Abstract
A paper by Dag Westerståhl and myself twenty years ago introduced operators that are both connectives and quantifiers. We introduced two binary operators that are classically interdefinable: one that fuses conjunction and existential quantification and one that fuses implication and universal quantification. We called the system PFO. A complete Gentzen-Prawitz style Natural Deduction axiomatization of classical PL was provided. For intuitionistic PL, however, it seemed that existential quantification should be fused with disjunction rather than with conjunction. Whether this was true, and if so why, were questions not answered at the time. Also, it seemed that there is no uniform definition of such a disjunctive-existential operator in classical PFO. This, too, remained a conjecture. In this paper, I return to these previously unresolved questions, and resolve them.
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- 1.
This problem was introduced into the modern literature by Geach (1962).
- 2.
It also turned out to be straightforward to extend the system to treating sub-sentential expression in a flexibly binding version of Montague Grammar. All that was needed was to subject the \(\lambda \) operator to the same binding principles. See Pagin and Westerståhl (1994).
- 3.
The following observations are added in the present paper.
References
Geach, P. T. (1962). Reference and generality. Ithaca: Cornell University Press.
Groenendijk, J., & Stokhof, M. (1991). Dynamic predicate logic. Linguistics and Philosophy, 14, 39–100.
Pagin, P., & Westerståhl, D. (1993). Predicate logic with flexibly binding operators and natural language semantics. Journal of Logic, Language and Information, 2, 89–128.
Pagin, P & Westerståhl, D. (1994). Flexible variable-binding and Montague grammar. In P. Dekker & M. Stokhof (Eds.), Proceedings of the ninth Amsterdam Colloquium(pp. 519–525)
Prawitz, D. (1965). Natural deduction. A proof-theoretic study. Stockholm: Almqvist and Wiksell International. Republished by Dover Publications, Mineola, NY, 2006.
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Pagin, P. (2015). Fusing Quantifiers and Connectives: Is Intuitionistic Logic Different? . In: Wansing, H. (eds) Dag Prawitz on Proofs and Meaning. Outstanding Contributions to Logic, vol 7. Springer, Cham. https://doi.org/10.1007/978-3-319-11041-7_11
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DOI: https://doi.org/10.1007/978-3-319-11041-7_11
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