ABSTRACT
Logic is the study of valid arguments. In the early chapters of this work we developed a propositional language and provided a definition of validity for arguments expressible in that language. We saw that there were valid arguments not expressible in the propositional language. Hence we enriched it to obtain a predicate language. We have yet to develop a definition of validity appropriate to this enriched language. This is a considerably more complex business than was the case for the propositional language. A valid argument is one which is such that if the premises are true the conclusion must be true. Thus a detailed definition of validity for the arguments expressible in a given language depends on a definition of truth for the sentences of that language. In the case of a propositional language it is relatively simple to explicate the notion of truth of an arbitrary sentence of that language by reference to the notion of the truth of a simple sentence taken together with the notion of a truth-table. That is, we arrive at the truth-value of a complex sentence from an assignment of truth-values to the simple sentences which are parts of the sentence with the aid of truth-tables. However, in the case of the predicate language, the parts of our complex sentences are not necessarily sentences. To see this compare the propositional language sentence ‘P → Q’ with the predicate language sentence ‘(∀x)(Fx → Gx)’. In the former case some parts, i.e. ‘P’, ‘Q’ are themselves sentences. In the latter case neither ‘(∀x)’, ‘Fx’, nor ‘Gx’ are sentences. The complexities that this produces will take us some time to explore and in this introductory work we will not be able to do more than indicate the direction to be followed in developing a rigorous definition of truth. The first step is to give a characterization of the predicate language analogous to that given in Chapter 4 for the propositional language.
