After giving a short summary of the traditional theory of the syllogism, it is shown how the square of opposition reappears in the much more powerful concept logic of Leibniz. Within Leibniz’s algebra of concepts, the categorical forms are formalized straightforwardly by means of the relation of concept-containment plus the operator of concept-negation as ‘S contains P’ and ‘S contains Not-P’, ‘S doesn’t contain P’ and ‘S doesn’t contain Not-P’, respectively. Next we consider Leibniz’s version of the so-called Quantification of (...) the Predicate which consists in the introduction of four additional forms ‘Every S is every P’, ‘Some S is every P’, ‘Every S isn’t some P’, and ‘Some S isn’t some P’. Given the logical interpretation suggested by Leibniz, these unorthodox propositions also form a Square of Opposition which, when added to the traditional Square, yields a “Cube of Opposition”. Finally it is shown that besides the categorical forms, also the non-categorical forms can be formalized within an extension of Leibniz’s logic where “indefinite concepts” X, Y, Z\ function as quantifiers and where individual concepts are introduced as maximally consistent concepts. (shrink)

It is a received view that Kant’s formal logic (or what he calls “pure general logic”) is thoroughly intensional. On this view, even the notion of logical extension must be understood solely in terms of the concepts that are subordinate to a given concept. I grant that the subordination relation among concepts is an important theme in Kant’s logical doctrine of concepts. But I argue that it is both possible and important to ascribe to Kant an objectual notion of logical (...) extension according to which the extension of a concept is the multitude of objects falling under it. I begin by defending this ascription in response to three reasons that are commonly invoked against it. First, I explain that this ascription is compatible with Kant’s philosophical reflections on the nature and boundary of a formal logic. Second, I show that the objectual notion of extension I ascribe to Kant can be traced back to many of the early modern works of logic with which he was more or less familiar. Third, I argue that such a notion of extension makes perfect sense of a pivotal principle in Kant’s logic, namely the principle that the quantity of a concept’s extension is inversely proportional to that of its intension. In the process, I tease out two important features of the Kantian objectual notion of logical extension in terms of which it markedly differs from the modern one. First, on the modern notion the extension of a concept is the sum of the objects actually falling under it; on the Kantian notion, by contrast, the extension of a concept consists of the multitude of possible objects—not in the metaphysical sense of possibility, though—to which a concept applies in virtue of being a general representation. While the quantity of the former extension is finite, that of the latter is infinite—as is reflected in Kant’s use of a plane-geometrical figure (e.g., circle, square), which is continuum as opposed to discretum, to represent the extension in question. Second, on the modern notion of extension, a concept that signifies exactly one object has a one-member extension; on the Kantian notion, however, such a concept has no extension at all—for a concept is taken to have extension only if it signifies a multitude of things. This feature of logical extension is manifested in Kant’s claim that a singular concept (or a concept in its singular use) can, for lack of extension, be figuratively represented only by a point—as opposed to an extended figure like circle, which is reserved for a general concept (or a concept in its general use). Precisely on account of these two features, the Kantian objectual extension proves vital to Kant’s theory of logical quantification (in universal, particular and singular judgments, respectively) and to his view regarding the formal truth of analytic judgments. (shrink)

According to the received view (Bocheński, Kneale), from the end of the fourteenth to the second half of nineteenth century, logic enters a period of decadence. If one looks at this period, the richness of the topics and the complexity of the discussions that characterized medieval logic seem to belong to a completely different world: a simplified theory of the syllogism is the only surviving relic of a glorious past. Even though this negative appraisal is grounded on good reasons, it (...) overlooks, however, a remarkable innovation that imposes itself at the beginning of the sixteenth century: the attempt to connect the two previously separated disciplines of logic and mathematics. This happens along two opposite directions: the one aiming to base mathematical proofs on traditional (Aristotelian) logic; the other attempting to reduce logic to a mathematical (algebraical) calculus. This second trend was reinforced by the claim, mainly propagated by Hobbes, that the activity of thinking was the same as that of performing an arithmetical calculus. Thus, in the period of what Bocheński characterizes as ‘classical logic’, one may find the seeds of a process which was completed by Boole and Frege and opened the door to the contemporary, mathematical form of logic. (shrink)

In this paper, the author develops a theory of concepts and shows that it captures many of the ideas about concepts that Leibniz expressed in his work. Concepts are first analyzed in terms of a precise background theory of abstract objects, and once concept summation and concept containment are defined, the axioms and theorems of Leibniz's calculus of concepts (in his logical papers) are derived. This analysis of concepts is then seamlessly connected with Leibniz's modal metaphysics of complete individual concepts. (...) The fundamental theorem of Leibniz's modal metaphysics of concepts is proved, namely, whenever an object x has F contingently, then (i) the individual concept of x contains the concept F and (ii) there is a (counterpart) complete individual concept y which doesn't contain the concept F and which `appears' at some other possible world. Finally, the author shows how the concept containment theory of truth can be made precise and made consistent with a modern conception of truth. (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)

This thesis studies the role of conceptual content in inference and reasoning. The first two chapters offer a theoretical and historical overview of the relation between inference and meaning in philosophy and psychology. In particular, a critical analysis of the formality thesis, i.e., the idea that rational inference is a rule-based and topic-neutral mechanism, is advanced. The origins of this idea in logic and its influence in philosophy and cognitive psychology are discussed. Chapter 3 consists of an analysis of the (...) relationship between inference and representation. It is argued that inference has to be studied from a pluralistic per- spective due to its dependence on different formats of representing information. The following four chapters apply conceptual spaces, a formal theory of concepts within cognitive semantics, to three concept-based inference-types. First, an explication of Sellars notion of material inference is advanced. Later, the model is extended to account for nonmonotonic inference by studying the role expectations in reasoning. Finally, a conceptual space-model of category-based induction is presented. This model predicts most of the empirical properties of this psychological phenomenon and subsumes some of the previous theories in psychology. It is stated that the explanatory fruitfulness of this new approach is evidence for the failure of the formality thesis and calls for a unified model of rational inference that puts semantics at center stage. The last chapter of the thesis discusses how inference and concepts interact in scientific reasoning, which makes constant use of hybrid symbolic structures for representing conceptual information. Stephen Toulmin’s notions of method of representation and inferential technique are developed and applied in a case study about the emergence of the notion of instantaneous speed during the passage from geometrical physics to analytical mechanics. It is claimed that this analysis provides support to the pluralistic perspective for theorizing about reasoning. (shrink)

Greek antiquity saw the development of two distinct systems of logic: Aristotle’s theory of the categorical syllogism and the Stoic theory of the hypothetical syllogism. Some ancient logicians argued that hypothetical syllogistic is more fundamental than categorical syllogistic on the grounds that the latter relies on modes of propositional reasoning such asreductio ad absurdum. Peripatetic logicians, by contrast, sought to establish the priority of categorical over hypothetical syllogistic by reducing various modes of propositional reasoning to categorical form. In the 17th (...) century, this Peripatetic program of reducing hypothetical to categorical logic was championed by Gottfried Wilhelm Leibniz. In an essay titledSpecimina calculi rationalis, Leibniz develops a theory of propositional terms that allows him to derive the rule ofreductio ad absurdumin a purely categorical calculus in which every proposition is of the formA is B. We reconstruct Leibniz’s categorical calculus and show that it is strong enough to establish not only the rule ofreductio ad absurdum, but all the laws of classical propositional logic. Moreover, we show that the propositional logic generated by the nonmonotonic variant of Leibniz’s categorical calculus is a natural system of relevance logic known as RMI$_{{}_ \to ^\neg }$. (shrink)

In this paper I present a formal language in which complex predicates stand for properties and relations, and assignments of denotations to complex predicates and assignments of extensions to the properties and relations they denote are both homomorphisms. This system affords a fresh perspective on several important philosophical topics, highlighting the algebraic features of properties and clarifying the sense in which properties can be represented by their extensions. It also suggests a natural modification of current logics of properties, one in (...) which some complex predicates stand for properties while others do not. (shrink)