This paper addresses questions of universality related to ontological engineering, namely aims at substantiating (negative) answers to the following three basic questions: (i) Is there a ‘universal ontology’?, (ii) Is there a ‘universal formal ontology language’?, and (iii) Is there a universally applicable ‘mode of reasoning’ for formal ontologies? To support our answers in a principled way, we present a general framework for the design of formal ontologies resting on two main principles: firstly, we endorse Rudolf Carnap’s principle of logical (...) tolerance by giving central stage to the concept of logical heterogeneity, i.e. the use of a plurality of logical languages within one ontology design. Secondly, to structure and combine heterogeneous ontologies in a semantically well-founded way, we base our work on abstract model theory in the form of institutional semantics, as forcefully put forward by Joseph Goguen and Rod Burstall. In particular, we employ the structuring mechanisms of the heterogeneous algebraic specification language HetCasl for defining a general concept of heterogeneous, distributed, highly modular and structured ontologies, called hyperontologies. Moreover, we distinguish, on a structural and semantic level, several different kinds of combining and aligning heterogeneous ontologies, namely integration, connection, and refinement. We show how the notion of heterogeneous refinement can be used to provide both a general notion of sub-ontology as well as a notion of heterogeneous equivalence of ontologies, and finally sketch how different modes of reasoning over ontologies are related to these different structuring aspects. (shrink)

The algebra of sets has, basically, two different types of symbols. One type of symbol (∩, ?, +, ?) defines another set from two other sets. A second type of symbol (?, ?, =, ?) makes a proposition about two sets. When the construction of these two types of symbols is based on the same four-dot matrix as the logic symbols described in a previous paper, the three symbol types then dovetail together into a harmonious whole that greatly simplifies derivation (...) in the algebra of sets. (shrink)

Logic diagrams have seen a resurgence in their application in a range of fields, including logic, biology, media science, computer science and philosophy. Consequently, understanding the history and philosophy of these diagrams has become crucial. As many current diagrammatic systems in logic are based on ideas that originated in the 18th and 19th centuries, it is important to consider what motivated the use of logic diagrams in the past and whether these reasons are still valid today. This paper proposes that (...) transcendental philosophy was a key inspiration for the development of logic diagrams and that such diagrams can be employed in transcendental arguments, even after the linguistic turn. (shrink)

What is the relationship between the world and logic, between intuition and language, between objects and their quantitative determinations? Rationalists, on the one hand, hold that the world is structured in a rational way. Representationalists, on the other hand, assume that language, logic, and mathematics are only the means to order and describe the intuitively given world. In World and Logic, Jens Lemanski takes up three surprising arguments from Arthur Schopenhauer’s hitherto undiscovered Berlin Lectures, which concern the philosophy of language, (...) logic, and mathematics. Based on these arguments, Lemanski develops a new position entitled ‘rational representationalism’: the world is always structured by human beings according to linguistic, logical, and mathematical principles, but the basic vocabulary of these structural descriptions already contains metaphors taken from the world around us. (shrink)

The recent wave of data on exoplanets lends support to METI ventures (Messaging to Extra-Terrestrial Intelligence), insofar as the more exoplanets we find, the more likely it is that “exominds” await our messages. Yet, despite these astronomical advances, there are presently no well-confirmed tests against which to check the design of interstellar messages. In the meantime, the best we can do is distance ourselves from terracentric assumptions. There is no reason, for example, to assume that all inferential abilities are language-like. (...) With that in mind, I argue that logical reasoning does not have to be couched in symbolic notation. In diagrammatic reasoning, inferences are underwritten, not by rules, but by transformations of self-same qualitative signs. I use the Existential Graphs of C. S. Peirce to show this. Since diagrams are less dependent on convention and might even be generalized to cover non-visual senses, I argue that METI researchers should add some form of diagrammatic representations to their repertoire. Doing so can shed light, not just on alien minds, but on the deepest structures of reasoning itself. (shrink)

6. Seeing With the Mind’s Eye 1: The Puzzle of Mental Imagery .................................................6-1 6.1 What is the puzzle about mental imagery?..............................................................................6-1 6.2 Content, form and substance of representations ......................................................................6-6 6.3 What is responsible for the pattern of results obtained in imagery studies?.................................6-8..

This essay discusses Peirce’s appeal to logical machines as an argument against psychologism. It also contends that some of Peirce’s anti-psychologistic remarks on logic contain interesting premonitions arising from his perception of the asymmetry of proof complexity in monadic and relational logical calculi that were only given full formulation and explication in the early twentieth century through Church’s Theorem and Hilbert’s broad-ranging Entscheidungsproblem.

In this paper, I aim to identify Peirce?s great contribution to logical diagrams and its limit.Peirce is the first person who believed that the same logical status can be given to diagrams as to symbolic systems.Even though this belief led him to invent his own graphical system, Existential Graphs, the success or failure of this system does not determine the value of Peirce?s general insights about logical diagrams.In order to make this point clear, I will show that Peirce?s revolutionary ideas (...) about diagrams not only overcame some important defects of Venn diagrams but opened a new horizon for logical diagrams.Finally, I will point out where Peirce?s new horizon for logical diagrams stopped and will claim that this limit is mainly responsible for the discrepancy between Peirce?s and others? estimates of his contribution to logical diagrams. (shrink)

In this paper we argue that there were several currents, ideas and problems in 19th-century logic that motivated John Venn to develop his famous logic diagrams. To this end, we first examine the problem of uncertainty or over-specification in syllogistic that became obvious in Euler diagrams. In the 19th century, numerous logicians tried to solve this problem. The most famous was the attempt to introduce dashed circles into Euler diagrams. The solution that John Venn developed for this problem, however, came (...) from a completely different area of logic: instead of orienting to syllogistic like Euler diagrams, Venn applied Boolean algebra to improve visual reasoning. Venn’s contribution to solving the problem of elimination also played an important role. The result of this development is still known today as the ‘Venn Diagram’. (shrink)

It is of common use in modern Venn diagrams to mark a compartment with a cross to express its non-emptiness. Modern scholars seem to derive this convention from Charles S. Peirce, with the assumption that it was unknown to John Venn. This paper demonstrates that Venn actually introduced several methods to represent existentials but felt uneasy with them. The resistance to formalize existentials was not limited to diagrammatic systems, as George Boole and his followers also failed to provide a satisfactory (...) symbolic representation for them. This difficulty points out issues that are inherent to the very nature of existentials. This paper assesses the various methods designed for the representation of existential statements with Venn diagrams. First, Venn’s own attempts are discussed and compared with other solutions proposed by his contemporaries and successors, notably Lewis Carroll and Peirce. Since disjunctives hold an important role in an effective representation of existentials, their representation is also discussed. Finally, recent methods for the diagrammatic representation of existing individuals, rather than mere existence, are surveyed. (shrink)

Logicians have strongly preferred first-order natural deductive systems over Peirce's Beta Graphs even though both are equivalent to each other. One of the main reasons for this preference, I claim, is that inference rules for Beta Graphs are hard to understand, and, therefore, hard to apply for deductions. This paper reformulates the Beta rules to show more fine-grained symmetries built around visual features of the Beta system, which makes the rules more natural and easier to use and understand. Noting that (...) the rules of a natural deductive system are natural in a different sense, this case study shows that the naturalness and the intuitiveness of rules depends on the type of representation system to which they belong. In a diagrammatic system, when visual features are discovered and fully used, we have a more efficacious deductive system. I will also show that this project not only helps us to apply these rules more easily but to understand the validity of the system at a more intuitive level. (shrink)

This article addresses some crucial assumptions that are rarely acknowledged when organisms and machines are compared. We begin by presenting a short historical reconstruction of the concept of “machine.” We show that there has never been a unique and widely accepted definition of “machine” and that the extant definitions are based on specific technologies. Then we argue that, despite the concept's ambiguity, we can still defend a more robust, specific, and useful notion of machine analogy that accounts for successful strategies (...) in connecting specific devices (or mechanisms) with particular living phenomena. For that purpose, we distinguish between what we call “generic identity” and proper “machine analogy.” We suggest that “generic identity”—which, roughly stated, presumes that some sort of vague similarity might exist between organisms and machines—is a source of the confusion haunting many persistent disagreements and that, accordingly, it should be dismissed. Instead, we endorse a particular form of “machine analogy” where the relation between organic phenomena and mechanical devices is not generic but specific and grounded on the identification of shared “invariants.” We propose that the machine analogy is a kind of analogy as proportion and we elucidate how this is used or might be used in scientific practices. We finally argue that while organisms are not machines in a generic sense, they might share many robust “invariants,” which justify the scientists' use of machine analogies for grasping living phenomena. (shrink)

Logicians have often suggested that the use of Euler-type diagrams has influenced the idea of the quantification of the predicate. This is mainly due to the fact that Euler-type diagrams display more information than is required in traditional syllogistics. The paper supports this argument and extends it by a further step: Euler-type diagrams not only illustrate the quantification of the predicate, but also solve problems of traditional proof theory, which prevented an overall quantification of the predicate. Thus, Euler-type diagrams can (...) be called the natural basis of syllogistic reasoning and can even go beyond. In the paper, these arguments are presented in connection with the book Nucleus Logicae Weisaniae by Johann Christian Lange from 1712. (shrink)

By way of a close reading of Boole and Frege’s solutions to the same logical problem, we highlight an underappreciated aspect of Boole’s work—and of its difference with Frege’s better-known approach—which we believe sheds light on the concepts of ‘calculus’ and ‘mechanization’ and on their history. Boole has a clear notion of a logical problem; for him, the whole point of a logical calculus is to enable systematic and goal-directed solution methods for such problems. Frege’s Begriffsschrift, on the other hand, (...) is a visual tool to scrutinize concepts and inferences, and is a calculus only in the thin sense that every possible transition between sentences is fully and unambiguously specified in advance. While Frege’s outlook has dominated much of philosophical thinking about logical symbolism, we believe there is value—particularly in light of recent interest in the role of notations in mathematics and logic—in reviving Boole’s idea of an intrinsic link between, as he put it, a ‘calculus’ and a ‘directive method’ to solve problems. (shrink)

Truth diagrams are introduced as a novel graphical representation for propositional logic. To demonstrate their epistemic efficacy a set of 28 concepts are proposed that any comprehensive representation for PL should encompass. TDs address all the criteria whereas seven other existing representations for PL only provide partial coverage. These existing representations are: the linear formula notation, truth tables, a PL specific interpretation of Venn Diagrams, Frege’s conceptual notation, diagrams from Wittgenstein’s Tractatus, Pierce’s alpha graphs and Gardner’s shuttle diagrams. The comparison (...) of the representations succeeds in distinguishing ideas that are fundamental to PL from features of common PL representations that are somewhat arbitrary. (shrink)

I critically review Raymond Turner’s Computational Artifacts – Towards a Philosophy of Computer Science by placing beside his position a rather different one, according to which computer science is a branch of, and is therefore subsumed by, immaterial formal logic.

This paper describes Peirce's systems of logic diagrams, focusing on the so-called ''existential'' graphs, which are equivalent to the first-order predicate calculus. It analyses their implications for the nature of mental representations, particularly mental models with which they have many characteristics in common. The graphs are intended to be iconic, i.e., to have a structure analogous to the structure of what they represent. They have emergent logical consequences and a single graph can capture all the different ways in which a (...) possibility can occur. Mental models share these properties. But, as the graphs show, certain aspects of propositions cannot be represented in an iconic or visualisable way. They include negation, and the representation of possibilities qua possibilities, which both require representations that do not depend on a perceptual modality. Peirce took his graphs to reveal the fundamental operations of reasoning, and the paper concludes with an analysis of different hypotheses about these operations. (shrink)

SummaryThe Science Museum, London, recently acquired some circular logic demonstrators by Charles Mahon, 3rd Earl Stanhope. A study of what exists of Stanhope's unpublished book, notes and letters on the subject allows the development of his demonstrators to be traced. A consideration of Stanhope's characters and interests reveals an Enlightenment figure with aspirations consonant with that era, the logic demonstrators representing the material culmination of his ideals. Yet the demonstrators were not made public during his lifetime, and the final form (...) was greeted with derision 60 years after his death. (shrink)

As with mathematics, logic is easier to do if its symbols and their rules are better. In a graphic way, the logic symbols introduced in thís paper show their truth-table values, their composite truth-functions, and how to say them as either ?or? or ?if ? then? propositions. Simple rules make the converse, add or remove negations, and resolve propositions.

We present a reading of the traditional syllogistics in a fragment of the propositional intuitionistic multiplicative linear logic and prove that with respect to a diagrammatic logical calculus that we introduced in a previous paper, a syllogism is provable in such a fragment if and only if it is diagrammatically provable. We extend this result to syllogistics with complemented terms à la De Morgan, with respect to a suitable extension of the diagrammatic reasoning system for the traditional case and a (...) corresponding reading of such De Morgan style syllogistics in the previously referred to fragment of linear logic. (shrink)