## Beyond the fringe

In the first chapter we said that we would construct a formal model of deductive reasoning. The result was first-order logic. The ambition of every model-builder is that their model should be as accurate a representation of the reality modelled as possible. There are, however, types of deductive reasoning that do not seem well modelled, if modelled at all, by first-order logic. One is modal reasoning, that is to say reasoning involving the modalities necessarily and possibly. The other has already had attention drawn to it, in Chapter 1: this is reasoning involving so-called counterfactual conditionals, i.e. conditional sentences whose grammatical structure indicates that the antecedent is false. This grammatical structure is typically manifested in the use of the subjunctive mood for the verbs involved, as, for example, in â€˜Had A been the case, so would B have been.â€™ In this chapter we shall briefly review attempts to extend the apparatus of first-order logic to accommodate counterfactuals. We shall adopt Lewisâ€™s notation, symbolising the counterfactual with antecedent A and consequent B as Aâ†’B. Since any formal language for counterfactuals is obtained simply by adding the new connective to the truth-functional ones, we can regard the language as propositional and A, B, etc. as sentence letters.