Journal of Logic, Language and Information

, Volume 19, Issue 3, pp 283–314 | Cite as

Reasoning Processes in Propositional Logic

  • Claes Strannegård
  • Simon Ulfsbäcker
  • David Hedqvist
  • Tommy Gärling
Article

Abstract

We conducted a computer-based psychological experiment in which a random mix of 40 tautologies and 40 non-tautologies were presented to the participants, who were asked to determine which ones of the formulas were tautologies. The participants were eight university students in computer science who had received tuition in propositional logic. The formulas appeared one by one, a time-limit of 45 s applied to each formula and no aids were allowed. For each formula we recorded the proportion of the participants who classified the formula correctly before timeout (accuracy) and the mean response time among those participants (latency). We propose a new proof formalism for modeling propositional reasoning with bounded cognitive resources. It models declarative memory, visual memory, working memory, and procedural memory according to the memory model of Atkinson and Shiffrin and reasoning processes according to the model of Newell and Simon. We also define two particular proof systems, T and NT, for showing propositional formulas to be tautologies and non-tautologies, respectively. The accuracy was found to be higher for non-tautologies than for tautologies (p < .0001). For tautologies the correlation between latency and minimum proof length in T was .89 and for non-tautologies the correlation between latency and minimum proof length in NT was .87.

Keywords

Bounded resources Proof system Propositional logic Psychological experiment Reasoning 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adler J. E., Rips L. J. (2008) Reasoning: Studies of human inference and its foundations. Cambridge University Press, CambridgeGoogle Scholar
  2. Anderson J., Lebiere C. (1998) The atomic components of thought. Lawrence Erlbaum, Mahwah NJGoogle Scholar
  3. Atkinson R. C., Shiffrin R. M. (1968) Human memory: A proposed system and its control processes. Academic Press, New York, pp 89–195Google Scholar
  4. Baddeley A. (2007) Working memory, thought and action. Oxford University Press, OxfordGoogle Scholar
  5. Braine M. D. S., O’Brien D. P. (1998) Mental logic. L. Erlbaum Associates, EnglandGoogle Scholar
  6. Braine, M. D. S., Reiser, B. J., & Rumain, B. (1998). Evidence for the theory: Predicting the difficulty of propositional logic inference problems. In Mental logic (pp. 91–144). England: L. Erlbaum Associates.Google Scholar
  7. Buss, S. (eds) (1998) Handbook of proof theory. Elsevier, AmsterdamGoogle Scholar
  8. Byrne R. M. J. (1989) Suppressing valid inferences with conditionals. Cognition 31: 61–83CrossRefGoogle Scholar
  9. Conway A. R. A., Kane M. J., Engle R. W. (2003) Working memory capacity and its relation to general intelligence. Trends in Cognitive Sciences 7(12): 547–552CrossRefGoogle Scholar
  10. Cowan, N. (2001). The magical number 4 in short-term memory: A reconsideration of mental storage capacity. Behavioral and Brain Sciences, 24(1), 87–114.CrossRefGoogle Scholar
  11. Fitch F. B. (1952) Symbolic logic: An introduction. Ronald Press, New YorkGoogle Scholar
  12. Gentzen G. (1969) Investigations into logical deductions. In: Szabo M. E. (eds) The collected papers of Gerhard Gentzen. North-Holland Publishing Co, Amsterdam, pp 68–131Google Scholar
  13. Geuvers H., Nederpelt R. (2004) Rewriting for fitch style natural deductions. In: Oostrom V. (eds) Rewriting techniques and applications, 15th international conference. Springer, New YorkGoogle Scholar
  14. Guglielmi A. (2007) A system of interaction and structure. ACM Transactions on Computational Logic 8(1): 1–64CrossRefGoogle Scholar
  15. Hedqvist, D. (2007). Human reasoning in propositional logic. Master’s thesis, Chalmers University of Technology.Google Scholar
  16. Holyoak, K. J. & Morrison, R. (Eds.). (2005). The Cambridge handbook of thinking and reasoning. Cambridge: Cambridge University Press.Google Scholar
  17. Jaśkowski, S. (1934). On the rules of suppositions in formal logic. Studia Logica, 1, 5–32. Reprinted in S. McCall (Ed.), Polish logic 1920–1939 (pp. 232–258). Oxford: Clarendon Press.Google Scholar
  18. Johnson-Laird P. N. (1983) Mental models. Harvard University Press, CambridgeGoogle Scholar
  19. Johnson-Laird P. N. (2008) How we reason. Oxford University Press, OxfordGoogle Scholar
  20. Johnson-Laird P. N. (2008) Mental models and deductive reasoning. In: Adler J. E., Rips L. J. (eds) Reasoning: Studies of human inference and its foundations. Cambridge University Press, CambridgeGoogle Scholar
  21. Laird J., Newell A., Rosenbloom P. (1987) Soar: An architecture for general intelligence. Artificial Intelligence 33(3): 1–64CrossRefGoogle Scholar
  22. van Lambalgen M., Stenning K. (2008) Interpretation, representation and deductive reasoning. In: Adler J. E., Rips L. J. (eds) Reasoning: Studies of human inference and its foundations. Cambridge University Press, CambridgeGoogle Scholar
  23. Lovett M. C., Anderson J. R. (2005) Thinking as a production system. In: Holyoak K. J., Morrison R. (eds) The Cambridge handbook of thinking and reasoning. Cambridge University Press, CambridgeGoogle Scholar
  24. Luck S. J., Hollingworth A. (2008) Visual memory systems. In: Luck S. J., Hollingworth A. (eds) Visual memory. Oxford University Press, OxfordCrossRefGoogle Scholar
  25. Miller G. A. (1956) The magical number seven, plus or minus two: Some limits on our capacity for processing information. Psychological Review 63: 81–97CrossRefGoogle Scholar
  26. Negri S., von Plato J. (2001) Structural proof theory. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  27. Newell, A., & Simon, H. A. (1956). The logic theory machine: A complex information processing system. IRE Transactions on Information Theory, IT-2(3), 61–79.Google Scholar
  28. Newell A., Simon H. A. (1961) GPS, a program that simulates human thought. In: Billing H. (eds) Lernende automaten. R. Oldenbourg, München, pp 109–124Google Scholar
  29. Newell A., Simon H. A. (1972) Human problem solving. Prentice-Hall, Englewood Cliffs, NJGoogle Scholar
  30. Newstead S. (1989) Interpretational errors in syllogistic reasoning. Journal of Memory and Language 28: 78–91CrossRefGoogle Scholar
  31. Newstead S. (1995) Gricean implicatures and syllogistic reasoning. Journal of Memory and Language 34: 644–664CrossRefGoogle Scholar
  32. Osherson D. N. (1976) Logical abilities in children (Vol. 1–4). Erlbaum, Hillsdale, NJGoogle Scholar
  33. Prawitz D. (1965) Natural deduction. A proof-theoretical study, Stockholm studies in philosophy (Vol. 3). Almqvist & Wiksell, StockholmGoogle Scholar
  34. Rips L. (1996) The psychology of proof. Bradford, CambridgeGoogle Scholar
  35. Rips L. J. (2008) Logical approaches to human reasoning. In: Adler J. E., Rips L. J. (eds) Reasoning: Studies of human inference and its foundations. Cambridge University Press, CambridgeGoogle Scholar
  36. Robinson A., Voronkov A. (2001) Handbook of automated reasoning. Elsevier Science, AmsterdamGoogle Scholar
  37. Schütte K. (1960) Beweistheorie Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete (Vol. 103). Springer, New YorkGoogle Scholar
  38. Sheeran M., Stålmarck G. (2000) A tutorial on Stålmarck’s proof procedure for propositional logic. Formal Methods in Systems Design 16(1): 23–58CrossRefGoogle Scholar
  39. Smith R. E., Passer M. W. (2008) Psychology: The science of mind and behavior. McGraw-Hill, New YorkGoogle Scholar
  40. Smullyan, R. M. (1995). First-order logic (2nd corrected ed.). Dover Publications, New York. First published 1968 by Springer.Google Scholar
  41. Strannegård C. (2006) A proof system for modeling reasoning processes in propositional logic. Bulletin of Symbolic Logic 12(5): 347Google Scholar
  42. Strannegård, C. (2007). Proving first-order sentences with bounded cognitive resources. Philosophical communications, Web series, no. 39, Göteborg University.Google Scholar
  43. Troelstra A. S., & van Dalen D. (1988). Constructivism in mathematics, vol 1. Studies in logic and the foundations of mathematics (Vol. 121). North Holland, Amsterdam.Google Scholar
  44. Troelstra A. S., Schwichtenberg H. (1996) Basic proof theory. Cambridge University Press, CambridgeGoogle Scholar
  45. Wason, P. C. (1966). Reasoning. In New horizons in psychology. Penguin.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • Claes Strannegård
    • 1
  • Simon Ulfsbäcker
    • 2
  • David Hedqvist
    • 2
  • Tommy Gärling
    • 3
  1. 1.Department of Applied Information TechnologyChalmers University of TechnologyGothenburgSweden
  2. 2.Department of Computer ScienceChalmers University of TechnologyGothenburgSweden
  3. 3.Department of PsychologyUniversity of GothenburgGothenburgSweden

Personalised recommendations