we shall take the elements of an arbitrary given nonempty set, possibly, but not necessarily, the set of natural numbers.. (shrink)

Model-theoretic 1-types overa given first-order theory T may be construed as natural metalogical miniatures of G. W. Leibniz' ``complete individual notions'', ``substances'' or ``substantial forms''. This analogy prompts this essay's modal semantics for an essentiallyundecidable first-order theory T, in which one quantifies over such ``substances'' in a boolean universe V(C), where C is the completion of the Lindenbaum-algebra of T.More precisely, one can define recursively a set-theoretic translate of formulae N of formulae of a normal modal theory Tm (...) based on T, such that the counterpart `i' of a the modal variable `xi' of L(Tm) in this translation-scheme ranges over elements of V(C) that are 1-types of T with value 1 (sometimes called `definite' C-valued 1-types of T). (shrink)

We characterize the modal logics of elementary classes of Kripke frames as precisely those modal logics that are axiomatized by modal axioms synthesized in a certain effective way from "quasi-positive" sentences of hybrid logic. These are pure positive hybrid sentences with arbitrary existential and relativized universal quantification over nominals. The proof has three steps. The first step is to use the known result that the modal logic of any elementary class of Kripke frames is also the (...) modal logic of the closure of this class under disjoint unions, generated subframes, bounded morphic images, and ultraroots. This latter class can be defined by the first-order sentences of a special syntactic form (called pseudo-equations by Goldblatt) that are valid in the former class. The second step is to translate these pseudo-equations into equivalent quasi-positive hybrid sentences. In the third and main step, we show that any quasi-positive sentence S generates an infinite set of modal formulas called "approximants," which together axiomatize a canonical modal logic that is sound and complete for the class of frames validating S. The proof is analogous to standard proofs of Sahlqvist's theorem. It generalizes to sets of quasi-positive sentences. The main result now follows. (shrink)

Following a proposal by Kooi and Tamminga, we introduce a conservative translation manual for every four-valued truth-functional propositional logic into a modal logic. However, the application of this translation does not preserve the intuitive reading of the truth-values for every four-valued logic. In order to solve this problem, we modify the translation manual and prove its conservativity by exploiting the method of generalized truth-values.

The Renaissance, known primarily for the art and literature that it produced, was also a period in which philosophical thought flourished. This two-volume anthology, which was originally published in 1997, contains forty translations of important works on moral and political philosophy written during the Renaissance and hitherto unavailable in English. The anthology is designed to be used in conjunction with The Cambridge History of Renaissance Philosophy, in which all of these texts are discussed. The works, originally written in Latin, Italian, (...) French, Spanish and Greek, cover such topics as: concepts of man, Aristotelian, Platonic, Stoic, and Epicurean ethics, scholastic political philosophy, theories of princely and republican government in Italy and northern European political thought. Each text is supplied with an introduction and a guide to further reading. (shrink)

We describe BDD-based decision procedures for the modal logic K. Our approach is inspired by the automata-theoretic approach, but we avoid explicit automata construction. Instead, we compute certain fixpoints of a set of types — which can be viewed as an on-the-fly emptiness of the automaton. We use BDDs to represent and manipulate such type sets, and investigate different kinds of representations as well as a “level-based” representation scheme. The latter turns out to speed up construction and reduce memory (...) consumption considerably. We also study the effect of formula simplification on our decision procedures. To prove the viability of our approach, we compare our approach with a representative selection of other approaches, including a translation of K to QBF. Our results indicate that the BDD-based approach dominates for modally heavy formulae, while search-based approaches dominate for propositionally heavy formulae. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Avicenna's Theory of Science: Logic, Metaphysics, Epistemology by Riccardo StrobinoThérèse-Anne DruartRiccardo Strobino. Avicenna's Theory of Science: Logic, Metaphysics, Epistemology. Oakland: University of California Press, 2021. Pp. xvi + 428. Hardback, $95.00.Strobino's remarkable book does not simply present Avicenna's theory of science; it also highlights the importance of demonstration not only for logic but also for metaphysics and epistemology. Hence, Strobino's work is essential to appreciate and better understand (...) Avicenna's philosophy as a whole.Strobino grounds his research on Avicenna's theory of science in his Book of Demonstration in the Shifā', which focuses on Aristotle's Posterior Analytics. Taking into account the lack of any translation into English or any other main modern language as well as the lack of a full medieval Latin translation (only chapter 7 of book 3 exists in a medieval Latin version) of Avicenna's text, Strobino kindly provides in appendix C (357–71) a detailed "map" of this text, useful not only to readers who cannot handle Arabic but even to those who can, as the [End Page 326] text is so rich and complex. A simple but careful look at this map confirms what Strobino makes clear and insists upon: Avicenna greatly develops, enriches, completes, and even outgrows some of Aristotle's views. The Book of Demonstration is far more than a commentary on the Posterior Analytics; it is a complete reworking of it.Strobino shows that Avicenna grounds his theory of demonstration and science on two basic notions—conception (tasawwur) and assertion (tasdīq, which some other scholars translate as "assent")—and so gives a unified approach to his theory. Scientific conceptions require better understanding of definition and description. Avicenna builds definitions and descriptions on essentialism and, therefore, grounds them in metaphysics. As for scientific assertions, in contradistinction to nonscientific ones, they must be necessary and give the cause. Therefore, Avicenna places great importance on modality and develops modal syllogisms, whereas Aristotle focused on categorical syllogisms and neglected hypothetical and disjunctive syllogisms. Avicenna also elaborates a sophisticated account of modality and its metaphysical implications. Furthermore, he broadens and enriches the appreciation of causality and explanation. These developments, according to Strobino, lead Avicenna to set forth an intricate division and hierarchy of the sciences under metaphysics, which provides their ultimate principles and allows for a unified understanding of the sciences while respecting their particularities.According to Strobino, Avicenna is keen to develop a theory of science that gives a good account of science as it is practiced. Strobino carefully shows how Avicenna's theory better reflects this practice than does Aristotle's.Strobino does not simply highlight how much Avicenna distances himself from Aristotle and goes far beyond what the Posterior Analytics yields, but also shows how much Avicenna relies on points adumbrated by his predecessor al-Fārābī in his own Book of Demonstration.The way Strobino through his analysis of demonstration links Avicenna's understanding of logic, epistemology, and metaphysics underscores the coherence and unity of Avicenna's philosophy. Scholars were aware of the importance of demonstration for Avicenna in various philosophical fields, but until the release of this book, they seemed not to have fully grasped the centrality and essential role played by the Avicennian understanding and focus on demonstration.Of course, Strobino does not limit himself to an analysis of passages in the Book of Demonstration. He has an encompassing knowledge of Avicenna's texts and uses them well to flesh out and enrich what remains terse in the Book of Demonstration. He even takes into account the way one of his successors, al-Tūsī, reads him. He also shows an extensive and wide-ranging knowledge of Aristotle's texts and of Aristotelian scholarship, as well as an interest in the Greek commentators and Galen and their relevance for interpreting Avicenna.Avicenna's Theory of Science is not an easy book to read, as it is often very technical, particularly when dealing with purely logical points or issues, but reading it is rewarding if one does so slowly and meditatively. The author, in fact, illuminates difficult matters but does not waste words. From... (shrink)

On this, the 97th anniversary of the year of his birth, thoughts turn naturally to Willard Van Orman Quine. Quine, known as ‘Van’ to his friends but ‘That putz with the beret’ to everyone else, was one of the great systematists of the last century. The range of topics he addressed is awesome: epistemology, confirmation, philosophical logic, set theory, analyticity, modality, and, perhaps most familiarly, the indeterminacy of translation. My focus in this, my final and most challenging address as (...) Chair of the Department of Philosophy, will be on this last topic. (shrink)

Classical modal logics, based on the neighborhood semantics of Scott and Montague, provide a generalization of the familiar normal systems based on Kripke semantics. This paper defines AGM revision operators on several first-order monotonic modal correspondents, where each first-order correspondence language is defined by Marc Pauly’s version of the van Benthem characterization theorem for monotone modal logic. A revision problem expressed in a monotone modal system is translated into first-order logic, the revision is performed, and the (...) new belief set is translated back to the original modal system. An example is provided for the logic of Risky Knowledge that uses modal AGM contraction to construct counter-factual evidence sets in order to investigate robustness of a probability assignment given some evidence set. A proof of correctness is given. (shrink)

The prime purpose of this paper is, first, to restore to discourse-bound occasion sentences their rightful central place in semantics and secondly, taking these as the basic propositional elements in the logical analysis of language, to contribute to the development of an adequate logic of occasion sentences and a mathematical (Boolean) foundation for such a logic, thus preparing the ground for more adequate semantic, logical and mathematical foundations of the study of natural language. Some of the insights elaborated in this (...) paper have appeared in the literature over the past thirty years, and a number of new developments have resulted from them. The present paper aims at providing an integrated conceptual basis for this new development in semantics. In Section 1 it is argued that the reduction by translation of occasion sentences to eternal sentences, as proposed by Russell and Quine, is semantically and thus logically inadequate. Natural language is a system of occasion sentences, eternal sentences being merely boundary cases. The logic has fewer tasks than is standardly assumed, as it excludes semantic calculi, which depend crucially on information supplied by cognition and context and thus belong to cognitive psychology rather than to logic. For sentences to express a proposition and thus be interpretable and informative, they must first be properly anchored in context. A proposition has a truth value when it is, moreover, properly keyed in the world, i.e. is about a situation in the world. Section 2 deals with the logical properties of natural language. It argues that presuppositional phenomena require trivalence and presents the trivalent logic ${\rm PPC}_{3}$ , with two kinds of falsity and two negations. It introduces the notion of Σ-space for a sentence A (or / A /, the set of situations in which A is true) as the basis of logical model theory, and the notion of / ${\bf P}^{A}$ / (the Σ-space of the presuppositions of A), functioning as a 'private' subuniverse for / A /. The trivalent Kleene calculus is reinterpreted as a logical account of vagueness, rather than of presupposition. ${\rm PPC}_{3}$ and the Kleene calculus are refinements of standard bivalent logic and can be combined into one logical system. In Section 3 the adequacy of ${\rm PPC}_{3}$ as a truth-functional model of presupposition is considered more closely and given a Boolean foundation. In a noncompositional extended Boolean algebra, three operators are defined: $1_{a}$ for the conjoined presuppositions of a, ã for the complement of a within $1_{a}$ , and â for the complement of $1_{a}$ within Boolean 1. The logical properties of this extended Boolean algebra are axiomatically defined and proved for all possible models. Proofs are provided of the consistency and the completeness of the system. Section 4 is a provisional exploration of the possibility of using the results obtained for a new discourse-dependent account of the logic of modalities in natural language. The overall result is a modified and refined logical and model-theoretic machinery, which takes into account both the discourse-dependency of natural language sentences and the necessity of selecting a key in the world before a truth value can be assigned. (shrink)

This work is divided in two papers (Part I and Part II). In Part I, we study a class of polymodal logics (herein called the class of "Rare-logics") for which the set of terms indexing the modal operators are hierarchized in two levels: the set of Boolean terms and the set of terms built upon the set of Boolean terms. By investigating different algebraic properties satisfied by the models of the Rare-logics, reductions for decidability are established by faithfully translating (...) the Rare-logics into more standard modal logics. The main idea of the translation consists in eliminating the Boolean terms by taking advantage of the components construction and in using various properties of the classes of semilattices involved in the semantics. The novelty of our approach allows us to prove new decidability results (presented in Part II), in particular for information logics derived from rough set theory and we open new perspectives to define proof systems for such logics (presented also in Part II). (shrink)

Alexius Meinong's Gegenstandstheorie is subject to a formal semantic paradox. The theory of defective objects originally developed by Meinong in response to Ernst Mally's paradox about self-referential thought is rejected as a general solution to paradox in the object theory. The intentionality thesis is also refuted by the counter-example of the unapprehended mountain. It is argued that despite these difficulties, an object theory is required in order to make intuitively correct sense of ontological commitment. ;A version of Meinong's theory is (...) formalized. The logic has non-standard truth value semantics and an objectual interpretation of quantifiers for a domain of existent and non-existent objects. Theories of intentional, referential, and extensional identity are offered, together with two theories of definite description, lambda abstraction, eight systems of alethic modal logic, and formal metatheory. The logic avoids the incompleteness and undecidability limitations of Godel and Church, and all logical, semantic, and set theoretical paradoxes, without type theory or Tarskian hierarchy of metalanguages. It is naively complete and mechanically decidable. ;Wittgenstein's private language argument is refuted as an obstacle to defining private mental objects in the logic. The argument is shown to be valid, but unsound. The object theory logic may therefore provide a mixed or impure private language, and its domain may contain private mental objects as well as ordinary public and non-existent objects. This means that it constitutes an object theory logic of intention, and can be used to represent the logical structure of intentional and specifically phenomenological philosophical theories that require the intelligible designation of private mental objects. ;Appendices include Henkin-type consistency, completeness, and compactness proofs ; criticism of Kazimierz Twardowski's arguments for the distinction between content and object in phenomenological psychology; translation of and commentary on Mally's essay, "Uber die Unabhangigkeit der Gegenstande vom Denken"; a discussion of the distinction between nuclear and extra-nuclear properties, and the eliminability of the modal moment; and a philosophical application of the logic to a solution of the paradox of analysis. (shrink)

John Burgess is the author of a rich and creative body of work which seeks to defend classical logic and mathematics through counter-criticism of their nominalist, intuitionist, relevantist, and other critics. This selection of his essays, which spans twenty-five years, addresses key topics including nominalism, neo-logicism, intuitionism, modal logic, analyticity, and translation. An introduction sets the essays in context and offers a retrospective appraisal of their aims. The volume will be of interest to a wide range of readers (...) across philosophy of mathematics, logic, and philosophy of language. (shrink)

A novel non-classical mereological concept built up by blending the Metaphysics of Xavier Zubiri and the Quantum Theory of Atoms in Molecules of R. F. W. Bader is proposed. It is argued that this philosophical concept is necessary to properly account for what happens in a chemical reaction. From the topology of the gradient of the laplacian of the electronic charge density, \\) within the QTAIM framework, different “atomic graphs” are found for each atom depending on the molecular context, (...) reflecting how the whole molecule affects each atom. In this way, the whole molecular system imposes certain geometry onto each atom, and every atom exhibits different ontological modality. On the Metaphysical side, for X. Zubiri real things are system of different notes. The physical essence is the subsystem of essential notes with a coherence unity. Every note is grounded on the essence. The unity of the system is present somehow in every note-of beforehand, and every essential note-of turns towards the other. The essence determines the position of each note within the system, and hence is the grounding for modality of the notes. By conflating both “theories” and taking the atoms as essential notes, we propose the concept of “Molecule in atoms-of” or “atoms-of in Molecules”. Within the course of a chemical reaction the atoms-of modified their “of” as required by the new molecular unity being formed, and eventually change their modality. The validity to the Zubiri’s statements, is attained by evaluating necessity and possibility in a set-up of finite accessible “possible worlds”. (shrink)

This paper deals with the translation of first order formulas to predicate S5 formulas. This translation does not bring the first order formula itself to a modal system, but modal interpretation of the first order formula can be given by the translation. Every formula can be translated, and the additional condition such as formula's having only one variable, or having both world domain and individual domain is not required. I introduce an indexical predicate 'E' for (...) the translation. The meaning that 'E(a)' is true is 'this world is 'a' '. Because of this meaning, I call 'E' an indexical predicate. 'E' plays an important role for the translation. In addition that the modal formulas can be translated into first order formulas, we can conclude that the first order logic and modal predicate logic isintertranslatable. (shrink)

We present a relational proof system in the style of dual tableaux for a multimodal propositional logic for order of magnitude qualitative reasoning to deal with relations of negligibility, non-closeness, and distance. This logic enables us to introduce the operation of qualitative sum for some classes of numbers. A relational formalization of the modal logic in question is introduced in this paper, i.e., we show how to construct a relational logic associated with the logic for order-of-magnitude reasoning and its (...) dual tableau system which is a validity checker for the modal logic. For that purpose, we define a validity preserving translation of the modal language into relational language. Then we prove that the system is sound and complete with respect to the relational logic defined as well as with respect to the logic for order of magnitude reasoning. Finally, we show that in fact relational dual tableau does more. It can be used for performing the four major reasoning tasks: verification of validity, proving entailment of a formula from a finite set of formulas, model checking, and verification of satisfaction of a formula in a finite model by a given object. (shrink)