David Miller

Kant is invariably read by his followers as antipathetic to all forms of nationalism. Yet he was interested in differences of national character and used an organic metaphor to explain why states should not be broken up or annexed (unfortunately he never commented explicitly on the dismemberment of Poland by Prussia and its allies). He favoured a plural world in which national differences of language and religion prevented the emergence of despotic world government. So his acknowledgement of a limited obligation to provide refuge to vulnerable people should not be amplified into an acceptance of culturally disruptive mass migration.

1. Despite the plain fact that there is nothing in this world that can be proved without reliance on some assumption or another, there is an inalienable difference between an argument that begins by assuming what it is designed to establish and one that begins by assuming the contradictory of what it is designed to establish. Arguments of the first kind are uncontroversially acknowledged to be circular, or question-begging; though valid they achieve nothing. Those of the second kind conform to the classical pattern of reductio ad absurdum or indirect proof. A typical example is the proof of the irrationality of √2, which begins by assuming that for some integral m, n the identity m2/n2 = 2, and arrives at a contradiction. Nothing whatever is established or justified by arguments of the first kind, even granted the truth of all the other assumptions, and the validity of the rules of inference; while the second may establish that, if those assumptions are true, then the additional assumption is false

If the open society is a society that ‘sets free the critical powers of man’ (Popper, 1945, Introduction), then the subject of critical thinking, now widely taught in universities in North America and at the level of further education in the UK, might seem to be a welcome innovation. Caution is advised. By mistakenly supposing that thinking intelligently is identical with thinking logically, critical thinking textbooks almost invariably regard the purpose of argument to be a combination of justification and persuasion, authoritarian goals that critical rationalists, and other supporters of the open society, must shun. If students do not learn the proper place of reason, and its limitations, they will be disappointed when it fails, and like many irrationalists in the past, may be induced to reason their way out of reason altogether.

Calling into service the theory of truth approximation of his (1997) and (2000), Kuipers defends the view that "beauty can be a road to the truth" and endorses the general conclusions of McAllister (1996) that aesthetic criteria reasonably play a role in theory selection in science. My comments pertain first to the general adequacy of Kuipers's theory of truth approximation; secondly to its methodological aspects; thirdly to the aetiolated role that aesthetic factors turn out to play in his account; and fourthly to the question before us, with a remark on McAllister's doctrine that scientific revolutions are characterized above all by novelty of aesthetic judgement.

In order to measure the degree of dissimilarity between elements of a Boolean algebra, the author’s proposed to use pseudometrics satisfying generalizations of the usual axioms for identity. The proposal is extended, as far as is feasible, from Boolean algebras to Brouwerian algebras. The relation between Boolean and Brouwerian geometries of logic turns out to resemble in a curious way the relation between Euclidean and non-Euclidean geometries of physical space. The paper ends with a brief consideration of the problem of the metrization of the algebra of theories.