Semantic Tableaux Methods for Modal Logics That Include the B(rowerische) and G(each) Axioms

As its name suggests, the method uses semantic facts about the language in an attempt to generate a countermodel for the argument at hand (or to show that there is no such counter-model and that the argument is therefore valid). In this, the method bears a strong similarity to Resolution, but it does not require any conversion to a normal form. The method is also different in being an analytic system: each non-initial formula in any proof is a subformula of some earlier formula in the proof. Natural deduction is not analytic in this sense, nor is any Resolution method other than Unit Resolution. There are various different formats for stating the method of semantic tableaux. Participants in this AAAI Symposium are doubtless familiar with the form given by Fitting (1988). The method he uses is quite similar to the method I wish to employ, and there should be no difficulty in transforming the one into the other. In this note, I am interested in extending my method to a certain class of modal ogics. In this extension we will only consider propositional logics a fact that lends some considerable simplification to the description of the tableaux method. (By not needing to describe quantification rules and their difficult integration into modal logics. See Garson 1988 for a survey of the difficulties). I will first lay out the tableaux method that I prefer, restricting my attention to classical propositional ogic. I then show how it is extended to a certain class of modal logics, the normal modal logics. The idea of