The Phenomenology of Second-Level Inference: Perfumes in The Deductive Garden

David Makinson

Bulletin of the Section of Logic (2020)

  • Volume: 49, Issue: 4, page 327-342
  • ISSN: 0138-0680

Abstract

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We comment on certain features that second-level inference rules commonly used in mathematical proof sometimes have, sometimes lack: suppositions, indirectness, goal-simplification, goal-preservation and premise-preservation. The emphasis is on the roles of these features, which we call 'perfumes', in mathematical practice rather than on the space of all formal possibilities, deployment in proof-theory, or conventions for display in systems of natural deduction.

How to cite

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David Makinson. "The Phenomenology of Second-Level Inference: Perfumes in The Deductive Garden." Bulletin of the Section of Logic 49.4 (2020): 327-342. <http://eudml.org/doc/296785>.

@article{DavidMakinson2020,
abstract = {We comment on certain features that second-level inference rules commonly used in mathematical proof sometimes have, sometimes lack: suppositions, indirectness, goal-simplification, goal-preservation and premise-preservation. The emphasis is on the roles of these features, which we call 'perfumes', in mathematical practice rather than on the space of all formal possibilities, deployment in proof-theory, or conventions for display in systems of natural deduction.},
author = {David Makinson},
journal = {Bulletin of the Section of Logic},
keywords = {second-level inference; suppositions; indirect inference; goal simplification; goal preservation; wlog; premise preservation},
language = {eng},
number = {4},
pages = {327-342},
title = {The Phenomenology of Second-Level Inference: Perfumes in The Deductive Garden},
url = {http://eudml.org/doc/296785},
volume = {49},
year = {2020},
}

TY - JOUR
AU - David Makinson
TI - The Phenomenology of Second-Level Inference: Perfumes in The Deductive Garden
JO - Bulletin of the Section of Logic
PY - 2020
VL - 49
IS - 4
SP - 327
EP - 342
AB - We comment on certain features that second-level inference rules commonly used in mathematical proof sometimes have, sometimes lack: suppositions, indirectness, goal-simplification, goal-preservation and premise-preservation. The emphasis is on the roles of these features, which we call 'perfumes', in mathematical practice rather than on the space of all formal possibilities, deployment in proof-theory, or conventions for display in systems of natural deduction.
LA - eng
KW - second-level inference; suppositions; indirect inference; goal simplification; goal preservation; wlog; premise preservation
UR - http://eudml.org/doc/296785
ER -

References

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  1. [1] G. Gentzen, Untersuchungen über das logische Schliessen, Mathematische Zeitschrift, vol. 23 (1934), pp. 176–210, 405–431, DOI: http://dx.doi.org/10.1007/BF01201353 English translation: Investigation into logical deduction, pp. 68–131 of The Collected Papers of Gerhard Gentzen, ed. M. E. Szabo. North-Holland, Amsterdam, 1969. 
  2. [2] J. Harrison, Without loss of generality, [in:] S. Berghofer, T. Nipkow (eds.), Theorem-Proving in Higher Order Logics, vol. 5674 of Lecture Notes in Computer Science, Springer, Berlin (2009), pp. 43–59, DOI: http://dx.doi.org/10.1007/978-3-642-03359-9_3 
  3. [3] A. Indrzejczak, Natural Deduction, Hybrid Systems and Modal Logic, Springer, Dordrecht (2010), DOI: http://dx.doi.org/10.1007/978-90481-8785-0 
  4. [4] A. Indrzejczak, Natural Deduction, Internet Encyclopedia of Philosophy, (2015), URL: https://iep.utm.edu/nat-ded/ consulted 16.07.2020. 
  5. [5] S. Ja_skowski, On the rules of suppositions in formal logic, Studia Logica, vol. 1 (1934), pp. 1–32. 
  6. [6] D. Kalish, R. Montague, Logic: Techniques of Formal Reasoning, Harcourt Brace, New York (1964), also second edition with co-author G. Mar, Oxford University Press, 2001. 
  7. [7] J. Lucas, Introduction to Abstract Mathematics, 2nd ed., Rowman & Little_eld, Maryland, USA (1990). 
  8. [8] D. Makinson, Sets, Logic and Maths for Computing, 3rd ed., Undergraduate Topics in Computer Science, Springer, London (2020), DOI: http://dx.doi.org/10.1007/978-1-4471-2500-6 
  9. [9] D. Makinson, Relevance-sensitive truth-trees, [in:] Alasdair Urquhart on Nonclassical and Algebraic Logic and Complexity of Proofs, Outstanding Contributions to Logic, Springer, Berlin (2021), to appear. 
  10. [10] J. Pelletier, A brief history of natural deduction, History and Philosophy of Logic, vol. 20 (1999), pp. 1–31, DOI: http://dx.doi.org/10.1080/014453499298165 
  11. [11] J. Pelletier, A. P. Hazen, A history of natural deduction, [in:] D. Gabbay, F. J. Pelletier, E. Woods (eds.), Handbook of the History of Logic, vol. 11, North-Holland, Amsterdam (2012), pp. 341–414, DOI: http://dx.doi.org/10.1016/B978-0-444-52937-4.50007-1 
  12. [12] E. Schechter, Constructivism is di_cult, American Mathematical Monthly, vol. 108 (2001), pp. 50–54, DOI: http://dx.doi.org/10.1080/00029890.2001.11919720 
  13. [13] R. Sikorski, Boolean Algebras, 2nd ed., Springer, Berlin (1964), DOI: http://dx.doi.org/10.1007/978-3-662-01507-0 

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