Saul Kripke once noted that there is a tight connection between computation and de re knowledge of whatever the computation acts upon. For example, the Euclidean algorithm can produce knowledge of which number is the greatest common divisor of two numbers. Arguably, algorithms operate directly on syntactic items, such as strings, and on numbers and the like only via how the numbers are represented. So we broach matters of notation. The purpose of this article is to explore the relationship between (...) the notations acceptable for computation, the usual idealizations involved in theories of computability, flowing from Alan Turing’s monumental work, and de re propositional attitudes toward numbers and other mathematical objects. (shrink)

This paper presents a diagrammatic notation for visualizing epistemic entities and relations. The notation was created during the Visualizing Worldviews project funded by the University of Toronto’s Jackman Humanities Institute and has been further developed by the scholars participating in the university’s Research Opportunity Program. Since any systematic diagrammatic notation should be based on a solid ontology of the respective domain, we first outline the current state of the scientonomic ontology. We then proceed to providing diagrammatic tools for visualizing the (...) epistemic entities and relations of this ontology. These basic diagramming techniques allow us to construct diagrams of various types for both synchronic and diachronic visualizations. The paper concludes by highlighting some future research directions. As the notation presented here is de facto accepted and used in scientonomy, the paper suggests no modifications. (shrink)

Ontological pluralism is the view that there are different ways to exist. It is a position with deep roots in the history of philosophy, and in which there has been a recent resurgence of interest. In contemporary presentations, it is stated in terms of fundamental languages: as the view that such languages contain more than one quantifier. For example, one ranging over abstract objects, and another over concrete ones. A natural worry, however, is that the languages proposed by the pluralist (...) are mere notational variants of those proposed by the monist, in which case the debate between the two positions would not seem to be substantive. Jason Turner has given an ingenious response to this worry, in terms of a principle that he calls ‘logical realism’. This paper offers a counter-response on behalf of the ‘notationalist’. I argue that, properly applied, the principle of logical realism is no threat to the claim that the languages in question are notational variants. Indeed, there seems to be every reason to think that they are. (shrink)

This paper argues that many so-called digital technologies can be construed as notational technologies, explored through the example of Monegraph, an art and digital asset management platform built on top of the blockchain system originally developed for the cryptocurrency bitcoin. As the paper characterizes it, a notational technology is the performance of syntactic notation within a field of reference, a technologized version of what Nelson Goodman called a “notational system.” Notational technologies produce abstracted entities through positive and reliable, or constitutive, (...) tests of socially acceptable meaning. Accordingly, this account deviates from typical narratives of blockchains, instead demonstrating that blockchain technologies are effective at managing digital assets because they produce abstracted identities through the performance of notation. Since notational technologies rely on configurations of socially acceptable meaning, this paper also provides a philosophical account of how blockchain technologies are socially embedded. (shrink)

The pragmatic notion of assertion has an important inferential role in logic. There are also many notational forms to express assertions in logical systems. This paper reviews, compares and analyses languages with signs for assertions, including explicit signs such as Frege’s and Dalla Pozza’s logical systems and implicit signs with no specific sign for assertion, such as Peirce’s algebraic and graphical logics and the recent modification of the latter termed Assertive Graphs. We identify and discuss the main ‘points’ of these (...) notations on the logical representation of assertions, and evaluate their systems from the perspective of the philosophy of logical notations. Pragmatic assertions turn out to be useful in providing intended interpretations of a variety of logical systems. (shrink)

In the study of modal and nonclassical logics, translations have frequently been employed as a way of measuring the inferential capabilities of a logic. It is sometimes claimed that two logics are “notational variants” if they are translationally equivalent. However, we will show that this cannot be quite right, since first-order logic and propositional logic are translationally equivalent. Others have claimed that for two logics to be notational variants, they must at least be compositionally intertranslatable. The definition of compositionality these (...) accounts use, however, is too strong, as the standard translation from modal logic to first-order logic is not compositional in this sense. In light of this, we will explore a weaker version of this notion that we will call schematicity and show that there is no schematic translation either from first-order logic to propositional logic or from intuitionistic logic to classical logic. (shrink)

We provide an intuitive motivation for the hyperreal numbers via electoral axioms. We do so in the form of a Socratic dialogue, in which Protagoras suggests replacing big-oh complexity classes by real numbers, and Socrates asks some troubling questions about what would happen if one tried to do that. The dialogue is followed by an appendix containing additional commentary and a more formal proof.

In contemporary natural languages semantics one will often see the use of special brackets to enclose a linguistic expression, e.g. ⟦carrot⟧. These brackets---so-called denotation brackets or semantic evaluation brackets---stand for a function that maps a linguistic expression to its "denotation" or semantic value (perhaps relative to a model or other parameters). Even though this notation has been used in one form or another since the early development of natural language semantics in the 1960s and 1970s, Montague himself didn't make use (...) of this notation in his series of groundbreaking papers on semantics. That raises the question: When was the ⟦.⟧-notation introduced to semantics? This note answers that question. (shrink)

The nature of time is perceived by intellectuals variedly. An attempt is made in this paper to reconcile such varied views in the light of the Upanishads and related Indian spiritual and philosophical texts. The complex analysis of modern mathematics is used to represent the nature and presentation physical and psychological times so differentiated. Also the relation between time and energy is probed using uncertainty relations, forms of energy and phases of matter.

Wittgenstein composed five original musical fragments during his transitional middle period, in which he employs musical notation as a means by which to convey his philosophical thoughts. This is an overlooked aspect of the importance of aesthetics, and musical thinking in particular, in the development of Wittgenstein’s philosophy. We explain and evaluate the way the music interlinks with Wittgenstein’s philosophical thoughts. We show the direct relation of these musical examples as precursors to some of Wittgenstein’s most celebrated ideas (the push (...) beyond the inner/outer picture, the problem of rule-following, and aspect perception). We also show how important broad aesthetic notions such as style and composition concerning the very possibility of philosophizing were to Wittgenstein. (shrink)

The nature of time is perceived by intellectuals variedly. An attempt is made in this paper to reconcile such varied views in the light of the Upanishads and related Indian spiritual and philosophical texts. The complex analysis of modern mathematics is used to represent the nature and presentation physical and psychological times so differentiated. Also the relation between time and energy is probed using uncertainty relations, forms of energy and phases of matter. Implications to time-dependent Schrodinger wave equation and uncertainty (...) principle are hinted. (shrink)

This study examines problems related to the representation of music. It constructs the sender/message/perceiver/result model, a prototype broad enough to incorporate a large variety of music and other notation systems, including those having to do with computers. The work defines music notation itself, describes various models for studying the subject--including the binary types prescriptive/descriptive, and symbolic/iconic--and assesses music notation as a contemporary practice. It encompasses a review of the actions and intentions of composers, performers, and audiences, and a consideration of (...) the message and how it is affected by media , reference systems , and meanings. Finally, through an examination of the possibilities of the self acting as a notation system within models posited by feminist and queer theory, I attempt to demonstrate the power of notation to affect other systems and worlds, including the worlds of music, art, and politics. ;The consideration of these topics draws on theories from linguistics, semiotics, structuralism, post-structuralism, cybernetics, and music, as well as the philosophies of John Cage, Jean-Jacques Nattiez, Nelson Goodman, and others. The work examines various languages and applications used to program computers for musical purposes, though this examination does not incorporate programs dedicated exclusively to printing music. Nor does the study focus on non-Western notations, suggestions for reforms of notation, or encyclopedic lists of notation devices. ;The expository writing is augmented by excerpts from material I gathered in numerous interviews. There are separate sections containing interviews of Leonard Meyer, Allan Kaprow, John Cage, Earle Brown, Roger Reynolds, and Heidi von Gunden. There is also a large, partially-annotated bibliography. ;My hope is that an inclusive, interdisciplinary exploration of music notation will be useful in broadening the theoretical and creative boundaries of the field, as well as bridging gaps among musicians using various music technologies, and artists and theorists from diverse fields and perspectives. (shrink)

The design process in Kashmiri carpet weaving is distributed over a number of actors and artifacts and is mediated by a weaving notation called talim. The script encodes entire design in practice-specific symbols. This encoded script is decoded and interpreted via design-specific conventions by weavers to weave the design embedded in it. The cognitive properties of this notational system are described in the paper employing cognitive dimensions (CDs) framework of Green (People and computers, Cambridge University Press, Cambridge, 1989) and Blackwell (...) et al. (Cognitive technology: instruments of mind—CT 2001, LNAI 2117, Springer, Berlin, 2001). After introduction to the practice, the design process is described in ‘The design process’ section which includes coding and decoding of talim. In ‘Cognitive dimensions of talim’ section, after briefly discussing CDs framework, the specific cognitive dimensions possessed by talim are described in detail. (shrink)

In Languages of Art, Nelson Goodman presents a general theory of symbolic notation. However, I show that his theory could not adequately explain possible cases of natural language notational uses, and argue that this outcome undermines, not only Goodman's own theory, but any broadly type versus token based account of notational structure.Given this failure, an alternative representational theory is proposed, in which different visual or perceptual aspects of a given physical inscription each represent a different letter, word, or other notational (...) item. Such a view is strongly supported by the completely conventional relation between inscriptions and notation, as shown by encryption techniques etc. (shrink)

Transformed RAVAL NOTATION solves Syllogism problems very quickly and accurately. This method solves any categorical syllogism problem with same ease and is as simple as ABC… In Transformed RAVAL NOTATION, each premise and conclusion is written in abbreviated form, and then conclusion is reached simply by connecting abbreviated premises.NOTATION: Statements (both premises and conclusions) are represented as follows: Statement Notation a) All S are P, SS-P b) Some S are P, S-P c) Some S are not P, S / PP (...) d) No S is P, SS / PP (- implies are and / implies are not) All is represented by double letters; Some is represented by single letter. No S is P implies No P is S so its notation contains double letters on both sides. -/- RULES: (1) Conclusions are reached by connecting Notations. Two notations can be linked only through common linking terms. When the common linking term multiplies (becomes double from single), divides (becomes single from double) or remains double then conclusion is arrived between terminal terms. (Aristotle’s rule: the middle term must be distributed at least once) -/- (2)If both statements linked are having – signs, resulting conclusion carries – sign (Aristotle’s rule: two affirmatives imply an affirmative) -/- (3) Whenever statements having – and / signs are linked, resulting conclusion carries / sign. (Aristotle’s rule: if one premise is negative, then the conclusion must be negative) -/- (4)Statement having / sign cannot be linked with another statement having / sign to derive any conclusion. (Aristotle’s rule: Two negative premises imply no valid conclusion) Syllogism conclusion by Tranformed Raval’s Notation is in accordance with Aristotle’s rules for the same. It is visually very transparent and conclusions can be deduced at a glance, moreover it solves syllogism problems with any number of statements and it is quickest of all available methods. By new Raval method for solving categorical syllogism, solving categorical syllogism is as simple as pronouncing ABC and it is just continuance of Aristotle work on categorical syllogism. It’s believed that Boole's system could handle multi-term propositions and arguments, whereas Aristotle could handle only two-termed subject-predicate propositions and arguments. For example, it’s claimed that Aristotle's system could not deduce: "No quadrangle that is a square is a rectangle that is a rhombus" from "No square that is a quadrangle is a rhombus that is a rectangle" or from "No rhombus that is a rectangle is a square that is a quadrangle". Above conclusion is reached at a glance with Raval's Notations (Symbolic Aristotle’s syllogism rules). Premise: "No (square that is a quadrangle) is a (rhombus that is a rectangle)" Raval's Representations: S – Q, S – Q / Rh – Re, Rh – Re Premise: "No (rhombus that is a rectangle) is a (square that is a quadrangle)". Raval's Representations: Rh – Re, Rh – Re / S – Q, S - Q Conclusion: "No (quadrangle that is a square) is a (rectangle that is a rhombus)" Raval’s Representations: Q – S, Q – S / Re – Rh, Re – Rh As “ Q – S” follows from “S – Q” and “Re – Rh” from “Rh – Re”. Given conclusion follows from the given premises. Author disregards existential fallacy, as subset of a null set has to be a null set. -/- . (shrink)

Basic abstraction principles are reached through ontology, which was traditionally conceived as a depiction of the world itself. Ontology is also described using conceptual modeling (CM) that defines fundamental concepts of reality. CM is one of the central activities in computer science, especially as it is mainly used in software engineering as an intermediate artifact for system construction. To achieve such a goal, we propose Stoic CM (SCM) as a description of what a system must do functionally with minimal ambiguity. (...) As a case study, we apply SCM to investigate the ontology of BPMN (business process modeling notation). Such an undertaking would demonstrate SCM notions and simultaneously may offer a viable ontological foundation for BPMN. SCM defines the being of things and actions in reality based on Stoic notions of existence and subsistence. It has two levels of specification: (1) a subsistence static model where things and actions subsist and (2) an existence dynamic model where things and actions exist in time. From the Stoic ontological point of view, while a thing existing has a clear denotation, subsistence indicates the thing is “being there,” but it is inactive (does not participate in an event). We apply SCM to BPMN processes that involve buying a new car with many notions, such as activity, task, event, and message. The result indicates that SCM produces a tighter representation of reality, thus providing the necessary description of the part in the application world to be used as requirements for developing the software system. (shrink)

Why study notations, diagrams, or more broadly the variety of nonverbal “representations” or “signs” that are used in mathematical practice? This chapter maps out recent work on the topic by distinguishing three main philosophical motivations for doing so. First, some work (like that on diagrammatic reasoning) studies signs to recover norms of informal or historical mathematical practices that would get lost if the particular signs that these practices rely on were translated away; work in this vein has the potential (...) to broaden our view of the specific normativity of mathematics beyond logical proof. Second, some work focuses on signs to highlight the open-ended and “nonrigorous” aspects of mathematical practice, and the role of these aspects in the historical development of mathematics. The third motivation is based on the following observation: the reason differences in signs matter in mathematics is that humans are finite agents, with cognitive and computational limitations. Accordingly, studying signs can be a way to study how human computational constraints shape the mathematics that we do, and to tackle the topic of mathematical understanding. (shrink)

The aim of this article is to investigate the roles of commutative diagrams (CDs) in a specific mathematical domain, and to unveil the reasons underlying their effectiveness as a mathematical notation; this will be done through a case study. It will be shown that CDs do not depict spatial relations, but represent mathematical structures. CDs will be interpreted as a hybrid notation that goes beyond the traditional bipartition of mathematical representations into diagrammatic and linguistic. It will be argued that one (...) of the reasons why CDs form a good notation is that they are highly mathematically tractable: experts can obtain valid results by ‘calculating’ with CDs. These calculations, take the form of ‘diagram chases’. In order to draw inferences, experts move algebraic elements around the diagrams. It will be argued that these diagrams are dynamic. It is thanks to their dynamicity that CDs can externalize the relevant reasoning and allow experts to draw conclusions directly by manipulating them. Lastly, it will be shown that CDs play essential roles in the context of proof as well as in other phases of the mathematical enterprise, such as discovery and conjecture formation. (shrink)

Mathematical diagrams are frequently used in contemporary mathematics. They are, however, widely seen as not contributing to the justificatory force of proofs: they are considered to be either mere illustrations or shorthand for non-diagrammatic expressions. Moreover, when they are used inferentially, they are seen as threatening the reliability of proofs. In this paper, I examine certain examples of diagrams that resist this type of dismissive characterization. By presenting two diagrammatic proofs, one from topology and one from algebra, I show that (...) diagrams form genuine notational systems, and I argue that this explains why they can play a role in the inferential structure of proofs without undermining their reliability. I then consider whether diagrams can be essential to the proofs in which they appear. (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)

There are many different ways to talk about the world. Some ways of talking are more expressive than others—that is, they enable us to say more things about the world. But what exactly does this mean? When is one language able to express more about the world than another? In my dissertation, I systematically investigate different ways of answering this question and develop a formal theory of expressive power, translation, and notational variance. In doing so, I show how these investigations (...) help to clarify the role that expressive power plays within debates in metaphysics, logic, and the philosophy of language. (shrink)

The objective of this article is to take into account the functioning of representational cognitive tools, and in particular of notations and visualizations in mathematics. In order to explain their functioning, formulas in algebra and logic and diagrams in topology will be presented as case studies and the notion of manipulative imagination as proposed in previous work will be discussed. To better characterize the analysis, the notions of material anchor and representational affordance will be introduced.

The “sign of consequence” is a notation for propositional logic that Peirce invented in 1886 and used at least until 1894. It substituted the “copula of inclusion” which he had been using since 1870.

In this article, I explore the relationship between dance and the work of Nelson Goodman, which is found primarily in his early book, Languages of Art. Drawing upon the book’s first main thread, I examine Goodman’s example of a dance gesture as a symbol that exemplifies itself. I argue that self-exemplifying dance gestures are unique in that they are often independent and internally motivated, or “meta-self-exemplifying.” Drawing upon the book’s second main thread, I retrace Goodman’s analysis of dance’s relationship to (...) both notation in general and also Labanotation in particular. My argument is that dance gives the false impression of being notational, or is “meta-notational.”. (shrink)

Usual narratives among prehistoric archaeologists consider typological approaches as part of a past and outdated episode in the history of research, subsequently replaced by technological, functional, chemical, and cognitive approaches. From a historical and conceptual perspective, this paper addresses several limits of these narratives, which (1) assume a linear, exclusive, and additive conception of scientific change, neglecting the persistence of typological problems; (2) reduce collective developments to personal work (e.g. the “Bordes’” and “Laplace’s” methods in France); and (3) presuppose the (...) coherence and identity of these “methods” over time. It explores the case of the “Structural and analytical typology” method, developed in France, Spain, and Italy from the 1950s to the 2000s by Georges Laplace and his collaborators for lithic implements. This paper (1) provides a detailed historical account of the evolving content of this collective endeavour over five decades; (2) it addresses the epistemological question of what makes the identity and unity of a scientific method, demonstrating that the core component of the “analytical typology” lies in its particular way to represent real-world phenomena through its notation system; and (3) it reveals how this little known but significant episode of advances in the methods and theory of archaeology, contemporary but independent of the “New Archaeology” trend in English-speaking archaeology, was instrumental in the continuation of evolutionary perspectives in France and in the development of quantitative and formal methods in archaeology in southwestern Europe, foreseeing crucial knowledge representation issues raised today by digital methods in archaeology and data curation. (shrink)

In this thesis, we will study the embedding of classical first-order logic in first-order S4, which is based on the translation originally introduced in Fitting (1970). The initial main part is dedicated to a detailed model-theoretic proof of the soundness of the embedding. This will follow the proof sketch in Fitting (1970). We will then outline a proof procedure for a proof-theoretic replication of the soundness result. Afterwards, a potential proof of faithfulness of the embedding, read in terms of soundness (...) and completeness, will be discussed. We will particularly highlight the many difficulties coming with it. In the final section, we will relate this discussion to the debate on notational variance in French (2019). We will do this by showing how a weaker version of French's notion of ‘expressive equivalence’ conforms to the model-theoretic soundness result. We will then conclude that the soundness result without completeness might contain rather little overall insight by relating it to the extensibility of classical logic to S4. (shrink)

Building on the notational principles of C. S. Peirce’s graphical logic, Pietarinen has tried to develop a propositional logic unfolding in the medium of sound. Apart from its intrinsic interest, this project serves as a concrete test of logic’s range. However, I argue that Pietarinen’s inaugural proposal, while promising, has an important shortcoming, since it cannot portray double-negation without thereby portraying a contradiction.

Silence is a multifaceted concept which is not merely as an absence of sound but a presence with significant ontological, existential, and phenomenological implications. Through a thematic analysis, this paper deconstructs silence into various dimensions—its ontology, linguistic universality, and its function as cessation of speech, a form of listening, an act of kenosis, a form of ascesis, and a way of life. The study employs philosophical discourse and mathematical notation to delve into these aspects, demonstrating that while each perspective sheds (...) light on certain facets of silence, none can comprehensively encapsulate its essence. The research underscores the paradoxical nature of silence. It is both a universal phenomenon recognised across cultures and an intimate, personal experience that resists definitive categorisation. The paper concludes that the true nature of silence, particularly when considering the state of "being silent," remains indefinable and elusive, inviting deeper contemplation on its role in human experience. (shrink)

Elementary patterns of resemblance notate ordinals up to the ordinal of Pi^1_1-CA_0. We provide ordinal multiplication and exponentiation algorithms using these notations.

This paper discusses two notational variance views with respect to indexical singular reference and content: the view that certain forms of Millianism are at bottom notational variants of a Fregean theory of reference, the Fregean Notational Variance Claim; and the view that certain forms of Fregeanism are at bottom notational variants of a direct reference theory, the Millian Notational Variance Claim. While the former claim rests on the supposition that a direct reference theory could be easily turned into a particular (...) version of a neo-Fregean one by showing that it is bound to acknowledge certain senselike entities, the latter claim is based upon the supposition that a neo-Fregean theory could be easily turned into a particular version of a Millian one by showing that De Re senses are theoretically superfluous and hence eliminable. The question how many accounts of singular reference and content are we confronted with here — Two different (and mutually antagonistic) theories? Or just two versions of what is in essence the same theory? — is surely of importance to anyone interested in the topic. And this question should be answered by means of a careful assessment of the soundness of each of the above claims. Before trying to adjudicate between the two accounts, one would naturally want to know whether or not there are indeed two substantially disparate accounts. Grosso modo, if the Fregean Claim were sound then we would have a single general conception of singular reference to deal with, viz. Fregeanism; likewise, if the Millian Claim were sound we would be facing a single general conception of singular reference, viz. Millianism. My view is that both the Fregean Notational Variance Claim and its Millian counterpart are wrong, though naturally on different grounds. I have argued elsewhere that the Fregean Notational Variance Claim - considered in its application to the semantics of propositional-attitude reports involving proper names — is unsound. I intend tosupplement in this paper such a result by trying to show that the Millian Claim - taken in its application to the semantics of indexical expressions — should also be rated as incorrect. I focus on a certain set of arguments for the Millian Claim, arguments which I take as adequately representing the general outlook of the Millian theorist with respect to neo-Fregeanism about indexicals and which involve issues about the cognitive significance of sentences containing indexical terms. (shrink)

Numbers are concepts whose content, structure, and organization are influenced by the material forms used to represent and manipulate them. Indeed, as argued here, it is the inclusion of multiple forms (distributed objects, fingers, single- and two-dimensional forms like pebbles and abaci, and written notations) that is the mechanism of numerical elaboration. Further, variety in employed forms explains at least part of the synchronic and diachronic variability that exists between and within cultural number systems. Material forms also impart characteristics (...) like linearity that may persist in the form of knowledge and behaviors, ultimately yielding numerical concepts that are irreducible to and functionally independent of any particular form. Material devices used to represent and manipulate numbers also interact with language in ways that reinforce or contrast different aspects of numerical cognition. Not only does this interaction potentially explain some of the unique aspects of numerical language, it suggests that the two are complementary but ultimately distinct means of accessing numerical intuitions and insights. The potential inclusion of materiality in contemporary research in numerical cognition is advocated, both for its explanatory power, as well as its influence on psychological, behavioral, and linguistic aspects of numerical cognition. (shrink)

This chapter examines the pioneering work of Gottfried Wilhelm Leibniz (1646-1716) on various number systems, in particular binary, which he independently invented in the mid-to-late 1670s, and hexadecimal, which he invented in 1679. The chapter begins with the oft-debated question of who may have influenced Leibniz’s invention of binary, though as none of the proposed candidates is plausible I suggest a different hypothesis, that Leibniz initially developed binary notation as a tool to assist his investigations in mathematical problems that were (...) exercising him at the time, namely those concerning the divisibility of composite numbers, primality, and perfect numbers. The chapter then explores Leibniz’s development of binary, his little-known work on binary fractions and expansions, his use of binary as a symbol for creation in his philosophical-theology, and his response to the suggestion that there was a correlation between binary numeration and the hexagrams of the ancient Chinese divinatory text, the Yijing. The chapter then focuses on Leibniz’s work on other number systems, in particular his invention and exploration of hexadecimal as well as his work on duodecimal. The chapter concludes by revealing a hitherto unknown practical application of binary that Leibniz devised in the last year of his life. (shrink)

This paper argues that the theory of structured propositions is not undermined by the Russell-Myhill paradox. I develop a theory of structured propositions in which the Russell-Myhill paradox doesn't arise: the theory does not involve ramification or compromises to the underlying logic, but rather rejects common assumptions, encoded in the notation of the $\lambda$-calculus, about what properties and relations can be built. I argue that the structuralist had independent reasons to reject these underlying assumptions. The theory is given both a (...) diagrammatic representation, and a logical representation in a novel language. In the latter half of the paper I turn to some technical questions concerning the treatment of quantification, and demonstrate various equivalences between the diagrammatic and logical representations, and a fragment of the $\lambda$-calculus. (shrink)

W. V. Quine famously defended two theses that have fallen rather dramatically out of fashion. The first is that intensions are “creatures of darkness” that ultimately have no place in respectable philosophical circles, owing primarily to their lack of rigorous identity conditions. However, although he was thoroughly familiar with Carnap’s foundational studies in what would become known as possible world semantics, it likely wouldn’t yet have been apparent to Quine that he was fighting a losing battle against intensions, due in (...) large measure to developments stemming from Carnap’s studies and culminating in the work of Kripke, Hintikka, and Bayart. These developments undermined Quine’s crusade against intensions on two fronts. First, in the context of possible world semantics, intensions could after all be given rigorous identity conditions by defining them (in the simplest case) as functions from worlds to appropriate extensions, a fact exploited to powerful and influential effect in logic and linguistics by the likes of Kaplan, Montague, Lewis, and Cresswell. Second, the rise of possible world semantics fueled a strong resurgence of metaphysics in contemporary analytic philosophy that saw properties and propositions widely, fruitfully, and unabashedly adopted as ontological primitives in their own right. This resurgence — happily, in my view — continues into the present day. -/- For a time, at any rate, Quine experienced somewhat better success with his second thesis: that higher-order logic is, at worst, confused and, at best, a quirky notational alternative to standard first-order logic. However, Quine notwithstanding, a great deal of recent work in formal metaphysics transpires in a higher-order logical framework in which properties and propositions fall into an infinite hierarchy of types of (at least) every finite order. Initially, the most philosophically compelling reason for embracing such a framework since Russell first proposed his simple theory of types was simply that it provides a relatively natural explanation of the paradoxes. However, since the seminal work of Prior there has been a growing trend to consider higher-order logic to be the most philosophically natural framework for metaphysical inquiry, many of the contributors to this volume being among the most important and influential advocates of this view. Indeed, this is now quite arguably the dominant view among formal metaphysicians. -/- In this paper, and against the current tide, I will argue in §1 that there are still good reasons to think that Quine’s second battle is not yet lost and that the correct framework for logic is first-order and type-free — properties and propositions, logically speaking, are just individuals among others in a single domain of quantification — and that it arises naturally out of our most basic logical and semantical intuitions. The data I will draw upon are not new and are well-known to contemporary higher-order metaphysicians. However, I will try to defend my thesis in what I believe is a novel way by suggesting that these basic intuitions ground a reasonable distinction between “pure” logic and non-logical theory, and that Russell-style semantic paradoxes of truth and exemplification arise only when we move beyond the purely logical and, hence, do not of themselves provide any strong objection to a type-free conception of properties and propositions. -/- Most of my arguments in §1 are largely independent of any specific account of the nature of properties and propositions beyond their type-freedom. However, I will in addition argue that there are good reasons to take propositions, at least, to be very fine-grained. My arguments are thus bolstered significantly if it can be shown that there are in fact well-defined examples of logics that are not only type-free but which comport with such a conception of propositions. It is the purpose of §2 to lay out a logic of this sort in some detail, drawing especially upon work by George Bealer and related work of my own. With the logic in place, it will be possible to generalize the line of argument noted above regarding Russell-style paradoxes and, in §3, apply it to two propositional paradoxes — the Prior-Kaplan paradox and the Russell-Myhill paradox — that are often taken to threaten the sort of account developed here. (shrink)

Inspired by the eminently successful physical theories and informed by commonplace experiences such as seeing a cat upon looking at a cat, conscious experience is thought of as a measurement or photocopy of given stimulus. Conscious experience, unlike a photocopy, is symbolic—like language—in that the relation between conscious experience and physical stimulus is analogous to that of the word "cat" and its meaning, i.e., arbitrary and yet systematic. We present arguments against the photocopy model and arguments for a symbolic conception (...) of conscious experience. Learning and the corresponding plasticity of the brain make a strong case for the symbolic conception of conscious experience, while many extra-ordinary conscious experiences argue against the formalisation of consciousness as a measuring device. The notions of place-value notation and grammar, which organise quantitative measurements and conscious experiences in the medium of numbers and language, respectively, are suggested as model systems for putting together a comprehensive theory of conscious experience. (shrink)

In “Semantics of Fictional Terms,” Garcia-Carpintero critically surveys the most recent literature on the topic of fictional names. One of his targets is realism about fictional discourse. Realists about fictional discourse believe that: (a) it contains true sentences that have fictional names as their subjects; (b) sentences containing names can be true only if those names have referents; (c) fictional names have fictional characters – abstract objects – as their referents. The fundamental problem that arises for realists is that not (...) all true sentences containing fictional names are plausibly about abstract objects. This leads to the need to introduce disjunctive conceptions of property attribution that Garcia-Carpintero claims are implausible, and that realism should therefore be rejected. He also maintains, however, that (a) is correct. I agree. Furthermore, I am also committed to anti-realism about fictional discourse – that fictional names have no referents. Garcia-Carpintero claims that my view is simply a notational variant of realism. I argue that this is false – that my view could not possibly be a notational variant of any extant realist theory. (shrink)

According to Russell, the notation in Principia Mathematica has been designed to avoid the vagueness endemic to our natural language. But what does Russell think vagueness is? My argument is an attempt to show that his views on vagueness evolved and that the final conception he adopts is not coherent. Three phases of his conception of vagueness are identified, the most significant being the view that he articulates on vagueness in his 1923 address to the Jowett Society. My central thesis (...) is that the 1912 conception of vagueness -- what I characterize as "semantic egalitarianism" -- seriously conflicts with the later paradoxical account of vagueness. (shrink)

The characterization of early token-based accounting using a concrete concept of number, later numerical notations an abstract one, has become well entrenched in the literature. After reviewing its history and assumptions, this article challenges the abstract–concrete distinction, presenting an alternative view of change in Ancient Near Eastern number concepts, wherein numbers are abstract from their inception and materially bound when most elaborated. The alternative draws on the chronological sequence of material counting technologies used in the Ancient Near East—fingers, tallies, (...) tokens, and numerical notations—as reconstructed through archaeological and textual evidence and as interpreted through Material Engagement Theory, an extended-mind framework in which materiality plays an active role (Malafouris 2013). (shrink)

The paper proposes a way for adherents of Fregean, structured propositions to designate propositions and other complex senses/concepts using a special kind of functor. I consider some formulations from Peacocke's works and highlight certain problems that arise as we try to quantify over propositional constituents while referring to propositions using "that"-clauses. With the functor notation, by contrast, we can quantify over senses/concepts with objectual, first-order quantifiers and speak without further ado about their involvement in propositions. The functor notation also turns (...) out to come with an important kind of expressive strengthening, and is shown to be neutral on several controversial issues. (shrink)

According to an influential view, when it comes to representing reality, some words are better suited for the job than others. This is elitism. There is reason to believe that the set of the best, or elite, words should not be redundant or arbitrary. However, we are often forced to choose between these two theoretical vices, especially in cases involving theories that seem to be mere notational variants. This is the riddle of redundancy: both redundancy and arbitrariness are vicious, but (...) there are cases in which one must be picked. Logical realists admit that there are some logical constants among the elite words. This leads to awkward questions, such as which among conjunction and disjunction is elite. In this paper, I show how the riddle of redundancy arises for logical realists, and offer a solution. This approach requires us to change how we represent negation. Instead of using some particular symbol, we represent negation by flipping formulae over the horizontal axis. (shrink)

In the course of daily life we solve problems often enough that there is a special term to characterize the activity and the right to expect a scientific theory to explain its dynamics. The classical view in psychology is that to solve a problem a subject must frame it by creating an internal representation of the problem’s structure, usually called a problem space. This space is an internally generable representation that is mathematically identical to a graph structure with nodes and (...) links. The nodes can be annotated with useful information, and the whole representation can be distributed over internal and external structures such as symbolic notations on paper or diagrams. If the representation is distributed across internal and external structures the subject must be able to keep track of activity in the distributed structure. Problem solving proceeds as the subject works from an initial state in mentally supported space, actively constructing possible solution paths, evaluating them and heuristically choosing the best. Control of this exploratory process is not well understood, as it is not always systematic, but various heuristic search algorithms have been proposed and some experimental support has been provided for them. (shrink)

This article considers two different ways of formulating a desire-satisfaction theory of prudential value. The first version of the theory (the object view) assigns basic prudential value to the state of affairs that is the object of a person’s desire. The second version (the combo view) assigns basic prudential value to the compound state of affairs in which (a) a person desires some state of affairs and (b) this state of affairs obtains. My aims in this article are twofold. First, (...) I aim to highlight that these are not mere notational variants, but in fact have quite different implications, so that this distinction is not one that the theorist of prudential value should ignore. More positively, I argue that the object view is better able to capture what is distinctive and appealing about subjective theories of prudential value, on any plausible account of what the central subjectivist insight is. (shrink)

In a review of Frege's Puzzle1, Graeme Forbes makes the claim that Salmon's account of belief might be seen, under certain conditions, as a mere notational variant of a neo-Fregean theory; and thus that such an account might be reduced to a neo-Fregean one simply by rewriting it in terms of Fregean terminology. With a view to supporting his claim, Forbes offers an outline of an account of belief which, according to him, would satisfy the following conditions: (i) it could (...) be directly obtained from Salmon's own analysis by means of a certain set of substitutions, which presumably would not affect the essential features of Salmon's view; (ii) it could naturally be described as Fregean, in the sense that it would preserve, (at least) the spirit of Frege's doctrines, especially his fundamental intuitions about belief. Of course, the upshot of Forbes's argument is that Salmon's theory would not, at bottom, constitute a genuine alternative to a Fregean semantics for belief ascriptions. In this paper I argue to the effect that Forbes's claim is not in general sound. It seems to me that the sort of indirect argument used by Forbes - that of trying to undermine Salmon's theory by showing that it is just a version of a neo-Fregean account - does not provide someone working within a Fregean framework with an adequate strategy to counter Salmon's neo-Russellian views. It would perhaps be better to concentrate a Fregean attack on certain apparently dubious and highly controversial theses and results which are constitutive of Salmon's view, e.g. the counterintuitive character of a substantial set of consequences which follow from his theory of belief, as well as the associated revisionist stand he is forced to take towards our current patterns of speaking about belief. (shrink)

The minimalist view of truth endorsed by Paul Horwich denies that truth has any underlying nature. According to minimalism, the truth predicate ‘exists solely for the sake of a certain logical need’; ‘the function of the truth predicate is to enable the explicit formulation of schematic generalizations’. Horwich proposes that all there really is to truth follows from the equivalence schema: The proposition that p is true iff p, or, using Horwich’s notation, ·pÒ is true ´ p. The (unproblematic) instances (...) of the schema form ‘the minimal theory of truth’. Horwich claims that all the facts involving truth can be explained on the basis of the minimal theory. However, it has been pointed out, e.g. by Gupta (1993), that the minimal theory is too weak to entail any general facts about truth, e.g. the fact that.. (shrink)

If the begriffsschrift from Frege does represent the logical form of natural language it either lacks a logical form itself, or its logical form is different to that of natural language. But Frege insists that his notation has a logical form. So the second disjunct holds. This suggests that Frege's notation will generate consequences different to those that can be derived with natural language, with its different logical form. For anyone looking for "a means of avoiding misunderstandings", as Frege does, (...) this would be unpalatable, to say the least. (shrink)

Some philosophers defend the view that epistemic agents believe by lending credence. Others defend the view that such agents lend credence by believing. It can strongly appear that the disagreement between them is notational, that nothing of substance turns on whether we are agents of one sort or the other. But that is demonstrably not so. Only one of these types of epistemic agent, at most, could manifest a human-like configuration of attitudes; and it turns out that not both types (...) of agent are possible. (shrink)

Although traditionally neglected, mathematical diagrams have recently begun to attract attention from philosophers of mathematics. By now, the literature includes several case studies investigating the role of diagrams both in discovery and justification. Certain preliminary questions have, however, been mostly bypassed. What are diagrams exactly? Are there different types of diagrams? In the scholarly literature, the term “mathematical diagram” is used in diverse ways. I propose a working definition that carves out the phenomena that are of most importance for a (...) taxonomy of diagrams in the context of a practice-based philosophy of mathematics, privileging examples from contemporary mathematics. In doing so, I move away from vague, ordinary notions. I define mathematical diagrams as forming notational systems and as being geometric/topological representations or two-dimensional representations. I also examine the relationship between mathematical diagrams and spatiotemporal intuition. By proposing an explication of diagrams, I explain certain controversies in the existing literature. Moreover, I shed light on why mathematical diagrams are so effective in certain instances, and, at other times, dangerously misleading. (shrink)

When people make sense of situations, illustrations, instructions and problems they do more than just think with their heads. They gesture, talk, point, annotate, make notes and so on. What extra do they get from interacting with their environment in this way? To study this fundamental problem, I looked at how people project structure onto geometric drawings, visual proofs, and games like tic tac toe. Two experiments were run to learn more about projection. Projection is a special capacity, similar to (...) perception, but less tied to what is in the environment. Projection, unlike pure imagery, requires external structure to anchor it, but it adds ‘mental’ structure to the external scene much like an augmented reality system adds structure to an outside scene. A person projects when they look at a chessboard and can see where a knight may be moved. Because of the cognitive costs of sustaining and extending projection, humans make some of their projections real. They create structure externally. They move the piece, they talk, point, notate, represent. Much of our interactivity during sense making and problem solving involves a cycle of projecting then creating structure. (shrink)