3 More about propositional logic
Thus far our formal logic has been solely a system of proofs. Although validity was defined in terms of truth, and although the rules such as RAA and →I were motivated by appeal to the conditions under which it would be correct to assert statements of given forms, their formulation and use make no reference to meaning. The proof system is pure syntax. Questions about truth, falsehood and the like belong to semantics or the theory of meaning, which gives us another way to define and investigate the validity and invalidity of sequents.
In this chapter, we treat propositional logic semantically. The logic is the same as before, but here the notion of validity makes no reference to derivations and is not tied to any set of procedures for generating proofs. In a way, this makes it more abstract; yet in another way it ties logic to its concrete applications, since these involve interpreting the symbols so as to give them some sort of reference to things in the world.
Before leaving the purely propositional part of logic, we examine some generally useful equivalences and regularities, noting especially that formulae can be re-expressed in simple forms.
- Truth tables Meanings of the connectives; evaluating formulae
- Semantic tableaux Systematic reasoning about truth and falsehood
- Transformations Normal forms and other regularities
- Extra (math) Equivalent formulae can replace each other