223 research outputs found
Proof Theory of Finite-valued Logics
The proof theory of many-valued systems has not been investigated to an extent comparable to the work done on axiomatizatbility of many-valued logics. Proof theory requires appropriate formalisms, such as sequent calculus, natural deduction, and tableaux for classical (and intuitionistic) logic. One particular method for systematically obtaining calculi for all finite-valued logics was invented independently by several researchers, with slight variations in design and presentation. The main aim of this report is to develop the proof theory of finite-valued first order logics in a general way, and to present some of the more important results in this area. In Systems covered are the resolution calculus, sequent calculus, tableaux, and natural deduction. This report is actually a template, from which all results can be specialized to particular logics
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
One-variable fragments of first-order logics
The one-variable fragment of a first-order logic may be viewed as an
"S5-like" modal logic, where the universal and existential quantifiers are
replaced by box and diamond modalities, respectively. Axiomatizations of these
modal logics have been obtained for special cases -- notably, the modal
counterparts S5 and MIPC of the one-variable fragments of first-order classical
logic and intuitionistic logic -- but a general approach, extending beyond
first-order intermediate logics, has been lacking. To this end, a sufficient
criterion is given in this paper for the one-variable fragment of a
semantically-defined first-order logic -- spanning families of intermediate,
substructural, many-valued, and modal logics -- to admit a natural
axiomatization. More precisely, such an axiomatization is obtained for the
one-variable fragment of any first-order logic based on a variety of algebraic
structures with a lattice reduct that has the superamalgamation property,
building on a generalized version of a functional representation theorem for
monadic Heyting algebras due to Bezhanishvili and Harding. An alternative
proof-theoretic strategy for obtaining such axiomatization results is also
developed for first-order substructural logics that have a cut-free sequent
calculus and admit a certain interpolation property.Comment: arXiv admin note: text overlap with arXiv:2209.0856
Neutrality and Many-Valued Logics
In this book, we consider various many-valued logics: standard, linear,
hyperbolic, parabolic, non-Archimedean, p-adic, interval, neutrosophic, etc. We
survey also results which show the tree different proof-theoretic frameworks
for many-valued logics, e.g. frameworks of the following deductive calculi:
Hilbert's style, sequent, and hypersequent. We present a general way that
allows to construct systematically analytic calculi for a large family of
non-Archimedean many-valued logics: hyperrational-valued, hyperreal-valued, and
p-adic valued logics characterized by a special format of semantics with an
appropriate rejection of Archimedes' axiom. These logics are built as different
extensions of standard many-valued logics (namely, Lukasiewicz's, Goedel's,
Product, and Post's logics). The informal sense of Archimedes' axiom is that
anything can be measured by a ruler. Also logical multiple-validity without
Archimedes' axiom consists in that the set of truth values is infinite and it
is not well-founded and well-ordered. On the base of non-Archimedean valued
logics, we construct non-Archimedean valued interval neutrosophic logic INL by
which we can describe neutrality phenomena.Comment: 119 page
A Simple Logic of Functional Dependence
This paper presents a simple decidable logic of functional dependence LFD,
based on an extension of classical propositional logic with dependence atoms
plus dependence quantifiers treated as modalities, within the setting of
generalized assignment semantics for first order logic. The expressive
strength, complete proof calculus and meta-properties of LFD are explored.
Various language extensions are presented as well, up to undecidable
modal-style logics for independence and dynamic logics of changing dependence
models. Finally, more concrete settings for dependence are discussed:
continuous dependence in topological models, linear dependence in vector
spaces, and temporal dependence in dynamical systems and games.Comment: 56 pages. Journal of Philosophical Logic (2021
Modal Hybrid Logic
This is an extended version of the lectures given during the 12-th Conference on Applications of Logic in Philosophy and in the Foundations of Mathematics in Szklarska Poręba (7–11 May 2007). It contains a survey of modal hybrid logic, one of the branches of contemporary modal logic. In the first part a variety of hybrid languages and logics is presented with a discussion of expressivity matters. The second part is devoted to thorough exposition of proof methods for hybrid logics. The main point is to show that application of hybrid logics may remarkably improve the situation in modal proof theory
The Logic of Exact Covers: Completeness and Uniform Interpolation
We show that all (not necessarily normal or monotone) modal logics that can be axiomatised in rank-1 have the interpolation property, and that in fact interpolation is uniform if the logics just have finitely many modal operators. As immediate applicatio
Frege systems for quantified Boolean logic
We define and investigate Frege systems for quantified Boolean formulas (QBF). For these new proof systems, we develop a lower bound technique that directly lifts circuit lower bounds for a circuit class C to the QBF Frege system operating with lines from C. Such a direct transfer from circuit to proof complexity lower bounds has often been postulated for propositional systems but had not been formally established in such generality for any proof systems prior to this work. This leads to strong lower bounds for restricted versions of QBF Frege, in particular an exponential lower bound for QBF Frege systems operating with AC0[p] circuits. In contrast, any non-trivial lower bound for propositional AC0[p]-Frege constitutes a major open problem. Improving these lower bounds to unrestricted QBF Frege tightly corresponds to the major problems in circuit complexity and propositional proof complexity. In particular, proving a lower bound for QBF Frege systems operating with arbitrary P/poly circuits is equivalent to either showing a lower bound for P/poly or for propositional extended Frege (which operates with P/poly circuits). We also compare our new QBF Frege systems to standard sequent calculi for QBF and establish a correspondence to intuitionistic bounded arithmetic.This research was supported by grant nos. 48138 and 60842 from the John Templeton Foundation, EPSRC grant
EP/L024233/1, and a Doctoral Prize Fellowship from EPSRC (third author). The second author was funded by the European
Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC grant agreement
no. 279611 and under the European Union’s Horizon 2020 Research and Innovation Programme/ERC grant agreement no.
648276 AUTAR. The fourth author was supported by the Austrian Science Fund (FWF) under project number P28699 and by
the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2014)/ERC Grant
Agreement no. 61507.
Part of this work was done when Beyersdorff and Pich were at the University of Leeds and Bonacina at Sapienza University
Rome.Peer ReviewedPostprint (published version
Understanding Gentzen and Frege Systems for QBF
Recently Beyersdorff, Bonacina, and Chew [10] introduced a natural class of Frege systems for quantified Boolean formulas (QBF) and showed strong lower bounds for restricted versions of these systems. Here we provide a comprehensive analysis of the new extended Frege system from [10], denoted EF + ∀red, which is a natural extension of classical extended Frege EF. Our main results are the following: Firstly, we prove that the standard Gentzen-style system G*1 p-simulates EF + ∀red and that G*1 is strictly stronger under standard complexity-theoretic hardness assumptions.
Secondly, we show a correspondence of EF + ∀red to bounded arithmetic: EF + ∀red can be seen as the non-uniform propositional version of intuitionistic S12. Specifically, intuitionistic S12 proofs of arbitrary statements in prenex form translate to polynomial-size EF + ∀red proofs, and EF + ∀red is in a sense the weakest system with this property. Finally, we show that unconditional lower bounds for EF + ∀red would imply either a major breakthrough in circuit complexity or in classical proof complexity, and in fact the converse implications hold as well. Therefore, the system EF + ∀red naturally unites the central problems from circuit and proof complexity.
Technically, our results rest on a formalised strategy extraction theorem for EF + ∀red akin to witnessing in intuitionistic S12 and a normal form for EF + ∀red proofs
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