1,169 research outputs found
From IF to BI: a tale of dependence and separation
We take a fresh look at the logics of informational dependence and
independence of Hintikka and Sandu and Vaananen, and their compositional
semantics due to Hodges. We show how Hodges' semantics can be seen as a special
case of a general construction, which provides a context for a useful
completeness theorem with respect to a wider class of models. We shed some new
light on each aspect of the logic. We show that the natural propositional logic
carried by the semantics is the logic of Bunched Implications due to Pym and
O'Hearn, which combines intuitionistic and multiplicative connectives. This
introduces several new connectives not previously considered in logics of
informational dependence, but which we show play a very natural role, most
notably intuitionistic implication. As regards the quantifiers, we show that
their interpretation in the Hodges semantics is forced, in that they are the
image under the general construction of the usual Tarski semantics; this
implies that they are adjoints to substitution, and hence uniquely determined.
As for the dependence predicate, we show that this is definable from a simpler
predicate, of constancy or dependence on nothing. This makes essential use of
the intuitionistic implication. The Armstrong axioms for functional dependence
are then recovered as a standard set of axioms for intuitionistic implication.
We also prove a full abstraction result in the style of Hodges, in which the
intuitionistic implication plays a very natural r\^ole.Comment: 28 pages, journal versio
Logical closure properties of propositional proof systems - (Extended abstract)
In this paper we define and investigate basic logical closure properties of propositional proof systems such as closure of arbitrary proof systems under modus ponens or substitutions. As our main result we obtain a purely logical characterization of the degrees of schematic extensions of EF in terms of a simple combination of these properties. This result underlines the empirical evidence that EF and its extensions admit a robust definition which rests on only a few central concepts from propositional logic
Making proofs without Modus Ponens: An introduction to the combinatorics and complexity of cut elimination
This paper is intended to provide an introduction to cut elimination which is
accessible to a broad mathematical audience. Gentzen's cut elimination theorem
is not as well known as it deserves to be, and it is tied to a lot of
interesting mathematical structure. In particular we try to indicate some
dynamical and combinatorial aspects of cut elimination, as well as its
connections to complexity theory. We discuss two concrete examples where one
can see the structure of short proofs with cuts, one concerning feasible
numbers and the other concerning "bounded mean oscillation" from real analysis
The Computational Complexity of Propositional Cirquent Calculus
Introduced in 2006 by Japaridze, cirquent calculus is a refinement of sequent
calculus. The advent of cirquent calculus arose from the need for a deductive
system with a more explicit ability to reason about resources. Unlike the more
traditional proof-theoretic approaches that manipulate tree-like objects
(formulas, sequents, etc.), cirquent calculus is based on circuit-style
structures called cirquents, in which different "peer" (sibling, cousin, etc.)
substructures may share components. It is this resource sharing mechanism to
which cirquent calculus owes its novelty (and its virtues). From its inception,
cirquent calculus has been paired with an abstract resource semantics. This
semantics allows for reasoning about the interaction between a resource
provider and a resource user, where resources are understood in the their most
general and intuitive sense. Interpreting resources in a more restricted
computational sense has made cirquent calculus instrumental in axiomatizing
various fundamental fragments of Computability Logic, a formal theory of
(interactive) computability. The so-called "classical" rules of cirquent
calculus, in the absence of the particularly troublesome contraction rule,
produce a sound and complete system CL5 for Computability Logic. In this paper,
we investigate the computational complexity of CL5, showing it is
-complete. We also show that CL5 without the duplication rule has
polynomial size proofs and is NP-complete
The Broadest Necessity
In this paper the logic of broad necessity is explored. Definitions of what it means for one modality to be broader than another are formulated, and it is proven, in the context of higher-order logic, that there is a broadest necessity, settling one of the central questions of this investigation. It is shown, moreover, that it is possible to give a reductive analysis of this necessity in extensional language. This relates more generally to a conjecture that it is not possible to define intensional connectives from extensional notions. This conjecture is formulated precisely in higher-order logic, and concrete cases in which it fails are examined. The paper ends with a discussion of the logic of broad necessity. It is shown that the logic of broad necessity is a normal modal logic between S4 and Triv, and that it is consistent with a natural axiomatic system of higher-order logic that it is exactly S4. Some philosophical reasons to think that the logic of broad necessity does not include the S5 principle are given
A new foundational crisis in mathematics, is it really happening?
The article reconsiders the position of the foundations of mathematics after
the discovery of HoTT. Discussion that this discovery has generated in the
community of mathematicians, philosophers and computer scientists might
indicate a new crisis in the foundation of mathematics. By examining the
mathematical facts behind HoTT and their relation with the existing
foundations, we conclude that the present crisis is not one. We reiterate a
pluralist vision of the foundations of mathematics. The article contains a
short survey of the mathematical and historical background needed to understand
the main tenets of the foundational issues.Comment: Final versio
Bayesian Probabilities and the Histories Algebra
We attempt a justification of a generalisation of the consistent histories
programme using a notion of probability that is valid for all complete sets of
history propositions. This consists of introducing Cox's axioms of probability
theory and showing that our candidate notion of probability obeys them. We also
give a generalisation of Bayes' theorem and comment upon how Bayesianism should
be useful for the quantum gravity/cosmology programmes.Comment: 10 pages, accepted by Int. J. Theo. Phys. Feb 200
A predicative variant of a realizability tripos for the Minimalist Foundation.
open2noHere we present a predicative variant of a realizability tripos validating
the intensional level of the Minimalist Foundation extended with Formal Church
thesis.the file attached contains the whole number of the journal including the mentioned pubblicationopenMaietti, Maria Emilia; Maschio, SamueleMaietti, MARIA EMILIA; Maschio, Samuel
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