The Gödel Completeness Theorem for Uncountable Languages

Schlöder, Julian J.; Koepke, Peter

doi:10.2478/v10037-012-0023-z

Cited by 7 publications

(5 citation statements)

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“…Braselmann and Koepke [6,5] formalized in Mizar a sequent calculus for classical first-order logic and proved it sound and complete. Schlöder and Koepke [21] proved it complete for also uncountable languages. Ilik, Lee and Herbelin [9] introduced a Kripke-style semantics for classical first-order logic, and Ilik [8] formalized in Coq the completeness of a sequent calculus with respect to this semantics.…”

Section: Related Workmentioning

confidence: 99%

Teaching a Formalized Logical Calculus

From

Jensen

Schlichtkrull

et al. 2020

Electron. Proc. Theor. Comput. Sci.

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Classical first-order logic is in many ways central to work in mathematics, linguistics, computer science and artificial intelligence, so it is worthwhile to define it in full detail. We present soundness and completeness proofs of a sequent calculus for first-order logic, formalized in the interactive proof assistant Isabelle/HOL. Our formalization is based on work by Stefan Berghofer, which we have since updated to use Isabelle's declarative proof style Isar (Archive of Formal Proofs, Entry FOL-Fitting, August 2007 / July 2018). We represent variables with de Bruijn indices; this makes substitution under quantifiers less intuitive for a human reader. However, the nature of natural numbers yields an elegant solution when compared to implementations of substitution using variables represented by strings. The sequent calculus considered has the special property of an always empty antecedent and a list of formulas in the succedent. We obtain the proofs of soundness and completeness for the sequent calculus as a derived result of the inverse duality of its tableau counterpart. We strive to not only present the results of the proofs of soundness and completeness, but also to provide a deep dive into a programming-like approach to the formalization of first-order logic syntax, semantics and the sequent calculus. We use the formalization in a bachelor course on logic for computer science and discuss our experiences. * Corresponding author: jovi@dtu.dk Teaching a Formalized Logical CalculusThe development described in this paper is available online: https://bitbucket.org/isafol/isafol/src/master/FOL_Berghofer/ 4253 lines FOL_Berghofer.thy 867 lines FOL_Tableau.thy A contribution of this paper 217 lines FOL_Sequent.thy A contribution of this paper 132 lines FOL_Appendix.thy A contribution of this paperThese numbers include blank lines and a few comments. All in all it takes around 5 seconds in real time to verify on a fairly standard computer. The entire formalization is based on the standard theory Main (the standard library of Isabelle/HOL which comes with many useful functions and facts about e.g. natural numbers and lists).We have recently named our system based on the formalization in Isabelle/HOL: SeCa  Sequent Calculus Verifier SeCaV formalizes first-order logic with constants and functionsSeCaV verifies one-sided sequent calculus proofs SeCaV uses the Isabelle proof assistantSeCaV is a tool for teaching logic Jørgen Villadsen Asta Halkjaer FromAlexander Birch Jensen Anders SchlichtkrullFor our bachelor course we focus on the fragment (499 lines) of the formalization available here:https://bitbucket.org/isafol/isafol/src/master/Sequent_Calculus/ Here only the soundness proof is included but in addition a small and a large proof in the sequent calculus is formalized. The structure of the paper is as follows. Section 2 explains the formalization of the syntax and the semantics of classical first-order logic. Section 3 describes the sequent calculus. Section 4 outlines the formalized soundness and completeness proof...

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Section: Related Workmentioning

confidence: 99%

Teaching a Formalized Logical Calculus

From

Jensen

Schlichtkrull

et al. 2020

Electron. Proc. Theor. Comput. Sci.

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“…• Braselmann and Koepke [11,12] formalized in Mizar soundness and completeness of a sequent calculus. • Schlöder and Koepke [65] formalized its completeness considering also uncountable languages. • A more exotic result is the formalization by Ilik [27] in Coq of completeness of a sequent calculus with respect to a Kripke-semantics for classical first-order logic [28].…”

Section: Related Workmentioning

confidence: 99%

Interactive Theorem Proving for Logic and Information

Villadsen¹,

From²,

Jensen³

et al. 2021

Studies in Computational Intelligence

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Automated reasoning is the study of computer programs that can build proofs of theorems in a logic. Such programs can be either automatic theorem provers or interactive theorem provers. The latter are also called proof assistants because the user constructs the proofs with the help of the system. We focus on the Isabelle proof assistant. The system ensures that the proofs are correct, in contrast to pen-and-paper proofs which must be checked manually. We present applications to logical systems and models of information, in particular selected modal logics extending classical propositional logic. Epistemic logic allows intelligent systems to reason about the knowledge of agents. Public announcements can change the knowledge of the system and its agents. In order to account for this, epistemic logic can be extended to public announcement logic. An axiomatic system consists of axioms and rules of inference for deriving statements in a logic. Sound systems can only derive valid statements and complete systems can derive all valid statements. We describe formalizations of sound and complete axiomatic systems for epistemic logic and public announcement logic, thereby strengthening the foundations of automated reasoning for logic and information.

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“…Schlöder and Koepke, in Mizar [45], formalize a Henkin-style argument for possibly uncountable languages. Building on an early insight by Krivine [30] concerning the expressibility of the completeness proof in intuitionistic second-order logic, Ilik [25] analyzes Henkin-style ar-guments for classical and intuitionistic logic with respect to standard and Kripke models and formalizes them in Coq (without employing codatatypes).…”

Section: Related Workmentioning

confidence: 99%

Soundness and Completeness Proofs by Coinductive Methods

2016

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Codatatypes are absent from many programming languages and theorem provers, reflecting their absence from the standard repertoire of most computer scientists and logicians. We make a case for their usefulness by showing how they can be employed to produce compact, high-level proofs of key results in logic: the soundness and completeness of proof systems for variations of first-order logic. For the classical completeness result, we first establish an abstract property of possibly infinite derivation trees. The abstract proof can be instantiated for a wide range of Gentzen and tableau systems as well as various flavors of first-order logic. Soundness is often easy to prove, but it becomes interesting as soon as one allows infinite proofs of first-order formulas. This forms the subject of several cyclic proof systems for first-order logic augmented with inductive predicate definitions studied in the literature. All the discussed results are formalized using Isabelle/HOL's recently introduced support for codatatypes and corecursion. The development illustrates some unique features of Isabelle/HOL's new coinductive specification language such as nesting through non-free types and mixed recursion-corecursion.

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The Gödel Completeness Theorem for Uncountable Languages

Cited by 7 publications

References 4 publications

Teaching a Formalized Logical Calculus

Teaching a Formalized Logical Calculus

Interactive Theorem Proving for Logic and Information

Soundness and Completeness Proofs by Coinductive Methods

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