Artículos con la etiqueta ‘Logic (math.LO)’

A proposed characterisation of the intrinsically justified reflection principles

Por • 4 abr, 2014 • Category: Opinion

Building on previous work of Tait, Koellner, and myself exploring the question of which reflection principles are intrinsically justified on the basis of the iterative conception of set, we formulate a new reflection principle, which subsumes all previously known reflection principles which have been proposed as intrinsically justified and are also known to be consistent, and which is itself consistent relative to an ω -Erd\»os cardinal. An open-ended family of strengthenings of this principle is tentatively proposed as exhausting everything that can be justified on the basis of the iterative conception of set.

A Note on the Fixed Points in Justification Logics

Por • 26 mar, 2014 • Category: Educacion

In this note we study the effect of adding fixed points to justification logics. We introduce two extensions of justification logics: extensions by fixed point (or diagonal) operators, and extensions by least fixed points. The former is a justification version of Smory\`{n}ski’s Diagonalization Operator Logic, and the latter is a justification version of Kozen’s modal μ -calculus. We also introduce fixed point extensions of Fitting’s quantified logic of proofs, and formalize the Knower Paradox and the Surprise Test Paradox in these extensions. By interpreting a surprise statement as a statement for which there is no justification, we give a solution to the self-reference version of the Surprise Test Paradox in quantified logic of proofs.

On Strongly First-Order Dependencies

Por • 19 mar, 2014 • Category: Ciencia y tecnología

We prove that the expressive power of first-order logic with team semantics plus contradictory negation does not rise beyond that of first-order logic (with respect to sentences), and that the totality atoms of arity k +1 are not definable in terms of the totality atoms of arity k. We furthermore prove that all first-order nullary and unary dependencies are strongly first order, in the sense that they do not increase the expressive power of first order logic if added to it.

Restructuring Logic

Por • 12 mar, 2014 • Category: Filosofía

The outline of a programme for restructuring mathematical logic. We explain what we mean by «restructuring» and carry out exemplary parts of the programme

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.

Creature forcing and five cardinal characteristics of the continuum

Por • 13 feb, 2014 • Category: Opinion

We use a (countable support) creature construction to show that consistently
d=ℵ 1 =cov(NULL)

Scheme representation for first-order logic

Por • 13 feb, 2014 • Category: Educacion

Although contemporary model theory has been called «algebraic geometry minus fields», the formal methods of the two fields are radically different. This dissertation aims to shrink that gap by presenting a theory of logical schemes, geometric entities which relate to first-order logical theories in much the same way that algebraic schemes relate to commutative rings. The construction relies on a Grothendieck-style representation theorem which associates every coherent or classical first-order theory with an affine scheme: a topological groupoid (the spectrum of the theory) together with a sheaf of (local) syntactic categories.

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 .

(In)dependence Logic and Abstract Independence Relations

Por • 30 ene, 2014 • Category: Opinion

We generalize the results of [15] and [16] to the framework of abstract independence relations for an arbitrary AEC. We give a model-theoretic interpretation of the independence atom and characterize under which conditions we can prove a completeness result with respect to the deductive system that axiomatizes independence in team semantics.

Category theory, logic and formal linguistics: some connections, old and new

Por • 30 ene, 2014 • Category: Crítica

We seize the opportunity of the publication of selected papers from the \emph{Logic, categories, semantics} workshop in the \emph{Journal of Applied Logic} to survey some current trends in logic, namely intuitionistic and linear type theories, that interweave categorical, geometrical and computational considerations. We thereafter present how these rich logical frameworks can model the way language conveys meaning.