There are several different systems of conditional logics. Amongst them we focus on the system CK and its standard extensions, namely CK + {ID, MP, CS, CEM}. These systems have a simple and useful semantics. One just needs to consider a set of possible worlds W, and a selection function f; for each world i and each formula A, f selects the set of worlds of W which are closer to i given the information A. Thus a counterfactual sentence A > B is true at a world i if, and only if, B is true at all those worlds that are closer to i given the information A.
In this talk we aim at presenting sequent calculi for the system CK and all of its extensions. These calculi are based on and fully exploit the simple semantics interpretation of such systems. Moreover, they are contraction-free, weakening-free and cut-free; finally, their logical rules are all invertible. As far as we know the only other sequent calculi that have been proposed for the system CK and its extensions are those of Olivetti and al. [4] (sequent calculi for other systems of conditional logics have been proposed by e.g. [1]). With respect to these calculi the main differences consist in a lighter formalisms and simpler logical rules to manipulate.
By using the same technique adopted for the sequent calculi, we will also briefly show how to construct natural deduction calculi for CK + {ID, MP, CS, CEM}.
References
[1] Crocco, G. and Lamarre, P. On the connection between non-monotonic inference systems and conditional logics. In Proceedings of the 3rd International Conference on Principles of Knowledge. Representation and Reasoning, B. Nebel and E. Sandewall Ed., 565-571, 1992.
[2] Lewis, D. Counterfactuals. Basil Blackwell, 1973.
[3] Nute, D. Topics in Conditional Logic. Reidel, Dordrecht, 1980.
[4] Olivetti, N., Pozzato, G. L., and Schwind, C. B. A sequent calculus and a
theorem prover for standard conditional logics, ACM Transactions on Computational Logic (TOCL) 8 : 557-590, 2007.
[5] Stalnaker, R. A theory of conditionals, American Philosophical Quarterly, Monograph Series no.2, Blackwell, Oxford, 98-112, 1968.
Speaker: Francesca Poggiolesi (CNRS-CEPERC)
Title: An alternative proof-theoretical approach to standard conditional logics
Date:
Time:
16:30
- 18:00
Location: Room F1.15, ILLC, Science Park 107, Amsterdam
Abstract: Conditional logics, which have a long and venerable history [5, 2, 3], have been introduced to capture counterfactual sentences, i.e. conditionals of the form “if A were the case, then B would be the case”, where A is false. If we interpret counterfactuals as material implications, we have that all counterfactuals are trivially true, and this is an unpleasant conclusion. By means of conditional logics, on the other hand, we can give a different and meaningful interpretation of counterfactual sentences.