Logical and historical determination of the Arrow and Sen impossibility theorems

General classification of mathematical statements divides them into universal, those of the form xA , and existential ЭxA ones. Common formulations of impossibility theorems of K. J. Arrow and A. K. Sen are represented by the statements of the form "there is no x such that A". Bearing in mind logical equivalence of formulae ¬ЭxA and x¬A, we come to the conclusion that the corpus of impossibility theorems, which appears in the theory of social choice, could make a specific and recognizable subclass of universal statements. In this paper, on the basis of the established logical and methodological criteria, we point to a sequence of extremely significant "impossibility theorems", reaching throughout the history of mathematics to the present days and the famous results of Arrow and Sen in field of mathematical economics. We close with specifying the context which makes it possible to formulate the results of Arrow and Sen accurately, presenting a new direct proof of Sen’s result, with no reliance on the notion of minimal liberalism.

The Political Economy of Post-Communist Autocracy: The Continuum Between Dictatorship and Democracy

The article deals with the economic behaviour of a hypothetical political leader in the institutionally underdeveloped democratic environment of a transitional society. By using Stigler-Peltzman’s model of economic regulation, it has been shown that there is no optimal solution for an autocratic leader. In the long run, wealth transfer to political supporters alone is not sufficient for the autocrat to stay in power. That’s why such regimes are generally unstable, and will ultimately be either overthrown or transformed into ‘hard’ dictatorships. The marginal costs of regime protection and opposition, crucial for the political behaviour of conflicted social groups, can be considered as political turning points responsible for sudden and unexpected social changes

Dictatorship, liberalism and the Pareto rule: Possible and impossible

The current economic crisis has shaken belief in the capacity of neoliberal 'free market' policies. Numerous supports of state intervention have arisen, and the interest for social choice theory has revived. In this paper we consider three standard properties for aggregating individual into social preferences: dictatorship, liberalism and the Pareto rule, and their formal negations. The context of the pure first-order classical logic makes it possible to show how some combinations of the above mentioned conditions, under the hypothesis of unrestricted domain, form simple and reasonable examples of possible or impossible social choice systems. Due to their simplicity, these examples, including the famous 'liberal paradox', could have a particular didactic value

Proving, Refuting, Improving—Looking for a Theorem

Exploring the proofs and refutations of an abstract statement, conjecture with the aim to give a formal syntactic treatment of its proving–refuting process, we introduce the notion of extrapolation of a possibly unprovable statement having the form if A, then B, and propose a procedure that should result in the new statement if A′, then B′, which is similar to the starting one, but provable. We think that this procedure, based on the extrapolation method, can be considered a basic methodological tool applicable to prove–refute–improve any conjecture. This new notion, extrapolation, presents a dual counterpart of the well-known interpolation introduced in traditional logic sixty-five years ago