Over the centuries, logic has developed precise methods that guide us in hypothetical reasoning: if we assume that certain things are true, then we can conclude that also some other things are true. But what if our assumptions are false, and we want to draw conclusions from assuming their truth, and this assumption is contrary to the present evidence but is nevertheless conceivable in some scenario alternative to the current state of affairs?
The basic element of such forms of reasoning is given by what is known as counterfactual conditionals, an if-then sentence in which the if-clause may or may not be true. Counterfactual conditionals occur all the time in everyday language in sentences such as "If Helsinki were not the capital of Finland, then Turku would be'', or "If Finland had not won the 1995 Ice Hockey World Championship then Sweden would have".
As shown by a series of fallacies, counterfactual conditionals escape a straightforward logical analysis based on the standard notion of implication. Difficulties in such analysis arise from the fact that such conditionals are not immediately truth-functional for the reason that in counterfactual scenarios, truth becomes a relative notion.
Many attempts have been made in the literature to capture the meaning of such conditionals and eventually develop a formal logical analysis as precise as the one achieved for the standard conditional of classical logic. Among these, we shall analyse in detail the approach developed by David Lewis in his book "Counterfactuals," based on a relation that orders possible worlds with respect to their similarity to the actual world. This sophisticated semantical analysis has not been accompanied by an equally developed proof-theoretical investigation of logical formalisms for reasoning with conditionals which latter have been limited, apart from isolated attempts, to axiomatic systems.
After reviewing Lewis' and other semantic approaches, we shall develop a formal method for reasoning with conditionals based on an abstract neighborhood semantics, a generalization of Kripke semantics. We shall establish its adequacy, completeness, and inferential properties, as well as a decision procedure.