248 research outputs found
Formal logic: Classical problems and proofs
Not focusing on the history of classical logic, this book provides discussions and quotes central passages on its origins and development, namely from a philosophical perspective. Not being a book in mathematical logic, it takes formal logic from an essentially mathematical perspective. Biased towards a computational approach, with SAT and VAL as its backbone, this is an introduction to logic that covers essential aspects of the three branches of logic, to wit, philosophical, mathematical, and computational
Advanced Proof Viewing in ProofTool
Sequent calculus is widely used for formalizing proofs. However, due to the
proliferation of data, understanding the proofs of even simple mathematical
arguments soon becomes impossible. Graphical user interfaces help in this
matter, but since they normally utilize Gentzen's original notation, some of
the problems persist. In this paper, we introduce a number of criteria for
proof visualization which we have found out to be crucial for analyzing proofs.
We then evaluate recent developments in tree visualization with regard to these
criteria and propose the Sunburst Tree layout as a complement to the
traditional tree structure. This layout constructs inferences as concentric
circle arcs around the root inference, allowing the user to focus on the
proof's structural content. Finally, we describe its integration into ProofTool
and explain how it interacts with the Gentzen layout.Comment: In Proceedings UITP 2014, arXiv:1410.785
Systematic construction of natural deduction systems for many-valued logics
A construction principle for natural deduction systems for arbitrary, finitely-many-valued first order logics is exhibited. These systems are systematically obtained from sequent calculi, which in turn can be automatically extracted from the truth tables of the logics under consideration. Soundness and cut-free completeness of these sequent calculi translate into soundness, completeness, and normal-form theorems for natural deduction systems
On the relative proof complexity of deep inference via atomic flows
We consider the proof complexity of the minimal complete fragment, KS, of
standard deep inference systems for propositional logic. To examine the size of
proofs we employ atomic flows, diagrams that trace structural changes through a
proof but ignore logical information. As results we obtain a polynomial
simulation of versions of Resolution, along with some extensions. We also show
that these systems, as well as bounded-depth Frege systems, cannot polynomially
simulate KS, by giving polynomial-size proofs of certain variants of the
propositional pigeonhole principle in KS.Comment: 27 pages, 2 figures, full version of conference pape
Generic Modal Cut Elimination Applied to Conditional Logics
We develop a general criterion for cut elimination in sequent calculi for
propositional modal logics, which rests on absorption of cut, contraction,
weakening and inversion by the purely modal part of the rule system. Our
criterion applies also to a wide variety of logics outside the realm of normal
modal logic. We give extensive example instantiations of our framework to
various conditional logics. For these, we obtain fully internalised calculi
which are substantially simpler than those known in the literature, along with
leaner proofs of cut elimination and complexity. In one case, conditional logic
with modus ponens and conditional excluded middle, cut elimination and
complexity were explicitly stated as open in the literature
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
Analytic Tableaux for Simple Type Theory and its First-Order Fragment
We study simple type theory with primitive equality (STT) and its first-order
fragment EFO, which restricts equality and quantification to base types but
retains lambda abstraction and higher-order variables. As deductive system we
employ a cut-free tableau calculus. We consider completeness, compactness, and
existence of countable models. We prove these properties for STT with respect
to Henkin models and for EFO with respect to standard models. We also show that
the tableau system yields a decision procedure for three EFO fragments
Simple Decision Procedure for S5 in Standard Cut-Free Sequent Calculus
In the paper a decision procedure for S5 is presented which uses a cut-free sequent calculus with additional rules allowing a reduction to normal modal forms. It utilizes the fact that in S5 every formula is equivalent to some 1-degree formula, i.e. a modally-flat formula with modal functors having only boolean formulas in its scope. In contrast to many sequent calculi (SC) for S5 the presented system does not introduce any extra devices. Thus it is a standard version of SC but with some additional simple rewrite rules. The procedure combines the proces of saturation of sequents with reduction of their elements to some normal modal form
Integrating a Global Induction Mechanism into a Sequent Calculus
Most interesting proofs in mathematics contain an inductive argument which
requires an extension of the LK-calculus to formalize. The most commonly used
calculi for induction contain a separate rule or axiom which reduces the valid
proof theoretic properties of the calculus. To the best of our knowledge, there
are no such calculi which allow cut-elimination to a normal form with the
subformula property, i.e. every formula occurring in the proof is a subformula
of the end sequent. Proof schemata are a variant of LK-proofs able to simulate
induction by linking proofs together. There exists a schematic normal form
which has comparable proof theoretic behaviour to normal forms with the
subformula property. However, a calculus for the construction of proof schemata
does not exist. In this paper, we introduce a calculus for proof schemata and
prove soundness and completeness with respect to a fragment of the inductive
arguments formalizable in Peano arithmetic.Comment: 16 page
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
- …