scholarly journals Collapsing Arguments for Facts and Propositions

2008 ◽  
Vol 6 ◽  
Author(s):  
John Howard Sobel
Keyword(s):  
Mathematical Logic ◽  
Definite Descriptions ◽  
Kurt Gödel ◽  
Stephen Neale

Kurt Gödel argues in “Russell’s Mathematical Logic” that on the assumption that, contrary to Russell, definite descriptions are terms, it follows given only several “apparently obvious axioms” that “all true sentences have the same signification (as well as all false ones).” Stephen Neale has written that this argument, and others by Church, Davidson, and Quine to similar conclusions, are of considerable philosophical interest. Graham Oppy, responding to this opinion, says they are of minimal interest. Falling between these is my opinion that implications of these arguments for propositions and facts are of moderate philosophical interest, and that these arguments provide occasions for reflection of possible interest on fine lines of several theories of definite descriptions and class–abstractions.

Gödel's Incompleteness Theorems

1992 ◽  
Author(s):  
Raymond M. Smullyan
Keyword(s):  
Mathematical Logic ◽  
Number Theory ◽  
Continuum Theory ◽  
Kurt Gödel ◽  
The World ◽  
Computer Scientists ◽  
Incompleteness Theorems ◽  
The Axiom Of Choice ◽  
Gödel’S Incompleteness Theorems ◽  
The Continuum

Kurt Godel, the greatest logician of our time, startled the world of mathematics in 1931 with his Theorem of Undecidability, which showed that some statements in mathematics are inherently "undecidable." His work on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum theory brought him further worldwide fame. In this introductory volume, Raymond Smullyan, himself a well-known logician, guides the reader through the fascinating world of Godel's incompleteness theorems. The level of presentation is suitable for anyone with a basic acquaintance with mathematical logic. As a clear, concise introduction to a difficult but essential subject, the book will appeal to mathematicians, philosophers, and computer scientists.

scholarly journals Kurt Gödel, 28 April 1906 - 14 January 1978

1980 ◽  
Vol 26 ◽  
pp. 148-224 ◽  
Keyword(s):  
Mathematical Logic ◽  
Set Theory ◽  
Current Practice ◽  
Mathematical Reasoning ◽  
Internal Development ◽  
Kurt Gödel ◽  
The Subject ◽  
Formal Rules

Kurt Gödel did not invent mathematical logic; his famous work in the thirties settled questions which had been clearly formulated in the preceding quarter of this century. Despite sensational presentations by crackpots, philosophers and journalists (or even in poems, for example, by H. M. Enzensberger, set to music by H. W. Henze), Gödel’s results have not revolutionized the silent majority’s conception of mathematics, let alone its practice; much less so than the internal development of the subject since then. Certainly, those results refuted most elegantly each of the grand foundational ‘theories’ current at the time, of which Hilbert’s, on the place of formal rules in mathematical reasoning, and those associated with Frege and Russell, on its reduction to universal systems like set theory, were most popular. (Gödel’s own and related results also deflate the particular ‘anti-formalist’ foundations of the time, Poincaré’s and Brouwer’s constructivist and Zermelo’s infinitistic schemes being extreme examples; they are taken up in the last sections of parts II-IV.) For obvious reasons, in his original publications Gödel made a point of formulating his work in terms acceptable to the theories mentioned, and to stress its bearing on them. But it is fair to say that they were suspect anyway, and—less trivially—that they can be refuted more convincingly by simple constatations rather than by (his) mathematical theorems as explained in more detail in part II. Further, as so often with very grand schemes, the refutations put nothing comparable in the place of the discredited foundational views which are, quite properly, simply ignored in current practice.

Jean van Heijenoort. Introductory note. From Frege to Gödel, A source book in mathematical logic, 1879–1931, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1967, pp. 1–5. Reprinted in Frege and Gödel, Two fundamental texts in mathematical logic, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1970, pp. 1–5. - Gottlob Frege. Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought. English translation of 491 by Stefan Bauer-Mengelberg. From Frege to Gödel, A source book in mathematical logic, 1879–1931, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1967, pp. 5–82. Reprinted in Frege and Gödel, Two fundamental texts in mathematical logic, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1970, pp. 5–82. - Jean van Heijenoort. Introductory note. From Frege to Gödel, A source book in mathematical logic, 1879–1931, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1967, pp. 592–595. Reprinted in Frege and Gödel, Two fundamental texts in mathematical logic, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1970, pp. 83–86. - Kurt Gödel. Some metamathematical results on completeness and consistency. English translation of 4181 by Stefan Bauer-Mengelberg. From Frege to Gödel, A source book in mathematical logic, 1879–1931, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1967, pp. 595–596. Reprinted in Frege and Gödel, Two fundamental texts in mathematical logic, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1970, pp. 86–87. - Kurt Gödel. On formally undecidable propositions of Principia mathematica and related systems I. English translation of 4183 by Jean van Heijenoort. From Frege to Gödel, A source book in mathematical logic, 1879–1931, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1967, pp. 596–616. Reprinted in Frege and Gödel, Two fundamental texts in mathematical logic, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1970, pp. 87–107. - Kurt Gödel. On completeness and consistency. English translation of 4188 by Jean van Heijenoort. From Frege to Gödel, A source book in mathematical logic, 1879–1931, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1967, pp. 616–617. Reprinted in Frege and Gödel, Two fundamental texts in mathematical logic, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1970, pp. 107–108.

1972 ◽  
Vol 37 (2) ◽  
pp. 405-405 ◽  
Author(s):  
Alonzo Church
Keyword(s):  
Mathematical Logic ◽  
English Translation ◽  
Harvard University ◽  
Gottlob Frege ◽  
Source Book ◽  
Kurt Gödel ◽  
Introductory Note ◽  
Jean Van Heijenoort

Closing the Circle: An Analysis of Emil Post's Early Work

2006 ◽  
Vol 12 (2) ◽  
pp. 267-289 ◽  
Author(s):  
Liesbeth de Mol
Keyword(s):  
Mathematical Logic ◽  
Decision Problem ◽  
Symbolic Logic ◽  
Negative Solution ◽  
Kurt Gödel ◽  
Historical Fact

AbstractIn 1931 Kurt Gödel published his incompleteness results, and some years later Church and Turing showed that the decision problem for certain systems of symbolic logic has a negative solution. However, already in 1921 the young logician Emil Post worked on similar problems which resulted in what he called an “anticipation” of these results. For several reasons though he did not submit these results to a journal until 1941. This failure ‘to be the first’, did not discourage him: his contributions to mathematical logic and its foundations should not be underestimated. It is the purpose of this article to show that an interest in the early work of Emil Post should be motivated not only by this historical fact, but also by the fact that Post's approach and method differs substantially from those offered by Gödel, Turing and Church. In this paper it will be shown how this method evolved in his early work and how it finally led him to his results.

scholarly journals The Gödel Editorial Project: A Synopsis

2005 ◽  
Vol 11 (2) ◽  
pp. 132-149 ◽  
Author(s):  
Solomon Feferman
Keyword(s):  
Mathematical Logic ◽  
Vantage Point ◽  
Critical Studies ◽  
Personal Involvement ◽  
Oxford University ◽  
Kurt Gödel ◽  
The Absolute ◽  
Significant Figures ◽  
Source Materials ◽  
Oxford University Press

The final two volumes, numbers IV and V, of the Oxford University Press edition of the Collected Works of Kurt Gödel [3]–[7] appeared in 2003, thus completing a project that started over twenty years earlier. What I mainly want to do here is trace, from the vantage point of my personal involvement, the at some times halting and at other times intense development of the Gödel editorial project from the first initiatives following Gödel's death in 1978 to its completion last year. It may be useful to scholars mounting similar editorial projects for other significant figures in our field to learn how and why various decisions were made and how the work was carried out, though of course much is particular to who and what we were dealing with.My hope here is also to give the reader who is not already familiar with the Gödel Works a sense of what has been gained in the process, and to encourage dipping in according to interest. Given the absolute importance of Gödel for mathematical logic, students should also be pointed to these important source materials to experience first hand the exercise of his genius and the varied ways of his thought and to see how scholarly and critical studies help to expand their significance.Though indeed much has been gained in our work there is still much that can and should be done; besides some indications below, for that the reader is referred to [2].

The Logical Form of Descriptions

Dialogue ◽  
1992 ◽  
Vol 31 (4) ◽  
pp. 677-684 ◽  
Author(s):  
Bernard Linsky
Keyword(s):  
Natural Language ◽  
Logical Form ◽  
Ordinary Language ◽  
Definite Descriptions ◽  
Systematic Theory ◽  
Referential Use ◽  
Central Interest ◽  
Theory Of Descriptions ◽  
The One ◽  
Stephen Neale

Stephen Neale defends Russell's famous theory of definite descriptions against more than 40 years' worth of criticisms beginning long before Strawson's “On Referring.” Ever since Strawson's parting shot in that paper (“… ordinary language has no exact logic”), the theory of descriptions has been a battleground for the larger issue of whether a systematic theory of the semantics of natural language is really possible. Neale provides us with a sketch of part of that project as it currently stands. All of the complexities and irregularities of the use of definite descriptions in natural language can be combined, after all, in a single theory based on an “exact logic.” Neale argues that one can give a Russellian account of “incomplete descriptions” (as in ‘The table is covered with books’), generic uses of ‘the’ (‘The whale is a mammal’), plural descriptions (‘The men carried the piano’) and, of central interest, the purportedly referential uses identified by Donnellan (as in ‘The murderer of Smith is insane’ when it is Jones the accused we have in mind). Neale follows familiar answers to these objections; incorporate demonstratives into the account (to get ‘The table over there …’), distinguish the proposition expressed from the one meant (the “referential” use is what was meant not said), and point out that the problem is not unique to definite descriptions and so cannot be a fault of any particular theory of them (many expressions have generic, plural and “referential” uses).

Definite descriptions, A reader, edited by Gary Ostertag, Bradford books, The MIT Press, Cambridge, Mass., and London, 1998, xii + 411 pp. - Gary Ostertag, Introduction, Pp. 1–34. - Bertrand Russell, On denoting, A reprint of 1119. Pp. 35–49. - A. N. Whitehead and Bertrand Russell, From Principia mathematica, A reprint of pp. 30–32, 66–71, 173–175 of 1941. Pp. 51–65. - Bertrand Russell, Descriptions, A reprint of pp. 167–180 of 11126. Pp. 67–77. - Stephen Neale, Grammatical form, logical form, and incomplete symbols. A reprint of LXI 1391. Pp. 79–121. - Rudolf Carnap, From Meaning and necessity, A reprint of pp. 32–42 of XIV 237. Pp. 123–133. - P. F. Strawson, On referring, A reprint of XVIII 87, Pp. 135–160. - Karel Lambert, A theory of definite descriptions, A revised reprint of XXXII 252(1, 3) with altered title, Pp. 161–171. (Reprinted from Philosophical applications of free logic, edited by Karel Lambert, Oxford University Press, New York and Oxford 1991, pp. 17–27). - Keith Donnellan, Reference and definite descriptions, A reprint of XL 276(12), Pp. 173–193. - H. P. Grice, From “Vacuous names,” A reprint of pp. 138–144 of XL 479(7), Pp. 195–200. - Christopher Peacocke, Proper names, reference, and rigid designation, Pp. 201–224. (Reprinted from Meaning, reference and necessity, New studies in semantics, edited by Simon Blackburn, Cambridge University Press, Cambridge etc. 1975, pp. 109–132.) - Saul Kripke, Speaker's reference and semantic reference, Pp. 225–256. (Reprinted from Contemporary perspectives in the philosophy of language, edited by Peter A. French, Theodore E. Uehling, Jr., and Howard K. Wettstein, University of Minnesota Press, Minneapolis 1979, pp. 6–27; also in Studies in the philosophy of language, edited by Peter A. French, Theodore E. Uehling, Jr., and Howard K. Wettstein, Midwest studies in philosophy, vol. 2, The University of Minnesota, Morris 1977, pp. 255–276.) - Howard Wettstein, Demonstrative reference and definite descriptions, Pp. 257–273. (Reprinted from Philosophical studies, vol. 40 (1981), pp. 241–257.) - Scott Soames, Incomplete definite descriptions, Pp. 275–308. (Reprinted from Notre Dame journal of formal logic, vol. 27 (1986), pp. 349–375.) - Stephen Neale, Context and communication, Pp. 309–368. (Reprinted from Stephen Neale, Descriptions, Bradford books, The MIT Press, Cambridge, Mass., and London, 1990, pp. 62–117.) - Stephen Schiffer, Descriptions, indexicals, and belief reports: some dilemmas (but not the ones you expect). Pp. 369–395. (Reprinted from Mind, n.s. vol. 104 (1995), pp. 107–131.)

1999 ◽  
Vol 64 (3) ◽  
pp. 1371-1374
Author(s):  
Delia Graff
Keyword(s):  
New York ◽  
Philosophy Of Language ◽  
Logical Form ◽  
Definite Descriptions ◽  
Bertrand Russell ◽  
University Of Minnesota ◽  
Simon Blackburn ◽  
Christopher Peacocke ◽  
The University ◽  
Stephen Neale

Some facts about Kurt Gödel

1981 ◽  
Vol 46 (3) ◽  
pp. 653-659 ◽  
Author(s):  
Hao Wang
Keyword(s):  
Mathematical Logic ◽  
Number Theory ◽  
Secondary School ◽  
Chinese Remainder Theorem ◽  
Doctoral Dissertation ◽  
Greek Philosophy ◽  
Kurt Gödel ◽  
Time Part ◽  
The University ◽  
Primitive Recursive

The text of this article was done together with Gödel in 1976 to 1977 and was approved by him at that time. The footnotes and section headings have been added much later.Gödel was born on April 28, 1906 at Brno (or Brünn in German), Czechoslovakia (at that time part of the Austro-Hungarian Monarchy). After completing secondary school there, he went, in 1924, to Vienna to study physics at the University. His interest in precision led him from physics to mathematics and to mathematical logic. He enjoyed much the lectures by Furtwangler on number theory and developed an interest in this subject which was, for example, relevant to his application of the Chinese remainder theorem in expressing primitive recursive functions in terms of addition and multiplication. In 1926 he transferred to mathematics and coincidentally became a member of the M. Schlick circle. However, he has never been a positivist, but accepted only some of their theses even at that time. Later on, he moved further and further away from them. He completed his formal studies at the University before the summer of 1929. He also attended during this period philosophical lectures by Heinrich Gomperz whose father was famous in Greek philosophy.At about this time he read the first edition of Hilbert-Ackermann (1928) in which the completeness of the (restricted) predicate calculus was formulated and posed as an open problem. Gödel settled this problem and wrote up the result as his doctoral dissertation which was finished and approved in the autumn of 1929. The degree was granted on February 6, 1930. A somewhat revised version of the dissertation was published in 1930 in the Monatshefte.

Sign in / Sign up

Export Citation Format

Share Document