Artículos con la etiqueta ‘Lógica modal’

Topos Semantics for Higher-Order Modal Logic

Por • 6 mar, 2014 • Category: Educacion

We define the notion of a model of higher-order modal logic in an arbitrary elementary topos E . In contrast to the well-known interpretation of (non-modal) higher-order logic, the type of propositions is not interpreted by the subobject classifier ΩE , but rather by a suitable complete Heyting algebra H . The canonical map relating H and ΩE both serves to interpret equality and provides a modal operator on H in the form of a comonad. Examples of such structures arise from surjective geometric morphisms f:F→E , where H=f∗ΩF . The logic differs from non-modal higher-order logic in that the principles of functional and propositional extensionality are no longer valid but may be replaced by modalized versions. The usual Kripke, neighborhood, and sheaf semantics for propositional and first-order modal logic are subsumed by this notion.

Two Kinds of Discovery: An Ontological Account

Por • 6 feb, 2014 • Category: Educacion

What can we discover? As the discussion in this paper is limited to ontological considerations, it does not deal with the discovery of new concepts. It raises the following question: What are the entities or existents that we can discover? There are two kinds of such entities: (1) actual entities, and (2) possible entities, which are pure possibilities. The paper explains why the first kind of discovery depends primarily on the second kind. The paper illustrates the discoveries of individual pure possibilities by presenting examples such as the Higgs particle, Dirac’s positron, and Pauli-Fermi’s neutrino.

Characterising intermediate tense logics in terms of Galois connections

Por • 4 feb, 2014 • Category: Educacion

We propose a uniform way of defining for every logic L intermediate between intuitionistic and classical logics, the corresponding intermediate minimal tense logic LK t . This is done by building the fusion of two copies of intermediate logic with a Galois connection LGC , and then interlinking their operators by two Fischer Servi axioms. The resulting system is called here L2GC+FS . In the cases of intuitionistic logic Int and classical logic Cl , it is noted that Int2GC+FS is syntactically equivalent to intuitionistic minimal tense logic IK t by W. B.Ewald and Cl2GC+FS equals classical minimal tense logic K t . This justifies to consider L2GC+FS as minimal L -tense logic LK t for any intermediate logic L . We define H2GC+FS-algebras as expansions of HK1-algebras, introduced by E. Or{\l}owska and I. Rewitzky. For each intermediate logic L , we show algebraic completeness of L2GC+FS and its conservativeness over L .

The Rule of Global Necessitation

Por • 5 oct, 2012 • Category: Crítica

For half a century, authors have weakened the rule of necessitation in various more or less ad hoc ways in order to make inconsistent systems consistent. More recently, necessitation was weakened in a systematic way, not for the purpose of resolving paradoxes but rather to salvage the deduction theorem for modal logic. We show how this systematic weakening can be applied to the older problem of paradox resolution. Four examples are given: a predicate symbol S4 consistent with arithmetic; a resolution of the surprise examination paradox; a resolution of Fitch’s paradox; and finally, the construction of a knowing machine which knows its own code. We discuss a technique for possibly finding answers to a question of P. \’Egr\’e and J. van Benthem.

Structural connections between a forcing class and its modal logic

Por • 5 ago, 2012 • Category: Crítica

The modal logic of forcing arises when one considers a model of set theory in the context of all its forcing extensions, interpreting necessity as “in all forcing extensions” and possibility as “in some forcing extension”. In this modal language one may easily express sweeping general forcing principles, such as the assertion that every possibly necessary statement is necessarily possible, which is valid for forcing, or the assertion that every possibly necessary statement is true, which is the maximality principle, a forcing axiom independent of but equiconsistent with ZFC (Stavi-V\”a\”an\”anen, Hamkins). Every definable forcing class similarly gives rise to the corresponding forcing modalities, for which one considers extensions only by forcing notions in that class. In previous work, we proved that if ZFC is consistent, then the ZFC-provably valid principles of the class of all forcing are precisely the assertions of the modal theory S4.2.

First-order Logic: Modality and Intensionality

Por • 23 ago, 2011 • Category: Ciencia y tecnología

Contemporary use of the term ‘intension’ derives from the traditional logical Frege-Russell’s doctrine that an idea (logic formula) has both an extension and an intension. From the Montague’s point of view, the meaning of an idea can be considered as particular extensions in different possible worlds. In this paper we analyze the minimal intensional semantic enrichment of the syntax of the FOL language, by unification of different views: Tarskian extensional semantics of the FOL, modal interpretation of quantifiers, and a derivation of the Tarskian theory of truth from unified semantic theory based on a single meaning relation.