Skip to main content
SpringerLink
Log in

Interpreting Connexive Principles in Coherence-Based Probability Logic

  • Conference paper
  • First Online:
Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU 2021)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 12897))

Abstract

We present probabilistic approaches to check the validity of selected connexive principles within the setting of coherence. Connexive logics emerged from the intuition that conditionals of the form If \(\mathord {\thicksim }A\), then A, should not hold, since the conditional’s antecedent \(\mathord {\thicksim }A\) contradicts its consequent A. Our approach covers this intuition by observing that for an event A the only coherent probability assessment on the conditional event \(A|\bar{A}\) is \(p(A|\bar{A})=0\). Moreover, connexive logics aim to capture the intuition that conditionals should express some “connection” between the antecedent and the consequent or, in terms of inferences, validity should require some connection between the premise set and the conclusion. This intuition is covered by a number of principles, a selection of which we analyze in our contribution. We present two approaches to connexivity within coherence-based probability logic. Specifically, we analyze connections between antecedents and consequents firstly, in terms of probabilistic constraints on conditional events (in the sense of defaults, or negated defaults) and secondly, in terms of constraints on compounds of conditionals and iterated conditionals. After developing different notions of negations and notions of validity, we analyze the following connexive principles within both approaches: Aristotle’s Theses, Aristotle’s Second Thesis, Abelard’s First Principle and selected versions of Boethius’ Theses. We conclude by remarking that coherence-based probability logic offers a rich language to investigate the validity of various connexive principles.

Both authors contributed equally to the article and are listed alphabetically.

N. Pfeifer is supported by the BMBF project 01UL1906X.

G. Sanfilippo is a member of the GNAMPA Research Group and partially supported by the INdAM–GNAMPA Project 2020 Grant U-UFMBAZ-2020-000819.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 99.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 129.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Institutional subscriptions

Notes

  1. 1.

    According to \(\varepsilon \)-semantics (see, e.g., [1, 38]) a default is interpreted by \(p(C|A)\ge 1-\varepsilon \), with \(\varepsilon >0\) and \(p(A)>0\). Gilio introduced a coherence-based probability semantics for defaults by also allowing \(\varepsilon \) and p(A) to be zero [15]. In this context, defaults in terms of probability 1 can be used to give a alternative definition of p-entailment which preserve the usual non-monotonic inference rules like those of System P [4, 15, 20, 21], see also [9, 10]. For the psychological plausibility of the coherence-based semantics of non-monotonic reasoning, see, e.g., [41,42,43, 47].

References

  1. Adams, E.W.: The logic of conditionals. An application of probability to deduction. Reidel, Dordrecht (1975)

    Book  Google Scholar 

  2. Berti, P., Regazzini, E., Rigo, P.: Well calibrated, coherent forecasting systems. Theory Probability Appl. 42(1), 82–102 (1998). https://doi.org/10.1137/S0040585X97975988

    Article  MathSciNet  MATH  Google Scholar 

  3. Biazzo, V., Gilio, A.: A generalization of the fundamental theorem of de Finetti for imprecise conditional probability assessments. Int. J. Approximate Reasoning 24(2–3), 251–272 (2000)

    Article  MathSciNet  Google Scholar 

  4. Biazzo, V., Gilio, A., Lukasiewicz, T., Sanfilippo, G.: Probabilistic logic under coherence, model-theoretic probabilistic logic, and default reasoning in System P. J. Appl. Non-Classical Logics 12(2), 189–213 (2002)

    Article  MathSciNet  Google Scholar 

  5. Calabrese, P.: Logic and Conditional Probability: A Synthesis. College Publications (2017)

    Google Scholar 

  6. Capotorti, A., Lad, F., Sanfilippo, G.: Reassessing accuracy rates of median decisions. Am. Stat. 61(2), 132–138 (2007)

    Article  MathSciNet  Google Scholar 

  7. Ciucci, D., Dubois, D.: Relationships between connectives in three-valued logics. In: Greco, S., Bouchon-Meunier, B., Coletti, G., Fedrizzi, M., Matarazzo, B., Yager, R.R. (eds.) IPMU 2012. CCIS, vol. 297, pp. 633–642. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31709-5_64

    Chapter  Google Scholar 

  8. Ciucci, D., Dubois, D.: A map of dependencies among three-valued logics. Inf. Sci. 250, 162–177 (2013). https://doi.org/10.1016/j.ins.2013.06.040

    Article  MathSciNet  MATH  Google Scholar 

  9. Coletti, G., Scozzafava, R.: Probabilistic Logic in a Coherent Setting. Kluwer, Dordrecht (2002)

    Book  Google Scholar 

  10. Coletti, G., Scozzafava, R., Vantaggi, B.: Possibilistic and probabilistic logic under coherence: Default reasoning and System P. Math. Slovaca 65(4), 863–890 (2015)

    Article  MathSciNet  Google Scholar 

  11. de Finetti, B.: Sul significato soggettivo della probabilitá. Fundam. Math. 17, 298–329 (1931)

    Article  Google Scholar 

  12. de Finetti, B.: Theory of Probability, vol. 1, 2. Wiley, Chichester (1970/1974)

    Google Scholar 

  13. Flaminio, T., Godo, L., Hosni, H.: Boolean algebras of conditionals, probability and logic. Artif. Intell. 286, 103347 (2020). https://doi.org/10.1016/j.artint.2020.103347

  14. Freund, M., Lehmann, D., Morris, P.: Rationality, transitivity, and contraposition. Artif. Intell. 52(2), 191–203 (1991)

    Article  MathSciNet  Google Scholar 

  15. Gilio, A.: Probabilistic reasoning under coherence in System P. Ann. Math. Artif. Intell. 34, 5–34 (2002)

    Article  MathSciNet  Google Scholar 

  16. Gilio, A., Pfeifer, N., Sanfilippo, G.: Transitivity in coherence-based probability logic. J. Appl. Log. 14, 46–64 (2016). https://doi.org/10.1016/j.jal.2015.09.012

    Article  MathSciNet  MATH  Google Scholar 

  17. Gilio, A., Pfeifer, N., Sanfilippo, G.: Probabilistic entailment and iterated conditionals. In: Elqayam, S., Douven, I., Evans, J.S.B.T., Cruz, N. (eds.) Logic and Uncertainty in the Human Mind: A Tribute to David E. Over, pp. 71–101. Routledge, Oxon (2020). https://doi.org/10.4324/9781315111902-6

  18. Gilio, A., Sanfilippo, G.: Conditional random quantities and iterated conditioning in the setting of coherence. In: van der Gaag, L.C. (ed.) ECSQARU 2013. LNCS (LNAI), vol. 7958, pp. 218–229. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39091-3_19

    Chapter  MATH  Google Scholar 

  19. Gilio, A., Sanfilippo, G.: Conjunction, disjunction and iterated conditioning of conditional events. In: Kruse, R., Berthold, M.R., Moewes, C., Gil, M.A., Grzegorzewski, P., Hryniewicz, O. (eds.) Synergies of Soft Computing and Statistics for Intelligent Data Analysis, AISC, vol. 190, pp. 399–407. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-33042-1_43

    Chapter  MATH  Google Scholar 

  20. Gilio, A., Sanfilippo, G.: Probabilistic entailment in the setting of coherence: the role of quasi conjunction and inclusion relation. Int. J. Approximate Reasoning 54(4), 513–525 (2013). https://doi.org/10.1016/j.ijar.2012.11.001

    Article  MathSciNet  MATH  Google Scholar 

  21. Gilio, A., Sanfilippo, G.: Quasi conjunction, quasi disjunction, t-norms and t-conorms: Probabilistic aspects. Inf. Sci. 245, 146–167 (2013). https://doi.org/10.1016/j.ins.2013.03.019

    Article  MathSciNet  MATH  Google Scholar 

  22. Gilio, A., Sanfilippo, G.: Conditional random quantities and compounds of conditionals. Stud. Logica. 102(4), 709–729 (2014). https://doi.org/10.1007/s11225-013-9511-6

    Article  MathSciNet  MATH  Google Scholar 

  23. Gilio, A., Sanfilippo, G.: Conjunction of conditional events and t-norms. In: Kern-Isberner, G., Ognjanović, Z. (eds.) ECSQARU 2019. LNCS (LNAI), vol. 11726, pp. 199–211. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-29765-7_17

    Chapter  MATH  Google Scholar 

  24. Gilio, A., Sanfilippo, G.: Generalized logical operations among conditional events. Appl. Intell. 49(1), 79–102 (2019). https://doi.org/10.1007/s10489-018-1229-8

    Article  MATH  Google Scholar 

  25. Gilio, A., Sanfilippo, G.: Algebraic aspects and coherence conditions for conjoined and disjoined conditionals. Int. J. Approximate Reasoning 126, 98–123 (2020). https://doi.org/10.1016/j.ijar.2020.08.004

    Article  MathSciNet  MATH  Google Scholar 

  26. Gilio, A., Sanfilippo, G.: Compound conditionals, Fréchet-Hoeffding bounds, and Frank t-norms. Int. J. Approximate Reasoning 136, 168–200 (2021). https://doi.org/10.1016/j.ijar.2021.06.006

    Article  MATH  Google Scholar 

  27. Gilio, A., Sanfilippo, G.: On compound and iterated conditionals. Argumenta 6(2), 241–266 (2021). https://doi.org/10.14275/2465-2334/202112.gil

  28. Goodman, I.R., Nguyen, H.T., Walker, E.A.: Conditional Inference and Logic for Intelligent Systems: A Theory of Measure-Free Conditioning. North-Holland (1991)

    Google Scholar 

  29. Holzer, S.: On coherence and conditional prevision. Bollettino dell’Unione Matematica Italiana 4(6), 441–460 (1985)

    MathSciNet  MATH  Google Scholar 

  30. Horn, L.R.: A Natural History of Negation. CSLI Publications, Stanford (2001)

    Google Scholar 

  31. Kaufmann, S.: Conditionals right and left: probabilities for the whole family. J. Philos. Log. 38, 1–53 (2009)

    Article  MathSciNet  Google Scholar 

  32. Kraus, S., Lehmann, D., Magidor, M.: Nonmonotonic reasoning, preferential models and cumulative logics. Artif. Intell. 44, 167–207 (1990)

    Article  MathSciNet  Google Scholar 

  33. Lad, F.: Operational Subjective Statistical Methods: A Mathematical, Philosophical, and Historical Introduction. Wiley, New York (1996)

    MATH  Google Scholar 

  34. Lenzen, W.: A critical examination of the historical origins of connexive logic. Hist. Philos. Logic 41(1), 16–35 (2020)

    Article  MathSciNet  Google Scholar 

  35. Lewis, D.: Probabilities of conditionals and conditional probabilities. Philos. Rev. 85, 297–315 (1976)

    Article  Google Scholar 

  36. McCall, S.: A history of connexivity. In: Gabbay, D.M., Pelletier, F.J., Woods, J. (eds.) Handbook of the History of Logic, vol. 11 (Logic: A History of its Central Concepts). Elsevier, Amsterdam (2012)

    Google Scholar 

  37. McGee, V.: Conditional probabilities and compounds of conditionals. Philos. Rev. 98, 485–541 (1989)

    Article  Google Scholar 

  38. Pearl, J.: Probabilistic semantics for nonmonotonic reasoning: a survey. In: Shafer, G., Pearl, J. (eds.) Readings in Uncertain Reasoning, pp. 699–711. Morgan Kaufmann, San Mateo (1990)

    MATH  Google Scholar 

  39. Pfeifer, N.: Experiments on Aristotle’s Thesis: towards an experimental philosophy of conditionals. Monist 95(2), 223–240 (2012)

    Article  Google Scholar 

  40. Pfeifer, N.: Probability logic. In: Knauff, M., Spohn, W. (eds.) Handbook of Rationality. MIT Press, Cambridge (in press)

    Google Scholar 

  41. Pfeifer, N., Kleiter, G.D.: Coherence and nonmonotonicity in human reasoning. Synthese 146(1–2), 93–109 (2005)

    Article  MathSciNet  Google Scholar 

  42. Pfeifer, N., Kleiter, G.D.: Framing human inference by coherence based probability logic. J. Appl. Log. 7(2), 206–217 (2009)

    Article  MathSciNet  Google Scholar 

  43. Pfeifer, N., Kleiter, G.D.: The conditional in mental probability logic. In: Oaksford, M., Chater, N. (eds.) Cognition and Conditionals: Probability and Logic in Human Thought, pp. 153–173. Oxford University Press, Oxford (2010)

    Chapter  Google Scholar 

  44. Pfeifer, N., Sanfilippo, G.: Probabilistic squares and hexagons of opposition under coherence. Int. J. Approximate Reasoning 88, 282–294 (2017). https://doi.org/10.1016/j.ijar.2017.05.014

    Article  MathSciNet  MATH  Google Scholar 

  45. Pfeifer, N., Sanfilippo, G.: Probabilistic semantics for categorical syllogisms of Figure II. In: Ciucci, D., Pasi, G., Vantaggi, B. (eds.) SUM 2018. LNCS (LNAI), vol. 11142, pp. 196–211. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-00461-3_14

    Chapter  Google Scholar 

  46. Pfeifer, N., Sanfilippo, G.: Probability propagation in selected Aristotelian syllogisms. In: Kern-Isberner, G., Ognjanović, Z. (eds.) ECSQARU 2019. LNCS (LNAI), vol. 11726, pp. 419–431. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-29765-7_35

    Chapter  Google Scholar 

  47. Pfeifer, N., Tulkki, L.: Conditionals, counterfactuals, and rational reasoning. An experimental study on basic principles. Minds Mach. 27(1), 119–165 (2017)

    Google Scholar 

  48. Ramsey, F.P.: General propositions and causality (1929). In: Mellor, D.H. (ed.) Philosophical Papers by F. P. Ramsey, pp. 145–163. Cambridge University Press, Cambridge (1929/1994)

    Google Scholar 

  49. Regazzini, E.: Finitely additive conditional probabilities. Rendiconti del Seminario Matematico e Fisico di Milano 55, 69–89 (1985)

    Article  MathSciNet  Google Scholar 

  50. Sanfilippo, G.: Lower and upper probability bounds for some conjunctions of two conditional events. In: Ciucci, D., Pasi, G., Vantaggi, B. (eds.) SUM 2018. LNCS (LNAI), vol. 11142, pp. 260–275. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-00461-3_18

    Chapter  Google Scholar 

  51. Sanfilippo, G., Gilio, A., Over, D.E., Pfeifer, N.: Probabilities of conditionals and previsions of iterated conditionals. Int. J. Approximate Reasoning 121, 150–173 (2020). https://doi.org/10.1007/978-3-030-00461-3_14

    Article  MathSciNet  MATH  Google Scholar 

  52. Walley, P., Pelessoni, R., Vicig, P.: Direct algorithms for checking consistency and making inferences from conditional probability assessments. J. Stat. Plann. Inference 126(1), 119–151 (2004)

    Article  MathSciNet  Google Scholar 

  53. Wansing, H.: Connexive logic. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy. Spring 2020 edn. (2020). https://plato.stanford.edu/entries/logic-connexive/

Download references

Acknowledgment

Thanks to four anonymous reviewers for useful comments.

Author information

Authors and Affiliations

  1. Department of Philosophy, University of Regensburg, Regensburg, Germany

    Niki Pfeifer

  2. Department of Mathematics and Computer Science, University of Palermo, Palermo, Italy

    Giuseppe Sanfilippo

Authors
  1. Niki Pfeifer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giuseppe Sanfilippo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Niki Pfeifer or Giuseppe Sanfilippo .

Editor information

Editors and Affiliations

  1. Institute of Information Theory and Automation, Prague, Czech Republic

    Jiřina Vejnarová

  2. Insight Centre for Data Analytics, School of Computer Science and IT University College Cork, Cork, Ireland

    Nic Wilson

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Pfeifer, N., Sanfilippo, G. (2021). Interpreting Connexive Principles in Coherence-Based Probability Logic. In: Vejnarová, J., Wilson, N. (eds) Symbolic and Quantitative Approaches to Reasoning with Uncertainty. ECSQARU 2021. Lecture Notes in Computer Science(), vol 12897. Springer, Cham. https://doi.org/10.1007/978-3-030-86772-0_48

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-86772-0_48

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-86771-3

  • Online ISBN: 978-3-030-86772-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation