Incomplete Symbols Research Articles

Superextension, Modal Assertability, and Hypothetical Discernibility:

We argue that Russell’s treatment of propositions containing non-existent denotations is the prelude of a strategy for discerning the predictive content of any proposition with descriptions (and incomplete symbols), insofar as its indeterminacy is classifiable as a possible extension. We argue, further, that Russell’s theory of propositional functions is dedicated to explaining the discernibility of hypothetical propositions, through representing (what we call) the superextension of assertoric modal propositions. The indeterminacy of the truth of hypothetical propositions is thus paired with a second- order classification condition, expressing its conjectural content as the projected scope of instantiation of a propositional function; which in turn is the superextension of a modal predicate. Russell’s theory is his non-pragmatic answer to the problem of assertability in unstable and uncertain conditions. It involves a sophisticated thesis about the discernibility of complex contents, such as those represented by analogical and approximate denotations (descriptions, fictions, etc.), which classify more than one possibility of instantiation in different modal and intensional contexts. We will conclude this article by testing the assumption that Russell’s theory favors a coherentist theory, seeing that as it admits assertability conditions that are super-mapped in a system, or that derive from different orders of referential layers.

On the incomplete second Appell hypergeometric matrix functions

ABSTRACT In this paper, we introduce the incomplete second Appell hypergeometric matrix functions in terms of the incomplete Pochhammer matrix symbols. We also derive certain properties such as matrix differential equation, integral formula, recurrence relation, recursion formula and differentiation formula of the incomplete second Appell hypergeometric matrix functions. This enriches the theory of special matrix functions. The results listed here are believed to be new.

STUDENTS' REPRESENTATION IN SOLVING WORD PROBLEM

The purpose of this research is to describe the representation of sixth grade students in solving mathematics word problems. The focus of the representation of this research is an external representation which is viewed from students with high mathematical abilities. The method used in this research is task-based interview, by giving a problem test of word problems. Students who have a high level of abilities, he makes pictures of all problems and successfully solve the problems. Students whose level of abilities is lacking, he only makes incomplete symbol / verbal representations, he has wrong when solving the problems. Various kinds of representations and increasing abilities in many problems such as multiplying exercises and solve the word pronlem. Applying various representations to students are very important to be improved by students in order to succeed in solving various mathematical word problems.

"Distinction of Reason" is an Incomplete Symbol

"Distinction of Reason" is an Incomplete Symbol

Susan Stebbing, Incomplete Symbols and Foundherentist Meta-Ontology

Susan Stebbing’s work on incomplete symbols and analysis was instrumental in clarifying, sharpening, and improving the project of logical constructions which was pivotal to early analytic philosophy. She dispelled use-mention confusions by restricting the term ‘incomplete symbol’ to expressions eliminable through analysis, rather than those expressions’ purported referents, and distinguished linguistic analysis from analysis of facts. In this paper I explore Stebbing’s role in analytic philosophy’s development from anti-holism, presupposing that analysis terminates in simples, to the more holist or foundherentist analytic philosophy of the later 20th century. I read Stebbing as a transitional figure who made room for more holist analytic movements, e.g., applications of incomplete symbol theory to Quinean ontological commitment. Stebbing, I argue, is part of a historical narrative which starts with the holism of Bradley, an early influence on her, to which Moore and Russell’s logical analysis was a response. They countered Bradley’s holist reservations about facts with the view that the world is built up out of individually knowable simples. Stebbing, a more subtle and sympathetic reader of the British idealists, defends analysis, but with important refinements and caveats which prepared the way for a return to foundherentism and holism within analytic philosophy.

Incomplete Symbols — Definite Descriptions Revisited

We investigate incomplete symbols, i.e. definite descriptions with scope-operators. Russell famously introduced definite descriptions by contextual definitions; in this article definite descriptions are introduced by rules in a specific calculus that is very well suited for proof-theoretic investigations. That is to say, the phrase ‘incomplete symbols’ is formally interpreted as to the existence of an elimination procedure. The last section offers semantical tools for interpreting the phrase ‘no meaning in isolation’ in a formal way.

Computer Vision Recognition of Incomplete Symbols in Russian Symbols

In the computer vision recognition of incomplete symbols in Russian symbols, the traditional identification methods can only identify a small number of complete Russian symbols, and have a low recognition rate of the incomplete Russian symbols. To this end, this paper presents a method for computer vision recognition of incomplete symbols in Russian symbols based on Hough transform algorithm. According to the mapping from the image space to the parameter space, the complex edge feature information in image space is transformed into the clustering problem in the parameter space, and the discrimination function and the rules are developed and employed to recognize the image need to be recognized. Experiments show that with Hough transform algorithm to identify incomplete symbols in Russian symbols, the incomplete symbol in Russian symbols can be identified quickly and effectively, which improves the performance of recognition method and meet the needs of many scholars.

An Integer Time Delay Estimation Algorithm Based on Zadoff–Chu Sequence in OFDM Systems

This paper presents a multiple signal classification (MUSIC)-based symbol integer delay estimation algorithm for orthogonal frequency-division multiplexing (OFDM) systems using the Zadoff-Chu (ZC) sequence. By exploiting a property of the ZC sequence, this algorithm is derived from the maximum-likelihood (ML) integer delay estimation, which retains the ability of coping with an incomplete OFDM symbol with reduced complexity. It can be realized without information on channel statistic and noise. Furthermore, taking into account the inefficiency of the MUSIC-based algorithm in estimating the integer delay if the fractional delay is compensated on local ZC, we present a real-valued delay-tracking scheme that iteratively adjusts the sampling clock phase of an analog-to-digital converter (ADC). In particular, the fractional delay is tracked via the ML scheme, whereas the integer delay is estimated via the MUSIC-based algorithm. The numerical results finally confirm the efficiency and the reliability of the proposed MUSIC-based algorithm, even in the case of a small number of samples.

Typos of Principia Mathematica

Principia Mathematic goes to great lengths to hide its order/type indices and to make it appear as if its incomplete symbols behave as if they are singular terms. But well-hidden as they are, we cannot understand the proofs in Principia unless we bring them into focus. When we do, some rather surprising results emerge – which is the subject of this paper.

Russell on Incomplete Symbols

AbstractRussell's notion of an incomplete symbol has become a standard against which philosophers compare their views on the relationship between language and the world. But Russell's exact characterization of incomplete symbols and the role they play in his philosophy are still disputed. In this paper, I trace the development of the notion of an incomplete symbol in Russell's philosophy. I suggest – against Kaplan, Evans, and others – that Russell's many characterizations of the notion of an incomplete symbol are compatible. To this end, I examine and reject arguments for the purported incompatibility between declaring an expression to be incomplete and incorporating that symbol into a compositional semantic theory. I then examine how Russell puts the notion of an incomplete symbol to use in metaphysics.

The paradoxes and Russell’s theory of incomplete symbols

Russell claims in his autobiography and elsewhere that he discovered his 1905 theory of descriptions while attempting to solve the logical and semantic paradoxes plaguing his work on the foundations of mathematics. In this paper, I hope to make the connection between his work on the paradoxes and the theory of descriptions and his theory of incomplete symbols generally clearer. In particular, I argue that the theory of descriptions arose from the realization that not only can a class not be thought of as a single thing, neither can the meaning/intension of any expression capable of singling out one collection (class) of things as opposed to another. If this is right, it shows that Russell’s method of solving the logical paradoxes is wholly incompatible with anything like a Fregean dualism between sense and reference or meaning and denotation. I also discuss how this realization lead to modifications in his understanding of propositions and propositional functions, and suggest that Russell’s confrontation with these issues may be instructive for ongoing research.

Principia Mathematica at 100

May 14, 2011 (10:00 am) C:\Users\Milt\WP data\TYPE3101\russell 31,1 078 red 002.wpd russell: the Journal of Bertrand Russell Studies n.s. 31 (summer 2011): 5–8 The Bertrand Russell Research Centre, McMaster U. issn 0036-01631; online 1913-8032 reface PRINCIPIA MATHEMATICAz AT 100 Nicholas Grif{f{in Russell Research Centre / McMaster U. Hamilton, on, Canada l8s 4l6 ngrif{f{in@mcmaster.ca Bernard Linsky Philosophy / U. of Alberta Edmonton, ab, Canada t6g 2e5 bernard.linsky@ualberta.ca A zll the papers in this special issue of Russellz are connected with the zPMy@z100 conference held at the Bertrand Russell Research zCentre at McMaster University to celebrate the centenary of the publication of the Wrst volume of Whitehead and Russell’s Principia Mathematica. The conference took place 21–24 May 2010, and over 60 people from ten diTerent countries attended to hear more than 30 papers coveringtheorigins,impact,philosophicalmotivationandmathematical content of that vast and still relatively unexplored work. All but two of the nine papers in this collection were presented at the PMy@z100 conference . Of the two exceptions, Anellis’s paper was included in the conference programme, but in the event Anellis was unable to attend and so the paper was never presented. Blackwell’s paper, by contrast, was presented as an after-dinner talk, with logicians in attendance, to the 2010 annual meeting of the Bertrand Russell Society which was held at McMaster over the same weekend at the PMy@z100 conference. The nine papers published here exemplify the range of the conference itself. Graham Stevens’ paper considers the view, much discussed by contemporary philosophers of language (cf.z notably Neale 1990, 1993, 2008), that all noun phrases are either semantically structured quantiWers or semantically unstructured singular terms. This view seems like a natural extension of the sharp distinction Russell made in his theory of descriptionsbetweendeWnitedescriptions ,whicharehandledquantiWcationally, May 14, 2011 (10:00 am) C:\Users\Milt\WP data\TYPE3101\russell 31,1 078 red 002.wpd 6 nicholas grif{f{in and bernard linsky and logically proper names, which are genuine singular terms. And yet there is a diUculty, which Stevens addresses in his paper. One reason Neale reformulates Russell’s theory of descriptions in terms of binary or restricted quantiWers is to narrow the gap Russell famously opened between the logical form of a sentence, as revealed after its analysis by the theory of descriptions, and the grammatical form that the unanalyzed sentence has in natural language. The outcome of this seems to be that Russell’s claim that deWnite descriptions are incomplete symbols is lost in the reformulation. Stevens argues that this is not the case, but only if “incomplete symbol” is properly understood, not as a symbol having no meaning, but as one which does not contribute an object to a proposition . Ray Perkins deals with another aspect of the very same issue, but with a quite diTerent approach. Perkins is concerned with a famous argument in Principiaz designed to show that deWnite descriptions are incomplete symbols. The argument has been frequently criticized, starting immediately with the book’s publication (Jones 1910–11), precisely because of its dependence on an alleged ambiguity in the notion of meaning it deploys. Perkins defends the argument against this charge and, addressing the concerns of the historical Russell rather than his modern followers , goes on to explore the implications of the argument for Russell’s epistemology, metaphysics and his conception of analysis. The theory of descriptions is one of Russell’s two best-known and most important contributions to Principiaz; the theory of types is the other. Yet the theory of types was never considered, even by its creator, as the triumphant success that the theory of descriptions was. It brought with it too many problems. One of them was the axiom of reducibility, discussed in Russell Wahl’s paper. The axiom was essential if the logicist project was to go forward; without it, for example, least upper bounds in real number theory would fall outside the sets they bounded. And yet the axiom’s status as an axiom of logic is very much in doubt. It requires the existence of very...

Incomplete Symbols in Principia Mathematica and Russell’s “Definite Proof”

Early in Principia Mathematica Russell presents an argument that “‘the author of Waverley means nothing”, an argument that he calls a “definite proof”. He generalizes it to claim that definite descriptions are incomplete symbols having meaning only in sentential context. This Principia “proof” went largely unnoticed until Russell reaffirmed a near-identical “proof” in his philosophical autobiography nearly 50 years later. The “proof” is important, not only because it grounds our understanding of incomplete symbols in the Principia programme, but also because failure to understand it fully has been a source of much unjustified criticism of Russell to the effect that he was wedded to a naïve theory of meaning and prone to carelessness and confusion in his philosophy of logic and language generally. In my paper, I (1) defend Russell’s “proof” against attacks from several sources over the last half century, (2) examine the implications of the “proof” for understanding Russell’s treatment of class symbols in Principia, and (3) see how the Principia notion of incomplete symbol was carried forward into Russell’s conception of philosophical analysis as it developed in his logical atomist period after 1910.

Logical form in Principia Mathematica and English

The theory of descriptions, presented informally in “On Denoting” and more formally in Principia Mathematica, has been endorsed by many linguists and philosophers of language as a contribution to natural-language semantics. However, the syntax of Principia’s formal language is far from ideal as a tool for the analysis of natural language. Stephen Neale has proposed a reconstruction of the theory of descriptions in a language of restricted quantification that gives a better approximation of the syntax of English (and, arguably, of other natural languages). This has led to resistance from some Russell scholars who object to the identification of descriptions with quantifiers at the level of logical form in this new language on the grounds that the identification fails to respect the Russellian conception of descriptions as incomplete symbols. I defend Neale’s reconstruction of the theory and argue that he has preserved everything essential to the theory, including the notion of an incomplete symbol. However, I then go on to argue, contrary to Neale and his objectors as well as Russell himself, that the doctrine of incomplete symbols is a superfluous and undesirable element of the theory that is best jettisoned from the theory.

Logical Form in <em>Principia Mathematica</em> and English

The theory of descriptions, presented informally in "On Denoting" and more formally in Principia Mathematica, has been endorsed by many linguists and philosophers of language as a contribution to natural-language semantics. However, the syntax of Principia’s formal language is far from ideal as a tool for the analysis of natural language. Stephen Neale has proposed a reconstruction of the theory of descriptions in a language of restricted quantification that gives a better approximation of the syntax of English (and, arguably, of other natural languages). This has led to resistance from some Russell scholars who object to the identification of descriptions with quantifiers at the level of logical form in this new language on the grounds that the identification fails to respect the Russellian conception of descriptions as incomplete symbols. I defend Neale’s reconstruction of the theory and argue that he has preserved everything essential to the theory, including the notion of an incomplete symbol. However, I then go on to argue, contrary to Neale and his objectors as well as Russell himself, that the doctrine of incomplete symbols is a superfluous and undesirable element of the theory that is best jettisoned from the theory.

Incomplete Symbols in <em>Principia Mathematica</em> and Russell’s “Definite Proof”

Early in Principia Mathematica Russell presents an argument that "‘the author of Waverley’ means nothing", an argument that he calls a "definite proof". He generalizes it to claim that definite descriptions are incomplete symbols having meaning only in sentential context. This Principia "proof" went largely unnoticed until Russell reaffirmed a near-identical "proof" in his philosophical autobiography nearly 50 years later. The "proof" is important, not only because it grounds our understanding of incomplete symbols in the Principia programme, but also because failure to understand it fully has been a source of much unjustified criticism of Russell to the effect that he was wedded to a naive theory of meaning and prone to carelessness and confusion in his philosophy of logic and language generally. In my paper, I (1) defend Russell’s "proof" against attacks from several sources over the last half century, (2) examine the implications of the "proof" for understanding Russell’s treatment of class symbols in Principia, and (3) see how the Principia notion of incomplete symbol was carried forward into Russell’s conception of philosophical analysis as it developed in his logical atomist period after 1910.

Russell on Substitutivity and the Abandonment of Propositions

One of the main arguments Russell offers for the Theory of Descriptions in “On Denoting” is that it can be used to solve a puzzle about substitutivity. This article argues that, owing to a widespread mischaracterization of the substitutivity principle Russell means to be defending, neither the puzzle itself nor Russell's solution to it have been well understood. The principle Russell seeks to defend concerns not the substitution of expressions in a sentence but rather the substitution of propositional constituents in a Russellian proposition. This article further argues that Russell's solution requires him to adopt certain substantive views about the nature of the referents of what are usually called “logically proper names.” In particular, Russell's solution will work only if the referents of such names are given to one who understands them in a manner that is entirely “aspect-free.” Russell's notion of acquaintance, since it fits this bill, is therefore motivated by his solution to the puzzle, as is his choice of sense-data to be the referents of genuine Russellian names of particulars. Finally, the article argues that since a version of the George IV puzzle arises for propositions, Russell's solution commits him to regarding some instances of sentences as incomplete symbols and to denying that we have acquaintance with propositions. This in turn leads Russell in Principia Mathematica to adopt the Multiple Relation Theory of Judgment, and thus to embrace agnosticism about propositions. His further move to the stronger position of full eliminativism concerning propositions is motivated only by his late-in-the-day espousal of the Principle of Sufficient Reason.

Sharvy's theory of descriptions: a paradigm subverted

By now his theory has attained the status of a new paradigm. Even a casual trawl of the literature throws up over a score of citations and endorsements, and no rebuttals of its central contention that the primary use of 'the' is to indicate totality (621). We shall argue that it is wrong for mass description, wrong for singular description, wrong for plural description and wrong for generic 'the'. Sharvy would have approved: his obituary says that his car displayed the slogan 'Subvert the Dominant Paradigm'. He follows Russell in treating descriptions as incomplete symbols, but nothing in his paper turns on this aspect of it, and we can bypass the issue by considering them in the context of complete sentences. Where Russell translates F(the G) into the predicate calculus as

Russell’s “paralysis”

January 28, 2009 (12:22 pm) G:\WPData\TYPE2802\russell 28,2 051red.wpd 182 Reviews RUSSELL’S “PARALYSIS” James Connelly Philosophy, Trent U./York U. Peterborough, on k9l 1z6 / Toronto, on m3j 1p3, Canada jamesconnelly@trentu.ca Rosalind Carey. Russell and Wittgenstein on the Nature of Judgement. London and New York: Continuum International, 2007. Pp. [viii], 150. isbn 978-0-82648811 -4. us$132.00; £65.00 (hb). Rosalind Carey’s new book, Russell and Wittgenstein on the Nature of Judgement , oTers a fresh look at Wittgenstein’s fateful but notoriously obscure criticisms of Russell’s so-called “multiple-relation theory of judgment”, oTered in May–June of 1913 as Russell was hastily composing the Theory of Knowledge manuscript. In light of the fact that this manuscript was abandoned and subsequently suppressed as a result of these criticisms, and given the fact that Russell nevertheless went on the develop and defend new versions of the multiple- January 28, 2009 (12:22 pm) G:\WPData\TYPE2802\russell 28,2 051red.wpd Reviews 183 1 In giving this reading, Carey follows Peter Hylton, Russell, Idealism, and the Emergence of Analytic Philosophy (Oxford, UK: Clarendon P., 1990), esp. pp. 266–7, and relation theory long after these events, Carey sets out to answer the question: “what objections to his theory of judgment could have allowed Russell to think that his theory of judgment was still viable, but suUced to cause him to abandon the manuscript?” (p. 2). Through a careful and detailed examination of extant correspondence, working notes and diagrams, as well as of both the manuscript itself and subsequent work of Russell’s, Carey attempts to reconstruct the events of May–June 1913 in such a way as to give a plausible answer to this question. In so doing, she makes a powerful case against the orthodox interpretation of Wittgenstein ’s criticisms and Russell’s reactions to them, an interpretation according to which the criticisms culminate in the so-called “postcard” objection of June 1913 and concern either issues of logical type, or the unity of the proposition. Carey downplays the signiWcance of the June letter and instead sees Russell’s paralysis as consisting in his having no ready analysis of the form of beliefz which will both account for the bipolarity of judgment as well as accommodate Wittgenstein ’s critique. Unwilling to eliminate belief and so succumb to Wittgenstein ’s more austere extensionalism, Russell had no choice but to cease writing the manuscript, while holding out hope that an analysis of the form of belief which is immune to Wittgenstein’s critique would someday be forthcoming. Carey develops her case over the course of four compact chapters (the whole of which make for a quite digestible 120 pages), each of which builds on the insights of those coming previously. In Chapter 1, Carey looks at the development of Russell’s epistemological and logical views over the period from 1903 through 1910, for the express purposes of showing “how the multiple relation theory of judgment emerges from and is inXuenced by a number of problems and doctrines that contribute to his eventual paralysis in the face of Wittgenstein’s objections” (p. 6). She begins by introducing Russell’s early conception of belief as involving a dual relation between a subject and a complex unity, a so-called “Russellian proposition”, and grounds this conception in Russell’s view to the eTect that word meanings consist in mind- and language-independent, often Platonic, objects and properties. The relation between these semantic doctrines and Russell’s characterization of logical truths as topically and contextually unrestricted , universal principles, is also probed. Carey then goes on to show how tensions within and amongst these views lead to important theoretical diUculties , which in turn motivate the development of the theory of incomplete symbols. Notably, Carey reads the theory of incomplete symbols as designed primarily to address the sorts of concerns outlined in the so-called “Grey’s Elegy argument” contained in “On Denoting” and having only secondarily to do with either issues of ontological parsimony, or diUculties surrounding non-referring singular terms, e.g. “The Present King of France”.1 That is, she reads the theory January...

Truth and Simplicity

‘Truth and Simplicity’ is the title we have supplied for a very remarkable nine page typescript of a talk that Ramsey gave to the Cambridge Apostles on April 29, 1922.1 It is a relatively early work but it already touches even in such brief compass on several topics of fundamental philosophical interest: truth, ontological simplicity, the theory of types, facts, universals, and probability, that are recurrent themes found in most of Ramsey’s papers, published and posthumous. The present paper has not appeared in any of the classic collections, and it comes as a pleasant surprise—just when we thought we had seen it all.2 In the paper, Ramsey develops two ideas he thinks are related: that ‘true’ is an incomplete symbol, and ‘the only things in whose existence we have reason to believe are simple, not complex’. The first relies on Russell’s notion of an incomplete symbol and the second he says is a view now (1922) held by Russell, and, as he generously says, he got it from Russell in conversation and doubts whether he would ever have thought of it by himself. Ramsey is very clear as to what simplicity requires: