A Revised Projectivity Calculus for Inclusion and Exclusion Reasoning

  • Ka-fat ChowEmail author


We present a Revised Projectivity Calculus (denoted RC) that extends the scope of inclusion and exclusion inferences derivable under the Projectivity Calculus (denoted C) developed by Icard (Stud Log 100(4):705–725, 2012). After pointing out the inadequacies of C, we introduce four opposition properties (OPs) which have been studied by Chow (in: Aloni et al (eds) Proceedings of the 18th Amsterdam Colloquium, Springer, Berlin, 2012; Beziau, Georgiorgakis (eds) New dimensions of the square of opposition, Philosophia Verlag GmbH, München, 2017) and are more appropriate for the study of exclusion reasoning. Together with the monotonicity properties (MPs), the OPs will form the basis of RC instead of the additive/multiplicative properties used in C. We also prove some important results of the OPs and their relation with the MPs. We then introduce a set of projectivity signatures together with the associated operations and conditions for valid inferences, and develop RC by inheriting the key features of C. We then show that under RC, we can derive some inferences that are not derivable under C. We finally discuss some properties of RC and point to possible directions of further studies.


Inclusion Exclusion Opposition properties Projectivity signatures Natural Logic 



  1. Beghelli, F. (1994). Structured quantifiers. In M. Kanazawa & C. Piñón (Eds.), Dynamics, polarity and quantification (pp. 119–143). Stanford: CSLI.Google Scholar
  2. Chow, K. F. (2012). Generalizing monotonicity inferences to opposition inferences. In M. Aloni, et al. (Eds.), Proceedings of the 18th Amsterdam Colloquium (pp. 281–290). Berlin: Springer.Google Scholar
  3. Chow, K. F. (2017). Opposition inferences and generalized quantifiers. In J.-Y. Beziau & S. Georgiorgakis (Eds.), New dimensions of the square of opposition (pp. 155–199). München: Philosophia Verlag GmbH.Google Scholar
  4. Icard, T. F. (2012). Inclusion and exclusion in natural language. Studia Logica, 100(4), 705–725.CrossRefGoogle Scholar
  5. Icard, T. F. (2014). Higher-order syllogistics. In G. Morrill, et al. (Eds.), Formal grammar (pp. 1–14). Berlin: Springer.Google Scholar
  6. Icard, T. F., & Moss, L. S. (2014). Recent progress on monotonicity. Linguistic Issues in Language Technology, 9(7), 167–194.Google Scholar
  7. Keenan, E. L. (2003). Excursions in natural logic. In C. Casadio, et al. (Eds.), Language and grammar: Studies in mathematical linguistics and natural language (pp. 31–52). Stanford: CSLI.Google Scholar
  8. Keenan, E. L. (2008). Further excursions in natural logic: The mid-point theorems. In F. Hamm & S. Kepser (Eds.), Logics for linguistic structures (pp. 87–104). Berlin: Mouton de Gruyter.Google Scholar
  9. Keenan, E. L., & Faltz, L. M. (1985). Boolean semantics for natural language. Dordrecht: Reidel.Google Scholar
  10. Keenan, E. L., & Westerståhl, D. (2011). Generalized quantifiers in linguistics and logic. In J. van Benthem & A. ter Meulen (Eds.), Handbook of logic and language (2nd ed., pp. 859–910). Amsterdam: Elsevier.CrossRefGoogle Scholar
  11. MacCartney, B. (2009). Natural language inference. Ph.D. dissertation, Stanford University.Google Scholar
  12. MacCartney, B., & Manning, C.D. (2009). An extended model of natural logic. In Proceedings of the eighth international conference on computational semantics (pp. 140–156).Google Scholar
  13. Moss, L. S. (2012). The soundness of internalized polarity marking. Studia Logica, 100(4), 683–704.CrossRefGoogle Scholar
  14. Peters, S., & Westerståhl, D. (2006). Quantifiers in language and logic. Oxford: Clarendon Press.Google Scholar
  15. Sánchez Valencia, V. (1991). Studies on natural logic and categorial grammar. Ph.D. dissertation, Universiteit van Amsterdam.Google Scholar
  16. van Benthem, J. (1986). Essays in logical semantics. Dordrecht: Reidel.CrossRefGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.The Hong Kong Polytechnic UniversityHung HomHong Kong

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