Skip to main content


  • GERHARD SCHURZ (a1) and PAUL D. THORN (a1)

Systems of logico-probabilistic (LP) reasoning characterize inference from conditional assertions that express high conditional probabilities. In this paper we investigate four prominent LP systems, the systems O, P, Z, and QC. These systems differ in the number of inferences they licence (O ⊂ P ⊂ ZQC). LP systems that license more inferences enjoy the possible reward of deriving more true and informative conclusions, but with this possible reward comes the risk of drawing more false or uninformative conclusions. In the first part of the paper, we present the four systems and extend each of them by theorems that allow one to compute almost-tight lower-probability-bounds for the conclusion of an inference, given lower-probability-bounds for its premises. In the second part of the paper, we investigate by means of computer simulations which of the four systems provides the best balance of reward versus risk. Our results suggest that system Z offers the best balance.

Corresponding author
Hide All
Adams E. W. (1965). The logic of conditionals. Inquiry, 8, 166197.
Adams E. W. (1971). A note on comparing probabilistic and modal logics of conditionals. Theoria, 43, 186194.
Adams E. W. (1974). On the logic of ‘almost all’. Journal of Philosophical Logic, 3, 317.
Adams E. W. (1975). The Logic of Conditionals. Dordrecht, The Netherlands: Reidel.
Adams E. W. (1986). On the logic of high probability. Journal of Philosophical Logic, 15, 255279.
Adams E. W. (1996). Four probability-preserving properties of inferences. Journal of Philosophical Logic, 25, 124.
Bamber D. (2000). Entailment with near surety of scaled assertions of high conditional probability. Journal of Philosophical Logic, 29, 174.
Bennett J. (2003). A Philosophical Guide to Conditionals. New York: Oxford University Press.
Bourne R., & Parsons S. (1998). Propagating probabilities in system P. In Proceedings of the 11th International FLAIRS Conference. pp. 440445.
Brewka G. (1991). Nonmonotonic Reasoning: Logical Foundations of Commonsense. Cambridge, UK: Cambridge University Press.
Carnap R. (1950). Logical Foundations of Probability. Chicago, IL: University of Chicago.
Carnap R. (1971). Inductive logic and rational decisions. In Carnap R., and Jeffrey R., editors. Studies in Inductive Logic and Probability I. Los Angeles, CA: University of California Press, pp. 532.
Delgrande J. P. (1988). An approach to default reasoning based on a 1st order conditional logic: Revised report. Artificial Intelligence, 36, 6390.
Edgington D. (1995). On conditionals. Mind, 104, 235329.
Edgington D. (2001). Conditionals. In Goble L., editor. The Blackwell Guide to Philosophical Logic. Oxford, UK: Blackwell, pp. 385414.
Evans J. S., Simon J. H., & Over D. E. (2003). Conditionals and conditional probability. Journal of Experimental Psychology: Learning, Memory, and Cognition, 29, 321335.
Gabbay D. (1984). Theoretical foundations for non-monotonic reasoning in expert systems. In Apt K. R., editor. Logics and Models for Concurrent Systems. Berlin: Springer, pp. 439458.
Gabbay D., Hogger C., & Robinson J., editors. (1994). Handbook of Logic in Artificial Intelligence, Vol. 3: Nonmonotonic Reasoning and Uncertain Reasoning. Oxford, UK: Clarendon Press.
Gärdenfors P., & Makinson D. (1994). Nonmonotonic inference based on expectation orderings. Artificial Intelligence, 65, 197245.
Gigerenzer G., & Hoffrage U. (1995). How to improve Bayesian reasoning without instruction. Psychological Review, 102, 684704.
Gilio A. (2002). Precise propagation of upper and lower probability bounds in system P. Annals of Mathematics and Artificial Intelligence, 34, 534.
Goldszmidt M., & Pearl J. (1996). Qualitative probabilities for default reasoning, belief revision and causal modeling. Artificial Intelligence, 84, 57112.
Good I. J. (1983). Good Thinking. Minneapolis, MN: University of Minnesota Press.
Halpern J. (2003). Reasoning about Uncertainty. Cambridge, MA: MIT Press.
Halpern J., & Koller D. (2004). Representation dependence in probabilistic inference. Journal of Artificial Intelligence Research, 21, 319356.
Harper W. (1976). Rational belief change, Popper-functions and counterfactuals. In Harper W., and Hooker C. A., editors. Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science. Dordrecht, The Netherlands: Reidel, pp. 73112.
Harper W., Stalnaker R., & Pearce G., editors. (1981). Ifs. Dordrecht, The Netherlands: Reidel.
Hawthorne J. (1996). On the logic of non-monotonic conditionals and conditional probabilities. Journal of Philosophical Logic, 25, 185218.
Hawthorne J. (2005). Degree-of-belief and degree-of-support: Why Bayesians need both notions. Mind, 114, 277320.
Hawthorne J., & Makinson D. (2007). The quantitative/qualitative watershed for rules of uncertain inference. Studia Logica, 86, 247297.
Horty J. (2002). Skepticism and floating conclusions. Artificial Intelligence, 135, 5572.
Horty J. (2007). Defaults with priorities. Journal of Philosophical Logic, 36, 367413.
Hill L. H., & Paris J. B. (2003). When maximizing entropy gives the rational closure. Journal of Logic and Computation, 13, 5168.
Howson C., & Urbach P. (2006). Scientific Reasoning: The Bayesian Approach (third edition). Chicago, IL: Open Court Publishing.
Jaynes E. (1968). Prior probabilities, IEEE Transactions On Systems Science and Cybernetics, 4, 227241.
Johnson M. P., & Parikh R. (2008). Probabilistic conditionals are almost monotonic. The Review of Symbolic Logic, 1, 17.
Kraus S., Lehmann D., & Magidor M. (1990). Nonmonotonic reasoning, preferential models and cumulative logics. Artificial Intelligence, 44, 167207.
Kyburg H. E. (1974). The Logical Foundations of Statistical Inference. Dordrecht, The Netherlands: Reidel.
Lehmann D., & Magidor M. (1992). What does a conditional knowledge base entail? Artificial Intelligence, 55, 160.
Leitgeb H. (2004). Inference on the Low Level. Dordrecht, The Netherlands: Kluwer.
Levi I. (1977). Direct inference. The Journal of Philosophy, 74, 529.
Lewis D. (1973). Counterfactuals. Oxford, UK: Blackwell.
Lewis D. (1976). Probabilities of conditionals and conditional probabilities. The Philosophical Review, 85, 297315.
Lukasiewicz T. (1999). Probabilistic deduction with conditional constraints over basic events. Journal of Artificial Intelligence Research, 10, 199241.
Makinson D. (2005). Bridges from Classical to Nonmonotonic Logic. London: College Publications.
Makinson D. (2011). Conditional probability logic in the light of qualitative belief change. Journal of Philosophical Logic, 40, 125153.
McDermott D., & Doyle J. (1980). Non-monotonic logic I. Artificial Intelligence, 25, 4172.
McGee V. (1989). Conditional probabilities and compounds of conditionals. The Philosophical Review, 98, 485541.
Moore R. C. (1985). Semantical considerations on nonmonotonic logic. Artificial Intelligence, 25, 7594.
Oberauer K., & Wilhelm O. (2003). The meaning(s) of conditionals: Conditional probabilities, mental models, and personal utilities. Journal of Experimental Psychology: Learning, Memory, and Cognition, 29, 680693.
Over D. E. (2003a). From massive modularity to metarepresentation: The evolution of higher cognition. In Over D. E., editor. Evolution and the Psychology of Thinking: The Debate, Hove, UK: Psychology Press, pp. 121144.
Over D. E., editor. (2003b). Evolution and the Psychology of Thinking: The Debate. Hove, UK: Psychology Press.
Paris J. B. (1994). The Uncertain Reasoner’s Companion. Cambridge: Cambridge University Press.
Paris J. B., & Simmonds R. (2009). O is not enough. Review of Symbolic Logic, 2, 298309.
Paris J. B., & Vencovská A. (1997). In defence of the maximum entropy inference process. International Journal of Approximate Reasoning, 17, 77103
Pearl J. (1988). Probabilistic Reasoning in Intelligent Systems. Santa Mateo, CA: Morgan Kaufmann.
Pearl J. (1990). System Z. In Proceedings of Theoretical Aspects of Reasoning about Knowledge, Santa Mateo, CA. pp. 21135.
Pelletier F. J., & Elio R. (1997). What should default reasoning be, by default? Computational Intelligence, 13, 165187.
Pollock J. (1990). Nomic Probability and the Foundations of Induction. Oxford: Oxford University Press.
Pollock J. (1994). Justification and defeat. Artificial Intelligence, 67, 377407.
Pollock J. (1995). Cognitive carpentry: A blueprint for how to build a person. Cambridge, MA: MIT Press.
Poole D. (1988). A logical framework for default reasoning. Artificial Intelligence, 36, 2747.
Poole D. (1991). The Effect of Knowledge on Belief. Artificial Intelligence, 49, 281307.
Reichenbach H. (1949). The Theory of Probability. Berkeley: University of California Press.
Reiter R. (1980). A logic for default reasoning. Artificial Intelligence, 13, 81132.
Schurz G. (1997). Probabilistic default reasoning based on relevance and irrelevance assumptions. In Gabbay D., et al. ., editors. Qualitative and Quantitative Practical Reasoning (LNAI 1244). Berlin: Springer, pp. 536553.
Schurz G. (1998). Probabilistic semantics for Delgrande’s conditional logic and a counterexample to his default logic. Artificial Intelligence, 102, 8195.
Schurz G. (2001). What is ‘normal’? An evolution-theoretic foundation of normic laws and their relation to statistical normality. Philosophy of Science, 68, 476497.
Schurz G. (2005a). Non-monotonic reasoning from an evolutionary viewpoint. Synthese, 146, 3751.
Schurz G. (2005b). Logic, matter of form, and closure under substitution. In Behounek L., and Bilkova M., editors. The Logica Yearbook 2004. Prague, Czech Republic: Filosofia, pp. 3346.
Schurz G. (2007). Human conditional reasoning explained by non-monotonicity and probability. In Vosniadou S., et al. ., editors. Proceedings of EuroCogSci07. The European Cognitive Science Conference 2007. New York: Lawrence Erlbaum Assoc., pp. 628633.
Schurz G. (2012). Tweety, or why probabilism (and even Bayesianism) need objective and evidential probabilities. In Dieks D., et al. ., editors. Probabilities, Laws and Structures. New York: Springer, pp. 5774.
Schurz G., & Leitgeb H. (2008). Finitistic and frequentistic approximations of probability measures with or without sigma-additivity. Studia Logica, 89/2, 258283.
Segerberg K. (1989). Notes on conditional logic. Studia Logica, 48, 157168.
Skyrms B. (1980). Causal Necessity. New Haven, CT: Yale University Press.
Spohn W. (1980). Stochastic independence, causal independence, and shieldability. Journal of Philosophical Logic, 9, 7399.
Spohn W. (1988). Ordinal conditional functions: A dynamic theory of epistemic states. In Harper W. L., and Skyrms B., editors. Causation in Decision, Belief Change and Statistics. Dordrecht, The Netherlands: Reidel, pp. 105134.
Stalnaker R. C. (1970). Probability and conditionals. Philosophy of Science, 37, 6480.
Strevens M. (2000). Do large probabilities explain better? Philosophy of Science, 67, 366390.
Suppes P. (1966). Probabilistic inference and the concept of total evidence. In Hintikka J., and Suppes P. editors. Aspects of Inductive Logic. Amsterdam: North-Holland Publ. Comp., pp. 4965.
Touretzky D., Horty J., & Thomason R. (1987). A clash of intuitions: The current state of monotonic multiple inheritance systems. In Proceedings of the Tenth international Joint Conference on Artificial Intelligence. pp. 476482.
Van Fraassen B. (1989). Laws and Symmetry. Oxford: Oxford University Press.
Williamson J. (2007). Motivating objective Bayesianism: From empirical constraints to objective probabilities. In Harper W. L., and Wheeler G. R., editors. Probability and Inference: Essays in Honor of Henry E. Kyburg Jr. London: College Publications, pp. 155183.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Review of Symbolic Logic
  • ISSN: 1755-0203
  • EISSN: 1755-0211
  • URL: /core/journals/review-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 18 *
Loading metrics...

Abstract views

Total abstract views: 167 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 24th November 2017. This data will be updated every 24 hours.