It is argued that contraposition is valid for a class of natural language conditionals, if some modifications are allowed to preserve the meaning of the original conditional. In many cases, implicit temporal indices must be considered, making a change in verb tense necessary. A suitable contrapositive for implicative counterfactual conditionals can also usually be found. In some cases, the addition of certain words is necessary to preserve meaning that is present in the original sentence and would be lost or changed (...) in the contrapositive without them. A distinction is made between adding new meaning and adding new words to preserve existing meaning. For concessive conditionals and relevance conditionals, however, no valid contrapositive can be found. They do not belong to the class of contraposable conditionals, which can be independently defined. Difficult cases are also discussed in which the contradictory of the consequent semantically entails the truth of the antecedent. In such cases the content of the antecedent is implicit in the meaning of the consequent. Contraposition becomes possible if what is implicit in the original consequent is made explicit in the contrapositive antecedent. (shrink)

A theory of necessary and sufficient conditions is presented, as well as a theory of necessary and sufficient causes and effects, viewed as a particular case of the former. Ambiguities of the terms 'condition' and 'necessary condition' are explored, and a neutral meaning for 'condition' is favoured. The relation between necessary and sufficient conditions and implicative conditionals (including counterfactuals) is also discussed. Two problems of counterfactual theories of causal explanation are indicated, concerning (i) how to account for causes that are (...) not necessary and (ii) how to interpret the counterfactual from the negation of the effect to the negation of the cause. Application of the theory of necessary and sufficient conditions to Euler’s explanation of the Königsberg bridge problem is discussed. The theory of necessary and sufficient causes and effects is then applied to the flagpole shadow explanation and to two biological explanations, one involving a sufficient but not necessary cause and the other a necessary but not sufficient cause. Causes that are neither sufficient nor necessary are also recognized. Finally, a probabilistic approach to causes that are either not sufficient or not necessary is explored, including its application to examples of such cases. (shrink)