Proof Theory: From Arithmetic to Set Theory
... largely through the copious writings of Chrysippus that the Stoic school became established, though many of these writings have been lost. • The patterns of reasoning described by Stoic logic are the patterns of interconnection between propositions that are completely independent of what those propo ...
... largely through the copious writings of Chrysippus that the Stoic school became established, though many of these writings have been lost. • The patterns of reasoning described by Stoic logic are the patterns of interconnection between propositions that are completely independent of what those propo ...
Propositional Logic and Methods of Inference
... The basic idea of normal form is to express wffs in a standard form that uses only the ^, v, and possibly ~ The resolution method is then applied to normal form wffs in which all other connectives and quantifiers have been eliminated Resolution is an operation on pairs of disjuncts, which produces n ...
... The basic idea of normal form is to express wffs in a standard form that uses only the ^, v, and possibly ~ The resolution method is then applied to normal form wffs in which all other connectives and quantifiers have been eliminated Resolution is an operation on pairs of disjuncts, which produces n ...
Principia Mathematica
The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. In 1927, it appeared in a second edition with an important Introduction To the Second Edition, an Appendix A that replaced ✸9 and an all-new Appendix C.PM, as it is often abbreviated, was an attempt to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could in principle be proven. As such, this ambitious project is of great importance in the history of mathematics and philosophy, being one of the foremost products of the belief that such an undertaking may be achievable. However, in 1931, Gödel's incompleteness theorem proved definitively that PM, and in fact any other attempt, could never achieve this lofty goal; that is, for any set of axioms and inference rules proposed to encapsulate mathematics, either the system must be inconsistent, or there must in fact be some truths of mathematics which could not be deduced from them.One of the main inspirations and motivations for PM was the earlier work of Gottlob Frege on logic, which Russell discovered allowed for the construction of paradoxical sets. PM sought to avoid this problem by ruling out the unrestricted creation of arbitrary sets. This was achieved by replacing the notion of a general set with the notion of a hierarchy of sets of different 'types', a set of a certain type only allowed to contain sets of strictly lower types. Contemporary mathematics, however, avoids paradoxes such as Russell's in less unwieldy ways, such as the system of Zermelo–Fraenkel set theory.PM is not to be confused with Russell's 1903 Principles of Mathematics. PM states: ""The present work was originally intended by us to be comprised in a second volume of Principles of Mathematics... But as we advanced, it became increasingly evident that the subject is a very much larger one than we had supposed; moreover on many fundamental questions which had been left obscure and doubtful in the former work, we have now arrived at what we believe to be satisfactory solutions.""The Modern Library placed it 23rd in a list of the top 100 English-language nonfiction books of the twentieth century.