Ekeland’s variational principle in weak and strong systems of arithmetic

We analyze Ekeland’s variational principle in the context of reverse mathematics. We find that that the full variational principle is equivalent to $$\Pi ^1_1\text{- }\mathsf {CA}_0$$ Π 1 1 - CA 0 , a strong theory of second-order arithmetic, while natural restrictions (e.g. to compact spaces or to continuous functions) yield statements equivalent to weak König’s lemma ($$\mathsf {WKL}_0$$ WKL 0 ) and to arithmetical comprehension ($$\mathsf {ACA}_0$$ ACA 0 ). We also find that the localized version of Ekeland’s variational principle is equivalent to $$\Pi ^1_1\text{- }\mathsf {CA}_0$$ Π 1 1 - CA 0 , even when restricted to continuous functions. This is a rare example of a statement about continuous functions having great logical strength.

Dealing with Incompatibilities among Procedural Goals under Uncertainty

By considering rational agents, we focus on the problem of selecting goals out of a set of incompatible ones. We consider three forms of incompatibility introduced by Castelfranchi and Paglieri, namely the terminal, the instrumental (or based on resources), and the superfluity. We represent the agent's plans by means of structured arguments whose premises are pervaded with uncertainty. We measure the strength of these arguments in order to determine the set of compatible goals. We propose two novel ways for calculating the strength of these arguments, depending on the kind of incompatibility thatexists between them. The first one is the logical strength value, it is denoted by a three-dimensional vector, which is calculated from a probabilistic interval associated with each argument. The vector represents the precision of the interval, the location of it, and the combination of precision and location. This type of representation and treatment of the strength of a structured argument has not been defined before by the state of the art. The second way for calculating the strength of the argument is based on the cost of the plans (regarding the necessary resources) and the preference of the goals associated with the plans. Considering our novel approach for measuring the strength of structured arguments, we propose a semantics for the selection of plans and goals that is based on Dung's abstract argumentation theory. Finally, we make a theoretical evaluation of our proposal.

THE LARGE STRUCTURES OF GROTHENDIECK FOUNDED ON FINITE-ORDER ARITHMETIC

The large-structure tools of cohomology including toposes and derived categories stay close to arithmetic in practice, yet published foundations for them go beyond ZFC in logical strength. We reduce the gap by founding all the theorems of Grothendieck’s SGA, plus derived categories, at the level of Finite-Order Arithmetic, far below ZFC. This is the weakest possible foundation for the large-structure tools because one elementary topos of sets with infinity is already this strong.

CALIBRATING DETERMINACY STRENGTH IN LEVELS OF THE BOREL HIERARCHY

We analyze the set-theoretic strength of determinacy for levels of the Borel hierarchy of the form$\Sigma _{1 + \alpha + 3}^0 $, forα<ω1. Well-known results of H. Friedman and D.A. Martin have shown this determinacy to requireα+ 1 iterations of the Power Set Axiom, but we ask what additional ambient set theory is strictly necessary. To this end, we isolate a family of weak reflection principles, Π1-RAPα, whose consistency strength corresponds exactly to the logical strength of${\rm{\Sigma }}_{1 + \alpha + 3}^0 $determinacy, for$\alpha < \omega _1^{CK} $. This yields a characterization of the levels ofLby or at which winning strategies in these games must be constructed. Whenα= 0, we have the following concise result: The leastθso that all winning strategies in${\rm{\Sigma }}_4^0 $games belong toLθ+1is the least so that$L_\theta \models {\rm{``}}{\cal P}\left( \omega \right)$exists, and all wellfounded trees are ranked”.

Uwagi o formach kontrargumentacji

The paper provides an elementary introduction to the problems of counterarguments. The ways of possible critical reaction are divided according to four objects of attack: the proponent, strength of argument, premises, conclusion. The most important strategies of attack are discussed and evaluated according to their logical strength.

What Do Reciprocals Mean?

Research on reciprocals has uncovered a variety of semantic contributions that the reciprocal can make, creating problems for proposals that the reciprocal unambiguously means something weak (e.g., Langendoen 1978). However, there is no real evidence that reciprocals are ambiguous, despite previous claims to the contrary (e.g., Fiengo and Lasnik 1973). First, we classify the apparently heterogeneous list of meanings proposed in previous research into a natural taxonomy, showing how they arise from a small stock of logical operations and predicates. Second, we exhibit a partial ordering of the various reciprocal meanings according to logical strength, which we make crucial use of in determining what reciprocals mean in each specific context where they appear. Third, we hypothesize that a reciprocal statement expresses the strongest candidate meaning that is consistent with known properties of the relation expressed by the scope of the reciprocal. This hypothesis is supported by analysis of a large collection of examples we have gathered from various corpora.

CONSISTENCY AND THE THEORY OF TRUTH

What is the logical strength of theories of truth? That is: If you take a theory${\cal T}$and add a theory of truth to it, how strong is the resulting theory, as compared to${\cal T}$? Once the question has been properly formulated, the answer turns out to be about as elegant as one could want: At least when${\cal T}$is finitely axiomatized theory, theories of truth act more or less as a kind of abstract consistency statement. To prove this result, however, we have to formulate truth-theories somewhat differently from how they have been and instead follow Tarski in ‘disentangling’ syntactic theories from object theories.

Improved Separations of Regular Resolution from Clause Learning Proof Systems

This paper studies the relationship between resolution and conflict driven clause learning (CDCL) without restarts, and refutes some conjectured possible separations. We prove that the guarded, xor-ified pebbling tautology clauses, which Urquhart proved are hard for regular resolution, as well as the guarded graph tautology clauses of Alekhnovich, Johannsen, Pitassi, and Urquhart have polynomial size pool resolution refutations that use only input lemmas as learned clauses. For the latter set of clauses, we extend this to prove that a CDCL search without restarts can refute these clauses in polynomial time, provided it makes the right choices for decision literals and clause learning. This holds even if the CDCL search is required to greedily process conflicts arising from unit propagation. This refutes the conjecture that the guarded graph tautology clauses or the guarded xor-ified pebbling tautology clauses can be used to separate CDCL without restarts from general resolution. Together with subsequent results by Buss and Kolodziejczyk, this means we lack any good conjectures about how to establish the exact logical strength of conflict-driven clause learning without restarts.