Landini discusses the second edition of Principia Mathematica, to show Russella (TM)s intellectual relationship with Wittgenstein and Ramsey.

EbertPhilip A and RossbergMarcus, eds.* * _ Essays on Frege’s Basic Laws of Arithmetic_. Oxford: Oxford University Press, 2019. Pp. xii + 673. ISBN: 978-0-19-871208-4 ; 978-0-19-102005-6, 978-0-19-178024-0. doi: 10.1093/oso/9780198712084.001.0001.

This book explores an important central thread that unifies Russell's thoughts on logic in two works previously considered at odds with each other, the Principles of Mathematics and the later Principia Mathematica. This thread is Russell's doctrine that logic is an absolutely general science and that any calculus for it must embrace wholly unrestricted variables. The heart of Landini's book is a careful analysis of Russell's largely unpublished "substitutional" theory. On Landini's showing, the substitutional theory reveals the unity (...) of Russell's philosophy of logic and offers new avenues for a genuine solution of the paradoxes plaguing Logicism. (shrink)

Wittgenstein's Tractatus has generated many interpretations since its publication in 1921, but over the years a consensus has developed concerning its criticisms of Russell's philosophy. In Wittgenstein's Apprenticeship with Russell, Gregory Landini draws extensively from his work on Russell's unpublished manuscripts to show that the consensus characterises Russell with positions he did not hold. Using a careful analysis of Wittgenstein's writings he traces the 'Doctrine of Showing' and the 'fundamental idea' of the Tractatus to Russell's logical atomist research (...) program, which dissolves philosophical problems by employing variables with structure. He argues that Russell and his apprentice Wittgenstein were allies in a research program that makes logical analysis and reconstruction the essence of philosophy. His sharp and controversial study will be essential reading for all who are interested in this rich period in the history of analytic philosophy. (shrink)

Gregory Landini offers a detailed historical account of Frege's notations and the philosophical views that led Frege from Begriffssscrhrift to his mature work Grundgesetze, addressing controversial issues that surround the notations.

This paper offers an interpretation of Russell's multiple-relation theory of judgment which characterizes it as direct application of the 1905 theory of definite descriptions. The paper maintains that it was by regarding propositional symbols (when occurring as subordinate clauses) as disguised descriptions of complexes, that Russell generated the philosophical explanation of the hierarchy of orders and the ramified theory of types of _Principia mathematica (1910). The interpretation provides a new understanding of Russell's abandoned book _Theory of Knowledge (1913), the 'direction (...) problems' and Wittgenstein's criticisms. (shrink)

In truth theory one aims at general formal laws governing the attribution of truth to statements. Gupta’s and Belnap’s revision-theoretic approach provides various well-motivated theories of truth, in particular T* and T#, which tame the Liar and related paradoxes without a Tarskian hierarchy of languages. In property theory, one similarly aims at general formal laws governing the predication of properties. To avoid Russell’s paradox in this area a recourse to type theory is still popular, as testified by recent work in (...) formal metaphysics by Williamson and Hale. There is a contingent Liar that has been taken to be a problem for type theory. But this is because this Liar has been presented without an explicit recourse to a truth predicate. Thus, type theory could avoid this paradox by incorporating such a predicate and accepting an appropriate theory of truth. There is however a contingent paradox of predication that more clearly undermines the viability of type theory. It is then suggested that a type-free property theory is a better option. One can pursue it, by generalizing the revision-theoretic approach to predication, as it has been done by Orilia with his system P*, based on T*. Although Gupta and Belnap do not explicitly declare a preference for T# over T*, they show that the latter has some advantages, such as the recovery of intuitively acceptable principles concerning truth and a better reconstruction of informal arguments involving this notion. A type-free system based on T# rather than T* extends these advantages to predication and thus fares better than P* in the intended applications of property theory. (shrink)

Frege's Grundgesetze der Arithmetik offers a conception of cpLogic as the study of functions. Among functions are included those that are concepts, i.e. characteristic functions whose values are the logical objects that are the True/the False. What, in Frege's view, are the objects the True/the False? Frege's stroke functions are themselves concepts. His stipulation introducing his negation stroke mentions that it yields [...]. But curiously no accommodating axiom is given, and there is no such theorem. Why is it that some (...) of Frege's informal stipulations never made appearances as axioms? I offer an explanation that sheds new light on the Grundgesetze. No axioms should over-determination the True as a logical object. Perhaps the True = 0, as would be common in the mathematics of characteristic functions. But the logical objects that are cardinal numbers are value ranges correlated with second-level numerical concepts by a non-homogeneous second-level value-range function [...]. The existence of concepts would be ontologically circular if the True is itself a number. We find this circularity perfectly agreeable to Frege, and suggest that he had accepted that the existence of functions that are concepts in his cpLogic may well be ontologically inseparable from the existence of his value-range function. His cpLogic itself stands or falls with the viability of some value-range function. (shrink)

This article compares the theory of Meinongian objects proposed by Edward Zalta with a theory of fiction formulated within an early Russellian framework. The Russellian framework is the second-order intensional logic proposed by Nino B. Cocchiarelly as a reconstruction of the form of Logicism Russell was examining shortly after writing The Principles of Mathematics. A Russellian theory of denoting concepts is developed in this intensional logic and applied as a theory of the "objects' of fiction. The framework retains the Orthodox (...) early Russellian ontology of existents, possible non-existents, and properties and relations in intension. This avoids the assumption, found in Meinongian theories, of impossible and incomplete objects. It also obviates the need to preserve consistency by distinguishing a new "mode of predication", or a "distinction in kinds of predicates". Thus, it is argued that an early Russellian theory forms a powerful rival to a Meinongian theory of objects. (shrink)

Unaware of Frege's 1879 Begriffsschrift, Russell's 1903 The Principles of Mathematics set out a calculus for logic whose foundation was the doctrine that any such calculus must adopt only one style of variables–entity (individual) variables. The idea was that logic is a universal and all-encompassing science, applying alike to whatever there is–propositions, universals, classes, concrete particulars. Unfortunately, Russell's early calculus has appeared archaic if not completely obscure. This paper is an attempt to recover the formal system, showing its philosophical background (...) and its semantic completeness with respect to the tautologies of a modern sentential calculus. (shrink)

There is a curious letter of 24 April 1928, reproduced in Russell’s Autobiography. It is from Russell to Max Newman. It is my thesis that there is a crucial “not” missing from the text and interpretations of the letter. This small point, if it is correct, has a very large impact for clarifying how Russell saw Newman’s challenge to his structural realism according to which all of our empirical knowledge in physics concerns structure alone.

This article compares the theory of Meinongian objects proposed by Edward Zalta with a theory of fiction formulated within an early Russellian framework. The Russellian framework is the second-order intensional logic proposed by Nino B. Cocchiarelly as a reconstruction of the form of Logicism Russell was examining shortly after writing The Principles of Mathematics. A Russellian theory of denoting concepts is developed in this intensional logic and applied as a theory of the "objects' of fiction. The framework retains the Orthodox (...) early Russellian ontology of existents, possible non-existents, and properties and relations in intension. This avoids the assumption, found in Meinongian theories, of impossible and incomplete objects. It also obviates the need to preserve consistency by distinguishing a new "mode of predication", or a "distinction in kinds of predicates". Thus, it is argued that an early Russellian theory forms a powerful rival to a Meinongian theory of objects. (shrink)

Confronted with Russell's Paradox, Frege wrote an appendix to volume II of his _Grundgesetze der Arithmetik_. In it he offered a revision to Basic Law V, and proclaimed with confidence that the major theorems for arithmetic are recoverable. This paper shows that Frege's revised system has been seriously undermined by interpretations that transcribe his system into a predicate logic that is inattentive to important details of his concept-script. By examining the revised system as a concept-script, we see how Frege imagined (...) that minor modifications of his former proofs would recover arithmetic. (shrink)

This is a critical discussion of Nino B. Cocchiarella’s book “Formal Ontology and Conceptual Realism.” It focuses on paradoxes of hyperintensionality that may arise in formal systems of intensional logic.

This book offers a comprehensive critical survey of issues of historical interpretation and evaluation in Bertrand Russell's 1918 logical atomism lectures and logical atomism itself. These lectures record the culmination of Russell's thought in response to discussions with Wittgenstein on the nature of judgement and philosophy of logic and with Moore and other philosophical realists about epistemology and ontological atomism, and to Whitehead and Russell’s novel extension of revolutionary nineteenth-century work in mathematics and logic. Russell's logical atomism lectures have had (...) a lasting impact on analytic philosophy and on Russell's contemporaries including Carnap, Ramsey, Stebbing, and Wittgenstein. Comprised of 14 original essays, this book will demonstrate how the direct and indirect influence of these lectures thus runs deep and wide. (shrink)

Frege seems to hold two incompatible theses:(i) that sentences differing in structure can yet express the same sense; and (ii) that the senses of the meaningful parts of a complex term are determinate parts of the sense of the term. Dummett offered a solution, distinguishing analysis from decomposition. The present paper offers an embellishment of Dummett?s distinction by providing a way of depicting the internal structures of complex senses?determinate structures that yield distinct decompositions. Decomposition is then shown to be adequate (...) as a foundation for the informativity and analyticity of logic. (shrink)

There are many wonderful puzzles concerning Principia Mathematica, but none are more striking than those arising from the crisis that befell Whitehead in November of 1910. Volume 1 appeared in December of 1910. Volume 2 on cardinal numbers and Russell's relation arithmetic might have appeared in 1911 but for Whitehead's having halted the printing. He discovered that inferences involving the typically ambiguous notation ‘Nc‘α’ for the cardinal number of α might generate fallacies. When the volume appeared in 1912, it was (...) extensively emended by Whitehead and accompanied by a Prefatory Statement of Symbolic Conventions. This paper endeavors to recover from Whitehead's bad emendations—including his bewildering thesis that since ‘‘α’ is ‘true whenever significant,’ ‘α is to be accepted. It is supposedly a fallacy to apply Modus Ponens and infer Nc‘α from ‘α and‘‘α. (shrink)

On investigating a theorem that Russell used in discussing paradoxes of classes, Graham Priest distills a schema and then extends it to form an Inclosure Schema, which he argues is the common structure underlying both class-theoretical paradoxes (such as that of Russell, Cantor, Burali-Forti) and the paradoxes of ?definability? (offered by Richard, König-Dixon and Berry). This article shows that Russell's theorem is not Priest's schema and questions the application of Priest's Inclosure Schema to the paradoxes of ?definability?.1 1?Special thanks to (...) Francesco Orilia for criticisms of an early draft of this article. (shrink)

This paper examines Russell's substitutional theory of classes and relations, and its influence on the development of the theory of logical types between the years 1906 and the publication of Principia Mathematica (volume I) in 1910. The substitutional theory proves to have been much more influential on Russell's writings than has been hitherto thought. After a brief introduction, the paper traces Russell's published works on type-theory up to Principia. Each is interpreted as presenting a version or modification of the substitutional (...) theory. New motivations for Russell's 1908 axiom of infinity and axiom of reducibility are revealed. (shrink)

Perhaps it is not overly pedantic to say that one will find Wittgenstein’s Tractatus Logico-Philosophicus very difficult even if one first understands Russell’s philosophical logic. But the question remains as to whether the work is intended in alliance with Russell’s research program for a scientific method in philosophy or splits from that program. This paper endeavors to answer the question by revealing new evidence that Wittgenstein held his Doctrine of Showing in 1913 and that it was a demand he imposed (...) in criticizing Russell’s multiple relation theory of judgment. (shrink)

Perhaps it is not overly pedantic to say that one will find Wittgenstein’s Tractatus Logico-Philosophicus very difficult even if one first understands Russell’s philosophical logic. But the question remains as to whether the work is intended in alliance with Russell’s research program for a scientific method in philosophy or splits from that program. This paper endeavors to answer the question by revealing new evidence that Wittgenstein held his Doctrine of Showing in 1913 and that it was a demand he imposed (...) in criticizing Russell’s multiple relation theory of judgment. (shrink)

In his new introduction to the 1925 second edition of Principia Mathematica, Russell maintained that by adopting Wittgenstein's idea that a logically perfect language should be extensional mathematical induction could be rectified for finite cardinals without the axiom of reducibility. In an Appendix B, Russell set forth a proof. Godel caught a defect in the proof at *89.16, so that the matter of rectification remained open. Myhill later arrived at a negative result: Principia with extensionality principles and without reducibility cannot (...) recover mathematical induction. The finite cardinals are indefinable in it. This paper shows that while Gödel and Myhill are correct, Russell was not wrong. The 1925 system employs a different grammar than the original Principia. A new proof for *89.16 is given and induction is recovered. (shrink)

Bernard Linsky, The Evolution of Principia Mathematica; Bertrand Russell's Manuscripts and Notes for the Second Edition. Cambridge: Cambridge University Press. 2011. 407 pp. + two plates. $150.00/£...

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.

Simple-type theory is widely regarded as inadequate to capture the metaphysics of mathematics. The problem, however, is not that some kinds of structure cannot be studied within simple-type theory. Even structures that violate simple-types are isomorphic to structures that can be studied in simple-type theory. In disputes over the logicist foundations of mathematics, the central issue concerns the problem that simple-type theory fails to assure an infinity of natural numbers as objects. This paper argues that the problem of infinity is (...) based on a metaphysical prejudice in favor of numbers as objects — a prejudice that mathematics can get along without. (shrink)

In his "Grundgesetze", Frege hints that prior to his theory that cardinal numbers are objects he had an "almost completed" manuscript on cardinals. Taking this early theory to have been an account of cardinals as second-level functions, this paper works out the significance of the fact that Frege's cardinal numbers is a theory of concept-correlates. Frege held that, where n > 2, there is a one—one correlation between each n-level function and an n—1 level function, and a one—one correlation between (...) each first-level function and an object. Applied to cardinals, the correlation offers new answers to some perplexing features of Frege's philosophy. It is shown that within Frege's concept-script, a generalized form of Hume's Principle is equivalent to Russell's Principle ofion — a principle Russell employed to demonstrate the inadequacy of definition by abstraction. Accordingly, Frege's rejection of definition of cardinal number by Hume's Principle parallels Russell's objection to definition by abstraction. Frege's correlation thesis reveals that he has a way of meeting the structuralist challenge that it is arithmetic, and not a privileged progression of objects, that matters to the finite cardinals. (shrink)

Hume's Principle, dear to neo-Logicists, maintains that equinumerosity is both necessary and sufficient for sameness of cardinal number. All the same, Whitehead demonstrated in Principia Mathematica's logic of relations that Cantor's power-class theorem entails that Hume's Principle admits of exceptions. Of course, Hume's Principle concerns cardinals and in Principia's ‘no-classes’ theory cardinals are not objects in Frege's sense. But this paper shows that the result applies as well to the theory of cardinal numbers as objects set out in Frege's Grundgesetze. (...) Though Frege did not realize it, Cantor's power-theorem entails that Frege's cardinals as objects do not always obey Hume's Principle. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:RUSSELL TO FREGE, 24 MAY 1903: "I BELIEVE I HAVE DISCOVERED THAT CLASSES ARE ENTIRELY SUPERFLUOUS" GREGORY LANDINI Philosophy / University of Iowa Iowa City, IA 52242, USA It was his consideration of Cantor's proof that there is no greatest cardinal, Russell recalls in My Philosophical Development, that led in the spring of 1901 to the discovery of the paradox of the class of all classes not (...) members of themselves. "Never glad confident morning again", were Whitehead's reported words (MPD, p. 58). Whitehead was, of course, wrong. Russell had many new confident mornings. One in particular apparently occurred on 19 May 1903. On 23 May Russell wrote in his journal: "Four days ago 1 solved the Contradiction -the relief of this is unspeakable" (Papm 12: 24). The "solution" was communicated to Whitehead, who responded by telegram: "Heartiest congratulations Aristoteles secundus. 1 am delighted" (Clark, p. III). But it was not long before Russell knew his proposal was inadequate. On the form he would scrawl: "A propos of solVing the Contradiction. [But the solution was wrong]" (ibid; Russell's brackets). The episode is intriguing. What was the proposal? Russell's correspondence with Frege sheds some light. On 20 October 1902 Frege had sent a letter suggesting a way to avoid the contradiction. Frege's suggestion, formulated in a great hurry so that it might appear in the Appendix of Vol. II of the Grundgesetze der Arithmetik, was to abandon his Basic Law V; zcj>z =zez. ==. (x)(x. ==. ex), and to replace it by Russell to Frege, 24 May I903 161 V' zz =zez. ==. (x)(x *zz & x* zez: ~ : x. ==. ex). The underlying idea was to accept that one concept may have the same extension as another even though the two concepts are not coextensive. Though he thought the underlying idea "probably correct",! Russell expressed reservation in a reply of 12 December:... I find it difficult to accept your solution even though it is probably correct. Do you deny, e.g., that all classes form a class? And if this is admitted, then it is possible that - ana. Moreover, the class of non-humans is a. non-human. Otherwise it must be admitted that not ail objects fall either under a or not-a; namely, if a is a range of values, then a fails neither under a nor under not-a. This contradicts the law of excluded middle, which will be inconvenient to say the least.2 (Frege I980, p. 151) Frege's response of 28 December does not mention the contradiction, and Volume II of his Grundgesetze appeared early in 1903. Russell's reservations concerning Frege's solution· continue in a letter to Frege of 20 February 1903: What you say about my contradiction is of the greatest interest to me. Do you believe that the range of values remains unchanged if some subclass of the class is assigned to it as a new member? Extension seems to fit this view better than intension. Bur I fe;el far from clear about this question. (Frege I980, p. 155) Frege replied on 21 May, explaining that in general a class does not remain unchanged when a particular subclass is added to it, but that two concepts may have the same extension (the same class) when the only difference between them is that this class falls under the first concept but not under the second (p. 157). Then on 24 May 1903 Russell excitedly wrote Frege of a solution of the paradox of classes: "I received your letter this morning, and 1 am replying to it at once, for russell: the Journal of the Berrrand Russdl Archives McMasrer University Library Press n.S.•2 (wincer 1992-93): 160-85 ISSN 0036-01631 • The same evaluation appears in Russell's Appendix to The Principles ofMathematics (p. 522). 2 Frege's '\- a II b" is Peano and Russell's "a E b". Russell should have put "~ana". 162 GREGORY LANDINI I believe I have discovered that classes are entirely superfluous. Your designation i(z) can be used for itself, and x E i(z) for (x).... this seems to me to avoid the contradiction" (p. 159). The proposal is also mentioned in a... (shrink)

HENRY LAYCOCK, Words Without Objects: Semantics, Ontology, and Logic for Non-Singularity. Oxford: Clarendon Press, 2006. xvi + 202pp. £35.00. ISBN 0‐19‐928171‐8. Gregory Landini, Department of Phil...

In lieu of an abstract, here is a brief excerpt of the content:January 22, 2009 (8:41 pm) G:\WPData\TYPE2802\russell 28,2 048red.wpd russell: the Journal of Bertrand Russell Studies n.s. 28 (winter 2008–09): 127–42 The Bertrand Russell Research Centre, McMaster U. issn 0036-01631; online 1913-8032 YABLO’S PARADOX AND RUSSELLIAN PROPOSITIONS Gregory Landini Philosophy / U. of Iowa Iowa City, ia 52242–1408, usa gregory-landini@uiowa.edu Is self-reference necessary for the production of Liar paradoxes? Yablo has given an argument that self-reference (...) is not necessary. He hopes to show that the indexical apparatus of self-reference of the traditional Liar paradox can be avoided by appealing to a list, a consecutive sequence, of sentences correlated one-one withnaturalnumbers.Yabloopens his “Paradox without Self-Reference” (Analysis, 1993) with the assumption that there is a sequence such that: Snz: “(;k)(k > n.!. True Sk)” Each sentence on Yablo’s list is supposed to be correlated one-one with number n. Each sentence is supposed to say that for every natural number kz greater than n, the k-th sentence on the list is not true. By comparing Yablo’s construction to an analogous construction with early Russellian propositions, we show that Yablo has failed to generate a paradox. 1. vicious circularity in yablo’s paradox S elf-reference has been taken to be the source of paradoxes. Poincaré and Russell are commonly associated with this thesis. In an amusing passage, the mathematician Jourdain recalls something of the dialogue between them: Nearly all mathematicians agreed that the way to solve these paradoxes was simply not to mention them; but there was some divergence of opinion as to how they were to be unmentioned. It was clearly unsatisfactory merely not to mention them. Thus Poincaré was apparently of the opinion that the best way of avoiding such awkward subjects was to mention that they were not to be mentioned. But [as Russell put it] “one might as well, in talking to a man with January 22, 2009 (8:41 pm) G:\WPData\TYPE2802\russell 28,2 048red.wpd 128 gregory landini 1 Philip Jourdain, The Philosophy of Mr. B*rtr*nd R*ss*llz (London: Allen & Unwin, 1918), p. 77. Russell is quoted from “Mathematical Logic as Based on the Theory of Types” (1908), LK, p. 63. 2 “On ‘Insolubilia’ and Their Solution By Symbolic Logic”, in B. Russell, Essays in Analysis, ed. Douglas Lackey (London: Allen & Unwin, 1973), p. 196. 3 I believe that this is Russell’s view in 1906. See Gregory Landini, “Russell’s Separation of the Logical and Semantic Paradoxes”, Revue internationale de philosophie 58 (2004): 257–94 (issue titled “Russell en héritage / Le centenaire des Principes”, ed. Philippe de Rouilhan). a long nose, say: ‘When I speak of noses, I except such as are inordinately long’, which would not be a very successful eTort to avoid a painful topic.”1 Poincaré maintained that one should exclude the oTending cases and therebyavoid“viciouscircles”ofself-referencewhichgenerateparadoxes. Russell objected: We may illustrate this by what M. Poincaré says concerning Richard’s paradox. Having Wrst put E = “all numbers deWnable in a Wnite number of words” we arrive at a paradox, due, says M. Poincaré, to our having included a number only deWnable in a Wnite number of words by means of E. This vicious circle he proposes to avoid by deWning E as “all numbers deWnable in a Wnite number of words without mentioning Ey”. To the uninitiated, this deWnition looks more circular than ever.2 Russell held that some paradoxes such as those of classes require, in order to exclude oTending cases without mentioning them, a “reconstruction of logical Wrst principles”. Others, such as Richard’s, are to be dismissed because they involve confused and viciously circular notions of “deWnability ”.3 Is self-reference necessary for paradoxes? The question is not well crafted. There are quite diTerent notions ofz “self-reference” involved in paradoxes. The self-reference involved in the Liar “This sentence is false” is provided by an apparatus of indexicals. This apparatus is quite distinct from the self-reference involved in Russell’s early ontology of propositions (as mind- and language-independent states of aTairs). This is an ontological self... (shrink)

This paper examines the quantification theory of *9 of Principia Mathematica. The focus of the discussion is not the philosophical role that section *9 plays in Principia's full ramified type-theory. Rather, the paper assesses the system of *9 as a quantificational theory for the ordinary predicate calculus. The quantifier-free part of the system of *9 is examined and some misunderstandings of it are corrected. A flaw in the system of *9 is discovered, but it is shown that with a minor (...) repair the system is semantically complete. Finally, the system is contrasted with the system of *8 of Principia's second edition. (shrink)

The quantification theory of propositions in Russell’s Principles of Mathematics has been the subject of an intensive study and in reconstruction has been found to be complete with respect to analogs of the truths of modern quantification theory. A difficulty arises in the reconstruction, however, because it presents universally quantified exportations of five of Russell’s axioms. This paper investigates whether a formal system can be found that is more faithful to Russell’s original prose. Russell offers axioms that are universally quantified (...) implications that have antecedent clauses that are conjunctions. The presence of conjunctions as antecedent clauses seems to doom the theory from the onset, it will be found that there is no way to prove conjunctions so that, after universal instantiation, one can detach the needed antecedent clauses. Amalgamating two of Russell’s axioms, this paper overcomes the difficulty. (shrink)

In his Tractatus, Wittgenstein sets out what he calls his N-operator notation which can be used to calculate whether an expression is a tautology. In his Laws of Form, George Spencer Brown offers what he calls a “primary algebra” for such calculation. Both systems are perplexing. But comparing two blurry images can reduce noise, producing a focus. This paper reveals that Spencer Brown independently rediscovered the quantifier-free part of the N-operator calculus. The comparison sheds a flood light on each and (...) from the letters of correspondence we shall find that Russell, as one might have surmised, was a catalyst for both. (shrink)

The second printing of Principia Mathematica in 1925 offered Russell an occasion to assess some criticisms of the Principia and make some suggestions for possible improvements. In Appendix A, Russell offered *8 as a new quantification theory to replace *9 of the original text. As Russell explained in the new introduction to the second edition, the system of *8 sets out quantification theory without free variables. Unfortunately, the system has not been well understood. This paper shows that Russell successfully antedates (...) Quine's system of quantification theory without free variables. It is shown as well, that as with Quine's system, a slight modification yields a quantification theory inclusive of the empty domain. (shrink)

Bertrand Russell: Logic For Russell, Aristotelian syllogistic inference does not do justice to the subject of logic. This is surely not surprising. It may well be something of a surprise, however, to learn that in Russell’s view neither Boolean algebra nor modern quantification theory do justice to the subject. For Russell, logic is a synthetic … Continue reading Russell: Logic →.

Poetic license is an essential feature of intentionality. The mind is free to think about any objects, even objects with logically incompatible properties. Some philosophers maintain that a theory that embraces an ontology of objects of thought is indispensable to any account of the nature of intentionality. Any such theory, however, must face paradoxes whose solutions conflict with poetic license. In this paper, I propose a theory which rejects the argument from indispensability. The theory maintains that the intentionality of thought (...) is produced by the quantificational nature of the apparatus of thought. All de re ascriptions of propositional attitudes must respect simple-type stratification and quantify over concepts with a predicable nature only. There are no fictional objects. There are concepts which, in the impredicative reflections of quantificational thought, are presented as if objects of thought. (shrink)

There is a rather famous “Fitch argument” that not everything that is true is knowable. There is a rather famous “Mary argument” that is often used to argue that reductive physicalism is false. This paper sets out the two side by side as the Fitch Knowability Paradox and the Mary Knowability Paradox. It is found that they have the same logical form and thus the question of validity can be evaluated with the same tools. Likening the two is useful, since (...) it avoids the problem that since the logical forms involved in intentional and experiential contexts are unknown, we cannot be in a position to evaluate whether the Mary is deductively valid without begging questions. (shrink)

This paper explores the thesis that de re quantification into propositional attitudes has been wrongly conceived. One must never bind an individual variable in the context of a propositional attitude. Such quantification fails to respect the quantificational scaffolding of discursive thinking. This is the lesson of the Meinong–Russell debate over whether there are objects of thought about which it is true to say they are not. Respecting it helps to see how to solve contingent Liar paradoxes of propositional attitudes such (...) as Kripke’s Nixon–Jones. (shrink)