Axioms and Machines

Abstract

We have said in preceding chapters that an important motivation in abstracting logical forms is to provide the basis for a test of validity which can be regarded as mechanical. In these last chapters, we will be discussing whether there are any grounds for supposing that the tests of validity embodied in our logical systems do not really achieve the objective of a mechanical test. It will be found that “mechanical” is a somewhat vague term as measured against the theoretical possibilities of testing procedures. But for the moment we may ask whether proofs can be carried out without dependence upon the significance of the symbolism in which the proofs are expressed. In other words, we are asking whether in developing proofs we need to depend implicitly upon the significance of the symbolism to guide the process of proof development. The test for this is to determine whether an appropriate machine could be devised which could. carry out the various proof procedures used to test for validity in the various logical systems. One may argue that machine computation cannot depend upon insight, or upon an understanding of the symbolism, in just the appropriate sense to answer our question.