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The aim of this chapter is to develop asemantics for Calculus CL. CL is a diagrammatic calculus based on a logic machine presented by Johann Christian Lange in 1714, which combines features of Euler-, Venn-type, tree diagrams, squares of oppositions etc. In this chapter, it is argued that a Boolean account of formal ontology in CL helps to deal with logical oppositions and inferences of extended syllogistics. The result is a combination of Lange’s diagrams with an algebraic (...) class='Hi'>semantics of terms: Bit-CL, in which any ordered objects are identified by characteristic bitstrings. Then, a number of objections to Bit-CL are answered to, and the process of inference is explained in this new logical framework. (shrink)

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Logical geometry systematically studies Aristotelian diagrams, such as the classical square of oppositions and its extensions. These investigations rely heavily on the use of bitstrings, which are compact combinatorial representations of formulas that allow us to quickly determine their Aristotelian relations. However, because of their general nature, bitstrings can be applied to a wide variety of topics in philosophical logic beyond those of logical geometry. Hence, the main aim of this paper is to present a systematic technique for assigning bitstrings (...) to arbitrary finite fragments of formulas in arbitrary logical systems, and to study the logical and combinatorial properties of this technique. It is based on the partition of logical space that is induced by a given fragment, and sheds new light on a number of interesting issues, such as the logic-dependence of the Aristotelian relations and the subtle interplay between the Aristotelian and Boolean structure of logical fragments. Finally, thebitstring technique also allows us to systematically analyze fragments from contemporary logical systems, such as public announcement logic, which could not be done before. (shrink)

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This paper explores the interplay between logic-sensitivity andbitstringsemantics in the square of opposition.Bitstringsemantics is a combinatorial technique for representing the formulas that appear in a logical diagram, while logic-sensitivity entails that such a diagram may depend, not only on the formulas involved, but also on the logic with respect to which they are interpreted. These two topics have already been studied extensively in logical geometry, and are thus well-understood by themselves. However, the (...) precise details of their interplay turn out to be far more complicated. In particular, the paper describes an elegant and natural interaction between bitstrings and logic-sensitivity, which makes perfect sense when bitstrings are viewed as purely combinatorial entities. However, when we view bitstrings as semantically meaningful entities (which is actually the standard perspective, cf. the term ‘bitstringsemantics’!), this interaction does not seem to have a full and equally natural counterpart. The paper describes some attempts to address this situation, but all of them are ultimately found wanting. For now, it thus remains an open problem to capture this interaction between bitstrings and logic-sensitivity from a semantic (rather than merely a combinatorial) perspective. (shrink)

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It has been recently argued that the well-known square of opposition is a gathering that can be reduced to a one-dimensional figure, an ordered line segment of positive and negative integers [3]. However, one-dimensionality leads to some difficulties once the structure of opposed terms extends to more complex sets. An alternative algebraicsemantics is proposed to solve the problem of dimensionality in a systematic way, namely: partition (orbitstring)semantics. Finally, an alternative geometry yields a new and (...) unique pattern of oppositions that proceeds with colored diagrams and an increasing set of bitstrings. (shrink)

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In recent years CL diagrams inspired by Lange’s Cubus Logicus have been used in various contexts of diagrammatic reasoning. However, whether CL diagrams can also be used as a formal system seemed questionable. We present a CL diagram as a formal system, which is a fragment of propositional logic. Syntax andsemantics are presented separately and a variant ofbitstringsemantics is applied to prove soundness and completeness of the system.

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In this paper we provide an analysis of the logical relations within the conceptual or lexical field of angles in 2D geometry. The basic tripartition into acute/right/obtuse angles is extended in two steps: first zero and straight angles are added, and secondly reflex and full angles are added, in both cases extending the logical space of angles. Within the framework of logical geometry, the resulting partitions of these logical spaces yieldbitstringsemantics of increasing complexity. Thesebitstring (...) analyses allow a straightforward account of the Aristotelian relations between angular concepts. In addition, also relational concepts such as complementary and supplementary angles receive a naturalbitstring analysis. (shrink)

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Over the past few years, _CL_ diagrams have gained popularity in diagrammatic reasoning, drawing inspiration from Lange’s _C_ubus _L_ogicus. The intuitive understanding of _CL_ diagrams is based on simple structures that are straightforward both to draw and to comprehend. This structure supports embedding of information and inferencing. Furthermore, these diagrams are more than just heuristic tools; _CL_ diagrams can actually be extended to full blown formal systems. The present paper shows that a formal system for _CL_ diagrams can have an (...) expressivity similar to standard Boolean algebra of propositions or propositional logic without losing their intuitive properties. It is also possible to combine this interpretation of _CL_ with elements from the class calculus. The syntax andsemantics are presented separately, and the system’s soundness and completeness are proven using a variant ofbitstringsemantics. (shrink)

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CL diagrams – the abbreviation of Cubus Logicus – are inspired by J.C. Lange’s logic machine from 1714. In recent times, Lange’s diagrams have been used for extended syllogistics,bitstringsemantics, analogical reasoning and many more. The paper presents a method for testing statistical syllogisms (also called proportional syllogisms or inductive syllogisms) by using CL diagrams.

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The paper examines Schopenhauer’s complex diagrams from the Berlin Lectures of the 1820 s, which show certain partitions of classes. Drawing upon ideas and techniques from logical geometry, we show that Schopenhauer’s partition diagrams systematically give rise to a special type of Aristotelian diagrams, viz. (strong) α -structures.

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The paper examines Schopenhauer’s complex diagrams from the Berlin Lectures of the 1820 s, which show certain partitions of classes. Drawing upon ideas and techniques from logical geometry, we show that Schopenhauer’s partition diagrams systematically give rise to a special type of Aristotelian diagrams, viz. (strong) α -structures.

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In this paper, we interpret a 19th century diagram, which is meant to visualise G.W.F. Hegel’s entire method of the `Science of Logic' on the basis of bitwise operations. For the interpretation of the diagram we use a binary numeral system, and discuss whether the anti-Hegelian argument associated with it is valid or not. The reinterpretation is intended to make more precise rules of construction, a stricter binary code and a review of strengths and weaknesses of the critique.

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The aim of this paper is to make sense of the Keynes–Johnson octagon of oppositions. We will discuss Keynes' logical theory, and examine how his view is reflected on this octagon. Then we will show how this structure is to be handled by means of asemantics of partition, thus computing logical relations between matching formulas with a semantic method that combines model theory and Boolean algebra.

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Aristotle’s philosophy is considered with respect to one central concept of his philosophy, viz. opposition. Far from being a mere side-effect of syllogistic, it is argued in the present paper that opposition helps to articulate ontology and logic through an account of what can be or cannot be in a systematic and structural way. The paper is divided into three main parts. In Section I, the notion of Being is scrutinized through Aristotle’s theory of categories. In Section II, the notion (...) of Non-Being is connected to Aristotle’s theory of oppositions. In Section III, the notion of essence is revisited in order to bring about a holist theory of meaning by individuating through opposite properties. In conclusion, the legacy of Aristotle is depicted as balanced between a powerful reflection around Being and a restrictive ontology of substance. (shrink)

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nated discussion of thesemantics of such verbs. I will call this view.

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The book is about logical analysis of natural language. Since we humans communicate by means of natural language, we need a tool that helps us to understand in a precise manner how the logical and formal mechanisms of natural language work. Moreover, in the age of computers, we need to communicate both with and through computers as well. Transparent Intensional Logic is a tool that is helpful in making our communication and reasoning smooth and precise. It deals with all kinds (...) of linguistic context in a fully compositional and anti-contextual way. (shrink)

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We explore some possibilities for developing epistemic logic using truthmakersemantics. We identify three possible targets of analysis for the epistemic logician. We then list some candidate epistemic principles and review the arguments that render some controversial. We then present the classic Hintikkan approach to epistemic logic and note—as per the ‘problem of logical omniscience’—that it validates all of the aforementioned principles, controversial or otherwise. We then lay out a truthmaker framework in the style of Kit Fine and present (...) six different ways of extending thissemantics with a conditional knowledge operator, drawing on notions of implication and content that are prominent in Fine’s work. We demonstrate that different logics are thereby generated, bearing on the aforementioned epistemic principles. Finally, we offer preliminary observations about the prospects for each logic. (shrink)

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In their delightfully provocative paper, “Semantics, Cross-Cultural Style,” Edouard Machery, Ron Mallon, Shaun Nichols, and Stephen Stich (2004),[1] make several striking claims about theories of reference. First, they claim: (I) Philosophical views about reference “are assessed by consulting one’s intuitions about the reference of terms in hypothetical situations” (p. B1). This claim is prompted by their observations of the role of intuitions in Saul Kripke’s refutation of the descriptivist view of proper names in favor of a causal-historical view (1980). (...) The particular intuitions they attend to are those aired in discussing Kripke’s cases of Gödel and Jonah. This prompts the next claim: (II) Those particular cases are “central” to Kripke’s refutation (p. B1). Indeed, Machery et al describe these cases as “some of the most influential thought experiments in the philosophy of reference” (p. B8). Inspired by recent work in psychology (e.g., Nisbett et al 2001) that shows “systematic cognitive differences between East Asians and Westerners” (p. B1), Machery et al predicted that there would be cultural differences in referential intuitions. They conducted some ingenious experiments on Gödel and Jonah cases to test this predication. The results in the Gödel cases, although not in the Jonah cases, confirmed their prediction, leading them to conclude: “Westerners are more likely than East Asians to report intuitions that are consistent with the causal-historical view” (p. B1). And, implicitly, they claim: (III) These results raise serious doubts about Kripke’s refutation, which relies solely on the intuitions of Westerners. They are explicit about the following bolder claim: (IV) The fact of these cultural differences “raises questions about the nature of the philosophical enterprise of developing a theory of reference” (p. B1); it points to “significant philosophical conclusions” (p.. (shrink)

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This paper establishes a sound and completesemantics for the impure logic of ground. Fine (Review of Symbolic Logic, 5(1), 1–25, 2012a) sets out a system for the pure logic of ground, one in which the formulas between which ground-theoretic claims hold have no internal logical complexity; and it provides a sound and completesemantics for the system. Fine (2012b) [§§6-8] sets out a system for an impure logic of ground, one that extends the rules of the original (...) pure system with rules for the truth-functional connectives, the first-order quantifiers, and λ-abstraction. However, nosemantics has yet been provided for this system. The present paper partly fills this lacuna by providing a sound and completesemantics for a system GG containing the truth-functional operators that is closely related to the truth-functional part of the system of Fine (2012b). (shrink)

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Recent developments in thesemantics of natural language seem to lead to a genuine synthesis of ideas from linguistics and logic, producing novel concepts and questions of interest to both parent disciplines. This book is a collection of essays on such new topics, which have arisen over the past few years. Taking a broad view, developments in formalsemantics over the past decade can be seen as follows. At the beginning stands Montague's pioneering work, showing how a rigorous (...)semantics can be given for complete fragments of natural language by creating a suitable fit between syntactic categories and semantic types. This very enterprise already dispelled entrenched prejudices concerning the separation of linguistics and logic. Having seen the light, however, there is no reason at all to stick to the letter of Montague's proposals, which are often debatable. Subsequently, then, many improvements have been made upon virtually every aspect of the enterprise. More sophisticated grammars have been inserted (lately, lexical-functional grammar and generalized phrase structure grammar), more sensitive model structures have been developed (lately, 'partial' rather than 'total' in their com position), and even the mechanism of interpretation itself may be fine-tuned more delicately, using various forms of 'representations' mediating between linguistic items and semantic reality. In addition to all these refinements of the semantic format, descriptive coverage has extended considerably. (shrink)

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A novel cognitive theory ofsemantics that proposes that the meanings of words can be described in terms of geometric structures.

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The literature on indicative conditionals contains two appealing views. The first is the selectional view: on this view, conditionals operate by selecting a single possibility, which is used to evaluate the consequent. The second is the informational view: on this view, conditionals don’t express propositions, but rather impose constraints on information states of speakers. Both views are supported by strong arguments, but they are incompatible on their standard formulations. Hence it appears that we have to choose between mutually exclusive options. (...) But the options are not exclusive. This paper develops a theory of thesemantics and assertability of conditionals that is both selectional and informational. The theory vindicates the signature inferences of both selectional and informational theories, including so-called Conditionals Excluded Middle and principles about the interplay between conditionals and ‘might’. It also predicts some interesting and puzzling facts about the assertion of conditionals. (shrink)

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José L. Zalabardo defends a pragmatist account of what grounds the meaning of central semantic discourses--ascriptions of truth, of propositional attitudes, and of meanings. He argues that it is the procedures that regulate acceptance and rejection that give the sentences of these discourses their meanings, and explores the application of the pragmatist template to ethical discourse. The pragmatist approach is presented as an alternative to representationalist accounts of the meaning grounds of declarative sentences, according to which a sentence has the (...) meaning it has as a result of links with the bits of the world that it purports to represent. Zalabardo develops a version of the open-question argument to support the claim that the meaning grounds of the discourses he focuses on cannot receive representationalist accounts. It is generally assumed that a declarative sentence cannot perform the function of representing the world unless it has a representationalist meaning ground. Zalabardo rejects this assumption, arguing that sentences with pragmatist meaning grounds can represent the world in exactly the same sense that sentences with representationalist meaning grounds do. This requires that there are states of affairs that the target sentences represent as obtaining, and Zalabardo develops an account of the nature of the states of affairs that can play this role for sentences with pragmatist meaning grounds. PragmatistSemantics concludes by developing the suggestion that the meaning grounds of all our representational discourses might be ultimately pragmatist. (shrink)

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In traditionalsemantics for classical logic and its extensions, such as modal logic, propositions are interpreted as subsets of a set, as in discrete duality, or as clopen sets of a Stone space, as in topological duality. A point in such a set can be viewed as a "possible world," with the key property of a world being primeness—a world makes a disjunction true only if it makes one of the disjuncts true—which classically implies totality—for each proposition, a world (...) either makes the proposition true or makes its negation true. This chapter surveys a more general approach to logicalsemantics, known as possibilitysemantics, which replaces possible worlds with possibly partial "possibilities." In classical possibilitysemantics, propositions are interpreted as regular open sets of a poset, as in set-theoretic forcing, or as compact regular open sets of an upper Vietoris space, as in the recent theory of "choice-free Stone duality." The elements of these sets, viewed as possibilities, may be partial in the sense of making a disjunction true without settling which disjunct is true. We explain how possibilities may be used insemantics for classical logic and modal logics and generalized tosemantics for intuitionistic logics. The goals are to overcome or deepen incompleteness results for traditionalsemantics, to avoid the nonconstructivity of traditionalsemantics, and to provide richer structures for the interpretation of new languages. (shrink)

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Linguistics. Artificial intelligence. Related fields. Computation.

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