On Mr. Kielkopf's Not So Sober Understanding of Standard Elementary Logic

Abstract

In his recent article 'The Binary Operation called Material Implication Soberly Understood' (Mind, lxxxi (I972), 338-347), Charles F. Kielkopf presents three cases of arguments valid within standard elementary logic, where true premisses allegedly lead to a false conclusion. This, he suggests, should induce us to burn all the familiar textbooks of logic, for, in the light of these examples, they appear to contain 'considerable portions of sophistry and illusion' (p. 338). Fortunately, he adds, a less inflammatory solution is readily available. It consists essentially in the following: proofs of validity supplied by standard elementary logic must be understood as proofs of relative validity; if q is a consequence under elementary logic of premiss p, and p is true, all we thereby know about q is that it cannot be falsified through certain procedures relevant to elementary logic-but it might very well be falsified through other procedures. A simple consideration will persuade us that Kielkopf's solution is indeed a conservative one-since it enables us to retain not only standard logic but a good deal else besides-but that, nevertheless, it is quite radical. Take any arbitrary physical hypothesis which is not directly verifiable by observation (in a very literal sense of direct verification); call it p. Suppose an empirically false sentence s is a logical consequence (by elementary logic) of our hypothesis p and the accepted truths ql, q2.... qr. This means that elementary logic enables us to infer the truth of not-p from the empirically true sentence not-s and the accepted truths ql, q2 .... qr. This would have sufficed in the old days to reject hypothesis p (unless we were willing to tamper with accepted truths). But it need no longer be so after Kielkopf's conservative revolution: The truth of the premisses not-s, ql, q2 .... qr guarantees thatwe cannot falsify not-p by any procedure relevant to standard elementary logic. But could we not falsify it eventually by some other procedure? Not until all such procedures have been tried, not until every developed or undeveloped branch of logic has been exhaustively examined, can we be sure that not-p is true, given that not-s, ql, q2, . . . qr are true. We may therefore entertain the wildest hypotheses until 'we have surveyed all methods for determining truth', which, as Kielkopf himself surmises on page 347, could very well not happen before the Greek Calends. The epistemological impact of Kielkopf's reform of logical theory is bound to be tremendous. Unfortunately, I find that his problems admit a less inspiring, indeed a rather trivial solution. In the following I assume that 'standard logic' or 'elementary logic' in Kielkopf's parlance is the first order predicate calculus with the standard syntactical and semantical rules (for the latter see, e.g. Schoenfield, Mathematical Logic, Reading, Mass., I967, pp. I8-I9, whose terminology I shall use).