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

The Peanese convention for the use of dots as brackets has the disadvantage that it gives only an awkward method for representing chains of indefinite length, such as the compound implicationSuch chains occur frequently in logical investigations of a metatheoretic nature, and it is convenient to have a systematic method of abbreviating them. The most obvious method of doing this would be to leave the parentheses out entirely, and to understand that in such cases the implication sign or other operation (...) appearing on the extreme left is the most inclusive; but this method, which has been followed by Quine, is at variance with our ordinary algebraic usage, in which we write a − b + c − d for ((a − b) + c) − d, and consequently leads to confusion. It is desirable to have a modification of the Peanese convention which gives a simple method of representing such chain implications, and at the same time avoids this difficulty.Such a modification is easily obtained by generalizing somewhat a procedure of Church. The latter author uses only a single dot, which he writes on the right of an operator to signify a bracket extending from that point to the end of the formula (or parenthesized expression). If dots are used only on the right of operations this is all that can be desired. But the essential idea of this device can be extended to the case where dots are used also on the left, as follows: let us suppose that a group of dots on the right of an operation or prefix denotes the beginning of a bracket which extends to the right until it meets a group with an equal or larger number of dots on the left of an operation; and that the scope of a group of dots on the left of an operation shall extend to the left until it reaches a larger group of dots on the right of some operation. (shrink)

"This book is valuable as expounding in full a theory of meaning that has its roots in the work of Frege and has been of the widest influence.... The chief virtue of the book is its systematic character. From Frege to Quine most philosophical logicians have restricted themselves by piecemeal and local assaults on the problems involved. The book is marked by a genial tolerance. Carnap sees himself as proposing conventions rather than asserting truths. However he provides plenty of matter (...) for argument."-Anthony Quinton, Hibbert Journal. (shrink)

By a `denoting phrase' I mean a phrase such as any one of the following: a man, some man, any man, every man, all men, the present King of England, the present King of France, the center of mass of the solar system at the first instant of the twentieth century, the revolution of the earth round the sun, the revolution of the sun round the earth. Thus a phrase is denoting solely in virtue of its form. We may distinguish (...) three cases: (1) A phrase may be denoting, and yet not denote anything; e.g., `the present King of France'. (2) A phrase may denote one definite object; e.g., `the present King of England' denotes a certain man. (3) A phrase may denote ambiguously; e.g. `a man' denotes not many men, but an ambiguous man. The interpretation of such phrases is a matter of considerably difficulty; indeed, it is very hard to frame any theory not susceptible of formal refutation. All the difficulties with which I am acquainted are met, so far as I can discover, by the theory which I am about to explain. (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)

The great three-volume Principia Mathematica is deservedly the most famous work ever written on the foundations of mathematics. Its aim is to deduce all the fundamental propositions of logic and mathematics from a small number of logical premisses and primitive ideas, and so to prove that mathematics is a development of logic. This abridged text of Volume I contains the material that is most relevant to an introductory study of logic and the philosophy of mathematics (more advanced students will wish (...) to refer to the complete edition). It contains the whole of the preliminary sections (which present the authors' justification of the philosophical standpoint adopted at the outset of their work); the whole of Part 1 (in which the logical properties of propositions, propositional functions, classes and relations are established); section 6 of Part 2 (dealing with unit classes and couples); and Appendices A and B (which give further developments of the argument on the theory of deduction and truth functions). (shrink)