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Three Logical Theories

Year 2019, Volume: 1 Issue: 1, 140 - 174, 01.07.2019

Abstract

This study concerns logical systems
considered as theories. By searching for the problems which the traditionally
given systems may reasonably be intended to solve, we clarify the rationales
for the adequacy criteria commonly applied to logical systems. From this point
of view there appear to be three basic types of logical systems: those
concerned with logical truth; those concerned with logical truth and with
logical consequence; and those concerned with deductionper se as well as with
logical truth and logical consequence. Adequacy criteria for systems of the
first two types include: effectiveness, soundness, completeness, Post
completeness, "strong soundness" and strong completeness.
Consideration of a logical system as a theory of deduction leads us to attempt
to formulate two adequacy criteria for systems of proofs. The first deals with
the concept of rigor or "gaplessness" in proofs. The second is a
completeness condition for a system of proofs. An historical note at the end of
the paper suggests a remarkable parallel between the above hierarchy of systems
and the actual historical development of this area of logic.

References

  • Anderson, J. M., and Johnstone, H. W., Natural Deduction, Belmont, California, 1962.
  • Bochenski, I. M., A History of Formal Logic (tr. Thomas, Ivo), Notre Dame, Indiana, 1961.
  • Church, A., Introduction to Mathematical Logic, Princeton, 1956.
  • Copi, I. M., and Gould, J. A., Readings on Logic, New York, 1964.
  • Gödel, K., "Die Vollstandigkeit der Axiome das logischen Functionenkalkuls," Monatshefte für Mathematik und Physik, vol. xxxvii, 1930, p. 349.
  • Henkin, L., "The completeness of the first order functional calculus," Journal of Symbolic Logic, vol. 14, 1949, p. 159.
  • Hilbert, D., and Ackermann, W., Principles of Mathematical Logic (tr. Hammond, Leckie, and Steinhardt), New York, 1950.
  • Hiż, H., "A warning about translating axioms," American Mathematical Monthly, vol. LXV, 1958, p. 613.
  • Kalish, D., and Montague, R., Logic: Techniques of Formal Reasoning, New York, 1964.
  • Kneale, W., and Kneale, M., The Development of Logic, Oxford, 1962.
  • Lewis, C. I., and Langford, C. H., Symbolic Logic, 2nd ed., New York, 1959.
  • Lightstone, A. H., The Axiomatic Method, Prentice-Hall, Englewood Cliffs, N.J., 1964.
  • Mates, B., Elementary Logic, New York, 1965.
  • Mendelson, Elliot, Introduction to Mathematical Logic, Princeton, 1964.
  • Parry, W. T., "Comments on a variant form of natural deduction," Journal of Symbolic Logic, vol. 30, 1965, p. 119.
  • Post, E. L., "Introduction to general theory of elementary propositions," American Journal of Mathematics, vol. 43, 1921, p. 163.
  • Quine, W. V. O., Methods of Logic (revised edition), New York, 1959.
  • Robinson, A., On the Metamathematics of Algebra, Amsterdam, 1951.
  • Tarski, A., Logic, Semantics and Metamathematics (tr. Woodger, J. H.), Oxford, 1956.
  • Whitehead, A. N., and Russell, B., Principia: Mathematica to 56, Cambridge, 1962.

Üç Mantıksal Teori

Year 2019, Volume: 1 Issue: 1, 140 - 174, 01.07.2019

Abstract

Bu çalışma, teori olarak düşünülen mantıksal sistemlerle ilgilidir.
Geleneksel olarak verilen sistemleri makul bir şekilde çözmenin hedeflendiği
sorunları araştırarak biz, mantıksal sistemlere yaygın olarak uygulanan
yeterlilik ölçütlerinin mantığını açıklığa kavuşturuyoruz. Bu açıdan mantıksal
sistemlerin üç temel türü var gibi görünüyor: bunlar mantıksal doğruluğa
ilişkin olanlar, mantıksal doğruluk ve mantıksal gerektirme ile ilgili olanlar
ve mantıksal doğruluk ve mantıksal gerektirme ile olduğu kadar, başlı başına
çıkarım ile de ilgili olanlardır. İlk iki tipteki sistemler için yeterlilik
ölçütleri şunları içerir: etkinlik, sağlamlık, tamamlanmışlık, Post
tamamlanmışlığı, "güçlü sağlamlık" ve güçlü tamamlanmışlık. Bir
mantıksal sistemin bir çıkarım teorisi olarak düşünülmesi, ispat sistemleri
için iki yeterlilik ölçütü formüle etmeye çalışmamıza sebep olur. Birincisi,
sıkılık kavramı veya delillerdeki boşluksuzluk(gaplessness) ile ilgilidir.
İkincisi, bir kanıt sistemi için bir tamamlanmışlık koşuludur. Makalenin
sonunda yer alan tarihsel bir not, üst sistem hiyerarşisi ile bu mantık
alanının gerçek tarihsel gelişimi arasında dikkate değer bir paralellik
olduğunu belirtmektedir.

References

  • Anderson, J. M., and Johnstone, H. W., Natural Deduction, Belmont, California, 1962.
  • Bochenski, I. M., A History of Formal Logic (tr. Thomas, Ivo), Notre Dame, Indiana, 1961.
  • Church, A., Introduction to Mathematical Logic, Princeton, 1956.
  • Copi, I. M., and Gould, J. A., Readings on Logic, New York, 1964.
  • Gödel, K., "Die Vollstandigkeit der Axiome das logischen Functionenkalkuls," Monatshefte für Mathematik und Physik, vol. xxxvii, 1930, p. 349.
  • Henkin, L., "The completeness of the first order functional calculus," Journal of Symbolic Logic, vol. 14, 1949, p. 159.
  • Hilbert, D., and Ackermann, W., Principles of Mathematical Logic (tr. Hammond, Leckie, and Steinhardt), New York, 1950.
  • Hiż, H., "A warning about translating axioms," American Mathematical Monthly, vol. LXV, 1958, p. 613.
  • Kalish, D., and Montague, R., Logic: Techniques of Formal Reasoning, New York, 1964.
  • Kneale, W., and Kneale, M., The Development of Logic, Oxford, 1962.
  • Lewis, C. I., and Langford, C. H., Symbolic Logic, 2nd ed., New York, 1959.
  • Lightstone, A. H., The Axiomatic Method, Prentice-Hall, Englewood Cliffs, N.J., 1964.
  • Mates, B., Elementary Logic, New York, 1965.
  • Mendelson, Elliot, Introduction to Mathematical Logic, Princeton, 1964.
  • Parry, W. T., "Comments on a variant form of natural deduction," Journal of Symbolic Logic, vol. 30, 1965, p. 119.
  • Post, E. L., "Introduction to general theory of elementary propositions," American Journal of Mathematics, vol. 43, 1921, p. 163.
  • Quine, W. V. O., Methods of Logic (revised edition), New York, 1959.
  • Robinson, A., On the Metamathematics of Algebra, Amsterdam, 1951.
  • Tarski, A., Logic, Semantics and Metamathematics (tr. Woodger, J. H.), Oxford, 1956.
  • Whitehead, A. N., and Russell, B., Principia: Mathematica to 56, Cambridge, 1962.
There are 20 citations in total.

Details

Primary Language Turkish
Subjects Logic
Journal Section Translated Articles
Authors

John Corcoran This is me

Translators

Fatmanur Berilğen This is me

Publication Date July 1, 2019
Submission Date May 29, 2019
Acceptance Date June 18, 2019
Published in Issue Year 2019 Volume: 1 Issue: 1

Cite

ISNAD Corcoran, John. “Üç Mantıksal Teori”. Mantık Araştırmaları Dergisi. Fatmanur BerilğenTrans 1/1 (July 2019), 140-174.