Effective completeness theorems for modal logic

Ganguli, Suman

doi:10.1016/s0168-0072(03)00134-9

Cited by 4 publications

(12 citation statements)

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“…[Har98,EGNR98,Ers98a,Ers98b,Mil99] revisit many classical concepts and theorems in classical model theory under the light of computability. Extending this approach to various non-classical logics has been carried out, e.g., in [GN04].…”

Section: Introductionmentioning

confidence: 99%

Effectiveness in RPL, with applications to continuous logic

Didehvar

Ghasemloo

Pourmahdian

2010

Annals of Pure and Applied Logic

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In this paper, we introduce a foundation for computable model theory of rational Pavelka logic (an extension of Lukasiewicz logic) and continuous logic, and prove effective versions of some related theorems in model theory. We show how to reduce continuous logic to rational Pavelka logic. We also define notions of computability and decidability of a model for logics with computable, but uncountable, set of truth values; we show that provability degree of a formula with respect to a linear theory is computable, and use this to carry out an effective Henkin construction. Therefore, for any effectively given consistent linear theory in continuous logic, we effectively produce its decidable model. This is the best possible, since we show that the computable model theory of continuous logic is an extension of computable model theory of classical logic. We conclude with noting that the unique separable model of a separably categorical and computably axiomatizable theory (such as that of a probability space or an L p Banach lattice) is decidable.

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Section: Introductionmentioning

confidence: 99%

Effectiveness in RPL, with applications to continuous logic

Didehvar

Ghasemloo

Pourmahdian

2010

Annals of Pure and Applied Logic

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“…SQML lacks internal adjointness in the expanded language including ∃ ∞ . 2 Proof. Once again the proof is to construct a counterexample.…”

Section: 4mentioning

confidence: 99%

“…There are of course many philosophical issues here which we will not discuss. Gangulia and Nerode [2] have shown that every decidable SQML theory has a decidable Kripke model. Before proving Theorem 4.3, we will briefly describe the difference between our results and theirs.…”

Section: Question Does Proposition 311 Hold For the Finitary Existementioning

confidence: 99%

“…Let S be the collection of pairs ( , v) where v is a finite variable assignment and is a computable set of formulas such that: (1) is consistent, (2) mentions exactly the variables in the domain of v, (3) x = y or x = y for each pair of variables x, y in the domain of v (depending on whether v(x) = v(y) or v(x) = v(y)), and 4∃ ≥n x for each n. 6 Note that the formulas in may have free variables. We will use, repeatedly, the fact that if the variablesx do not appear freely in , then ϕ(x,ȳ) if and only if (∀x)ϕ(x,ȳ).…”

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confidence: 99%

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First-Order Possibility Models and Finitary Completeness Proofs

Harrison-Trainor

2019

The Review of Symbolic Logic

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This article builds on Humberstone’s idea of defining models of propositional modal logic where total possible worlds are replaced by partial possibilities. We follow a suggestion of Humberstone by introducing possibility models for quantified modal logic. We show that a simple quantified modal logic is sound and complete for our semantics. Although Holliday showed that for many propositional modal logics, it is possible to give a completeness proof using a canonical model construction where every possibility consists of finitely many formulas, we show that this is impossible to do in the first-order case. However, one can still construct a canonical model where every possibility consists of a computable set of formulas and thus still of finitely much information.

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“…Others are not concerned directly with the deduction theorem as such, but only as a means of proving some other properties of the systems under study. For example, Ganguli and Nerode (2004) are interested in effective completeness theorems. For systems of first-order modal logics these theorems state that there is a decidable model for every decidable theory.…”

Section: The Deduction Theorem In Modal Logicmentioning

confidence: 99%

Does the deduction theorem fail for modal logic?

Hakli

Negri

2011

Synthese

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Abstract.Various sources in the literature claim that the deduction theorem does not hold for normal modal or epistemic logic, whereas others present versions of the deduction theorem for several normal modal systems. It is shown here that the apparent problem arises from an objectionable notion of derivability from assumptions in an axiomatic system. When a traditional Hilbert-type system of axiomatic logic is generalized into a system for derivations from assumptions, the necessitation rule has to be modified in a way that restricts its use to cases in which the premiss does not depend on assumptions. This restriction is entirely analogous to the restriction of the rule of universal generalization of first-order logic. A necessitation rule with this restriction permits a proof of the deduction theorem in its usual formulation. Other suggestions presented in the literature to deal with the problem are reviewed, and the present solution is argued to be preferable to the other alternatives. A contractionand cut-free sequent calculus equivalent to the Hilbert system for basic modal logic shows the standard failure argument untenable by proving the underivability of 2A from A.

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Effective completeness theorems for modal logic

Cited by 4 publications

References 0 publications

Effectiveness in RPL, with applications to continuous logic

Effectiveness in RPL, with applications to continuous logic

First-Order Possibility Models and Finitary Completeness Proofs

Does the deduction theorem fail for modal logic?

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