Logical Systems

Abstract

In earlier chapters, we described the set of PL forms, and we defined certain sequences of PL forms as most sophisticated PL argument consistency forms. These sequences were thought of as abstracting valid arguments if and only if the proofs which were developed from them blocked. A similar strategy was followed in presenting the other logical systems. Now, there are two interesting and distinct notions actually involved, a fact which we have not explicitly noted so far. One is the notion of implication. We can say that the premises of an argument imply its conclusion if and only if the premises cannot be jointly true in any possible world in which the conclusion is false. This notion of implication as a relationship between premises and conclusion of certain arguments is simply another way of looking at the property of validity which we attribute to certain arguments, and it was this notion that provided the grounds for the development of a logical symbolism. After a symbolism has been constructed and abstraction to it clarified, we can say that implication holds between the premises and the conclusion of an argu?ment if and only if the symbolic forms “P₁,” “P₂,” … , “P _n ” appropriately abstracting the n premises cannot all abstract true statements in some possible world while the form appropriately abstracting the conclusion abstracts a false statement.