The aim of this chapter is to develop a semantics 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)

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, the bitstring technique also allows us to systematically analyze fragments from contemporary logical systems, such as public announcement logic, which could not be done before. (shrink)

This paper explores the interplay between logic-sensitivity and bitstring semantics in the square of opposition. Bitstring semantics 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 ‘bitstring semantics’!), 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)

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 algebraic semantics is proposed to solve the problem of dimensionality in a systematic way, namely: partition (or bitstring) 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)

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 yield bitstring semantics of increasing complexity. These bitstring (...) 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 natural bitstring analysis. (shrink)

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 and semantics are presented separately and a variant of bitstring semantics is applied to prove soundness and completeness of the system.

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, bitstring semantics, 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.

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.

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.

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.

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 a semantics of partition, thus computing logical relations between matching formulas with a semantic method that combines model theory and Boolean algebra.

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)

"Semantic Truth Theories" uses the techniques of mathematical logic to develop a new semantic treatment of the concept of truth based on ideas of Saul Kripke. Yael Cohen goes on to solve the Liar paradox, Hempel's raven paradox in the philosophy of science, and other classical problems of philosophy. She does this by enlarging the scope of formal logic to include concepts of presupposition besides the usual implication. The book thus provides a unified treatment of many topics having to do (...) with truth, topics whose deep and unsuspected interconnections are not visible without the insights of mathematical knowledge which the author elaborates. The early chapters of the book contain an accessible introduction to semantic paradoxes which should be useful to students. (shrink)

Introduction -- Basic concepts -- Previous approaches -- Semantic expressions: introduction -- Formal issues -- Semantic expressions: basic features -- Advanced features -- Applications: capture -- Three little pigs -- Applications: creation.

It is widely assumed in philosophy that there is a tight connection between semantics and metaphysics. Semantic theories about the meanings of natural language terms and phrases are taken to provide evidence for and against various metaphysical theses about the nature of non-linguistic parts of the world. Call this view the widespread thesis. I argue that the widespread thesis is mistaken: semantic theories do not generally have robust metaphysical consequences. I contend that the best arguments for the widespread thesis (...) turn on an interpretation of semantic theories according to which they are metaphysically illuminating: the metaphysical structure of the world is directly mirrored in the structure of the theory. Terms and existential quantification in the theory signal the existence of entities in the world, predicates signal the existence of properties, and the structures of set theoretic objects invoked by the theory directly correspond to structural features of the world. Such an interpretation is not the only one available, however, and it is a substantive philosophical (and linguistic) issue which interpretation is correct. Accordingly, I develop and defend an alternative interpretation of semantic theories according to which the structural features of these theories and of their theorems reflect biological and computational constraints on the architecture of the language faculty and historical contingencies in its evolutionary development. Supposedly metaphysically committing features of the theory—including the appearance of quantification and (ostensibly) referring terms—may thus be arepresentational artifacts of these constraints, rather than representational features that reflect metaphysical reality. To determine the metaphysical consequences of a semantic theory then, we must determine which of its features represent language-independent characteristics of the world and which are arepresentational consequences of the structure of the language faculty. To do this, however, we must have some prior idea what the language-independent world is like—that is, we must engage in prior metaphysical theorizing that is not beholden to the semantic theory itself. (shrink)

_Insensitive Semantics_ is an overview of and contribution to the debates about how to accommodate context sensitivity within a theory of human communication, investigating the effects of context on communicative interaction and, as a corollary, what a context of utterance is and what it is to be in one. Provides detailed and wide-ranging overviews of the central positions and arguments surrounding contextualism Addresses broad and varied aspects of the distinction between the semantic and non-semantic content of language Defends a distinctive (...) and explanatorily powerful combination of semantic minimalism and speech act pluralism Confronts core problems which not only run to the heart of philosophy of language and linguistics, but which arise in epistemology, metaphysics, and moral philosophy as well. (shrink)

Minimal Semantics asks what a theory of literal linguistic meaning is for - if you were to be given a working theory of meaning for a language right now, what would you be able to do with it? Emma Borg sets out to defend a formal approach to semantic theorising from a relatively new type of opponent - advocates of what she call 'dual pragmatics'. According to dual pragmatists, rich pragmatic processes play two distinct roles in linguistic comprehension: as (...) well as operating in a post-semantic capacity to determine the implicatures of an utterance, they also operate prior to the determination of truth-conditional content for a sentence. That is to say, they have an integral role to play within what is usually thought of as the semantic realm. Borg believes dual pragmatic accounts constitute the strongest contemporary challenge to standard formal approaches to semantics since they challenge the formal theorist to show not merely that there is some role for formal processes on route to determination of semantic content, but that such processes are sufficient for determining content. Minimal Semantics provides a detailed examination of this school of thought, introducing readers who are unfamiliar with the topic to key ideas like relevance theory and contextualism, and looking in detail at where these accounts diverge from the formal approach. Borg's defence of formal semantics has two main parts: first, she argues that the formal approach is most naturally compatible with an important and well-grounded psychological theory, namely the Fodorian modular picture of the mind. Then she argues that the main arguments adduced by dual pragmatists against formal semantics - concerning apparent contextual intrusions into semantic content - can in fact be countered by a formal theory. The defence holds, however, only if we are sensitive to the proper conditions of success for a semantic theory. Specifically, we should reject a range of onerous constraints on semantic theorizing (e.g., that it answer epistemic or metaphysical questions, or that it explain our communicative skills) and instead adopt a quite minimal picture of semantics. (shrink)

In his latest book, Frank Ankersmit proposes an original theory of historical representation. In this review I focus on what I take to be his most important semantic points with respect to representation, meaning, truth, and reference. First, I provide a short summary of the book. Second, I explore his semantics in terms of aspects and compare it with a different account inspired by the Fregean notion of mode of presentation. As my examination shows, Ankersmit’s analysis faces the problem (...) of “loosing indirectness” and, moreover, there seems to exist an alternative and plausible view that does not suffer from this problem. Finally, I conclude with a couple of comments on the Copernican Revolution advocated in the book. (shrink)

This paper gives a semantics for schematic logic, proving soundness and completeness. The argument for soundness is carried out in ontologically innocent fashion, relying only on the existence of formulae which are actually written down in the course of a derivation in the logic. This makes the logic available to a nominalist, even a nominalist who does not wish to rely on modal notions, and who accepts the possibility that the universe may in fact be finite.

Those who naturalize semantics concentrate on avoiding difficulties in getting the right sort of cause for the biological item which is to possess semantic properties (to be ‘true of or to be ‘about’ some physical item). Using an analogy with sense‐data, I argue that the real difficulties will be trying to get any proposed neural representation to be the right sort of effect of natural processes. The idea of a biological item which can be a semantic ‘primitive’ is as (...) bankrupt as the idea of a mental item which could be a perceptual ‘primitive’. Eventually I propose a technique for finding naturalized representations without these problems. (shrink)