Movatterモバイル変換


[0]ホーム

URL:


Skip to main content

Advertisement

Springer Nature Link
Log in

Boole’s Criteria for Validity and Invalidity

  • Chapter

Part of the book series:Synthese Library ((SYLI,volume 291))

  • 278Accesses

  • 1Citation

Abstract

It is one thing for a given proposition to follow or to not follow from a given set of propositions and it is quite another thing for it to beshown either that the given proposition follows or that it does not follow.* Using a formal deduction to show that a conclusionfollows and using a countermodel to show that a conclusiondoes not follow are both traditional practices recognized by Aristotle and used down through the history of logic. These practices presuppose, respectively, a criterion of validity and a criterion of invalidity each of which has been extended and refined by modern logicians: deductions are studied in formal syntax (proof theory) and coun-termodels are studied in formal semantics (model theory).

The mathematics we have to construct are the mathematics of the human intellect. —Boole, 1847

This is a preview of subscription content,log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
JPY 3498
Price includes VAT (Japan)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
JPY 17159
Price includes VAT (Japan)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
JPY 21449
Price includes VAT (Japan)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide -see info
Hardcover Book
JPY 21449
Price includes VAT (Japan)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide -see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Anselm,De Grammatico [XIth c], Latin text and English translation in D. Henry,Commentary on De Grammatico, Reidel Publishing Co., Dordrecht, Holland, 1974.

    Google Scholar 

  2. Bacon, J., “Natural-deduction Rules for Syllogistic” (abstract),The Journal of Symbolic Logic, vol.31 (1966), pp. 686–687.

    Google Scholar 

  3. Boole, G.,The Mathematical Analysis of Logic, Being an Essay Towards a Calculus of Deductive Reasoning, Macmillan, Barclay & Macmillan, Cambridge and George Bell, London, 1847, also reprinted 1948 by Basil Blackwell, London.

    Google Scholar 

  4. Boole, G.,An Investigation of the Laws of Thought on Which Are Founded the Mathematical Theories of Logic and Probabilities, Macmillan, London, 1854. Pagination referred to here is from the 1954 Dover reprint.

    Google Scholar 

  5. Boole, G.,A Treatise on Differential Equations, Cambridge 1859. Pagination referred to from reprint of the fifth edition by Chelsea Publishing Co., New York.

    Google Scholar 

  6. Boole, G.,Studies in Logic and Probability, edited by R. Rhees, Open Court Publishing Company, LaSalle, Illinois and Watts & Co., London, 1952.

    Google Scholar 

  7. Church, A.introduction to Mathematical Logic, Princeton University Press, Princeton, N.J., 1956.

    Google Scholar 

  8. Cohen, P. and R. Hersh, “Non-Cantorian Set Theory”,Scientific American, vol. 217 (1967), pp. 104–116.

    Article  Google Scholar 

  9. Corcoran, J., “Three Logical Theories”,Philosophy of Science, vol. 36 (1969), pp. 153–177.

    Article  Google Scholar 

  10. Corcoran, J., “Completeness of an Ancient Logic”,The Journal of Symbolic Logic, vol. 37 (1972), pp. 696–702.

    Article  Google Scholar 

  11. Corcoran, J., “Aristotle’s Natural Deduction System” inAncient Logic and Its Modern Interpretations, ed., J. Corcoran, Reidel Publishing Co., Dordrecht, Holland, 1974.

    Chapter  Google Scholar 

  12. Corcoran, J. and S. Wood, “Validity and Invalidity: Boole’s Criteria” (abstract),The Journal of Symbolic Logic, vol. 44 (1979), p. 662.

    Google Scholar 

  13. Craig, W.,Logic in Algebraic Form, North-Holland, Amsterdam, 1974.

    Google Scholar 

  14. DeMorgan, A.,Formal Logic, ed., A.E. Taylor, Open Court Co., London, 1926, orig. publ. Taylor and Walton, London, 1847.

    Google Scholar 

  15. Dummett, M., Review of [6],The Journal of Symbolic Logic, vol. 24 (1959), pp. 203–209. [Reprinted in this volume, pp. 79–85.]

    Article  Google Scholar 

  16. Hailperin, T.,Boole’s Logic and Probability, North-Holland, Amsterdam, 1976.

    Google Scholar 

  17. Henry, D.,Commentary on De Grammatico, Reidel Publishing Co., Dordrecht, Holland, 1974.

    Book  Google Scholar 

  18. Kneale, W. and M. Kneale,The Development of Logic, Oxford University Press, Oxford 1962.

    Google Scholar 

  19. Langer, R.E., “Differential Equations”,Encyclopedia Brittanica, 1953 edition, vol. 7, pp. 358–363.

    Google Scholar 

  20. Mill, John Stuart,A System of Logic, Longmans, Green and Company, London, 1879, originally published in 1843.

    Google Scholar 

  21. Smiley, T., “What Is a Syllogism?”,Journal of Philosophical Logic, vol. 2 (1973), pp. 136–154.

    Article  Google Scholar 

  22. Tarski, A.,Introduction to Logic and to the Methodology of Deductive Sciences, tr., O. Helmer, Oxford University Press, New York, 1941.

    Google Scholar 

  23. Van Evra, J., “A Reassessment of George Boole’s Theory of Logic”,Notre Dame Journal of Formal Logic, vol. 18 (1977), pp. 363–377. [Reprinted in this volume, pp.87–99.]

    Article  Google Scholar 

  24. Whately, R.,Elements of Logic, James Munroe & Co., Boston, 1845, originally published in London, 1825.

    Google Scholar 

  25. Whitehead, A.N.,Universal Algebra, vol. 1, Cambridge University Press, Cambridge, 1898.

    Google Scholar 

  26. Wood, S.,George Boole’s Theory of Propositional Forms, Ph.D. Dissertation in Philosophy, SUNY/Buffalo, 1976.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. State University of New York at Buffalo, Buffalo, New York, 14260, USA

    John Corcoran & Susan Wood

Authors
  1. John Corcoran
  2. Susan Wood

Editor information

Editors and Affiliations

  1. University of Lausanne, Switzerland

    James Gasser

Rights and permissions

Copyright information

© 2000 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Corcoran, J., Wood, S. (2000). Boole’s Criteria for Validity and Invalidity. In: Gasser, J. (eds) A Boole Anthology. Synthese Library, vol 291. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9385-4_7

Download citation

Publish with us

Access this chapter

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
JPY 3498
Price includes VAT (Japan)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
JPY 17159
Price includes VAT (Japan)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
JPY 21449
Price includes VAT (Japan)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide -see info
Hardcover Book
JPY 21449
Price includes VAT (Japan)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide -see info

Tax calculation will be finalised at checkout

Purchases are for personal use only


[8]ページ先頭

©2009-2025 Movatter.jp