Although formal thought disorder (FTD) has been for long a clinical label in the assessment of some psychiatric disorders, in particular of schizophrenia, it remains a source of controversy, mostly because it is hard to say what exactly the “formal” in FTD refers to. We see anomalous processing of terminological knowledge, a core construct of human knowledge in general, behind FTD symptoms and we approach this anomaly from a strictly formal perspective. More specifically, we present here a (...) symbolic computational model of storage in, and activation of, a human semantic network, or semantic memory, whose core element is logical form; this is normalized by description logic (DL), namely by CL, a DL-based language – Conception Language – designed to formalize conceptualization from the viewpoint of individual cognitive agency. In this model, disruptions in the rule-based implementation of the logical form account for the apparently semantic anomalies symptomatic of FTD, which are detected by means of a CL-based algorithmic assessment. (shrink)

This paper attempts to illustrate a process of analysis that will hopefully open a path to more complete and useful definitions of sign and symbol. It applies a form-content analysis to the metaphysical properties of these two concepts. The objective is to locate criteria necessary and sufficient to derive formal definitions for these terms. Wittgenstein’s concept of “forms of representation” is analyzed and applied to the topic. Criteria are outlined that determine the appropriateness of the sign and (...) class='Hi'>symbol to be applied as labels. Criteria of definition are then developed using gesture, metaphor, and several other example types to illustrate the use of the criteria in distinguishing between sign and symbol. The structural organization of these two concepts proved to be especially complex and led to what some readers may find somewhat obscure. It is not our intention to be purposefully obscure. (shrink)

Recent experimental evidence from developmental psychology and cognitive neuroscience indicates that humans are equipped with unlearned elementary mathematical skills. However, formal mathematics has properties that cannot be reduced to these elementary cognitive capacities. The question then arises how human beings cognitively deal with more advanced mathematical ideas. This paper draws on the extended mind thesis to suggest that mathematical symbols enable us to delegate some mathematical operations to the external environment. In this view, mathematical symbols are not only used (...) to express mathematical concepts—they are constitutive of the mathematical concepts themselves. Mathematical symbols are epistemic actions, because they enable us to represent concepts that are literally unthinkable with our bare brains. Using case-studies from the history of mathematics and from educational psychology, we argue for an intimate relationship between mathematical symbols and mathematical cognition. (shrink)

It is common to distinguish two great families of representation. Symbolic representations include logical and mathematical symbols, words, and complex linguistic expressions. Iconic representations include dials, diagrams, maps, pictures, 3-dimensional models, and depictive gestures. This essay describes and motivates a new way of distinguishing iconic from symbolic representation. It locates the difference not in the signs themselves, nor in the contents they express, but in the semantic rules by which signs are associated with contents. The two kinds of rule have (...) divergent forms, occupying opposite poles on a spectrum of naturalness. Symbolic rules are composed entirely of primitive juxtapositions of sign types with contents, while iconic rules determine contents entirely by uniform natural relations with sign types. This distinction is marked explicitly in the formal semantics of familiar sign systems, both for atomic first-order representations, like words, pixel colors, and dials, and for complex second-order representations, like sentences, diagrams, and pictures. (shrink)

(This is for the Cambridge Handbook of Analytic Philosophy, edited by Marcus Rossberg) In this handbook entry, I survey the different ways in which formal mathematical methods have been applied to philosophical questions throughout the history of analytic philosophy. I consider: formalization in symbolic logic, with examples such as Aquinas’ third way and Anselm’s ontological argument; Bayesian confirmation theory, with examples such as the fine-tuning argument for God and the paradox of the ravens; foundations of mathematics, with examples such (...) as Hilbert’s programme and Gödel’s incompleteness theorems; social choice theory, with examples such as Condorcet’s paradox and Arrow’s theorem; ‘how possibly’ results, with examples such as Condorcet’s jury theorem and recent work on intersectionality theory; and the application of advanced mathematics in philosophy, with examples such as accuracy-first epistemology. (shrink)

Formalizing Euclid’s first axiom. Bulletin of Symbolic Logic. 20 (2014) 404–5. (Coauthor: Daniel Novotný) -/- Euclid [fl. 300 BCE] divides his basic principles into what came to be called ‘postulates’ and ‘axioms’—two words that are synonyms today but which are commonly used to translate Greek words meant by Euclid as contrasting terms. -/- Euclid’s postulates are specifically geometric: they concern geometric magnitudes, shapes, figures, etc.—nothing else. The first: “to draw a line from any point to any point”; the last: the (...) parallel postulate. -/- Euclid’s axioms are general principles of magnitude: they concern geometric magnitudes and magnitudes of other kinds as well even numbers. The first is often translated “Things that equal the same thing equal one another”. -/- There are other differences that are or might become important. -/- Aristotle [fl. 350 BCE] meticulously separated his basic principles [archai, singular archê] according to subject matter: geometrical, arithmetic, astronomical, etc. However, he made no distinction that can be assimilated to Euclid’s postulate/axiom distinction. -/- Today we divide basic principles into non-logical [topic-specific] and logical [topic-neutral] but this too is not the same as Euclid’s. In this regard it is important to be cognizant of the difference between equality and identity—a distinction often crudely ignored by modern logicians. Tarski is a rare exception. The four angles of a rectangle are equal to—not identical to—one another; the size of one angle of a rectangle is identical to the size of any other of its angles. No two angles are identical to each other. -/- The sentence ‘Things that equal the same thing equal one another’ contains no occurrence of the word ‘magnitude’. This paper considers the problem of formalizing the proposition Euclid intended as a principle of magnitudes while being faithful to the logical form and to its information content. (shrink)

*NEWEST VERSION OF THIS RESOURCE ONLINE @ Philosop-her dotcom This textbook has developed over the last few years of teaching introductory symbolic logic and critical thinking courses. It has been truly a pleasure to have benefited from such great students and colleagues over the years. As we have become increasingly frustrated with the costs of traditional logic textbooks (though many of them deserve high praise for their accuracy and depth), the move to open source has become more and more attractive. (...) We're happy to provide it free of charge for educational use. With that being said, there are always improvements to be made here and we would be most grateful for constructive feedback and criticism. We have chosen to write this text in LaTex and have adopted certain conventions with symbols. Certainly many important aspects of critical thinking and logic have been omitted here, including historical developments and key logicians, and for that we apologize. Our goal was to create a textbook that could be provided to students free of charge and still contain some of the more important elements of critical thinking and introductory logic. To that end, an additional benefit of providing this textbook as a Open Education Resource (OER) is that we will be able to provide newer updated versions of this text more frequently, and without any concern about increased charges each time. We are particularly looking forward to expanding our examples, and adding student exercises. We will additionally aim to continually improve the quality and accessibility of our text for students and faculty alike. We have included a bibliography that includes many admirable textbooks, all of which we have benefited from. The interested reader is encouraged to consult these texts for further study and clarification. These texts have been a great inspiration for us and provide features to students that this concise textbook does not. We would both like to thank the philosophy students at numerous schools in the Puget Sound region for their patience and helpful suggestions. In particular, we would like to thank our colleagues at Green River College, who have helped us immensely in numerous different ways. Please feel free to contact us with comments and suggestions. We will strive to correct errors when pointed out, add necessary material, and make other additional and needed changes as they arise. Please check back for the most up to date version. (shrink)

A series of representations must be semantics-driven if the members of that series are to combine into a single thought: where semantics is not operative, there is at most a series of disjoint representations that add up to nothing true or false, and therefore do not constitute a thought at all. A consequence is that there is necessarily a gulf between simulating thought, on the one hand, and actually thinking, on the other. A related point is that a popular doctrine (...) - the so-called 'computational theory of mind' (CTM) - is based on a confusion. CTM is the view that thought-processes consist in 'computations', where a computation is defined as a 'form-driven' operation on symbols. The expression 'form-driven operation' is ambiguous, as it may refer either to syntax-driven operations or to morphology-driven operations. Syntax-driven operations presuppose the existence of operations that are driven by semantic and extra-semantic knowledge. So CTM is false if the terms 'computation' and 'form-driven operation' are taken to refer to syntax-driven operations. Thus, if CTM is to work, those expressions must be taken to refer to morphology-driven operations. CTM therefore fails, given that an operation must be semantics-driven if it is to qualify as a thought. CTM therefore fails on each possible disambiguation of the expressions 'formal operation' and 'computation,' and it is therefore false. (shrink)

Modern semiotics is a branch of logics that formally defines symbol-based communication. In recent years, the semiotic classification of signs has been invoked to support the notion that symbols are uniquely human. Here we show that alarm-calls such as those used by African vervet monkeys (Cercopithecus aethiops), logically satisfy the semiotic definition of symbol. We also show that the acquisition of vocal symbols in vervet monkeys can be successfully simulated by a computer program based on minimal semiotic and (...) neurobiological constraints. The simulations indicate that learning depends on the tutor-predator ratio, and that apprentice-generated auditory mistakes in vocal symbol interpretation have little effect on the learning rates of apprentices (up to 80% of mistakes are tolerated). In contrast, just 10% of apprentice-generated visual mistakes in predator identification will prevent any vocal symbol to be correctly associated with a predator call in a stable manner. Tutor unreliability was also deleterious to vocal symbol learning: a mere 5% of “lying” tutors were able to completely disrupt symbol learning, invariably leading to the acquisition of incorrect associations by apprentices. Our investigation corroborates the existence of vocal symbols in a non-human species, and indicates that symbolic competence emerges spontaneously from classical associative learning mechanisms when the conditioned stimuli are self-generated, arbitrary and socially efficacious. We propose that more exclusive properties of human language, such as syntax, may derive from the evolution of higher-order domains for neural association, more removed from both the sensory input and the motor output, able to support the gradual complexification of grammatical categories into syntax. (shrink)

Although Kant (1998) envisaged a prominent role for logic in the argumentative structure of his Critique of Pure Reason, logicians and philosophers have generally judged Kantgeneralformaltranscendental logics is a logic in the strict formal sense, albeit with a semantics and a definition of validity that are vastly more complex than that of first-order logic. The main technical application of the formalism developed here is a formal proof that Kants logic is after all a distinguished subsystem of first-order logic, (...) namely what is known as geometric logic. (shrink)

Philosophical questions about minds and computation need to focus squarely on the mathematical theory of Turing machines (TM's). Surrogate TM's such as computers or formal systems lack abilities that make Turing machines promising candidates for possessors of minds. Computers are only universal Turing machines (UTM's)—a conspicuous but unrepresentative subclass of TM. Formal systems are only static TM's, which do not receive inputs from external sources. The theory of TM computation clearly exposes the failings of two prominent critiques, Searle's (...) Chinese room (1980) and arguments from Gödel's Incompleteness theorems (e.g., Lucas, 1961; Penrose, 1989), both of which fall short of addressing the complete TM model. Both UTM-computers and formal systems provide an unsound basis for debate. In particular, their special natures easily foster the misconception that computation entails intrinsically meaningless symbol manipulation. This common view is incorrect with respect to full-fledged TM's, which can process inputs non-formally, i.e., in a subjective and dynamically evolving fashion. To avoid a distorted understanding of the theory of computation, philosophical judgments and discussions should be grounded firmly upon the complete Turing machine model, the proper model for real computers. (shrink)

Kant uses the concept of the symbol to show the complicated relationship between the autonomy of beauty and its systematic function as a transition from nature to freedom, which are the two most important topics in the third Critique. Beauty’s symbolism of morality lies in the analog between aesthetic reflection and moral disposition; concretely, it lies in the purity or disinterestedness and self-legislation as negative and positive freedom in both subjective states of mind. In this scenario, beauty’s symbolism does (...) not refer to aesthetic ideas that either involve intelligent interests (in the beauty of nature) or presuppose an end (in the beauty of art); it also cannot be grounded in the supersensible substrate, which is an elevated and metaphysical principle of the judgment of taste given in the Dialectic but not the original principle of subjective purposiveness in the Analytic. With this formal relationship, beauty and morality accelerate each other in the empirical-anthological sense—but they are also not a sufficient or necessary condition for each other. Furthermore, through symbolism, taste looks toward the intelligible and serves as a transition from nature to freedom from the transcendental perspective. (shrink)

Dans un texte désormais célèbre, Ferdinand de Saussure insiste sur l’arbitraire du signe dont il vante les qualités. Toutefois il s’avère que le symbole, signe non arbitraire, dans la mesure où il existe un rapport entre ce qui représente et ce qui est représenté, joue un rôle fondamental dans la plupart des activités humaines, qu’elles soient scientifiques, artistiques ou religieuses. C’est cette dimension symbolique, sa portée, son fonctionnement et sa signification dans des domaines aussi variés que la chimie, la théologie, (...) les mathématiques, le code de la route et bien d’autres qui est l’objet du livre La Pointure du symbole. -/- Jean-Yves Béziau, franco-suisse, est docteur en logique mathématique et docteur en philosophie. Il a poursuivi des recherches en France, au Brésil, en Suisse, aux États-Unis (UCLA et Stanford), en Pologne et développé la logique universelle. Éditeur-en-chef de la revue Logica Universalis et de la collection Studies in Universal Logic (Springer), il est actuellement professeur à l’Université Fédérale de Rio de Janeiro et membre de l’Académie brésilienne de Philosophie. SOMMAIRE -/- PRÉFACE L’arbitraire du signe face à la puissance du symbole Jean-Yves BÉZIAU La logique et la théorie de la notation (sémiotique) de Peirce (Traduit de l’anglais par Jean-Marie Chevalier) Irving H. ANELLIS Langage symbolique de Genèse 2-3 Lytta BASSET -/- Mécanique quantique : quelle réalité derrière les symboles ? Hans BECK -/- Quels langages et images pour représenter le corps humain ? Sarah CARVALLO Des jeux symboliques aux rituels collectifs. Quelques apports de la psychologie du développement à l’étude du symbolisme Fabrice CLÉMENT Les panneaux de signalisation (Traduit de l’anglais par Fabien Shang) Robert DEWAR Remarques sur l’émergence des activités symboliques Jean LASSÈGUE Les illustrations du "Songe de Poliphile" (1499). Notule sur les hiéroglyphes de Francesca Colonna Pierre-Alain MARIAUX Signes de vie Jeremy NARBY Visualising relations in society and economics. Otto Neuraths Isotype-method against the background of his economic thought Elisabeth NEMETH Algèbre et logique symboliques : arbitraire du signe et langage formel Marie-José DURAND – Amirouche MOKTEFI Les symboles mathématiques, signes du Ciel Jean-Claude PONT La mathématique : un langage mathématique ? Alain M. ROBERT. (shrink)

One of the most expected properties of a logical system is that it can be algebraizable, in the sense that an algebraic counterpart of the deductive machinery could be found. Since the inception of da Costa's paraconsistent calculi, an algebraic equivalent for such systems have been searched. It is known that these systems are non self-extensional (i.e., they do not satisfy the replacement property). More than this, they are not algebraizable in the sense of Blok-Pigozzi. The same negative results hold (...) for several systems of the hierarchy of paraconsistent logics known as Logics of Formal Inconsistency (LFIs). Because of this, these logics are uniquely characterized by semantics of non-deterministic kind. This paper offers a solution for two open problems in the domain of paraconsistency, in particular connected to algebraization of LFIs, by obtaining several LFIs weaker than C1, each of one is algebraizable in the standard Lindenbaum-Tarski's sense by a suitable variety of Boolean algebras extended with operators. This means that such LFIs satisfy the replacement property. The weakest LFI satisfying replacement presented here is called RmbC, which is obtained from the basic LFI called mbC. Some axiomatic extensions of RmbC are also studied, and in addition a neighborhood semantics is defined for such systems. It is shown that RmbC can be defined within the minimal bimodal non-normal logic E+E defined by the fusion of the non-normal modal logic E with itself. Finally, the framework is extended to first-order languages. RQmbC, the quantified extension of RmbC, is shown to be sound and complete w.r.t. BALFI semantics. (shrink)

This short paper grew out of an observation—made in the course of a larger research project—of a surprising convergence between, on the one hand, certain themes in the work of Mary Hesse and Nelson Goodman in the 1950/60s and, on the other hand, recent work on the representational resources of science, in particular regarding model-based representation. The convergence between these more recent accounts of representation in science and the earlier proposals by Hesse and Goodman consists in the recognition that, in (...) order to secure successful representation in science, collective representational resources must be available. Such resources may take the form of (amongst others) mathematical formalisms, diagrammatic methods, notational rules, or—in the case of material models—conventions regarding the use and manipulation of the constituent parts. More often than not, an abstract characterization of such resources tells only half the story, as they are constituted equally by the pattern of (practical and theoretical) activities—such as instances of manipulation or inference—of the researchers who deploy them. In other words, representational resources need to be sustained by a social practice; this is what renders them collective representational resources in the first place. (shrink)

Cognitive, or knowledge, agents, who are in some way aware of describing their own view of the world (based on their mental concepts), need to become concerned with the expressions of their own conceptions. My main supposition is that agents’ conceptions are mainly expressed in the form of linguistic expressions that are spoken, written, and represented based on e.g. letters, numbers, or symbols. This research especially focuses on symbolic conceptions (that are agents’ conceptions that are manifested in the form of (...) their symbols). I attempt to logically (and using ????LSYM) analyse symbolic conceptions in terminological systems. ????LSYM is an assertional fragment of my developed Conception Language (????L), which is utilised as a formal-logical system for representing and explicating agents’ conceptions of the world. Keywords: assertional logic; cognitive agent; knowledge agent; conception language; conception; symbol; symbolic conception. (shrink)

Because formal systems of symbolic logic inherently express and represent the deductive inference model formal proofs to theorem consequences can be understood to represent sound deductive inference to deductive conclusions without any need for other representations.

This note presents an analysis of Symbolic Knowledge from Leibniz to Husserl, a collection of works from some members of The Southern Cone Group for the Philosophy of Formal Sciences. The volume delineates an outlook of the philosophical treatments presented by Leibniz, Kant, Frege, and the Booleans, as well as by Husserl, of some questions related to the conceptual singularities of symbolic knowledge –whose standard we find in the arts of algebra and arithmetic. The book’s unity of themes and (...) (at least in part) style is examined with the aim of showing the articulation of its parts. (shrink)

Because formal systems of symbolic logic inherently express and represent the deductive inference model formal proofs to theorem consequences can be understood to represent sound deductive inference to true conclusions without any need for other representations such as model theory.

To eliminate incompleteness, undecidability and inconsistency from formal systems we only need to convert the formal proofs to theorem consequences of symbolic logic to conform to the sound deductive inference model. -/- Within the sound deductive inference model there is a (connected sequence of valid deductions from true premises to a true conclusion) thus unlike the formal proofs of symbolic logic provability cannot diverge from truth.

Tarski "proved" that there cannot possibly be any correct formalization of the notion of truth entirely on the basis of an insufficiently expressive formal system that was incapable of recognizing and rejecting semantically incorrect expressions of language. -/- The only thing required to eliminate incompleteness, undecidability and inconsistency from formal systems is transforming the formal proofs of symbolic logic to use the sound deductive inference model.

A suprise may occur if we use a similar strategy to the Liar's paradox to mathematically formalize "This sentence does not contain the symbol X".

Theory of Computation -- Computation by Abstracts Devices.

Exactly how do the sign/symbol/token systems of endo- and exo-biosemiosis differ from those of cognitive semiosis? Do the biological messages that integrate metabolism have conceptual meaning? Semantic information has two subsets: Descriptive and Prescriptive. Prescriptive information instructs or directly produces nontrivial function. In cognitive semiosis, prescriptive information requires anticipation and “choice with intent” at bona fide decision nodes. Prescriptive information either tells us what choices to make, or it is a recordation of wise choices already made. Symbol systems (...) allow recordation of deliberate choices and the transmission of linear digital prescriptive information. Formal symbol selection can be instantiated into physicality using physical symbol vehicles (tokens). Material symbol systems (MSS) formally assign representational meaning to physical objects. Even verbal semiosis instantiates meaning into physical sound waves using an MSS. Formal function can also be incorporated into physicality through the use of dynamically-inert (dynamically-incoherent or -decoupled) configurable switch-settings in conceptual circuits. This paper examines the degree to which biosemiosis conforms to the essential formal criteria of prescriptive semiosis and cybernetic management. (shrink)

There is a prevalent notion among cognitive scientists and philosophers of mind that computers are merely formal symbol manipulators, performing the actions they do solely on the basis of the syntactic properties of the symbols they manipulate. This view of computers has allowed some philosophers to divorce semantics from computational explanations. Semantic content, then, becomes something one adds to computational explanations to get psychological explanations. Other philosophers, such as Stephen Stich, have taken a stronger view, advocating doing away (...) with semantics entirely. This paper argues that a correct account of computation requires us to attribute content to computational processes in order to explain which functions are being computed. This entails that computational psychology must countenance mental representations. Since anti-semantic positions are incompatible with computational psychology thus construed, they ought to be rejected. Lastly, I argue that in an important sense, computers are not formal symbol manipulators. (shrink)

John Searle once said: "The Chinese room shows what we knew all along: syntax by itself is not sufficient for semantics. (Does anyone actually deny this point, I mean straight out? Is anyone actually willing to say, straight out, that they think that syntax, in the sense of formal symbols, is really the same as semantic content, in the sense of meanings, thought contents, understanding, etc.?)." I say: "Yes". Stuart C. Shapiro has said: "Does that make any sense? Yes: (...) Everything makes sense. The question is: What sense does it make?" This essay explores what sense it makes to say that syntax by itself is sufficient for semantics. (shrink)

Viewed in the light of the remarkable performance of ‘Watson’ - IBMs proprietary artificial intelligence computer system capable of answering questions posed in natural language - on the US general knowledge quiz show ‘Jeopardy’, we review two experiments on formal systems - one in the domain of quantum physics, the other involving a pictographic languaging game - whereby behaviour seemingly characteristic of domain understanding is generated by the mere mechanical application of simple rules. By re-examining both experiments in the (...) context of Searle’s Chinese Room Argument, we suggest their results merely endorse Searle’s core intuition: that ‘syntactical manipulation of symbols is not sufficient for semantics’. Although, pace Watson, some artificial intelligence practitioners have suggested that more complex, higher-level operations on formal symbols are required to instantiate understanding in computational systems, we show that even high-level calls to Google translate would not enable a computer qua ‘formal symbol processor’ to understand the language it processes. We thus conclude that even the most recent developments in ‘quantum linguistics’ will not enable computational systems to genuinely understand natural language. (shrink)

The use of the symbol ∨for disjunction in formal logic is ubiquitous. Where did it come from? The paper details the evolution of the symbol ∨ in its historical and logical context. Some sources say that disjunction in its use as connecting propositions or formulas was introduced by Peano; others suggest that it originated as an abbreviation of the Latin word for “or,” vel. We show that the origin of the symbol ∨ for disjunction can be (...) traced to Whitehead and Russell’s pre-Principia work in formal logic. Because of Principia’s influence, its notation was widely adopted by philosophers working in logic (the logical empiricists in the 1920s and 1930s, especially Carnap and early Quine). Hilbert’s adoption of ∨ in his Grundzüge der theoretischen Logik guaranteed its widespread use by mathematical logicians. The origins of other logical symbols are also discussed. (shrink)

his article will focus on the mechanistic origins of the computer metaphor, which forms the conceptual framework for the methodology of the cognitive sciences, some areas of artificial intelligence and the philosophy of mind. The connection between the history of computing technology, epistemology and the philosophy of mind is expressed through the metaphorical dictionaries of the philosophical discourse of a particular era. The conceptual clarification of this connection and the substantiation of the mechanistic components of the computer metaphor is the (...) main goal of this article. The statement is substantiated that the invention of mechanical computing devices, having a long history in the European engineering tradition, formed the prerequisites for the emergence of machine functionalism in the modern philosophy of mind. The idea of multiple implementation stems from the principle that a formal symbol system prescribes rules for the use of rational abstractions through the physical architecture of a computational engine. The article considers the reasons for the conceptual shift and reveals the semantic foundations for the metaphorical transfer of the properties of abstract objects from the theory of automata to the field of modern philosophy of mind. The criticism and ways of protecting the philosophical program of machine functionalism are analyzed by changing the content of the metaphor “Mind as machine”. The reasons for the stability of the information-computer approach in cognitive sciences are also disclosed and explained. (shrink)

The formal ontology here presented is what we might call a typed combinatorial Meinongian mereology. Its author seeks to formulate the laws, here called ‘canons’, regulating how entities can combine together in wholes of different sorts. The method, as in Bergmann’s earlier works, involves the construction of an ideal language of such a sort that the analysis of complex wholes can be achieved by transforming our natural-language representations of reality into what we might think of as artificial characteristic maps (...) or diagrams which allow the relevant ontological structures to be read off immediately from the symbolic representations which results. In former works Bergmann had held that the symbolic language of Principia Mathematica could serve as the appropriate diagrammatic device for the standard first-order functional calculus and develops instead a new sort diagrammatic language. (shrink)

Throughout what is now the more than 50-year history of the computer many theories have been advanced regarding the contribution this machine would make to changes both in the structure of society and in ways of thinking. Like other theories regarding the future, these should also be taken with a pinch of salt. The history of the development of computer technology contains many predictions which have failed to come true and many applications that have not been foreseen. While we must (...) reserve judgment as to the question of the impact on the structure of society and human thought, there is no reason to wait for history when it comes to the question: what are the properties that could give the computer such far-reaching importance? The present book is intended as an answer to this question. The fact that this is a theoretical analysis is due to the nature of the subject. No other possibilities are available because such a description of the properties of the computer must be valid for any kind of application. An additional demand is that the description should be capable of providing an account of the properties which permit and limit these possible applications, just as it must make it possible to characterize a computer as distinct from a) other machines whether clocks, steam engines, thermostats, or mechanical and automatic calculating machines, b) other symbolic media whether printed, mechanical, or electronic and c) other symbolic languages whether ordinary languages, spoken or written or formal languages. This triple limitation, however, (with regard to other machines, symbolic media and symbolic languages) raises a theoretical question as it implies a meeting between concepts of mechanical-deterministic systems, which stem from mathematical physics, and concepts of symbolic systems which stem from the description of symbolic activities common to the humanities. The relationship between science and the humanities has traditionally been seen from a dualistic perspective, as a relationship between two clearly separate subject areas, each studied on its own set of premises and using its own methods. In the present case, however, this perspective cannot be maintained since there is both a common subject area and a new - and specific - kind of interaction between physical and symbolic processes. (shrink)

CAT4 is proposed as a general method for representing information, enabling a powerful programming method for large-scale information systems. It enables generalised machine learning, software automation and novel AI capabilities. It is based on a special type of relation called CAT4, which is interpreted to provide a semantic representation. This is Part 1 of a five-part introduction. The focus here is on defining the key mathematical structures first, and presenting the semantic-database application in subsequent Parts. We focus in Part 1 (...) on general axioms for the structures, and introduce key concepts. Part 2 analyses the CAT2 sub-relation of CAT4 in more detail. The interpretation of fact networks is introduced in Part 3, where we turn to interpreting semantics. We start with examples of relational and graph databases, with methods to translate them into CAT3 networks, with the aim of retaining the meaning of information. The full application to semantic theory comes in Part 4, where we introduce general functions, including the language interpretation or linguistic functions. The representation of linear symbolic languages, including natural languages and formal symbolic languages, is a function that CAT4 is uniquely suited to. In Part 5, we turn to software design considerations, to show how files, indexes, functions and screens can be defined to implement a CAT4 system efficiently. (shrink)

Departing from and closing with reflections on issues regarding teaching practices of philosophy of mathematics, I propose a comparison between the main features of the Leibnizian notion of symbolic knowledge and some passages from the Tractatus on arithmetic. I argue that this reading allows (i) to shed a new light on the specificities of the Tractarian definition of number, compared to those of Frege and Russell; (ii) to highlight the understanding of the nature of mathematical knowledge as symbolic or (...) class='Hi'>formal knowledge that Wittgenstein mobilizes in his book; (iii) to offer reasons for the claim that Wittgenstein can be considered the philosopher of mathematical practice avant la lettre. The paper ends with an overview, a return to the initial reflection on the connections between research and teaching, and a defense of the reading key used here in terms of its potential for the research in philosophy of mathematics. (shrink)

Fifty years of effort in artificial intelligence (AI) and the formalization of legal reasoning have produced both successes and failures. Considerable success in organizing and displaying evidence and its interrelationships has been accompanied by failure to achieve the original ambition of AI as applied to law: fully automated legal decision-making. The obstacles to formalizing legal reasoning have proved to be the same ones that make the formalization of commonsense reasoning so difficult, and are most evident where legal reasoning has to (...) meld with the vast web of ordinary human knowledge of the world. Underlying many of the problems is the mismatch between the discreteness of symbol manipulation and the continuous nature of imprecise natural language, of degrees of similarity and analogy, and of probabilities. (shrink)

It is here proposed an analysis of symbolic and sub-symbolic models for studying cognitive processes, centered on emergence and logical openness notions. The Theory of logical openness connects the Physics of system/environment relationships to the system informational structure. In this theory, cognitive models can be ordered according to a hierarchy of complexity depending on their logical openness degree, and their descriptive limits are correlated to Gödel-Turing Theorems on formal systems. The symbolic models with low logical openness describe cognition by (...) means of semantics which fix the system/environment relationship, while the sub-symbolic ones with high logical openness tends to seize its evolutive dynamics. An observer is defined as a system with high logical openness. In conclusion, the characteristic processes of intrinsic emergence typical of “bio-logic” - emerging of new codes-require an alternative model to Turing- computation, the natural or bio-morphic computation, whose essential features we are going here to outline. (shrink)

The paper has three objectives: to expound a set-theoretical triplet model of concepts; to introduce some triplet relations (symbolic, logical, and mathematical formalization; equivalence, intersection, disjointness) between object concepts, and to instantiate them by relations between certain physical object concepts.

The Genuine Process Logic described here (abbreviation: GPL) places the object-bound process itself at the center of formalism. It should be suitable for everyday use, i.e. it is not primarily intended for the formalization of computer programs, but instead, as a counter-conception to the classical state logics. The new and central operator of the GPL is an action symbol replacing the classical state symbols, e.g. of equivalence or identity. The complete renunciation of object-language state expressions also results in a (...) completely new metalinguistic framework, both regarding the axioms and the expressive possibilities of this system. A mixture with state logical terms is readily possible. (shrink)

Description Logics (DLs) are a family of well-known terminological knowledge representation formalisms in modern semantics-based systems. This research focuses on analysing how our developed Occurrence Logic (OccL) can conceptually and logically support the development of a description logic. OccL is integrated into the alternative theory of natural language syntax in `Deviational Syntactic Structures' under the label `EFA(X)3' (or the third version of Epi-Formal Analysis in Syntax, EFA(X), which is a radical linguistic theory). From the logical point of view, OccL (...) is a formal logic that mainly deals with the occurrences of symbols as well as with their priorities within linguistic descriptions, i.e. natural language syntax, semantics and phonology. In this article---based on our OccL-based definitions of the concepts of `strong implication' and `occurrence value' as well as of the logical concept `identical occurrence constructor (IDOC)' that is the most fundamental logical concept in our formalism---we will model Occurrence Description Logic ($\mathcal{ODL}$). Accordingly, we will formally-logically analyse `occurrence(s) of symbol(s)' within descriptions of the world in $\mathcal{ODL}$. In addition, we will analyse and assess the logical concepts of `occurrence' and `occurrence priority' in $\mathcal{ODL}$. This research can make a strong logical background for our future research in the development of a Modal Occurrence Description Logic. (shrink)

What does it look like when a group of scientists set out to re-envision an entire field of biology in symbolic and formal terms? I analyze the founding and articulation of Numerical Taxonomy between 1950 and 1970, the period when it set out a radical new approach to classification and founded a tradition of mathematics in systematic biology. I argue that introducing mathematics in a comprehensive way also requires re-organizing the daily work of scientists in the field. Numerical taxonomists (...) sought to establish a mathematical method for classification that was universal to every type of organism, and I argue this intrinsically implicated them in a qualitative re-organization of the work of all systematists. I also discuss how Numerical Taxonomy’s re-organization of practice became entrenched across systematic biology even as opposing schools produced their own competing mathematical methods. In this way, the structure of the work process became more fundamental than the methodological theories that motivated it. (shrink)

As noted in 1962 by Timothy Smiley, if Aristotle’s logic is faithfully translated into modern symbolic logic, the fit is exact. If categorical sentences are translated into many-sorted logic MSL according to Smiley’s method or the two other methods presented here, an argument with arbitrarily many premises is valid according to Aristotle’s system if and only if its translation is valid according to modern standard many-sorted logic. As William Parry observed in 1973, this result can be proved using my 1972 (...) proof of the completeness of Aristotle’s syllogistic. (shrink)

K denotes both the knowledge predicate satisfied by every currently known theorem and the finite set of all currently known theorems. The set K is time-dependent, publicly available, and contains theorems both from formal and constructive mathematics. Any theorem of any mathematician from past or present forever belongs to K. Mathematical statements with known constructive proofs exist in K separately and form the set K_c⊆K. We assume that mathematical sets are atemporal entities. They exist formally in ZFC theory although (...) their properties can be time-dependent (when they depend on K) or informal. The true statement "K is non-empty" is outside K forever because K is not a formal set. Paul Cohen proved in 1963 that the equality 2^{\aleph_0}=\aleph_1 is independent of ZFC. Before 1963, the statement "There is a known constructively defined integer n⩾1 such that 2^{\aleph_0}=\aleph_n" was outside K. Since 1963, this statement is outside K forever. The true statement "There exists a set X⊆{1,...,49} such that card(X)=6 and X never occurred as the winning six numbers in the Polish Lotto lottery" refers to the current non-mathematical knowledge and is outside K forever. In Martin-Löf's terminology, every currently known theorem is actually true whereas every theorem (known or unknown) is potentially true. Actual truth is knowledge dependent and tensed. Potential truth is knowledge independent and tenseless. Algorithms always terminate. We explain the distinction between algorithms whose existence is provable in ZFC and constructively defined algorithms which are currently known. By using this distinction, we obtain non-trivially true statements on decidable sets X⊆N that belong to constructive and informal mathematics and refer to the current mathematical knowledge on X. For every such statement, its truth concerning the current mathematical knowledge is implied by a true statement of the form: (the conjunction of i conditions of the form α∈K)∧(the conjunction of j conditions of the form β∉K)∧(the conjunction of k conditions of the form γ∈K_c)∧(the conjunction of l conditions of the form δ∉K_c), where i,j,k,l∈N and α,β,γ,δ are mathematical statements. In constructive mathematics and the traditional Brouwerian intuitionism, the truth of a mathematical statement means that we know a constructive proof. Therefore, the truth of a mathematical statement depends on time, where the statement is formally stated in the classical mathematics without the predicate K. In this article, mathematical statements on decidable sets X⊆N refer to time because they refer to the current mathematical knowledge on X. They cannot be formally stated in the classical mathematics without the predicate K and their logical values may change in time. For a set X⊆N whose infiniteness is false or unproven, we define which elements of X are classified as known. No known set X⊆N satisfies Conditions (1)-(4) and is widely known in number theory or naturally defined, where this term has only informal meaning. (1) A known algorithm with no input returns an integer n satisfying card(X)<ω ⇒ X⊆(-∞,n]. (2) A known algorithm for every k∈N decides whether or not k∈X. (3) For every known algorithm A with no input, the statement "A returns the logical value of the statement card(X)=ω" is outside K. (4) There are many elements of X and it is conjectured, though so far unproven, that X is infinite. (5) X is naturally defined. The infiniteness of X is false or unproven. X has the simplest definition among known sets Y⊆N with the same set of known elements. No set X⊆N will satisfy Conditions (1)-(4) forever, if for every algorithm with no input, at some future day, a computer will be able to execute this algorithm in 1 second or less. The physical limits of computation disprove the assumption of the above statement. We prove that the set X={n∈N: the interval [-1,n] contains more than 29.5+(11!/(3n+1))∙sin(n) primes of the form k!+1} satisfies Conditions (1)-(5) except the requirement that X is naturally defined. We present a table that shows satisfiable conjunctions of the form #(Condition1)∧(Condition 2)∧#(Condition 3)∧(Condition 4)∧#(Condition 5), where # denotes the negation ¬ or the absence of any symbol. We prove that there exists a naturally defined set C⊆N which satisfies the following conditions (6)-(11). (6) A known and simple algorithm for every k∈N decides whether or not k∈C. (7) For every known algorithm A with no input, the statement "A returns the logical value of the statement card(C)=ω" is outside K. (8) For every known algorithm A with no input, the statement "A returns the logical value of the statement card(N\C)=ω" is outside K. (9) It is conjectured, though so far unproven, that C is infinite. (10) For every known algorithm A with no input, the statement "A returns an integer n satisfying card(C)<ω ⇒ C⊆(-∞,n]" is outside K. (11) For every known algorithm A with no input, the statement "A returns an integer m satisfying card(N\C)<ω ⇒ N\C⊆(-∞,m]" is outside K. The set C={k∈N: 2^{2^k}+1 is composite} is naturally defined and satisfies conditions (6)-(11). (shrink)

I present a general metaphysical framework for any formal system that works with truth-values. To establish such a framework, I start with the notion of absolute nothingness, from which I construct a nothingness which is akin to the notion of an empty set in mathematics. Then I provide a formal system that its ability to produce symbols is an integral property and an inseparable part of its metaphysics.

"This book explores the relationship between performance and activism in Chile as a form of political expression and citizen participation during the period 2010-2020. Since the student mobilizations of 2006, the social movements that have taken place in Chile are characterized, in many cases, by the appropriation of public space and the political use of the body. This became particularly evident during the social outbreak of October 2019. The social upheaval was accompanied by a cultural explosion, where the arts in (...) all their dimensions played a crucial role as new strategies of collective action. The research proposes two hypotheses about this cultural revolution. Firstly, that this explosion of artistic activism had been fermenting for at least a decade (2010-2020) within the social movements, using performance and art as expressions of collective action. These artistic drivers, catalysts of this cultural revolution, have been mobilized around three fundamental axes of social demands: human rights, education, and feminism. The second hypothesis is that artistic activism in Chile between 2010-2020 emerged to exercise citizenship and strengthen the diversity of cultural identities. Chilean artistic activism today, unlike other historical moments, is a direct expression of citizen participation. It is no longer just artists who contribute to social and protest movements with their work: now, it is the citizens who use artistic expression as a practice of political participation. From an intersectional and feminist perspective, this book is an invitation to study and reflect on the aesthetic and political dimension of performance and its potential to establish justice and symbolic reparation when the State is absent. Furthermore, it incentivises a conversation about the ritual, pedagogical, and visionary dimension of art and its infinite possibilities of expressive and discursive resources. Performance art reimagines possible futures embodying the political demand, always loaded with memory, reclaiming the public space. Performance as the 'art of action' conquers the realm of non-formal education, permeating public space and shaping a Pedagogy of memory and hope." -/- . (shrink)

A uniform theory of conditionals is one which compositionally captures the behavior of both indicative and subjunctive conditionals without positing ambiguities. This paper raises new problems for the closest thing to a uniform analysis in the literature (Stalnaker, Philosophia, 5, 269–286 (1975)) and develops a new theory which solves them. I also show that this new analysis provides an improved treatment of three phenomena (the import-export equivalence, reverse Sobel-sequences and disjunctive antecedents). While these results concern central issues in the study (...) of conditionals, broader themes in the philosophy of language and formal semantics are also engaged here. This new analysis exploits a dynamic conception of meaning where the meaning of a symbol is its potential to change an agent’s mental state (or the state of a conversation) rather than being the symbol’s content (e.g. the proposition it expresses). The analysis of conditionals is also built on the idea that the contrast between subjunctive and indicative conditionals parallels a contrast between revising and consistently extending some body of information. (shrink)

Peer Review Book Description - Maria Seykora (female, published age 28) -/- -/- Classical Logic will attempt to give a comprehensive and rigorous introduction and more advanced overview of the area of logic widely known as “classical logic,” as distinguished from modern-day “non-classical logic,” for undergraduate students in general. It will cover the topics of Informal Logic (including logical fallacies, deduction, induction, and abductive reasoning) and Formal Logic. (Because it aims to cover these two topics, the title may change (...) to Informal and Classical Logic for the first edition.) Within the category of Formal Logic, it will cover Syllogistic or Categorical Logic and Symbolic Logic. Furthermore, within Symbolic Logic, it will cover Propositional Logic, Predicate Logic, and Modal Logic. The Formal Logic component is currently unfinished, and the Informal Logic component is somewhat unfinished. The unique selling proposition is “the first comprehensive Classical Logic textbook.” It seems that many textbooks for college students today, including Patrick J. Hurley’s, are loosely organized around Classical Logic and other various areas of logic, yet there is not a clear-cut, comprehensive summation of Classical Logic in textbook form. For instance, Hurley’s textbook does not have a section on Modal Logic, and does not address some fallacies such as Hypothesis Contrary to Fact, the Genetic Fallacy, Appeal to Nature, No True Scotsman Fallacy, Emotionally Loaded Language fallacies, etc. Stan Baronett’s Logic textbook also does not include discussions of Modal Logic or these additional fallacies and others like them, and Siu-Fan Lee’s textbook does not include Modal Logic either. Additionally, abductive reasoning, which involves making an inference to the best explanation, is important in real life, yet it seems absent from most logic textbooks. (shrink)

The mapping of numbers onto space is fundamental to measurement and to mathematics. Is this mapping a cultural invention or a universal intuition shared by all humans regardless of culture and education? We probed number-space mappings in the Mundurucu, an Amazonian indigene group with a reduced numerical lexicon and little or no formal education. At all ages, the Mundurucu mapped symbolic and nonsymbolic numbers onto a logarithmic scale, whereas Western adults used linear mapping with small or symbolic numbers and (...) logarithmic mapping when numbers were presented nonsymbolically under conditions that discouraged counting. This indicates that the mapping of numbers onto space is a universal intuition and that this initial intuition of number is logarithmic. The concept of a linear number line appears to be a cultural invention that fails to develop in the absence of formal education. (shrink)

“Second-order Logic” in Anderson, C.A. and Zeleny, M., Eds. Logic, Meaning, and Computation: Essays in Memory of Alonzo Church. Dordrecht: Kluwer, 2001. Pp. 61–76. -/- Abstract. This expository article focuses on the fundamental differences between second- order logic and first-order logic. It is written entirely in ordinary English without logical symbols. It employs second-order propositions and second-order reasoning in a natural way to illustrate the fact that second-order logic is actually a familiar part of our traditional intuitive logical framework and (...) that it is not an artificial formalism created by specialists for technical purposes. To illustrate some of the main relationships between second-order logic and first-order logic, this paper introduces basic logic, a kind of zero-order logic, which is more rudimentary than first-order and which is transcended by first-order in the same way that first-order is transcended by second-order. The heuristic effectiveness and the historical importance of second-order logic are reviewed in the context of the contemporary debate over the legitimacy of second-order logic. Rejection of second-order logic is viewed as radical: an incipient paradigm shift involving radical repudiation of a part of our scientific tradition, a tradition that is defended by classical logicians. But it is also viewed as reactionary: as being analogous to the reactionary repudiation of symbolic logic by supporters of “Aristotelian” traditional logic. But even if “genuine” logic comes to be regarded as excluding second-order reasoning, which seems less likely today than fifty years ago, its effectiveness as a heuristic instrument will remain and its importance for understanding the history of logic and mathematics will not be diminished. Second-order logic may someday be gone, but it will never be forgotten. Technical formalisms have been avoided entirely in an effort to reach a wide audience, but every effort has been made to limit the inevitable sacrifice of rigor. People who do not know second-order logic cannot understand the modern debate over its legitimacy and they are cut-off from the heuristic advantages of second-order logic. And, what may be worse, they are cut-off from an understanding of the history of logic and thus are constrained to have distorted views of the nature of the subject. As Aristotle first said, we do not understand a discipline until we have seen its development. It is a truism that a person's conceptions of what a discipline is and of what it can become are predicated on their conception of what it has been. (shrink)

This is an open-access introductory logic textbook, prepared by Jonathan Ichikawa, based on P.D. Magnus's forallx. This (v2.0, July 2020) is intended as a stable, ready-for-teaching edition.

The Foundational Model of Anatomy (FMA) symbolically represents the structural organization of the human body from the macromolecular to the macroscopic levels, with the goal of providing a robust and consistent scheme for classifying anatomical entities that is designed to serve as a reference ontology in biomedical informatics. Here we articulate the need for formally clarifying the is-a and part-of relations in the FMA and similar ontology and terminology systems. We diagnose certain characteristic errors in the treatment of these relations (...) and show how these errors can be avoided through adoption of the formalism we describe. We then illustrate how a consistently applied formal treatment of taxonomy and partonomy can support the alignment of ontologies. (shrink)

The currently standard philosophical conception of existence makes a connection between three things: certain ways of talking about existence and being in natural language; certain natural language idioms of quantification; and the formal representation of these in logical languages. Thus a claim like ‘Prime numbers exist’ is treated as equivalent to ‘There is at least one prime number’ and this is in turn equivalent to ‘Some thing is a prime number’. The verb ‘exist’, the verb phrase ‘there is’ and (...) the quantifier ‘some’ are treated as all playing similar roles, and these roles are made explicit in the standard common formalization of all three sentences by a single formula of first-order logic: ‘(∃ x )[P( x ) & N( x )]’, where ‘P( x )’ abbreviates ‘ x is prime’ and ‘N( x )’ abbreviates ‘ x is a number’. The logical quantifier ‘∃’ accordingly symbolizes in context the role played by the English words ‘exists’, ‘some’ and ‘there is’. (shrink)