**Topic:**

**GENERAL LOGIC**

## Modal Deduction in Second-Order Logic and Set Theory-I

By Johan van Benthem, Giovanna D'Agostino, Angelo Montanari, Alberto Policriti

We investigate modal deduction through translation into standard logic and set theory. Derivability in the minimal modal logic is captured precisely by translation into a weak, computationally attractive set theory \Omega. This approach is shown equivalent to working with standard first-order translations of modal formulas in a theory of general frames. Next, deduction in a more powerful second?order logic of general frames is shown equivalent with set?theoretic derivability in an `admissible variant' of \Omega. Our methods are mainly model?theoretic and set?theoretic, and they admit extension to richer languages than that of basic modal logic.Read more[1] [2]; [3]

Appeared in Journal of Logic and Computation 1997 7(2):251-265## Modal Deduction in Second-Order Logic and Set Theory - II

By Johan van Benthem, Giovanna D'Agostino, Angelo Montanari, Alberto Policriti

In this paper, we generalize the set-theoretic translation method for poly-modal logic introduced in [11] to extended modal logics. Instead of devising an ad-hoc translation for each logic, we develop a general framework within which a number of extended modal logics can be dealt with. We first extend the basic set-theoretic translation method to weak monadic second-order logic through a suitable change in the underlying set theory that connects up in interesting ways with constructibility; then, we show how to tailor such a translation to work with specific cases of extended modal logics.Keywords:Modal Logic; Modal Deduction; Translation Methods; Second-Order Logic; Set Theory

Appeared in Studia Logica, Volume 60, Number 3, May 1998, pp. 387-420(34)

## Modal Logic and Set Theory: a Set-Theoretic Interpretation of Modal Logic

Giovanna D’Agostino, Angelo Montanari, and Alberto Policriti

In this paper, we describe a novel set-theoretic interpretation of modal logic and show how it allows us to build promising bridges between modal deduction and set -theoretic reasoning. More specifically, we describe a translation technique that maps modal formulae into set-theoretic terms, thus making it possible to successfully exploit derivability in first-order set theories to implement derivability in modal logic.

Appeared in JFAK, a collection of essays dedicated to Johan van Benthem on the occasion of his 50th birthday, on June 12, 1999 (Edited by Jelle Gerbrandy et al.)

Posted by Tony Marmo
at 00:01 BST

Updated: Saturday, 10 June 2006 03:38 BST