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### Extended normal form theorems for logical proofs from axioms

Toshiyasu Arai, Grigori Mints
2000 Theoretical Computer Science
We present an exact modeling of cut-free arithmetic by cut-free predicate logic. Exact prooftheoretic analysis reveals connections with omega-consistency and re ection.  ...  First, we present here a short proof of these results which extends the proof in [4] . Second, we prove this normal form theorem by a transÿnite induction.  ...  we prove the extended normal form theorem, Theorem 2 by a transÿnite induction up to the nth epsilon number n .  ...

### General Recursive Functions in the Number-Theoretic Formal System

Sh^|^ocirc;ji MAEHARA
1957 Annals of the Japan Association for Philosophy of Science
For any LK or LJ-provable sequent, there exists an LK-or LJ-proof of the sequent which contains no cut, respectively. 1. 52. EXTENDED NORMAL FORM THEOREM ON LK.  ...  By ƒ¡a we denote the following axiom system: By Gentzen's Consistency theorem and Extended normal form theorem on LK, it can be immediately proved that Pa is consistent .  ...

### Proof systems for structured algebraic specifications: An overview [chapter]

Rolf Hennicker, Martin Wirsing
1997 Lecture Notes in Computer Science
proof systems for the underlying logic, may be used for deriving theorems of the specification.  ...  In the normal form approach of Bergstra, Hering and Klint, a flat set of axioms is constructed for each structured specification, whereas in the second approach not only individual axioms but also individual  ...  can be derived from the axioms of its normal form nf(SP) (using some standard proof system of the underlying logic).  ...

### The foundation of a generic theorem prover

Lawrence C. Paulson
1989 Journal of automated reasoning
Examples illustrate the use of this meta-logic to formalize logics and proofs. Axioms for first-order logic are shown to be sound and complete.  ...  Section 6 extends the backwards proof methods to handle quantifiers and unification, with examples from first-order logic. The representation of eigenvariables differs from Isabelle-86.  ...  The normal form does not take account of the logical rules. No effective procedure can reduce every theorem to some unique true formula. There is also a normalization procedure for HOL proofs.  ...

### Page 217 of Mathematical Reviews Vol. 33, Issue 2 [page]

1967 Mathematical Reviews
for arith- metic from a normal-form theorem.  ...  With the second version of deduction, a normal-form theorem results, none with the first. The system of relevant implication is extended to the minimal and classical logics.  ...

### Adding the everywhere operator to propositional logic

D Gries
1998 Journal of Logic and Computation
First, introducing an inference rule textual substitution allows seamless integration of the propositional and modal parts of the logic, giving a more practical system for writing formal proofs.  ...  Second, the two following approaches to axiomatizing a logic are shown to be not equivalent: (i) give axiom schemes that denote an infinite number of axioms and (ii) write a finite number of axioms in  ...  Leino, and members of the Eindhoven Tuesday Afternoon Club for their extremely helpful comments on drafts of this paper.  ...

### Realization Theorems for Justification Logics: Full Modularity [chapter]

Annemarie Borg, Roman Kuznets
2015 Lecture Notes in Computer Science
A uniform realization theorem for all the modal logics of the socalled modal cube, i.e., for the extensions of the basic modal logic K with any subset of the axioms d, t, b, 4, and 5, has been proven using  ...  A constructive proof of a realization theorem typically relies on a cut-free sequent-style proof system for the corresponding modal logic.  ...  The authors are indebted to Agata Ciabattoni for making this research possible.  ...

### Extracting Herbrand Disjunctions by Functional Interpretation

Philipp Gerhardy, Ulrich Kohlenbach
2003 BRICS Report Series
Carrying out a suggestion by Kreisel, we adapt Gödel's functional interpretation to ordinary first-order predicate logic(PL) and thus devise an algorithm to extract Herbrand terms from PL-proofs.  ...  The algorithm consists of two main steps: first we extract a functional realizer, next we compute the beta-normal-form of the realizer from which the Herbrand terms can be read off.  ...  The theorem also extends to tuples ∃x of quantifiers. Proof. The theorem follows from the above propositions and lemmas.  ...

### Algebraic semantics for a modal logic close to S1

Steffen Lewitzka
2014 Journal of Logic and Computation
Suszko's non-Fregean logic adapted to the language of modal logic (we call these axioms the axioms of propositional identity).  ...  Lewis as logics for strict implication. While there are Kripke semantics for S2 and S3, there is no known natural semantics for S1.  ...  Proof. Symmetry of ≈ Φ follows from propositional logic. Since ϕ → ϕ is an axiom, we get (ϕ → ϕ) ∈ Φ by rule AN. Thus, ≈ Φ is reflexive. Transitivity follows from the scheme of axioms (iii).  ...

### On Natural Deduction for Herbrand Constructive Logics III: The Strange Case of the Intuitionistic Logic of Constant Domains [article]

Federico Aschieri (Institut für Logic, Computation Technische Universität Wien)
2018 arXiv   pre-print
The logic of constant domains is intuitionistic logic extended with the so-called forall-shift axiom, a classically valid statement which implies the excluded middle over decidable formulas.  ...  Surprisingly, this logic is constructive and so far this has been proved by cut-elimination for ad-hoc sequent calculi.  ...  theorem for CD easily follows from Proposition 6.  ...

### Proof normalization for a first-order formulation of higher-order logic [chapter]

Gilles Dowek
1997 Lecture Notes in Computer Science
We show that the proof of the normalization theorem for the usual formulation of higher-order logic can be adapted to prove normalization for its rst-order formulation.  ...  Thus, from the point of view of proof normalization, de ning higher-order logic as a di erent logic or as a rst-order theory does not matter.  ...  The so called proof normalization theorem for rst-order logic is only a proof normalization theorem for the empty theory in rst-order logic.  ...

### An Axiomatization for Cylinder Computation Model [chapter]

Nan Zhang, Zhenhua Duan, Cong Tian
2014 Lecture Notes in Computer Science
To model and verify multi-core parallel programs, the paper proposes an axiom system for Propositional Projection Temporal Logic with Cylinder Computation Model (CCM-PPTL).  ...  Moreover, the axiom system of CCM-PPTL is established by extending that of PPTL with some axioms and inference rules of CCM operators.  ...  Further, to provide a highly automatical verification approach, the existing tool for PPTL theorem proving will be extended to support CCM operators.  ...

### Expanding the Realm of Systematic Proof Theory [chapter]

Agata Ciabattoni, Lutz Straßburger, Kazushige Terui
2009 Lecture Notes in Computer Science
This paper is part of a general project of developing a systematic and algebraic proof theory for nonclassical logics.  ...  Our method also works as a heuristic principle for finding appropriate rules for axioms located at levels higher than P 3 .  ...  Proof We have φ •− −• j k ( l ξ j,k,l ) 1 where ξ j,k,l is N 2 -normal, From N 2 -axioms to Sequent Rules In this section we provide an algorithm for transforming N 2 axioms into equivalent sequent calculus  ...

### An Arithmetical Interpretation of Verification and Intuitionistic Knowledge [chapter]

Tudor Protopopescu
2015 Lecture Notes in Computer Science
The BHK interpretation of intuitionistic logic has a precise formulation in the Logic of Proofs and its arithmetical semantics.  ...  We show here that this interpretation can be extended to the notion of verification upon which intuitionistic knowledge is based, thereby providing the systems of intuitionistic epistemic logic extended  ...  The question about exact interpretations for other intuitive readings of these logics is left for further investigation.  ...

### Classical Modal Display Logic in the Calculus of Structures and Minimal Cut-free Deep Inference Calculi for S5

R. Gore, A. Tiu
2007 Journal of Logic and Computation
Since every CMDL calculus enjoys cut-elimination, we obtain a cut-elimination theorem for all corresponding CoS calculi.  ...  We then show how our result leads to a minimal cut-free CoS calculus for modal logic S5. No other existing CoS calculi for S5 enjoy both these properties simultaneously.  ...  Acknowledgement We thank Kai Bru¨nnler and the anonymous referees for their comments on an earlier version of the article.  ...
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