Some ideologies, such as liberalism, conservatism, and socialism, are complex symbolic structures. Mastering them requires specialization, and because we are all amateurs in almost all symbolically rich domains, most people are not ideologically sophisticated. Instead, they reason about politics in a maturationally natural way, via friend-foe representations and inferences based on those representations (for example, friends of foes are foes). However, even complex ideologies are much simpler than the political, economic, and social systems that they are supposed to represent. Hence, all ideologies are inaccurate to varying degrees. More subtly, all are incomplete in various ways; in particular, they fail to anticipate some crucial events (for example, global warming), which leads to unanticipated value trade-offs and strategic conundrums. Ideologists sometimes adapt to incompleteness via recombinant innovation, producing hybrid ideologies (for example, ecosocialism). In turn, this tends to produce inconsistencies between new and old parts of an ideology. Thus, inaccuracy, incompleteness, and inconsistency are not pathologies of political belief systems. They are the inevitable result of ideologists grappling with a reality that is far more complex than their symbolic constructions can be. Therefore, evaluations of ideologies that identify errors, incompleteness, or inconsistency at a single point in time are often unenlightening. Following Imre Lakatos, evaluations should focus on how a sequence of ideologies—an ideological tradition—evolves.

We present an analogue of Gödel’s second incompleteness theorem for systems of second-order arithmetic. Whereas Gödel showed that sufficiently strong theories that are $\Pi ^0_1$-sound and $\Sigma ^0_1$-definable do not prove their own $\Pi ^0_1$-soundness, we prove that sufficiently strong theories that are $\Pi ^1_1$-sound and $\Sigma ^1_1$-definable do not prove their own $\Pi ^1_1$-soundness. Our proof does not involve the construction of a self-referential sentence but rather relies on ordinal analysis.

In his paper on the incompleteness theorems, Gödel seemed to say that a direct way of constructing a formula that says of itself that it is unprovable might involve a faulty circularity. In this note, it is proved that ‘direct’ self-reference can actually be used to prove his result.

Rudolf Carnap’s principle of tolerance states that there is no need to justify the adoption of a logic by philosophical means. Carnap uses the freedom provided by this principle in his philosophy of mathematics: he wants to capture the idea that mathematical truth is a matter of linguistic rules by relying on a strong metalanguage with infinitary inference rules. In this paper, I give a new interpretation of an argument by E. W. Beth, which shows that the principle of tolerance does not suffice to remove all obstacles to the employment of infinitary rules.

There has been a recent interest in hierarchical generalizations of classic incompleteness results. This paper provides evidence that such generalizations are readily obtainable from suitably formulated hierarchical versions of the principles used in the original proofs. By collecting such principles, we prove hierarchical versions of Mostowski’s theorem on independent formulae, Kripke’s theorem on flexible formulae, Woodin’s theorem on the universal algorithm, and a few related results. As a corollary, we obtain the expected result that the formula expressing “ $\mathrm {T}$ is $\Sigma _n$ -ill” is a canonical example of a $\Sigma _{n+1}$ formula that is $\Pi _{n+1}$ -conservative over $\mathrm {T}$ .

In this article, we tell a story about incompleteness in modal logic. The story weaves together an article of van Benthem (1979), “Syntactic aspects of modal incompleteness theorems,” and a longstanding open question: whether every normal modal logic can be characterized by a class of completely additive modal algebras, or as we call them, ${\cal V}$-baos. Using a first-order reformulation of the property of complete additivity, we prove that the modal logic that starred in van Benthem’s article resolves the open question in the negative. In addition, for the case of bimodal logic, we show that there is a naturally occurring logic that is incomplete with respect to ${\cal V}$-baos, namely the provability logic $GLB$ (Japaridze, 1988; Boolos, 1993). We also show that even logics that are unsound with respect to such algebras do not have to be more complex than the classical propositional calculus. On the other hand, we observe that it is undecidable whether a syntactically defined logic is ${\cal V}$-complete. After these results, we generalize the Blok Dichotomy (Blok, 1978) to degrees of ${\cal V}$-incompleteness. In the end, we return to van Benthem’s theme of syntactic aspects of modal incompleteness.

This paper presents a logic framework for modeling the interaction among deductive databases in a peer-to-peer (P2P) environment. Each peer joining a P2P system provides or imports data from its neighbors by using a set of mapping rules, that is, a set of semantic correspondences to a set of peers belonging to the same environment. By using mapping rules, as soon as it enters the system, a peer can participate and access all data available in its neighborhood, and through its neighborhood it becomes accessible to all the other peers in the system. A query can be posed to any peer in the system and the answer is computed by using locally stored data and all the information that can be consistently imported from the neighborhood. Two different types of mapping rules are defined: mapping rules allowing to import a maximal set of atoms not leading to inconsistency (called maximal mapping rules) and mapping rules allowing to import a minimal set of atoms needed to restore consistency (called minimal mapping rules). Implicitly, the use of maximal mapping rules states it is preferable to import as long as no inconsistencies arise; whereas the use of minimal mapping rules states that it is preferable not to import unless a inconsistency exists. The paper presents three different declarative semantics of a P2P system: (i) the Max Weak Model Semantics, in which mapping rules are used to import as much knowledge as possible from a peer’s neighborhood without violating local integrity constraints; (ii) the Min Weak Model Semantics, in which the P2P system can be locally inconsistent and the information provided by the neighbors is used to restore consistency, that is, to only integrate the missing portion of a correct, but incomplete database; (iii) the Max-Min Weak Model Semantics that unifies the previous two different perspectives captured by the Max Weak Model Semantics and Min Weak Model Semantics. This last semantics allows to characterize each peer in the neighborhood as a resource used either to enrich (integrate) or to fix (repair) the knowledge, so as to define a kind of integrate–repair strategy for each peer. For each semantics, the paper also introduces an equivalent and alternative characterization, obtained by rewriting each mapping rule into prioritized rules so as to model a P2P system as a prioritized logic program. Finally, results about the computational complexity of P2P logic queries are investigated by considering brave and cautious reasoning.

Cet article propose une analyse de la coopération dans les relations contractuelles à partir d'une hypothèse de rationalité procédurale. Suivant la théorie du framing de Lindenberg, nous définissons l'égoïsme éclairé comme la rationalité mise en œuvre lors des relations contractuelles. La coopération s'explique alors par la volonté de maintenir la relation qui diminue la saillance du cadre de gain; cette volonté est entretenue par l'échange de si-gnaux relationnels entre les parties. Si les possibilités d'opportunisme sont importantes, la coopération nécessite d'être soutenue, d'abord par la formalisation d'un contrat puis,plus les enjeux augmentent, par l'encastrement dans un réseau. Des mécanismes de réputation et de sélection peuvent alors être mobilisés. Nous montrons que dans une telle approche, l'acceptation par les parties de l'incomplétude inévitable de leur contrat est un signal positif, nécessaire à l'établissement de la coopération.