We define infinitary extensions to classical epistemic logic systems, and add also a common belief modality, axiomatized in a finitary, fixed-point manner. In the infinitary K system, common belief turns to be provably equivalent to the conjunction of all the finite levels of mutual belief. In contrast, in the infinitary monotonic system, common belief implies every transfinite level of mutual belief but is never implied by it. We conclude that the fixed-point notion of common belief is more powerful than the (...) iterative notion of common belief. (shrink)

It is known that a theory in S5‐epistemic logic with several agents may have numerous models. This is because each such model specifies also what an agent knows about infinite intersections of events, while the expressive power of the logic is limited to finite conjunctions of formulas. We show that this asymmetry between syntax and semantics persists also when infinite conjunctions (up to some given cardinality) are permitted in the language. We develop a strengthened S5‐axiomatic system for such infinitary logics, (...) and prove a strong completeness theorem for them. Then we show that in every such logic there is always a theory with more than one model. (shrink)

The French philosopher Simone Weil (1909-1943) thought of geometry and algebra not as complementary modes of mathematical investigation, but rather as constituting morally opposed approaches: whereas geometry is the sine qua non of inquiry leading from ruthless passion to temperate perception, in accord with the human condition, algebra leads in the reverse direction, to excess and oppression. We explore the constituents of this argument, with their roots in classical Greek thought, and also how Simone Weil came to qualify it following (...) her exchange with her brother, the mathematician André Weil. (shrink)

In Economics individuals are defined by their preferences over the consequences of their own actions and the actions carried out by others. In contrast, Simone Weil depicts the individual as continuously re-constituted by the contact that he establishes with reality via his action. Such an action is aimed at achieving an effect in the physical world, but what makes it human is not success per se, but rather the fact that it stems from reasoning and planning. Affliction is caused by (...) effort carried out mechanically like that of a beast of burden, when the individual has no opportunity to exercise reason for choosing how to confront reality's ever-challenging hazards and necessity. (shrink)

John Rawls and Simone Weil presented two distinct conceptions of political justice, aimed at articulating a common ethos in an inherently heterogeneous society. The terms of the former, chiefly concerned with the distribution of primary goods, underwrite much of today's Western democracies political liberalism. The terms of the latter, chiefly concerned with the way interaction is organised in social activities in view of the body and soul's balancing pairs of needs, are less well known. We explain the sense in which (...) the overlapping consensus in Weil's notion of political justice is “thicker”, and may thus deserve more attention – alongside that of Rawls – for substantiating a democratic ethos within political liberalism. (shrink)