8 August 1959. Cogitations

In what way does logic show itself? What signs betray its presence or absence? How can it be seen that the signs employed in logic, whether peculiar to logic or not, are being logically employed? In view of Quine's criticisms (Mathematical Logic, pp. 31, 32, 33) of Whitehead and Russell (1927), it seems evident that even the best logicians have great difficulty in being logical.

Is it possible that there is no necessary, inescapable deduction from any premise that is not merely a repetition of a definitory formula? For example, if ‘two’ means two, and ‘and’ means and, and ‘two’ means two, and ‘equals’ means equals, and ‘four’ means four, then two and two equals four.

In the instances quoted by Quine it seems as if the symbol or sign, –)–, has been so abstracted that it has lost precision necessary for the purpose for which Quine wishes to use it. What is the relationship between ‘implies’, ‘–)–’, used by Whitehead and Russell, and ‘–)–’ used by Quine? By what process is ‘implies’ denuded of its ordinary speech meanings so that it becomes precise? How is it determined when the sign has been sufficiently abstracted from its ordinary speech meanings to give it the required precision, yet not so much that it has become too generalized and so open to Quine's kind of criticism when scientifically employed? It looks as if the ordinary meanings of ‘implies’ have crept back into Whitehead and Russell's use of the sign, –)–. If this is so, then the fault does not lie in the sign but in the fact that the sign has to be employed by human beings who are governed, obviously, by the limitations of their mental equipment. This is similar to the situation with other tools. The tool can be made, be it bomb or –)–, but there still remains the problem of the use to which it is to be put and the motives impelling the person who employs it.

Why have ordinary meanings (if ____ is true then ____ is true; ____ is false or ____ is true; ____ implies ____; if ____ then ____) crept back into the