ABSTRACT
For the construction of a calculus the statement of the trans formation rules, as well as of the formation rules, as given for Language I, is essential. By means of the former we determine under what conditions a sentence is a consequence of another sentence or sentences (the premisses). But the fact that S 2 is a consequence of S x does not mean that the thought of S x will be accompanied by the thought of S 2. It is not a question of a psychological but of a logical relation between sentences. In the statement of S x, the statement of S 2 is already objectively in volved. We shall see that the relationship which is here indicated in a material way can be purely formally conceived. [Example: Let S i be ‘ (#)5 (Red (#))’, and S 2 ‘ Red(3)’ ; given that all positions up to 5 are red, then it is also given (implicitly) that the position 3 is red. In this particular case, perhaps, S 2 will have been “ thought” simultaneously with S x; but in other cases, where the transformation is more complicated, the consequence will not necessarily be thought coincidently with the premisses.]
It is impossible by the aid of simple methods to frame a de finition for the term ‘ consequence’ in its full comprehension. Such a definition has never yet been achieved in modern logic (nor, of course, in the older logic). But we shall return to this subject later. At present, we shall determine for Language I, instead of the term ‘ consequence ’, the somewhat narrower term ‘ derivable \ [In constructing systems of logic, it is generally customary to use only the latter narrower concept, and it is not usually clearly understood that the concept of derivability is not the general concept of consequence.] For this purpose, the term 1 directly derivable ’ will be defined, or-as it is more commonly expressedrules of inference will be laid down. [S 3 is called ‘ directly de rivable ’ from S i or from S x and S 2, when, with the help of one of the rules of inference, S 3 can be obtained from S x> or from S x
and S 2.]
By a derivation with the premisses 0 l5 S 2,... ©m (°f which the number is always finite, and may also be 0), we understand a series of sentences of any finite length, such that every sentence of the series is either one of the premisses, or a definition-sentence, or directly derivable from one or more (in our object-languages I and II, at most two) of the sentences which precede it in the series. If is the final sentence of a derivation with the pre misses ... S w, then <Zn is called derivable from (5ly ... (5W.
