One of the most striking claims advanced by T. Kuhn and P. Feyerabend, both classified as non-rationalists among philosophers of science, is that Newtonian mechanics and Einsteinian mechanics neither contradict nor agree with one another; they are just incommensurable. This means that in the transition from one to the other there has been a shift in meaning so extreme that the assertions of two respective proponents would simply pass one another by without any collision leading to either agreement or conflict. If the Newtonian says, for instance, "The ball will fall in the hole if its diameter is smaller", no reply by the Einsteinian can be construed as compatible or incompatible; the two persons are merely equivocating.

The thesis of incommensurability seems attractive to many because they don’t take it seriously and place it in the file "Exotic claims". Yet this frivolous attitude is unjustified; there is a sense in which Newtonian mechanics and Einsteinian mechanics are incommensurable. Let us return to the Newtonian’s assertion that the ball will fall in the hole. What could the Einsteinian reply? Needless to say, some initial conditions should be given – e.g. we have a horizontal plane with a hole in it and the ball is rolling towards the hole with a constant speed. There is an even better set-up. A rod is moving with a constant speed in a horizontal tube towards a place where a segment of the tube has been removed. The rod is shorter than the removed segment (when they are in the same inertial frame). Will the rod fall out of the tube when reaching the vacancy?

Needless to say, the Newtonian’s answer is yes. As for the Einsteinian, he notices that there are two different inertial frames – that of the tube and that of the rod. Part of the Einsteinian’s large soul serves as an observer in the tube’s frame. That part sees a very short rod and wholeheartedly agrees with the Newtonian. Yes, the rod will fall out of the tube. However the other part is an observer in the rod’s frame and sees things differently. If the speed of the rod is high enough, that second part of the Einsteinian’s soul sees the vacancy in the tube being much shorter than the rod and categorically disagrees with the Newtonian. No, the rod will by no means fall out of the tube.

Clearly, Kuhn and Feyerabend are right about incommensurability (although their arguments are different from mine). In Newtonian mechanics assertions are straightforward. In Einsteinian mechanics assertions are accompanied by their negations. Proponents of the two theories can neither agree nor disagree – they can only equivocate.

A theory where assertions are accompanied by their negations is called the inconsistency. A useful, although somewhat misleading, description of this malignant theoretical construction is given by Newton-Smith (W. H. Newton-Smith, The rationality of science, Routledge, London, 1981, p. 229):

"A theory ought to be internally consistent. The grounds for including this factor are a priori. For given a realist construal of theories, our concern is with verisimilitude, and if a theory is inconsistent it will contain every sentence of the language, as the following simple argument shows. Let ‘q’ be an arbitrary sentence of the language and suppose that the theory is inconsistent. This means that we can derive the sentence ‘p and not-p’. From this ‘p’ follows. And from ‘p’ it follows that ‘p or q’ (if ‘p’ is true then ‘p or q’ will be true no matter whether ‘q’ is true or not). Equally, it follows from ‘p and not-p’ that ‘not-p’. But ‘not-p’ together with ‘p or q’ entails ‘q’. Thus once we admit an inconsistency into our theory we have to admit everything. And no theory of verisimilitude would be acceptable that did not give the lowest degree of verisimilitude to a theory which contained each sentence of the theory’s language and its negation."

The deduction performed by Newton-Smith is unacceptable to a physicist since « from ‘p’ it follows that ‘p or q’ » is not a physical argument (see a tentative definition of a physical argument in http://www.wbabin.net/valev/valev4.htm ). Still the central idea – that the lowest degree of verisimilitude should be given to an inconsistency – is correct. We should only add that the lowest degree of commensurability should be given to two theories if in at least one of them assertions are accompanied by their negations.