Decision methods for linearly ordered Heyting algebras

Dyckhoff, Roy; Negri, Sara

doi:10.1007/s00153-005-0321-z

Cited by 11 publications

(5 citation statements)

References 29 publications

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“…Since then several other such calculi which employ ordinary sequents have been proposed (see [11,1,12,5,13]). All these calculi have the drawback of using some ad-hoc rules of a nonstandard form, in which several occurrences of connectives are involved.…”

Section: Introductionmentioning

confidence: 99%

A semantic proof of strong cut-admissibility for first-order Godel logic

Lahav¹,

Avron²

2012

Journal of Logic and Computation

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We provide a constructive direct semantic proof of the completeness of the cut-free part of the hypersequent calculus HIF for the standard first-order Gödel logic (thereby proving both completeness of the calculus for its standard semantics, and the admissibility of the cut rule in the full calculus). The results also apply to derivations from assumptions (or "non-logical axioms"), showing in particular that when the set of assumptions is closed under substitutions, then cuts can be confined to formulas occurring in the assumptions. The methods and results are then extended to handle the (Baaz) Delta connective as well. 2 Standard First-Order Gödel Logic Let L be a first-order language with ∧, ∨, ⊃ as binary connectives, ⊥ as a propositional constant, and ∀ and ∃ as unary quantifiers. We assume that the set of free variables and the set of bounded variables are disjoint (thus in a well-formed formula, the use of the bound variables is always in the scope of a quantification of the same variables). We use the metavariables a, b to range over the free variables, x to range over the bounded variables, p to range over the predicate symbols of L, c to range over its constant symbols, and f to range over its function symbols. The sets of L-terms and L-formulas are defined as usual, and are denoted by trm L and frm L , respectively. We mainly use t as a metavariable standing for L-terms, ϕ, ψ for L-formulas, Γ, ∆ for sets of L-formulas, and E, F for sets of L-formulas which are either singletons or empty. Given an L-term t, a free variable a, and another L-term t , we denote by t{t /a} the L-term obtained from t by replacing all occurrences of a by t. This notation is extended to formulas, set of formulas, etc. in the obvious way. Notation 1 To improve readability we use square parentheses in the metalanguage , and reserve round parentheses to the first-order language. As usual, there are two main approaches to define the standard first-order Gödel logic: Proof-theoretically The logic is defined using an Hilbert-style calculus. Such a calculus is obtained by adding the following axioms to any Hilbert-style calculus for first-order intuitionistic logic (see e.g. [7]): • The "linearity" axiom (ϕ ⊃ ψ) ∨ (ψ ⊃ ϕ). • The "quantifier-shifting" axiom (∀x(ϕ{x/a} ∨ ψ)) ⊃ (∀x(ϕ{x/a}) ∨ ψ). We shall denote such an Hilbert-style calculus by sG, and sG will stand for the consequence relation which is naturally associated with it (note that this is a relation between sets of formulas and formulas). Model-theoretically Here there are two options. First, the standard first-order Gödel logic can be defined as a first-order fuzzy logic based on the real interval [0, 1]. Second, it can be seen as an intermediate logic, defined in terms of Kripke-style semantics. In the next two subsections, we briefly review the many-valued semantics and the Kripke-style semantics of this logic. 2.1 Many-Valued Semantics Definition 2 An L-algebra is a pair D, I where D is a non-empty domain and I is an interpretation of constants and function symbols of L ...

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Section: Introductionmentioning

confidence: 99%

A semantic proof of strong cut-admissibility for first-order Godel logic

Lahav¹,

Avron²

2012

Journal of Logic and Computation

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“…and corresponding sequent calculus rules. Dyckhoff & Negri [6] developed this approach and gave a simple decision method, based on terminating proof search in a suitable sequent calculus, for the fragment of positively quantified formulae of the first-order theory of linearly ordered Heyting algebras. (Positively quantified formulae are those in which the universal quantifier occurs only in positive position, and the existential quantifier only in negative position.)…”

Section: C Lin1mentioning

confidence: 99%

From mathematical axioms to mathematical rules of proof: recent developments in proof analysis

Negri

Plato

2019

Phil. Trans. R. Soc. A.

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A short text in the hand of David Hilbert, discovered in Göttingen a century after it was written, shows that Hilbert had considered adding a 24th problem to his famous list of mathematical problems of the year 1900. The problem he had in mind was to find criteria for the simplicity of proofs and to develop a general theory of methods of proof in mathematics. In this paper, it is discussed to what extent proof theory has achieved the second of these aims. This article is part of the theme issue ‘The notion of ‘simple proof’ - Hilbert's 24th problem’.

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“…It can nevertheless be faced with sequent calculus by leaving the inequality between terms as atomic formulas in the logic and translating the axioms of the algebraic or ordered structure under consideration as rules that govern the behaviour of such relations, in the same way as transitivity or reflexivity were added as "nonlogical" sequent rules. The method has been employed to solve the uniform word problem for linear order (Negri, von Plato and Coquand, 2004), lattices von Plato, 2002, 2004), linear lattices (Negri, 2005a, and by a di↵erent route in section 7.2 of Negri and von Plato, 2011), groupoids (Negri and von Plato, 2011), linear Heyting algebras (Dyckho↵ and Negri, 2006), and ortholattices (Meinander, 2010).…”

Section: Back To Representations and Applications To Word Problemsmentioning

confidence: 99%

The intensional side of algebraic-topological representation theorems

Negri

2017

Synthese

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Stone representation theorems are a central ingredient in the metatheory of philosophical logics and are used to establish modal embedding results in a general but indirect and non-constructive way. Their use in logical embeddings will be reviewed and it will be shown how they can be circumvented in favour of direct and constructive arguments through the methods of analytic proof theory, and how the intensional part of the representation results can be recovered from the syntactic proof of those embeddings. Analytic methods will also be used to establish the embedding of subintuitionistic logics into the corresponding modal logics. Finally, proof-theoretic embeddings will be interpreted as a reduction of classes of word problems.

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Decision methods for linearly ordered Heyting algebras

Cited by 11 publications

References 29 publications

A semantic proof of strong cut-admissibility for first-order Godel logic

A semantic proof of strong cut-admissibility for first-order Godel logic

From mathematical axioms to mathematical rules of proof: recent developments in proof analysis

The intensional side of algebraic-topological representation theorems

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