In recent years, Gregory Landini has defended an increasingly influential interpretation of
Principia Mathematica (
PM) according to which Russell and Whitehead recognize no typed ontology at all, and instead hold that all entities — including all particulars and universals — are “individuals”. In particular, there are, according to Landini two styles of variable in
PM : “individual variables”, which are understood “objectually”, and which take absolutely all entities, including all particulars and universals, as their values; and “predicate variables”, which are understood “substitutionally” (in the contemporary sense), as “dummy schematic letters”, which are to be replaced by open sentences. I present a number of considerations against this sort of interpretation. First, Landini and those following him, including Kevin Klement and Graham Stevens, cite passages from Russell’s writings at the time of
PM in which he holds (as in his earlier
Principles of Mathematics) that predicates and relations can function either “as subjects” or “as adjectives” or “verbs” (respectively) in order to support the view that Russell also holds that “individual variables” are absolutely unrestricted. I argue, however, that one may regard universals as capable of such a “two-fold” use without also holding that any variable occurring in a subject-position in a propositional function is absolutely unrestricted; that in his 1913 manuscript
Theory of Knowledge, Russell clearly accepts such a position; and also that there are some indications — albeit not as clear as the 1913 manuscript — that this is also Russell’s view in his writings immediately following
PM in 1910–11. Moreover, I argue that comparing passages from Russell’s 1906 manuscript “The Paradox of the Liar” with those writings immediately following the publication of
PM supports the view that at that time, Russell does not regard predicates and relations as “individuals”, but instead regards them as entities of different types than particulars. With regard to the view that in
PM, Russell and Whitehead accept the “substitutional” interpretation of predicate variables, and simply identify “propositions” with sentences and “propositional functions” with open sentences, I argue that it not only would make it impossible for Russell (who would not countenance infinitary sentences) to sustain his commitment to Cantor’s theory of the transfinite, but also attributes to Russell views he could accept only after Wittgenstein had convinced him to reject the “multiple-relation” view of judgment. However, I conclude by arguing that even though Russell does not in
PM simply identify propositions with open sentences, he himself recognizes a problem in reconciling his “multiple-relation” view of judgment with Cantor’s theory of the transfinite.<\/div>";
abstracts_content += "
";
abstracts_content += "
Bernard Linsky<\/div>";
abstracts_content += "
University of Alberta<\/div>";
abstracts_content += "
THE SECOND EDITION OF
PRINCIPIA MATHEMATICA<\/div>";
abstracts_content += "
The Bertrand Russell Archives hold all of the manuscripts and a large body of notes for the material that Russell added for the 1925–27 second edition of
Principia Mathematica. Using this material, and the correspondence from this period, it is possible to reconstruct Russell’s preparation of the “second edition”. Russell, after consulting with Whitehead, began with a long manuscript, “The Hierarchy of Propositions and Functions”, which Russell had intended to simply insert into the original Introduction. That manuscript was revised and divided into the eventual “Introduction to the Second Edition” and three Appendices; A, B and C. A significant issue of technical logic concerning Appendix B can be resolved by study of the manuscript of the page where an error occurs, and broader interpretive questions about the second edition can be illuminated by consideration of the draft and unused notes. <\/div>";
abstracts_content += "
";
abstracts_content += "
Ed Mares<\/div>";
abstracts_content += "
Victoria University of Wellington<\/div>";
abstracts_content += "
REAL VARIABLES<\/div>";
abstracts_content += "
In
Principles of Mathematics, Russell holds that propositional variables (or open formulae as we would now say), when asserted, express classes of propositions. This view is retained in the first edition of
Principia — in the wake of the rejection of classes and propositions — in which asserted open formulae are said to be ambiguous in very much the same sense. But in a manuscript in 1912 Russell claims that open formulae express logical forms, and in the introduction to the second edition of
PM he reinterprets all of the open formulae asserted in
PM as standing for closed formulae. This paper investigates why Russell abandoned his original view, a view which seems from a narrowly logical point of view quite coherent regardless of which of the main interpretations of propositional functions and the other elements (or non-elements) of Russell’s ontology.<\/div>";
abstracts_content += "
";
abstracts_content += "
Conor Mayo-Wilson<\/div>";
abstracts_content += "
Carnegie Mellon University<\/div>";
abstracts_content += "
RUSSELL ON LOGICISM AND COHERENCE<\/div>";
abstracts_content += "
Until recently, many philosophers had suggested that Russell’s logicist project, if successful, would be important primarily because it implies that mathematical theorems possess desirable epistemic properties often attributed to logical theorems, such as a prioricity, necessity, and certainty. However, Andrew Irvine and Martin Godwyn argue that such philosophers have failed to understand Russell’s primary motivations for logicism, namely, that in reducing mathematics to logic one (a) facilitates mathematical discovery through the creation of new concepts and systematization of existing ones, and (b) improves mathematical explanations. This paper takes Irvine and Godwyn’s work as a starting point, and argues that Russell never intended to argue that logicism implies mathematical theorems inherit the traits of a prioricity, necessity, and/or certainty from logical ones. Contra Irvine and Godwyn, however, I argue that Russell thought a systematic reduction of mathematics increases the certainty of known mathematical theorems (even basic arithmetic) facts by showing mathematical knowledge to be coherently organized. The paper outlines Russell’s theory of coherence, and discusses its relevance to logicism and the certainty attributed to mathematics.<\/div>";
abstracts_content += "
";
abstracts_content += "
Roman Murawski<\/div>";
abstracts_content += "
Adam Mickiewicz University<\/div>";
abstracts_content += "
ON CHWISTEK’S PHILOSOPHY OF MATHEMATICS<\/div>";
abstracts_content += "
Leon Chwistek (1894–1944) is known mainly for his logical works, in particular for his simplification of Whitehead and Russell’s theory of types. His logical investigations however were — as it was the case by some Polish logicians — connected with his philosophical ideas concerning logic and mathematics. Moreover, they were in a sense motivated by those ideas. Building semantics, he wanted to overcome philosophical idealism and was against the conception of an absolute truth. He did not content himself with solving particular definite fragmentary problems but — similarly to Lesniewski — attempted to construct a system containing the whole of mathematics. The paper is devoted to the presentation of Chwistek’s philosophical ideas concerning logic and mathematics. According to him human knowledge is neither full nor absolute. He accepted the principle of rationalism of knowledge and was against irrationalism. A way from the difficulties caused by irrationalism and simultaneously a weapon in a struggle against it is formal logic, in particular rational metamathematics founded by him. His epistemological views were close to the neopositivism. According to Chwistek an object of a cognition can be only that what is given in an experience. There are however various types of experience. In this way we come to the best known original philosophical conception of Chwistek, namely to his theory of the plurality of realities. The main feature of his philosophy was nominalism which found full expression in his philosophy of mathematics. He claimed that the object of deductive sciences, hence in particular of mathematics, are expressions being constructed in them according to accepted rules of construction. Geometry is for Chwistek an experimental discipline. He clearly and categorically rejected conventionalism in geometry. Similarly as geometry one should treat also arithmetic, mathematical analysis and other mathematical theories, obtaining in this way a nominalistic interpretation of them. The fate of the philosophical conceptions of Chwistek was similar to the fate of his logical conceptions. The system of rational metamathematics has not been developed by him in detail. He worked on his own conceptions and ideas without any collaboration with other logicians, mathematicians or philosophers. His investigations were not in the mainstream of the development of logic and philosophy of mathematics.<\/div>";
abstracts_content += "
";
abstracts_content += "
Raymond Perkins, Jr.<\/div>";
abstracts_content += "
Plymouth State University<\/div>";
abstracts_content += "
INCOMPLETE SYMBOLS IN
PRINCIPIA MATHEMATICA AND RUSSELL’S “DEFINITE PROOF”<\/div>";
abstracts_content += "
Early in
Principia Mathematica Russell presents an informal argument that definite descriptions are incomplete symbols — that they function differently than proper names, and that they have meaning only in sentential context. A few pages later he refers to this argument as a “definite proof”. This “proof” is significant not only because it is central to understanding incomplete symbols so vital to Russell’s logicism, but also because it has been taken as evidence of Russell’s supposed carelessness and confusion in philosophy of logic and language. This paper will (1) defend Russell’s “proof” against several charges and thereby rectify some long-standing criticisms of Russell’s semantics, (2) bring out some of Russell’s epistemic and metaphysical ideas connected with his doctrine of names and incomplete symbols and (3) offer some observations on how his
Principia definitions of descriptions and class-symbols fit into his more general conception of philosophical analysis during his atomistic period.<\/div>";
abstracts_content += "
";
abstracts_content += "
Ori Simchen<\/div>";
abstracts_content += "
University of British Columbia<\/div>";
abstracts_content += "
POLYADIC QUANTIFICATION VIA DENOTING CONCEPTS<\/div>";
abstracts_content += "
The question of the origin of polyadic expressivity is explored and the results are brought to bear on Bertrand Russell’s 1903 theory of denoting concepts, which is the main object of criticism in Russell’s “On Denoting”. It is shown that, appearances to the contrary notwithstanding, the background ontology of the earlier theory of denoting enables the full-blown expressive power of first-order polyadic quantification theory without any syntactic accommodation of scopal differences among denoting phrases such as “all ϕ”, “every ϕ”, and “any ϕ” on the one hand, and “some ϕ” and “a ϕ” on the other. The case provides an especially vivid illustration of the general point that structural (or ideological) austerity can be paid for in the coin of ontological extravagance.<\/div>";
abstracts_content += "
";
abstracts_content += "
Peter Simons<\/div>";
abstracts_content += "
Trinity College, Dublin<\/div>";
abstracts_content += "
+c , OR WHY WHITEHEAD AND RUSSELL DIDN’T NEED THEIR FINGERS TO ADD UP<\/div> ";
abstracts_content += "
Proving wrong Kant’s claim that we need intuition in order to do simple addition was a key task for logicism. Showing elementary arithmetical propositions are truths of logic should have been child’s play, but Whitehead and Russell take a very long way round, defining addition of cardinal numbers as late as *112. Like multiplication and exponentiation, they require it to be defined for arbitrary classes, even if these are of different types, or overlap. The motivation for this liberality is maximum generality. Tracing the definitions back to their basics enables us to see the ingenious way they do it, but the tortuous route renders addition so opaque that the use of fingers appears much closer to the logic of addition than the
Principia definition, with implications for the status of Whitehead and Russell’s logicism.<\/div>";
abstracts_content += "
";
abstracts_content += "
Graham Stevens<\/div>";
abstracts_content += "
Manchester University<\/div>";
abstracts_content += "
LOGICAL FORM IN
PRINCIPIA MATHEMATICA AND ENGLISH<\/div>";
abstracts_content += "
TBA<\/div>";
abstracts_content += "
";
abstracts_content += "
Dustin Tucker<\/div>";
abstracts_content += "
University of Michigan<\/div>";
abstracts_content += "
OUTLINE OF A THEORY OF QUANTIFICATION<\/div>";
abstracts_content += "
Historically, Russell’s ramified theory of types has been somewhat neglected, but with good reason. As a foundation of mathematics, it required the infamous axiom of reducibility. As a solution to paradoxes, it seemed unnecessary in light of Ramsey’s distinction between semantical and set-theoretical paradoxes. As a general theory, it was motivated by the vicious-circle principle, which Gödel convincingly attacked. But several authors, including Alonzo Church, David Kaplan, and Rich Thomason, have suggested ramification as a resolution of a family of paradoxes that fall through the cracks of Ramsey’s division. These paradoxes have been largely neglected as well, and what little existing work there is often leaves much to be desired. But so does ramification. Like Tarski’s hierarchy of languages, it is much too heavy-handed to do justice to the paradoxes. I thus provide a refinement of ramification’s restrictions on propositional quantification, analogous to Kripke’s refinement of Tarski’s truth predicates. Along the way, I develop a more flexible logic of propositions and an alternate resolution of the paradoxes, using truth-value gaps.<\/div>";
abstracts_content += "
";
abstracts_content += "
Alasdair Urquhart<\/div>";
abstracts_content += "
University of Toronto<\/div>";
abstracts_content += "
PRINCIPIA MATHEMATICA AND ITS INFLUENCE<\/div>";
abstracts_content += "
This paper investigates the influence that
Principia Mathematica exerted on the course of logical studies in the century since its publication. It will demonstrate the waning influence of the treatise in the course of the twentieth century, as well as discussing its impact on major logicians such as Tarski, Herbrand and Gödel.<\/div>";
abstracts_content += "
";
abstracts_content += "
Russell Wahl<\/div>";
abstracts_content += "
Idaho State University<\/div>";
abstracts_content += "
THE AXIOM OF REDUCIBILITY<\/div>";
abstracts_content += "
The axiom of reducibility plays an important role in the logic of
Principia Mathematica, but has generally been condemned as an ad hoc non-logical axiom which was added simply because the ramified type theory without it would not yield all the required theorems. In this paper I examine the status of the axiom of reducibility. Whether the axiom can plausibly be included as a logical axiom will depend in no small part on the understanding of propositional functions. If we understand propositional functions as constructions of the mind, it is clear that the axiom is clearly not a logical axiom and in fact makes an implausible claim. I look at two other ways of understanding propositional functions, a nominalist interpretation along the lines of Landini (1998) and a realist interpretation along the lines of Linsky (1999) and Mares (2007). I argue that while on either of these interpretations it is not easy to see the axiom as a non-logical claim about the world, there are also appear to be difficulties in accepting it as a purely logical axiom.<\/div>";
abstracts_content += "
";
abstracts_content += "
Jan Woleński<\/div>";
abstracts_content += "
Jagiellonian University<\/div>";
abstracts_content += "
PRINCIPIA MATHEMATICA IN POLAND<\/div>";
abstracts_content += "
Russell says in his
My Philosophical Development, p. 86: “I used to know of only six people who had read the later parts of the book. Three of these were Poles, subsequently (I believe) liquidated by Hitler. The three other were Texans, subsequently successfully assimilated.”";
abstracts_content += "
As far as I know nobody has identified the three readers of “later parts” of PM from Texas. As far as the matter concerns Poles, we have at least four candidates, namely Jan Śleszyński (1854–1931), Leon Chwistek (1884–1944), Stanisław Leśniewski (1886–1939) and Alfred Tarski (1901–1983). The dates show that none of them was liquidated by Hitler (Leśniewski died in May 1939, Chwistek died in Soviet Union). On the other hand, the Nazis killed several other Polish logicians who could read PM, even later parts of this great work, for example Adolf Lindenbaum (1904–about 1941) or Mordechaj Wajsberg (1902–about 1943). PM was recommended as an advanced monograph for students of mathematics specializing in mathematical logic and the foundations of mathematics (see guides by Warsaw University). In fact, Leśniewski wanted to translate PM into Polish. These facts document how the treatise of Whitehead and Russell was popular in Poland.
";
abstracts_content += "
As far as the matter concerns Polish investigations related to Principia Mathematica, the following lines are relevant: (1) work on the Russell paradox (Chwistek, Leśniewski, Czeżowski, Tarski); ";
abstracts_content += "(2) simplifications of the theory of types (Chwistek, Wilkosz); ";
abstracts_content += "(3) criticism of PM (Leśniewski); ";
abstracts_content += "(4) the theory of syntactic categories and levels of language (Leśniewski, Tarski). Looking from another axis, logical and foundational systems (grand logics) of Chwistek and Leśniewski were a continuation (Chwistek) or response (Leśniewski) to PM. Moreover, some more concrete constructions, in particular, the semantic definition of truth, arose in the framework of Principia Mathematica.
<\/div>";
abstracts_content += "
";
abstracts_content += "
Byeong-Uk Yi<\/div>";
abstracts_content += "
University of Toronto<\/div>";
abstracts_content += "
CLASS AND NUMBER IN
PRINCIPIA MATHEMATICA<\/div>";
abstracts_content += "
This paper examines the no-class theory of
Principia Mathematica. It clarifies an anomaly in the logic of classes that results from the theory, and gives a reformulation of the theory that removes the anomaly and yields a complete parallelism between the logic of classes and the logic of individuals and propositional functions. The reformulation results from giving a refined account of the notion of predicative function using plural languages. The account yields as theorems of plural logic the strengthened version of the Axiom of Reducibility that implies that formally equivalent predicative functions are identical. The reformulation of the no-class theory suggests that natural numbers and classes can be identified as predicative functions of a special kind.<\/div>";
abstracts_content += "
";
abstracts_content += "
Richard Zach<\/div>";
abstracts_content += "
University of Calgary<\/div>";
abstracts_content += "
PRINCIPIA MATHEMATICA AND THE DEVELOPMENT OF LOGIC<\/div>";
abstracts_content += "
Even if by the 1920s and 1930s foundational research in mathematics itself was dominated by work in set theory, Whitehead and Russell’s work continued to have a profound influence on the development of logic and its applications in philosophy. In mathematical logic, this influence can be traced in particular through the reception of PM in David Hilbert’s school, which produced not only fundamental work in logic itself, where the influence of PM was decisive, but also two textbooks, Hilbert and Ackermann’s Grundzüge der theoretischen Logik (1928) and Hilbert and Bernays’ Grundlagen der Mathematik (1934, 1939), which remained standard texts for many years. In philosophy, PM was influential in particular through the work of Carnap, who wrote an introduction to Whitehead and Russell’s “new logic” (Abriss der Logistik, 1929) and made crucial use of the techniques developed in PM in his Aufbau (1928) and related works. These developments were not unrelated, and also not isolated.<\/div>";