The main doctrines of the traditional logic of terms (logica termini) are standardly organized according to the formation structure of syllogistic inferences. A syllogistic inference is analyzed into propositions, and propositions, in the last instance, to terms; the logic of terms, in line with this formation structure, is partitioned into the doctrines, respectively, of terms (terminorum), of propositions (propositionum) or judgments (judiciorum), and of immediate and mediate inferences or followings (consequentiarum immediatarum/mediatarum). This structure communicates the idea that a (...) given doctrine could be treated, to a certain extent, independently of the next one to come, so that propositions, for instance, could be examined, at least to a certain extent, independently of syllogistic inferences. Some classical philosophers such as Immanuel Kant have developed approaches that break in part this order of analysis which finds its clearest expression in the Port-Royal logic (Arnauld & Nicole’s Logic or the Art of Thinking). In Kant’s case the order is changed in favor of the doctrine of judgements. This paper examines the French philosopher Jules Lachelier’s (1832-1918) approach which similarly breaks the same doctrinal order, and which, more specifically, refines the typification of propositions according to the roles to be played by propositions in syllogisms. What lies at the core of this approach is the identification of the semantic values of categorical propositions (or, as Lachelier calls them, of propositions of inherence) in terms of modal statuses which are to determine these roles. Lachelier seems to use the set of values he assigns to the categorical forms as a tool for testing logical implication among types of predication, that is, as some sort of semantic theory in the modern sense. Lachelier also discusses the three-figured categorical syllogistic along with a method, effectively used by G. W. Leibniz, of transforming syllogistic moods into each other, and tries to confirm his semantic theory in terms of such syllogistic. This paper explains Lachelier’s analysis of meaning by treating it as a non-formal semantics of predication, and tries to make some modern sense of Lachelier’s general approach to syllogistic. (shrink)

Aristotle was the founder not only of logic but also of modal logic. In the Prior Analytics he developed a complex system of modal syllogistic which, while influential, has been disputed since antiquity--and is today widely regarded as incoherent. Combining analytic rigor with keen sensitivity to historical context, Marko Malink makes clear that the modal syllogistic forms a consistent, integrated system of logic, one that is closely related to other areas of Aristotle's philosophy. Aristotle's modal syllogistic differs (...) significantly from modern modal logic. Malink considers the key to understanding the Aristotelian version to be the notion of predication discussed in the Topics--specifically, its theory of predicables and the ten categories. The predicables introduce a distinction between essential and nonessential predication. In contrast, the categories distinguish between substantial and nonsubstantial predication. Malink builds on these insights in developing a semantics for Aristotle's modal propositions, one that verifies the ancient philosopher's claims of the validity and invalidity of modal inferences. While it acknowledges some limitations of this reconstruction, Aristotle's Modal Syllogistic brims with bold ideas, richly supported by close readings of the Greek texts. (shrink)

Ever since ?ukasiewicz, it has been opinio communis that Aristotle's modal syllogistic is incomprehensible due to its many faults and inconsistencies, and that there is no hope of finding a single consistent formal model for it. The aim of this paper is to disprove these claims by giving such a model. My main points shall be, first, that Aristotle's syllogistic is a pure term logic that does not recognize an extra syntactic category of individual symbols besides syllogistic (...) terms and, second, that Aristotelian modalities are to be understood as certain relations between terms as described in the theory of the predicables developed in the Topics. Semantics for modal syllogistic is to be based on Aristotelian genus-species trees. The reason that attempts at consistently reconstructing modal syllogistic have failed up to now lies not in the modal syllogistic itself, but in the inappropriate application of modern modal logic and extensional set theory to the modal syllogistic. After formalizing the underlying predicable-based semantics (Section 1) and having defined the syllogistic propositions by means of its term logical relations (Section 2), this paper will set out to demonstrate in detail that this reconstruction yields all claims on validity, invalidity and inconclusiveness that Aristotle maintains in the modal syllogistic (Section 3 and 4). (shrink)

This article investigates the prospect of giving de dicto- and de re-necessity a uniform treatment. The historical starting point is a puzzle raised by Aristotle's claim, advanced in one of the modal chapters of his Prior Analytics, that universally privative apodeictic premises simply convert. As regards the Prior and the Posterior Analytics, the data suggest a representation of propositions of the type in question by doubly modally qualified formulae of modal predicate logic that display a necessity operator in two distinct (...) positions. Can the N-operator occurring in these positions be given a unified semantical treatment (which would justify dispensing with a notational differentiation)? A positive answer, based on a suitably shaped truth condition for N-formulae, is given, and is supported in the final section with an alternative proof theoretically based conception of a property's essential belonging to an individual. (shrink)

This project is in the first place an attempt to clarify what transcendental logic is and how Kant uses it in order to achieve his goals. I use two keys in unlocking transcendental logic: Kant's philosophy of mathematics and his account of modality. I argue that Kant's categorical separation of philosophical and mathematical cognition in his reflections on method is too sweeping and undifferentiated to account for his practice in transcendental logic. On the basis of an examination of what it (...) would take in order to sustain Kant's claim for the peculiarity of modal judgments---their subjective syntheticity---I argue that the positive claims that transcendental logic makes regarding objects are legitimated by the construction of an object as such, whose concept is provided by the sum total of the categories. I suggest that the schemata play a role analogous to that of the a priori constructed figure in a mathematical proof and provide a reconstruction of the Transcendental Doctrine of the Capacity of Judgment which makes sense of Kant's decision to label the modal principles 'postulates' in the precise sense of the Euclidean mathematicians. Part Two argues that Kant's practical philosophy is grounded on something which deserves to be called a transcendental logic every bit as much as the logic that we find in the Critique of Pure Reason. Instead of being based upon our capacity for theoretical judgment, in which we refer a concept to an ontic object via intuition, practical transcendental logic is based upon our capacity for practical judgment , in which we refer a concept to a deontic object via feeling. I show how the central arguments of Kant's practical philosophy rely implicitly upon the construction of a good will as such, which plays a semantic role analogous to that of the object as such in theoretical philosophy. One part of my case is contained in the demonstration that 'good' functions as a practical modal concept and that predications of goodness have a peculiar logical form, analogous to that exhibited by theoretical predications of existence. (shrink)

In the paper we introduce a wide range of Anderson-like variants of Gödel's theory and prove for each of them strong completeness theorem wrt. corresponding class of modal structures.These theories — all formulated in the 2nd order modal language with a 2nd order unary predicate of positiveness — differ among themselves with respect of: properties of the necessity operator and of the predicate of positiveness, axioms characterizing identity between 1st sort terms, definitions of identity between 2nd sort terms, the treatment (...) of Godlikeness as a term of the 2nd order, the treatment of necessary existence as a term of the 2nd order if there exists the necessity of its presence, the treatment of Ix (Ix(y) is read: terms x, y are identical), for every 1st sort term x, as a term of the 2nd order if there exists the necessity of its presence.Concerning to semantical commitments, the following are essential: (i) 1st order terms order receive only rigid extensions in the constant objectual 1st order domain; (ii) 2nd order terms receive non-rigid extensions in the constant objectual domain of 2nd order and rigid intensions in the constant conceptual 2nd order domain; (iii) The behavior of the identity between 1st sort terms shifts from possible worlds to possible worlds. (shrink)

“Modal logic was conceived in sin: the sin of confusing use and mention.” So quips Quine. The stigma stuck with modal logic for a while. But by the mid-sixties, a whole cluster of mathematically elegant interpretations of modal logic became available. All are natural extensions of the classical Tarskian semantics of predicate logic. By the mid-seventies, Quine’s criticisms seemed obsolete. Today, we teach the model theory of modal logic as a matter of course. Quine’s “interpretive problem” is just forgotten. The (...) purpose of my dissertation is to revamp Quine’s thesis: the very foundations of modal logic—simple propositional modal logic, even before quantifiers are added—are shaky. Quine holds that the natural interpretation of the operator “necessarily” is in terms of a full-blooded metaphysical notion of real necessity. Such a notion, Quine conjectures, is committed to the “metaphysical jungle of Aristotelian essentialism”. Hence, the intended interpretation is to be rejected. Thus Quine: so much the worse for real necessity. An alternative comes to mind: a meta-level, formal rather than real, semantic interpretation of the operator of necessity. In the late forties, Carnap developed this line of interpretation which culminates in the aforementioned model theory (“Kripke’s possible world semantics”) in the sixties. On this basis a formal interpretation is introduced: it grounds necessity in the notion of model theoretic validity. As pointed by friends of the formal interpretation (Marcus, Parsons and Kaplan), no invidious essentialist claim is verified. But have Quine’s concerns been answered? Yes and no. Technical problems regarding quantification across modal operators, substitutivity, matters of scope, and the basis of cross-world identification have indeed been solved. However, no real interpretation of “necessarily” has been provided. I argue that the seeds of such an interpretation are present in Kripke’s philosophical discussion of de re necessities in Naming and Necessity. It is not model theoretic, and it does not reduce necessity to some non-modal notion, just as Quine predicted. Even more in his vein, the real interpretation is (i) metaphysically, grounded indeed in essentialist theses, and (ii) epistemologically, forced to give up Kant’s ideal that all necessities are known a priori. (shrink)

In the article we study the existence predicate \(\varepsilon\) in the context of semantics for first-order modal logic. For a formula \(\varphi\) we define \(\varphi^{\varepsilon}\) - the so called existence relativization. We point to a gap in the work of Fitting and Mendelsohn concerning the relationship between the truth of \(\varphi\) and \(\varphi^{\varepsilon}\) in classes of varying- and constant-domain models. We introduce operations on models which allow us to fill the gap and provide a more general perspective on the issue. (...) As a result we obtain a series of theorems describing the logical connection between the notion of truth of a formula with the existence predicate in constant-domain models and the notion of truth of a formula without the existence predicate in varying-domain models. (shrink)

There is perhaps no more famous objection to the Ontological Argument than Kant’s contention that existence is not a predicate. However, this is not his only objection against the Ontological Argument. It is rather part of a more comprehensive attack on the OA, one that contains at least four distinct arguments, only one of which involves. It is the purpose of this paper to explore Kant’s case for, consider three contemporary strategies used to reinforce it, assess their merits, and then (...) finally briefly present Kant’s three further arguments. (shrink)

The aim of this paper is to provide a complete Natural Kind Semantics for an Essentialist Theory of Kinds. The theory is formulated in two-sorted first order monadic modal logic with identity. The natural kind semantics is based on Rudolf Willes Theory of Concept Lattices. The semantics is then used to explain several consequences of the theory, including results about the specificity (species–genus) relations between kinds, the definitions of kinds in terms of genera and specific differences and the existence of (...) negative kinds. First, I show under which conditions the Hierarchy principle, which has been subjected to counterexamples in the literature, holds. I also show that a different principle about the species–genus relations between kinds, namely Kant’s Law, follows from the essentialist theory. Second, I introduce two new operations for kinds and show that they can be used to provide traditional definitions of kinds in terms of genera and specific differences. Finally, I show that these operations of specific difference induce, for each kind, a uniquely specified contrary kind and a uniquely specified subcontrary kind, which can be used as semantic values for non-classical predicate negations of kind terms. (shrink)

Ockham has a very particular definition of modality: every term that is predicable of a whole sentence is a modal term. His definition reaches well beyond “necessary,” “possible,” “contingent” and “impossible,” including predicates such as “known” and “believed,” but also “written” and “spoken.” He provides a general framework for inferences including every term covered by his definition of modality. However, there is a proper modal syllogistic in Ockham only for the Aristotelian modalities, that combines two distinct semantic apparatus, constituted (...) by Ockhamist modal predicates, on the one hand, and by modal and modally modified copulae, on the other. For the development of a syllogistic for the Aristotelian modalities, both are needed. There is no similar apparatus for other modalities in Ockham. (shrink)

This is the first of three essays which use Edmund Husserl's dependence ontology to formulate a non-Diodorean and non-Kantian temporal semantics for two-valued, first-order predicate modal languages suitable for expressing ontologies of experience (like physics and cognitive science). This essay's primary desideratum is to formulate an adequate dependence-ontological account of order. To do so it uses primitive (proper) part and (weak) foundation relations to formulate seven axioms and 28 definitions as a basis for Husserl's dependence ontological theory of relating moments. (...) The essay distinguishes between dependence v. independence, pieces v. moments, mediate v. immediate pieces and moments, maximal v. non-maximal pieces, founded v. unfounded qualities, integrative v. disintegrative dependence, and defines the concepts of the completion of an object, the adumbrational equivalence relation of objects, moments of unity which unify objects, and relating moments which relate objects. The eight theorems [CUT90]-[CUT97] show that relating moments of unity provide an adequate account of order in terms of primitive (proper) part and (weak) foundation relations. (shrink)

Extending the project of analysis -- Elaborating abilities : the expressive role of logic -- Artificial intelligence and analytic pragmatism -- Modality and normativity : from Hume and Quine to Kant and Sellars -- Incompatibility, modal semantics, and intrinsic logic -- Intentionality as a pragmatically mediated semantic relation -- Afterword : philosophical analysis and analytic philosophy.

This chapter begins with a discussion of Kant's theory of judgment-forms. It argues that it is not true in Kant's logic that assertoric or apodeictic judgments imply problematic ones, in the manner in which necessity and truth imply possibility in even the weakest systems of modern modal logic. The chapter then discusses theories of judgment-form after Kant, the theory of quantification, Frege's Begriffsschrift, C. I. Lewis and the beginnings of modern modal logic, the proof-theoretic approach to modal logic, possible world (...) semantics, correspondence theory, and modality and quantification. (shrink)

If □ is conceived as an operator, i.e., an expression that gives applied to a formula another formula, the expressive power of the language is severely restricted when compared to a language where □ is conceived as a predicate, i.e., an expression that yields a formula if it is applied to a term. This consideration favours the predicate approach. The predicate view, however, is threatened mainly by two problems: Some obvious predicate systems are inconsistent, and possible-worlds semantics for predicates of (...) sentences has not been developed very far. By introducing possible-worlds semantics for the language of arithmetic plus the unary predicate □, we tackle both problems. Given a frame (W, R) consisting of a set W of worlds and a binary relation R on W, we investigate whether we can interpret □ at every world in such a way that □ $\ulcorner A \ulcorner$ holds at a world ᵆ ∊ W if and only if A holds at every world $\upsilon$ ∊ W such that ᵆR $\upsilon$ . The arithmetical vocabulary is interpreted by the standard model at every world. Several 'paradoxes' (like Montague's Theorem, Gödel's Second Incompleteness Theorem, McGee's Theorem on the ω-inconsistency of certain truth theories, etc.) show that many frames, e.g., reflexive frames, do not allow for such an interpretation. We present sufficient and necessary conditions for the existence of a suitable interpretation of □ at any world. Sound and complete semi-formal systems, corresponding to the modal systems K and K4, for the class of all possible-worlds models for predicates and all transitive possible-worlds models are presented. We apply our account also to nonstandard models of arithmetic and other languages than the language of arithmetic. (shrink)

The abstract status of Kant's account of his ‘general logic’ is explained in comparison with Gödel's general definition of a formal logical system and reflections on ‘abstract’ (‘absolute’) concepts. Thereafter, an informal reconstruction of Kant's general logic is given from the aspect of the principles of contradiction, of sufficient reason, and of excluded middle. It is shown that Kant's composition of logic consists in a gradual strengthening of logical principles, starting from a weak principle of contradiction that tolerates a sort (...) of contradictions in predication, and then proceeding to the (constructive) principle of sufficient reason, and to a classical-like logic, which includes the principle of excluded middle. A first-order formalisation is applied to this reconstruction, which reveals implicit modalities in Kant's account of logic, and confirms the implementability of Kant's logic into a sound and complete formal system. (shrink)

Recent studies in semantics of modal and superintuitionistic predicate logics provided many examples of incompleteness, especially for Kripke semantics. So there is a problem: to find an appropriate possible- world semantics which is equivalent to Kripke semantics at the propositional level and which is strong enough to prove general completeness results. The present paper introduces a new semantics of Kripke metaframes' generalizing some earlier notions. The main innovation is in considering "n"-tuples of individuals as abstract "n"-dimensional vectors', together with some (...) transformations of these vectors. Soundness of the semantics is proved to be equivalent to some non- logical properties of metaframes; and thus we describe the maximal semantics of Kripke- type. (shrink)

This is the second in a sequence of three essays which axiomatize and apply Edmund Husserl's dependence ontology of parts and wholes as a non-Diodorean, non-Kantian temporal semantics for first-order predicate modal languages. The Ontology of Intentionality I introduced enough of Husserl's dependence-ontology of parts and wholes to formulate his account of order as effected by relating moments of unity, and The Ontology of Intentionality II extends that axiomatic dependence-ontology far enough to enable its semantic application. Formalizing the compatibility [Vereinbarkeit] (...) relation implicated in Husserl's notorious doctrine of impossible meanings, the essay introduces a compatibility restriction on relations to formulate Husserl's distinction between singular [einheitliche] and plural [mehrheitliche] objects, using plural relating moments to define first-order versions of Husserl's notions of relation complexes (i.e. Sachverhalte), abstracta of n-ary relation complexes, categorial relations, abstract eide as unifications of categorial relations, semantic domains as completions of abstract eide, and material regions as semantic domains which are compatibility upper bounds of categorial relations. These concepts will enable the formal dependence-ontological noetic semantics for two-valued, first-order modal languages introduced in the sequel Two-Valued Logics of Intentionality, the third essay in the sequence. (shrink)

In his precritical period Kant developed a demonstration of the existence of God that he believed to be the only possible one. Whether this argument is valid is a much-disputed question. To give an answer to this question I will reconstruct Kant’s argument with the means of modern logic and semantics and show which are—in my view—it’s weak points. To achieve this, I will use a combination of the formalism of predicate-, propositional and modal logic without, thought, introducing such a (...) combined logic. (shrink)

Aristotle’s viewpoints on modal propositions have been subjects of debate and exploration among scholars since his immediate successors. This study delves into how Aristotle defines assertoric and modal propositions and elucidates the correlations between these propositions and assertoric ones. Traditionally, commentators have interpreted propositions based on the premise that what belongs to the subject also belongs to the predicate. However, in this research, propositions are interpreted as ones in which what is predicated on the predicate will also be predicated on (...) the subject. The outcome of this investigation demonstrates the modal premise process through the Barbara and Celarent syllogism forms in the first figure, which can be reduced to encompass all other forms. (shrink)

The problem of completeness for predicate modal logics is still under investigation, although some results have been obtained in the last few years . As far as we know, the case of multimodal logics has not been addressed at all. In this paper, we study the combination of modal logics in terms of combining their semantics. We demonstrate by a simple example that in this sense predicate modal logics are not so easily manipulated as propositional ones: mixing two Kripke-complete predicate (...) modal logics results in a Kripke-incomplete system. (shrink)

This dissertation develops a substitutional semantics for first-order (modal) logic which, unlike truth-value semantics, allows a fine-grained analysis of the semantical behaviour of the terms and predicates from which atomic formulae are composed. Moreover, it proposes a nondenotational philosophical foundation for the semantics of substitutional quantified (modal) logic.

In this paper, the predicate counterparts, defined both axiomatically and semantically by means of Kripke frames, of the modal propositional logics $\textbf {GL}$, $\textbf {Grz}$, $\textbf {wGrz}$ and their extensions are considered. It is proved that the set of semantical consequences on Kripke frames of every logic between $\textbf {QwGrz}$ and $\textbf {QGL.3}$ or between $\textbf {QwGrz}$ and $\textbf {QGrz.3}$ is $\Pi ^1_1$-hard even in languages with three (sometimes, two) individual variables, two (sometimes, one) unary predicate letters, and a single (...) proposition letter. As a corollary, it is proved that infinite families of modal predicate axiomatic systems, based on the classical first-order logic and the modal propositional logics $\textbf {GL}$, $\textbf {Grz}$, $\textbf {wGrz}$ are not Kripke complete. Both $\Pi ^1_1$-hardness and Kripke incompleteness results of the paper do not depend on whether the logics contain the Barcan formula. (shrink)

What precisely are fragments of classical first-order logic showing “modal” behaviour? Perhaps the most influential answer is that of Gabbay 1981, which identifies them with so-called “finite-variable fragments”, using only some fixed finite number of variables (free or bound). This view-point has been endorsed by many authors (cf. van Benthem 1991). We will investigate these fragments, and find that, illuminating and interesting though they are, they lack the required nice behaviour in our sense. (Several new negative results support this claim.) (...) As a counterproposal, then, we define a large fragment of predicate logic characterized by its use of only bounded quantification. This so-called guarded fragment enjoys the above nice properties, including decidability, through an effectively bounded finite model property. (These are new results, obtained by generalizing notions and techniques from modal logic.) Moreover, its own internal finite variable hierarchy turns out to work well. Finally, we shall make another move. The above analogy works both ways. Modal operators are like quantifiers, but quantifiers are also like modal operators. This observation inspires a generalized semantics for first-order predicate logic with accessibility constraints on available assignments (cf. N´emeti 1986, 1992) which moves the earlier quantifier restrictions into the semantics. This provides a fresh look at the landscape of possible predicate logics, including candidates sharing various desirable features with basic modal logic – in particular, its decidability. (shrink)

According to indexical contextualism, the perspectival element of taste predicates and epistemic modals is part of the content expressed. According to nonindexicalism, the perspectival element must be conceived as a parameter in the circumstance of evaluation, which engenders “thin” or perspective-neutral semantic contents. Echoing Evans, thin contents have frequently been criticized. It is doubtful whether such coarse-grained quasi-propositions can do any meaningful work as objects of propositional attitudes. In this paper, I assess recent responses by Recanati, Kölbel, Lasersohn and MacFarlane (...) to the “incompleteness worry”. None of them manages to convince. Particular attention is devoted to an argument by John MacFarlane, which states that if perspectives must be part of the content, so must worlds, which would make intuitively contingent propositions necessary. I demonstrate that this attempt to defend thin content views such as nonindexical contextualism and relativism conflates two distinct notions of necessity, and that radical indexicalist accounts of semantics, such as Schaffer’s necessitarianism, are in fact quite plausible. (shrink)

This paper gives an outline of truthmaker semantics for natural language against the background of standard possible-worlds semantics. It develops a truthmaker semantics for attitude reports and deontic modals based on an ontology of attitudinal and modal objects and on a semantic function of clauses as predicates of such objects. It also présents new motivations for 'object-based truthmaker semantics' from intensional transitive verbs such as ‘need’, ‘look for’, ‘own’, and ‘buy’ and gives an outline of their semantics. This paper is (...) a commissioned 'target' article, with commentaries by W. Davis, B. Arsenijevic, K. Moulton, K. Liefke, M. Kaufmann, R. Matthews, P. Portner and A. Rubinstein, P. Elliott, G. Ramchand and my reply. (shrink)

In this work we extend results from [4], [3] and [2] about propositional calculi with graded modalities to the predicative level. Our semantic is based on Kripke models with a single domain of interpretation for all the worlds. Therefore the axiomatic system will need a suitable generalization of the Barcan formula. We haven't considered semantics with world-relative domains because they don't present any new difficulties with respect to classical case. Our language will have, as in [1], constant and function symbols, (...) but they will have a rigid interpretation. In this instance the terms will be considered rigid designators, that is, their interpretations will be the same in all possible worlds (cfr. [5]). Nevertheless we will not consider languages with identity: in this way we avoid the problems of identity in modal contexts. On the other hand, we think that in that context would arise essentially the same problems which occur in classical predicative modal logic. Finally we will consider only denumerable languages because the extension to non denumerable ones isn't conceptually problematic. (shrink)

This article deals with an alternative interpretation of syllogistics, different from the classical (extensional) one: an intensional one, in which subject and predicate are not associated with a set of individuals (the extension of the concept) but a set of attributes (the content of the concept). The authors of the paper draw attention to the fact that this approach was first proposed by Leibniz in works on logical calculus, which for a long time remained in the shadow of his other (...) philosophical works. Currently, the intensional approach is gaining more and more popularity due to the development of non-classical logics, and the article will present several existing intensional formal syllogistic semantics. The paper will also consider another historical approach to syllogistics, associated with the name of the Russian logician Nikolai Vasiliev, who is not only one of the founders of non-classical (non-Aristotelian logic) but also of a different intensional interpretation of such logic. The authors, along with the already known formalizations of Vasiliev’s ideas, present two new systems. One of them is a reconstruction of one type of imaginary logic with statements of three qualities: affirmative and two types of negative statements (with absolute and ordinary negation). The second system is the one that is adequate to semantics, in which instead of the four classical ones, only three types of statements are presented (two particular statements are replaced by one - accidental), and their significance is determined through the relation of the classical logical entailment. Both of them are interpreted intensionally. The intensional approach in logic and, in particular, in syllogistics allows us to expand the class of accepted principles (which occurs due to the expansion of the class of correct moods of syllogisms). (shrink)

This monograph explores the logical systems of early logicians in the Arabic tradition from a theoretical perspective, providing a complete panorama of early Arabic logic and centering it within an expansive historical context. By thoroughly examining the writings of the first Arabic logicians, al-Fārābī, Avicenna and Averroes, the author analyzes their respective theories, discusses their relationship to the syllogistics of Aristotle and his followers, and measures their influence on later logical systems. Beginning with an introduction to the writings of the (...) most prominent Arabic logicians, the author scrutinizes these works to determine their categorical logic, as well as their modal and hypothetical logics. Where most other studies written on this subject focus on the Arabic logicians’ epistemology, metaphysics, and theology, this volume takes a unique approach by focusing on the actual technical aspects and features of their logics. The author then moves on to examine the original texts as closely as possible and employs the symbolism of modern propositional, predicate, and modal logics, rendering the arguments of each logician clearly and precisely while clarifying the theories themselves in order to determine the differences between the Arabic logicians’ systems and those of Aristotle. By providing a detailed examination of theories that are still not very well-known in Western countries, the author is able to assess the improvements that can be found in the Arabic writings, and to situate Arabic logic within the breadth of the history of logic. This unique study will appeal mainly to historians of logic, logicians, and philosophers who seek a better understanding of the Arabic tradition. It also will be of interest to modern logicians who wish to delve into the historical aspects and progression of their discipline. Furthermore, this book will serve as a valuable resource for graduate students who wish to complement their general knowledge of Arabic culture, logic, and sciences. (shrink)

The complexity of any logical modeling reflects both the intrinsic structure of a topic described and the weight of the formal tools. Some of this weight seems inherent in even the most basic logical systems. Notably, standard predicate logic is undecidable. In this paper, we investigate ‘lighter’ versions of this general purpose tool, by modally ‘deconstructing’ the usual semantics, and locating implicit choice points in its set up. The first part sets out the interest of this program and the modal (...) techniques employed, while the second part provides technical elaborations demonstrating its viability. (shrink)

In this paper we systematically derive a predicate transformer semantics from a direct semantics for a simple probabilistic-nondeterministic programming language . This goal is achieved by exhibiting the direct semantics as isomorphic to a continuation semantics from which the predicate transformer semantics can be read off immediately. This isomorphism allows one to identify nonempty convex compact saturated sets of valuations on the set S of states with certain “good” functionals from to in a way similar to the one how H. (...) Minkowski in 1903 related nonempty convex compact subsets of to what is nowadays called Minkowski functionals. (shrink)

The semantical framework is fundamentally intensional: neither possible worlds nor sets as basic entities, but rather, besides individuals, propositions, properties and relations (in intension). Logical truth is defined in terms of logical form (without mentioning this notion) without employing sets of models and the concept of truth in a model. Truth itself is explicitly defined (without recursion); the truth-conditions for the logical constants of the object-language become theorems derivable from the axioms for "to intend"--the basic semantical relation.

I propose a semantics for a class of English predicates characteristically associated with possibility. The central idea is that such predicates are typically associated with an ordering source, and that differences among them are due to differences in their ordering sources. The ‘dispositional predicates’ that have been central to philosophical discussions are shown to be derivable as a special case from this more general class.

This paper develops an update semantics for weak necessity modals like ‘ought’ and ‘should’. I start with the basic approach to the weak/strong necessity modal distinction developed in Silk 2018: Strong necessity modals are given their familiar semantics of necessity, predicating the necessity of the prejacent of the actual world (evaluation world). The apparent “weakness” of weak necessity modals derives from their bracketing the assumption that the relevant worlds in which the prejacent is necessary (deontically, epistemically, etc.) need be candidates (...) for actuality. ‘Should φ’ can be accepted without needing to settle that the relevant considerations (norms, preferences, expectations, etc.) that actually apply, given the facts, verify the necessity of φ. I formalize these ideas within an Update with Centering framework. The meaning of ‘Should φ’ is explained, fundamentally, in terms of how its use updates attention toward possibilities in which φ is necessary. The semantics is also extended to deontic conditionals. The proposed analyses capture various contrasting logical properties and discourse properties of ‘should’ and ‘must’ — e.g., in sensitivities to standing contextual assumptions, entailingness, and force — and provide an improved treatment of largely neglected data concerning information-sensitivity. (shrink)

This is another contribution to the unending controversy over the two Barbaras. My approach to the problem is hopefully quite new: I wish to view the issue through the prism of modal upgrading. Modal upgrading occurs when a subject term that has only been predicated of assertorically in the premises is predicated of apodictically either: i) in the conclusion of a given syllogism, or; ii) in some proposition that is derived from either the premises of the given syllogism alone or (...) the premises in combination with other propositions that do not refer to the proposition’s subject term. I call the proposition after it has been upgraded the upgraded proposition. When a conclusion is the upgraded proposition, it is obviously a different predicate being predicated than was predicated in the premises. Aristotle endorses this kind of upgrading; it is effectively what happens in any valid mixed modal syllogism when the minor premise is not apodictic (e.g., Barbara LXL). In other cases the upgraded proposition is not a conclusion but still follows from the premises alone. In these cases it is the same predicate being predicated in the upgraded proposition as in the premises, although the quantity of the propositions are different (one is universal, another particular). Aristotle rejects this kind of upgrading and takes its occurrence as sufficient to deny the validity of the given syllogism (e.g., Barbara XLL). I will describe a third type where both the predicate remains the same and the quantity of the proposition remains the same as in the premise, e.g., the upgrading of “All C are B” to “All C are necessarily B”. In these cases it will turn out that the upgraded proposition is not derived from the premises alone, or at least, not syllogistically from the premises alone. This kind of upgrading too is reason for denying the validity of any syllogism from which the upgraded proposition follows as a consequence. I will show that Barbara LXL entails this kind of modal upgrading and should be rejected for this reason. Armed with this notion of modal upgrading I want to attack the problem of the two Barbaras in Aristotle’s apodictic syllogistic. Aristotle himself endorses mixed modal Barbara when the major is necessary and the minor is assertoric, thereby endorsing the first kind of modal upgrading, but rejects Barbara when the minor is necessary and the major is assertoric on the grounds that it leads to the second kind of modal upgrading. Theophrastus endorses the peioram rule which rejects both Barbaras on the grounds that the conclusion can only be as strong as the weakest premise. Łukasiewicz endorses both Barbaras. I will argue that both Barbaras lead to unacceptable modal upgrading and should be taken to be invalid for that reason. Hence, I agree with Theophrastus about the two Barbaras; however, I do not endorse the peioram rule because I think that the negative mixed modal syllogisms generally avoid this problem and is mostly correct. (shrink)

In recent years combinations of tense and modality have moved intothe focus of logical research. From a philosophical point of view, logical systems combining tense and modality are of interest because these logics have a wide field of application in original philosophical issues, for example in the theory of causation, of action, etc. But until now only methods yielding completeness results for propositional languages have been developed. In view of philosophical applications, analogous results with respect to languages of predicate logic (...) are desirable, and in this paper I present two such results. The main developments in this area can be split into two directions, differing in the question whether the ordering of time is world-independent or not. Semantically, this difference appears in the discussion whether T x W-frames or Kamp-frames (resp. Ockham-frames) provide a suitable semantics for combinations of tense and modality. Here, two calculi are presented, the first adequate with respect to Kamp-semantics, the second to T x Wsemantics. (Both calculi contain an appropriate version of Gabbay's irreflexivity rule.) Furthermore, the proposed constructions of canonical frames simplify some of those which have hitherto been discussed. (shrink)

This paper argues for a largely unnoted distinction between relational and modal components in the lexical semantics of verbs. Wehypothesize that many verbs encode two kinds of semantic information:a relationship among participants in a situation and a subset ofcircumstances or time indices at which this relationship isevaluated. The latter we term sublexical modality.We show that linking regularities between semantic arguments andsyntactic functions provide corroborating evidence in favor of thissemantic distinction, noting cases in which the semantic groundingof linking through participant-role properties (...) apparently fails. Thissemantic grounding can be preserved, however, once we abstractaway from sublexical modality in lexical semantic representations.Semantically-based linking constraints are insensitive to the sublexicalmodality component of lexical entries and depend only on informationin a predicator's situational core. (shrink)

In the first part of this paper we analyzed finite non-deterministic matrix semantics for propositional non-normal modal logics as an alternative to the standard Kripke possible world semantics. This kind of modal system characterized by finite non-deterministic matrices was originally proposed by Ju. Ivlev in the 70s. The aim of this second paper is to introduce a formal non-deterministic semantical framework for the quantified versions of some Ivlev-like non-normal modal logics. It will be shown that several well-known controversial issues of (...) quantified modal logics, relative to the identity predicate, Barcan’s formulas and de re and de dicto modalities, can be tackled from a new angle within the present framework. (shrink)

The standard view of clauses embedded under attitude verbs or modal predicates is that they act as terms standing for propositions, a view that faces a range of philosophical and linguistic difficulties. Recently an alternative has been explored according to which embedded clauses act semantically as predicates of content-bearing objects. This paper argues that this approach faces serious problems when it is based on possible worlds-semantics. It outlines a development of the approach in terms of truthmaker theory instead.

When it comes to Kripke-style semantics for quantified modal logic, there’s a choice to be made concerning the interpretation of the quantifiers. The simple approach is to let quantifiers range over all possible objects, not just objects existing in the world of evaluation, and use a special predicate to make claims about existence. This is the constant domain approach. The more complicated approach is to assign a domain of objects to each world. This is the varying domain approach. Assuming that (...) all terms denote, the semantics of predication on the constant domain approach is obvious: either the denoted object has the denoted property in the world of evaluation, or it hasn’t. On the varying domain approach, there’s a third possibility: the object in question doesn’t exist. Terms may denote objects not included in the domain of the world of evaluation. The question is whether an atomic formula then should be evaluated as true or false, or if its truth value should be undefined. This question, however, cannot be answered in isolation. The consequences of one’s choice depends on the interpretation of molecular formulas. Should the negation of a formula whose truth value is undefined also be undefined? What about conjunction, universal quantification and necessitation? The main contribution of this paper is to identify two partial semantics for logical operators, a weak and a strong one, which uniquely satisfy a list of reasonable constraints. I also show that, provided that the point of using varying domains is to be able to make certain true claims about existence without using any existence predicate, this result yields two possible partial semantics for quantified modal logic with varying domains. (shrink)

Many systems of quantified modal logic cannot be characterised by Kripke's well-known possible worlds semantic analysis. This book shows how they can be characterised by a more general 'admissible semantics', using models in which there is a restriction on which sets of worlds count as propositions. This requires a new interpretation of quantifiers that takes into account the admissibility of propositions. The author sheds new light on the celebrated Barcan Formula, whose role becomes that of legitimising the Kripkean interpretation of (...) quantification. The theory is worked out for systems with quantifiers ranging over actual objects, and over all possibilia, and for logics with existence and identity predicates and definite descriptions. The final chapter develops a new admissible 'cover semantics' for propositional and quantified relevant logic, adapting ideas from the Kripke-Joyal semantics for intuitionistic logic in topos theory. This book is for mathematical or philosophical logicians, computer scientists and linguists. (shrink)

Aristotle’s modal syllogistic is his study of patterns of reasoning about necessity and possibility. Many scholars think the modal syllogistic is incoherent, a ‘realm of darkness’. Others think it is coherent, but devise complicated formal modellings to mimic Aristotle’s results. This volume provides a simple interpretation of Aristotle’s modal syllogistic using standard predicate logic. Rini distinguishes between red terms, such as ‘horse’, ‘plant’ or ‘man’, which name things in virtue of features those things must have, and green (...) terms, such as ‘moving’, which name things in virtue of their non-necessary features. By applying this distinction to the Prior Analytics, Rini shows how traditional interpretive puzzles about the modal syllogistic melt away and the simple structure of Aristotle’s own proofs is revealed. The result is an applied logic which provides needed links between Aristotle’s views of science and logical demonstration. The volume is particularly valuable to researchers and students of the history of logic, Aristotle’s theory of modality, and the philosophy of logic in general. (shrink)

Aristotle's modal syllogistic has been an object of study ever since the time of Theophrastus; but these studies have been somewhat desultory. Remarkably, in the 1990s several new lines of research have appeared, with series of original publications by Fred Johnson, Richard Patterson and Ulrich Nortmann. Johnson presented for the first time a formal semantics adequate to a de re reading of the apodeictic syllogistic; this was based on a simple intuition linking the modal syllogistic to Aristotelian (...) metaphysics. Nortmann developed an ingenious de dicto analysis. Patterson articulated the links between the modal syllogistic and the metaphysics, using an analysis which strictly speaking is neither de re nor de dicto. My own studies in this field date from 1976, when my colleague Peter Roeper and I jointly wrote a paper "Aristotle's apodeictic syllogisms" for the XXIInd History of Logic Conference in Krakow. This paper contained the disjunctive reading of particular affirmative apodeictic propositions, which I still favour. Nonetheless, I did not consider that paper's results decisive or comprehensive enough to publish, and my 1981 book The Syllogism contained no treatment of the modal syllogism. The paper's ideas lay dormant till 1989, when I read Johnson's and Patterson's initial articles. I began publishing on the topic in 1991. Gradually my thoughts acquired a certain comprehensiveness and systematicity, till in 1993 I was able to take a semester's sabbatical to write up a draft of this book. (shrink)

In this text, a variety of modal logics at the sentential, first-order, and second-order levels are developed with clarity, precision and philosophical insight. All of the S1-S5 modal logics of Lewis and Langford, among others, are constructed. A matrix, or many-valued semantics, for sentential modal logic is formalized, and an important result that no finite matrix can characterize any of the standard modal logics is proven. Exercises, some of which show independence results, help to develop logical skills. A separate sentential (...) modal logic of logical necessity in logical atomism is also constructed and shown to be complete and decidable. On the first-order level of the logic of logical necessity, the modal thesis of anti-essentialism is valid and every de re sentence is provably equivalent to a de dicto sentence. An elegant extension of the standard sentential modal logics into several first-order modal logics is developed. Both a first-order modal logic for possibilism containing actualism as a proper part as well as a separate modal logic for actualism alone are constructed for a variety of modal systems. Exercises on this level show the connections between modal laws and quantifier logic regarding generalization into, or out of, modal contexts and the conditions required for the necessity of identity and non-identity. Two types of second-order modal logics, one possibilist and the other actualist, are developed based on a distinction between existence-entailing concepts and concepts in general. The result is a deeper second-order analysis of possibilism and actualism as ontological frameworks. Exercises regarding second-order predicate quantifiers clarify the distinction between existence-entailing concepts and concepts in general. Modal Logic is ideally suited as a core text for graduate and undergraduate courses in modal logic, and as supplementary reading in courses on mathematical logic, formal ontology, and artificial intelligence. (shrink)