# De Re, De Dicto, and Binding Modalities

• Melvin Fitting
Conference paper
Part of the Logic in Asia: Studia Logica Library book series (LIAA)

## Abstract

In classical logic, the move from propositional to quantificational is profound but essentially takes one route, following a direction we are all familiar with. In modal logic, such a move shoots off in many directions at once. One can quantify over things or over intensions. Quantifier domains can be the same from possible world to possible world, shrink or grow as one moves from a possible world to an accessible one, or follow no pattern whatsoever. A long time ago, Kripke showed us how shrinking or growing domains are related to validity of the Barcan and the converse Barcan formulas, bringing some semantic order into the situation. But when it comes to proof theory, things get somewhat strange. Nested sequents for shrinking or growing domains, or for constant domains or completely varying domains, are relatively straightforward. But axiomatically some oddities are quickly apparent. A simple combination of propositional modal axioms and rules with standard quantificational axioms and rules proves the converse Barcan formula, making it impossible to investigate its absence. Kripke showed how one could avoid this, at the cost of using a less common axiomatization of the quantifiers. But things can be complicated and even here an error crept into Kripke’s work that wasn’t pointed out until 20 years later, by Fine. Justification logic was created by Artemov, with a system called $$\textsf {LP}$$ which is related to propositional $$\textsf {S4}$$. This was extended to a quantified version by Artemov and Yavorskaya, for which a possible world semantics was supplied by Fitting. Subsequently Artemov and Yavorskaya introduced what they called binding modalities, by transferring ideas back from quantified $$\textsf {LP}$$ to $$\textsf {S4}$$. In this paper, we continue the investigation of binding modalities, but for $$\textsf {K}$$, and show that they provide a natural intuition for Kripke’s non-standard axiomatization, and relate directly to the distinction between de re and de dicto. Unlike in Kripke’s treatment, the heavy lifting is done through a generalization of the modal operator, instead of a restriction on quantifier axiomatizations.

## References

1. Artemov, Sergei, and Melvin Fitting. 2019. Justification logic: reasoning with reasons. Cambridge tracts in mathematics book 216. Cambridge: Cambridge University Press.Google Scholar
2. Artemov, Sergei N. 2001. Explicit Provability and Constructive Semantics. Bulletin of Symbolic Logic 7 (1): 1–36.
3. Artemov, Sergei N., and Melvin C. Fitting. 2011. Justification logic. In ed. Edward N. Zalta. Revised 2015. http://plato.stanford.edu/entries/logic-justification/.
4. Artemov, Sergei N., and Tatiana Yavorskaya (Sidon). 2011. First-order logic of proofs. Technical report, TR-2011005. City University of New York.Google Scholar
5. Artemov, Sergei N., and Tatiana Yavorskaya (Sidon). 2012. Binding modalities. Technical report, TR-2012011. City University of New York.Google Scholar
6. Artemov, Sergei N., and Tatiana Yavorskaya (Sidon). 2013. Binding modalities. Journal of Logic and Computation 26 (1): 451–461. Published online 07 Oct 2013.Google Scholar
7. Corsi, Giovanna. 2002. A unified completeness theorem for quantified modal logics. Journal of Symbolic Logic 67 (4): 1483–1510.
8. Feferman, Solomon, et al. (eds.). 1986–2003. Kurt Gödel collected works, vol. 5. Oxford: Oxford University Press.Google Scholar
9. Fine, Kit. 1983. The permutation principle in quantificational logic. Journal of Philosophical Logic 12: 33–37.
10. Fitting, Melvin. 2014. Possible world semantics for first-order logic of proofs. Annals of Pure and Applied Logic 165 (1): 225–240. Published online in Aug 2013.Google Scholar
11. Fitting, Melvin C., and Felipe Salvatore. 2017. First-order justification logic with constant domain semantics. To appear in Annals of Pure and Applied Logic.Google Scholar
12. Göodel, Kurt. 1933. Eine Interpretation des intuistionistischen Aussagenkalküls. In Ergebnisse eines mathematischen Kolloquiums, vol. 4. Translated as An interpretation of the intuitionistic propositional calculus in [8] I, 296–301, 39–40.Google Scholar
13. Kripke, Saul. 1963. Semantical considerations on modal logics. In Acta philosophica fennica, modal and many-valued logics, 83–94.Google Scholar
14. Mendelson, Elliott. 1997. Introduction to mathematical logic, 4th ed. London: Chapman and Hall.Google Scholar
15. Prior, Arthur. 1956. Modality and quantification in S5. Journal of Symbolic Logic 21: 60–62.
16. Quine, Willard Van Orman. 1940. Mathematical logic, 2nd ed. Cambridge: Harvard University Press. Revised 1951.Google Scholar