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Added by: Viviane FairbankAbstract:
In Remarks on the Foundations of Mathematics, and elsewhere, Wittgenstein made some general statements about mathematical proof which can only be described as paradoxical. What he says is directed to proofs for which there is no general method of solution such as there is for statements of the form a x b = c where a simple calculation is sufficient. His concern is with propositions for which one cannot in advance of proof describe any procedure which will demonstrate them... In an earlier study,' 'Proof and the Theorem Proved', I singled out a group of propositions for which his account of proof seemed correct, and not paradoxical.... But I did not succeed in making his account of deductions of connections between established concepts paradox-free. I should like to return to his account as applied to such demonstrations, with the aim of explaining what justification there is for it, and what his account comes to.
Comment: This paper is a follow-up to Ambrose's earlier "Proof and the Theorem Proved" (which is also listed in the DRL).
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Added by: Viviane FairbankAbstract:
This book is intended for the newcomer to formal logic and is designed to provide an introduction for the intelligent general reader as well as for the student in the classroom. Its primary purpose is to present in a clear, natural, and orderly fashion the fundamentals of a subject that has now acquired the status of an exact science... Chapters I through VII are devoted to the exposition of material necessary to the construction of the deductive system of truth-functions... Thanks to the special notation that has come to be characteristic of symbolic logic, it is possible to express the logical material with clarity and exactness and to organize it, as in Chapter VIII, into a deductive system. Chapters IX through XI are given over to the study of general propositions and of forms of inference involving them... Care has been taken to show how Aristotelian logic is accommodated within the framework of the extended, present-day subject: by introducing the reader to quantification theory and to the current notation for expressing forms of general propositions, and finally, by presenting, in Chapter XII, the calculus of classes.Comment: This introductory textbook to formal logic was widely used in America in the mid-20th Century. Although some of the methods and notations are now outdated, it still provides a useful introduction to central ideas in formal and informal logic, and could be used as a reference for those preparing an introductory logic course (or for those interested in the history of logic education).
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Added by: Viviane FairbankAbstract:
There is something curiously opaque about Wittgenstein's assertions concerning the connection between proof of a proposition p, sense of the sentence " p ", and understanding " p ". The difficulties inherent in his account are apparent, and he has
himself mentioned them, while leaving their resolution obscure. I shall now detail them, so that it will be clear what difficulties must be got round and what clarification is required.Comment: This is a useful work of Wittgenstein exegesis by one of his students and collaborators, concerning his views on mathematical proof with discussion of several examples. It could be used in an intermediate/advanced course on the philosophy of mathematics or on Wittgenstein's philosophy more generally.