Epistemic virtues, metavirtues, and computational complexity

I argue that considerations about computational complexity show that all finite agents need characteristics like those that have been called epistemic virtues. The necessity of these virtues follows in part from the nonexistence of shortcuts, or efficient ways of finding shortcuts, to cognitively expensive routines. It follows that agents must possess the capacities – metavirtues –of developing in advance the cognitive virtues they will need when time and memory are at a premium

P vs NP

In this paper, we discuss the $\mathcal{NP}$ problem using the Henkin's Theory and the Herbrand Theory in the first-order logic, and prove that $\mathcal{P}$ is a proper subset of $\mathcal{NP}$

Complete Additivity and Modal Incompleteness

In this paper, we tell a story about incompleteness in modal logic. The story weaves together a paper of van Benthem, `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, 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 paper 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 V-BAOs, namely the provability logic GLB. 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 V-complete. After these results, we generalize the Blok Dichotomy to degrees of V-incompleteness. In the end, we return to van Benthem's theme of syntactic aspects of modal incompleteness

Logic in the Tractatus

I present a reconstruction of the logical system of the Tractatus, which differs from classical logic in two ways. It includes an account of Wittgenstein’s “form-series” device, which suffices to express some effectively generated countably infinite disjunctions. And its attendant notion of structure is relativized to the fixed underlying universe of what is named. There follow three results. First, the class of concepts definable in the system is closed under finitary induction. Second, if the universe of objects is countably infinite, then the property of being a tautology is \Pi^1_1-complete. But third, it is only granted the assumption of countability that the class of tautologies is \Sigma_1-definable in set theory. Wittgenstein famously urges that logical relationships must show themselves in the structure of signs. He also urges that the size of the universe cannot be prejudged. The results of this paper indicate that there is no single way in which logical relationships could be held to make themselves manifest in signs, which does not prejudge the number of objects

Deciding First-Order Satisfiability when Universal and Existential Variables are Separated

We introduce a new decidable fragment of first-order logic with equality, which strictly generalizes two already well-known ones -- the Bernays-Sch\"onfinkel-Ramsey (BSR) Fragment and the Monadic Fragment. The defining principle is the syntactic separation of universally quantified variables from existentially quantified ones at the level of atoms. Thus, our classification neither rests on restrictions on quantifier prefixes (as in the BSR case) nor on restrictions on the arity of predicate symbols (as in the monadic case). We demonstrate that the new fragment exhibits the finite model property and derive a non-elementary upper bound on the computing time required for deciding satisfiability in the new fragment. For the subfragment of prenex sentences with the quantifier prefix $\exists^* \forall^* \exists^*$ the satisfiability problem is shown to be complete for NEXPTIME. Finally, we discuss how automated reasoning procedures can take advantage of our results.Comment: Extended version of our LICS 2016 conference paper, 23 page

Decidability of E*A-sentences in Membership Theories

3The problem is addressed of establishing the satisfiability of prenex formulas involving a single universal quantifier, in diversified axiomatic set theories. A rather general decision method for solving this problem is illustrated through the treatment of membership theories of increasing strength, ending with a subtheory of Zermelo-Fraenkel which is already complete with respect to the There Exists*For All class of sentences. NP-hardness and NP-completeness results concerning the problems under study are achieved and a technique for restricting the universal quantifier is presented.opennoneopenOMODEO E.; PARLAMENTO F; POLICRITI A.Omodeo, E.; Parlamento, Franco; Policriti, Albert

Teoria tradicional da informação semântica sem escândalo da dedução : uma reavaliação moderadamente externalista do tópico baseada em semântica urna e uma aplicação paraconsistente

Orientador: Walter Alexandre CarnielliTese (doutorado) - Universidade Estadual de Campinas, Instituto de Filosofia e Ciências HumanasResumo: A presente tese mostra que é possível reestabelecer a teoria tradicional da informação semântica (no que segue apenas TSI, originalmente proposta por Bar-Hillel e Carnap (1952, 1953)) a partir de uma descrição adequada das condições epistemológicas de nossa competência semântica. Uma consequência clássica de TSI é o assim chamado escândalo da dedução (no que segue SoD), tese segundo a qual verdades lógicas têm quantidade nula de informação. SoD é problemático dado que conflita com o caráter ampliativo do conhecimento formal. Baseado nisso, trabalhos recentes (e.g., Floridi (2004)) rejeitam TSI apesar de suas boas intuições sobre a natureza da informação semântica. Por outro lado, esta tese reconsidera a estratégia de assumir a semântica urna (RANTALA, 1979) como o pano de fundo metateórico privilegiado para o reestabelecimento de TSI sem SoD. A presente tese tem o seguinte plano. O capítulo 1 introduz o plano geral da tese. No capítulo 2, valendo-se fortemente de trabalhos clássicos sobre o externalismo semântico, eu apresento algum suporte filosófico para essa estratégia ao mostrar que a semântica urna corretamente caracteriza as condições epistemológicas de nossa competência semântica no uso de quantificadores. O capitulo 3 oferece uma descrição precisa da semântica urna a partir da apresentação de suas definições básicas e alguns de seus teoremas mais funda- mentais. No capítulo 4, eu me concentro mais uma vez no tema da informação semântica ao formalizar TSI em semântica urna e provar que nesse contexto SoD não vale. Finalmente, nos capítulos 5 e 6 eu considero resultados modelo-teóricos mais avançados sobre semântica urna e exploro uma possível aplicação paraconsistente das ideias principais dessa tese, respectivamenteAbstract: This thesis shows that it is possible to reestablish the traditional theory of semantic information (TSI, originally proposed by Bar-Hillel and Carnap (1952, 1953)) by providing an adequate account of the epistemological conditions of our semantic competence. A classical consequence of TSI is the so-called scandal of deduction (hereafter SoD) according to which logical truths have null amount of information. SoD is problematic since it does not make room for the ampliative character of formal knowledge. Based on this, recent work on the subject (e.g., Floridi (2004)) rejects TSI despite its good insights on the nature of semantic information. On the other hand, this work reconsiders the strategy of taking urn semantics (RANTALA, 1979) as a privileged metatheoretic framework for the formalization of TSI without SoD. The present thesis is planned in the following way. Chapter 1 introduces the thesis¿ overall plan. In chapter 2, relying heavily on classical works on semantic externalism, I present some philosophical support for this strategy by showing that urn semantics correctly characterizes the epistemological conditions of our semantic competence in the use of quantifiers. Chapter 3 offers a precise description of urn semantics by characterizing its basic definitions and some of its most fundamental theorems. In chapter 4, turning the focus once again to semantic information, I formalize TSI in urn semantics and show that in this context SoD does not hold. Finally, in chapter 5 and 6 I consider more advanced model-theoretic results on urn semantics and explore a paraconsistent possible application of the present idea, respectivelyDoutoradoFilosofiaDoutor em Filosofia142038/2014-8CNP

The Ontological Import of Adding Proper Classes

In this article, we analyse the ontological import of adding classes to set theories. We assume that this increment is well represented by going from ZF system to NBG. We thus consider the standard techniques of reducing one system to the other. Novak proved that from a model of ZF we can build a model of NBG (and vice versa), while Shoenfield have shown that from a proof in NBG of a set-sentence we can generate a proof in ZF of the same formula. We argue that the first makes use of a too strong metatheory. Although meaningful, this symmetrical reduction does not equate the ontological content of the theories. The strong metatheory levels the two theories. Moreover, we will modernize Shoenfields proof, emphasizing its relation to Herbrands theorem and that it can only be seen as a partial type of reduction. In contrast with symmetrical reductions, we believe that asymmetrical relations are powerful tools for comparing ontological content. In virtue of this, we prove that there is no interpretation of NBG in ZF, while NBG trivially interprets ZF. This challenges the standard view that the two systems have the same ontological content

A Theory of Structured Propositions

This paper argues that the theory of structured propositions is not undermined by the Russell-Myhill paradox. I develop a theory of structured propositions in which the Russell-Myhill paradox doesn't arise: the theory does not involve ramification or compromises to the underlying logic, but rather rejects common assumptions, encoded in the notation of the $\lambda$-calculus, about what properties and relations can be built. I argue that the structuralist had independent reasons to reject these underlying assumptions. The theory is given both a diagrammatic representation, and a logical representation in a novel language. In the latter half of the paper I turn to some technical questions concerning the treatment of quantification, and demonstrate various equivalences between the diagrammatic and logical representations, and a fragment of the $\lambda$-calculus