Such a question and such a suggestion may seem preposterous. The scholarly tradition represents Plato as the first historical figure to construct a utopia, and as one who stimulated Th. More, Rousseau and others to similar efforts at construction. While I agree with this tradition, I do not think that its view of Plato's intention can be taken for granted; such a view needs arguing and demonstrating--arguing against important objections. The question is certainly not preposterous, as will be obvious from (...) a consideration of relevant passages. (shrink)

In ‘Semantic Foundations for the Logic of Preference’ (Rescher, ed.,The Logic of Decision and Action, University Press, Pittsburgh, 1967), Nicholas Rescher claims that, on the semantics developed in that paper, a certain principle - call it ‘Q’ turns out to be ‘unacceptable’. I argue, however, that, given certain assumptions that Rescher invokes in that same paper,Q can in fact be shown to be a ‘preference-tautology’, and henceQ should be classified as ‘acceptable’ on Rescher's theory.

One of the main results in topological social choice states the non-existence of a continuous, anonymous, and unanimous aggregation rule on spheres. This note provides a proof based upon simple methods such as integration.

Rauszer and Sabalski proved in [2] that distributivity with respect to infi- nite joins and meets is a sucient and necessary condition making the RasiowaSikorski Lemma valid in distributive lattices. The main part of their proof is a direct construction of a required filter under distributivity. In this note we show that a generalization of the result can be obtained from the Rasiowa-Sikorski Lemma for Boolean algebras by using Gornemann’s result in [1] instead of a direct con- ¨ struction. (...) Suppose A is a distributive lattice and Q; R 2 A f;g. We call A complete if 8M 2 Q9 M 2 A and 8N 2 R9 F N 2 A. M 2 Q is -dis if 9 M 2 A and 8a 2 A a t M = m2M. N 2 R is F -dis if 9 F N 2 A and 8a 2 A a u F N = F n2N. is called distributive if every M 2 Q is -dis and every N 2 R is F -dis. Suppose A is complete and C; D A. By rC we mean the filter in A generated by C, in particular, let r; = ; = ;. is called complete if and , where 8M 2 Q 9m 2 MrC \ D[fmg = ;) 8N 2 R 9n 2 NrC[fng \ D = ;). (shrink)

Gentzen’s height measure of the 1938 consistency proof is a cumulative complexity measure for sequents that is measured bottom-up in a derivation. By a factorisation of the ordinal assignment a top-down ordinal assignment can be given that does not depend on information occurring below the sequent to which the ordinal is assigned. Furthermore, an ordinal collapsing function is defined in order to collapse the top-down ordinal to the one assigned by Gentzen’s own ordinal assignment. A direct definition of the factorised (...) assignment follows as a corollary. This extraction of an ordinal collapsing function hopes to provide a formal or conceptual clarification of Gentzen’s ordinal assignment and its height-line argument. (shrink)

Literally, προσβάλλον and προσβαλλόμενον mean, respectively, "that which strikes" and "that which is struck." The first suggests activity; the second passivity. Consequently, it would seem that the προσβάλλον should be said to emanate from the agent and the προσβαλλόμενον from the patient. And, since Plato explicitly identifies the agent with the perceptual object and the patient with the sensing organ, we should, it would seem, identify the προσβάλλον with the motion from the perceptual object and the προσβαλλόμενον with the motion (...) from the sensing organ, which is the reverse of what I did in the article. (shrink)

In 2002, Dvurečenskij extended Mundici’s equivalence between unital abelian l -groups and MV-algebras to the non-commutative case. We analyse the relationship to Bosbach’s cone algebras and clarify the rôle of the corresponding pair of L -algebras. As a consequence, it follows that one of the two L -algebra axioms can be dropped.

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