*Russell.* N.s. Vol.
31, no. 1.
Summer 2011.

Editor’s Notes | |

“ PRINCIPIA MATHEMATICA @
100”Edited by Nicholas Griffin, Bernard Linsky and Kenneth Blackwell
TABLE OF CONTENTS | |
---|---|

Preface by Nicholas Griffin and Bernard Linsky | |

Graham Stevens | “Logical Form in Principia
Mathematica and English”ABSTRACT: 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. |

Ray Perkins, Jr. | “Incomplete Symbols in
Principia
Mathematica and Russell’s ‘Definite
Proof’”ABSTRACT: 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 Wahl | “The Axiom of
Reducibility” ABSTRACT: 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 and a realist interpretation along the
lines of
Linsky and Mares. 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. |

Conor Mayo-Wilson | “Russell on Logicism and
Coherence” ABSTRACT: According to Quine, Charles Parsons, Mark Steiner, and others, Russell’s logicist project is important because, if successful, it would show that mathematical theorems possess desirable epistemic properties often attributed to logical theorems, such as aprioricity, necessity, and certainty. Unfortunately, Russell never attributed such importance to logicism, and such a thesis contradicts Russell’s explicitly stated views on the relationship between logic and mathematics. This raises the question: what did Russell understand to be the philosophical importance of logicism? Building on recent work by Andrew Irvine and Martin Godwyn, I argue that Russell thought a systematic reduction of mathematics increases the certainty of known mathematical theorems (even basic arithmetical 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. |

Brice Halimi | “Generality of Logical
Types” ABSTRACT: My aim is to examine logical types in Principia
Mathematica
from two (partly independent) perspectives. The first one pertains
to the
ambiguity of the notion of logical type as introduced in the
Introduction
(to the first edition). I claim that a distinction has to be made
between
types as called for in the context of paradoxes, and types as
logical
prototypes. The second perspective bears on typical ambiguity as
described in
Russell and Whitehead’s “Prefatory Statement of
Symbolic
Conventions”, inasmuch as it lends itself to a comparison
with specific
systems of modern typed λ-calculus. In particular, a recent
paper shows
that the theory of logical types can be formalized in the way of a
λ-calculus. This opens the way to an interesting
reconciliation between
type theories in the Russellian sense of the word, and type
theories in the
modern sense. But typical ambiguity is left aside in the paper. I
would like
to extend the suggestion by taking up the question of typical
ambiguity, still
in the realm of typed λ-calculus. |

Ryan Christensen | “Propositional
Quantification” ABSTRACT: Ramsey defined truth in the following way: x is
true if and
only if ∃p(x = [p] & p). This
definition
is ill-formed in standard first-order logic, so it is normally
interpreted
using substitutional or some kind of higher-order quantifier. I
argue that
these quantifiers fail to provide an adequate reading of the
definition, but
that, given certain adjustments, standard objectual quantification
does
provide an adequate reading. |

Roman Murawski | “On Chwistek’s
Philosophy of
Mathematics” ABSTRACT: The paper is devoted to the presentation of Chwistek’s philosophical ideas concerning logic and mathematics. The main feature of his philosophy was nominalism, which found full expression in his philosophy of mathematics. He claimed that the object of the deductive sciences, hence in particular of mathematics, is the expression being constructed in them according to accepted rules of construction. He treated geometry, arithmetic, mathematical analysis and other mathematical theories as experimental disciplines, and obtained in this way a nominalistic interpretation of them. The fate of Chwistek’s philosophical conceptions was similar to the fate of his logical conceptions. The system of rational meta-mathematics was not developed by him in detail. He worked on his own 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. |

Irving H. Anellis | “Did Principia
Mathematica
Precipitate a ‘Fregean Revolution’?”ABSTRACT: I begin by asking whether there was a Fregean revolution in logic, and, if so, in what did it consist. I then ask whether, and if so, to what extent, Russell played a decisive role in carrying through the Fregean revolution, and, if so, how. A subsidiary question is whether it was primarily the influence of The Principles of Mathematics or
Principia
Mathematica, or perhaps both, that stimulated and helped
consummate the
Fregean revolution. Finally, I examine cases in which logicians
sought to
integrate traditional logic into the Fregean paradigm, focusing on
the case
of Henry Bradford Smith. My proposed conclusion is that there were
different
means adopted for rewriting the syllogism, in terms of the logic of
relations,
in terms of the propositional calculus, or as formulas of the
monadic
predicate calculus. This suggests that the changes implemented as a
result of
the adoption of the Russell–Fregean conception of logic could
more
accurately be called by Grattan-Guinness’s term
convolution, rather
than revolution. |

Kenneth Blackwell | “The Wit and Humour of
Principia
Mathematica”ABSTRACT: Except for belatedly proving that “1 + 1 = 2”, Principia Mathematica doesn’t feature in studies of
mathematical
humour. Yet there is humour in that work, despite the inauspicious
conditions
under which it was written. Russell, to take one of the authors,
had an
irrepressible talent for enlivening any subject matter. This paper
reports
the results of exploring even the “obscure corners” of
PM
to uncover its humour and wit. |