Mathematical Logic

Ebbinghaus, Heinz-Dieter; Flum, Jörg; Thomas, Wolfgang

doi:10.1007/978-1-4757-2355-7

Cited by 274 publications

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“…Note that, at this point, Lindström's (first) theorem [15] can be applied to show that higher-order logic with the Henkin semantics is essentially just a variant of first-order logic. (Informally, Lindström's theorem states that a logical system (satisfying some weak technical conditions) that is at least as strong as first-order logic and satisfies conditions corresponding to the compactness theorem and Löwenheim-Skolem theorem is equally strong as first-order logic; here, 'at least as strong' and 'equally strong' are technical, model-theoretic notions.)…”

Section: Discussionmentioning

confidence: 99%

Probabilistic modelling, inference and learning using logical theories

Lloyd

Uther

2008

Ann Math Artif Intell

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This paper provides a study of probabilistic modelling, inference and learning in a logic-based setting. We show how probability densities, being functions, can be represented and reasoned with naturally and directly in higher-order logic, an expressive formalism not unlike the (informal) everyday language of mathematics. We give efficient inference algorithms and illustrate the general approach with a diverse collection of applications. Some learning issues are also considered.

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Section: Discussionmentioning

confidence: 99%

Probabilistic modelling, inference and learning using logical theories

Lloyd

Uther

2008

Ann Math Artif Intell

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“…Very similar system for classical logic with ¬, ∨, ∃ and linear proofs contains [15]. The following differences should be noticed:…”

Section: γ ϕ Obtained By Substitution From γ ϕmentioning

confidence: 99%

“…15 In 40ties Suszko proposed in his Ph.D. thesis [61] (see also [60] and [62]) the original SC operating on intuitionistic sequents with sequences in the antecedents. In order to understand properly his motivations one should note that in his system sequents are seen as expressing inference rules and neither antecedent (a list of premises) nor succedent (a conclusion) may be empty.…”

Section: Suszko's Systemmentioning

confidence: 99%

A Survey of Nonstandard Sequent Calculi

Indrzejczak

2014

Stud Logica

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Abstract. The paper is a brief survey of some sequent calculi (SC) which do not follow strictly the shape of sequent calculus introduced by Gentzen. We propose the following rough classification of all SC: Systems which are based on some deviations from the ordinary notion of a sequent are called generalised; remaining ones are called ordinary. Among the latter we distinguish three types according to the proportion between the number of primitive sequents and rules. In particular, in one of these types, called Gentzen's type, we have a subtype of standard SC due to Gentzen. Hence by nonstandard ones we mean all these ordinary SC where other kinds of rules are applied than those admitted in standard Gentzen's sequent calculi. We describe briefly some of the most interesting or important nonstandard SC belonging to the three abovementioned types.Keywords: Sequent Calculi, Natural Deduction, Proof Methods. IntroductionThe notion of sequent calculus (SC) is commonly, and rightly, connected with the name of Gerhard Gentzen. Quite automatically, we think of calculi with structural and logical rules introducing constants either to antecedent or to succedent of a sequent. However, one should be aware that nowadays this term may be applied not only to such calculi like original Gentzen's LJ, LK or their variants, but also to a variety of calculi which use sequents in a significantly different way than they are used in Gentzen's like calculi. Some of them are even older than Gentzen's LK (or LJ) -like Hertz's calculus -but most were invented after Gentzen, mainly for providing more flexible or natural systems for actual proof search than original Gentzen's calculus constructed rather for theoretical purposes. In this paper we survey some of the most important or interesting proposals. The paper is based on the 10th chapter of Indrzejczak [31].Let us define an ordinary sequent calculus (SC) as a finite collection of (schemata) of (primitive) sequent rules of the form: Sequents on the left of / are premises of the rule, whereas S n+1 is its conclusion. Note that n, i.e. the number of premises, may be 0 -in this case we will say on (primitive) axiomatic sequent; in case n > 0 we will simply say on rules. Note that every SC have a nonempty set of primitive sequents and a nonempty set of rules 1 .So far, the only restriction we put on the notion of a sequent calculus is that rules have unique conclusions. The next restriction is connected with the very notion of a sequent. We define it in a standard way as an ordered pair Γ ⇒ Δ, where Γ is the antecedent and Δ the succedent of a sequent. Hence we exclude from our considerations such calculi which operate on sequents having more arguments 2 or being structures of different character 3 or using more types of sequents in one system 4 .One should also specify what kind of structures are denoted by arguments of a sequent. We allow sequences, multisets or sets of formulae but exclude from further considerations other kinds of structures. Hence we do consider neither display calculi ...

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“…(We speak here explicitly of function symbols and predicate symbols, which we call in the other sections also briefly functions and predicates.) Moreover, an interpretation I, β represents a conventional second-order interpretation (Ebbinghaus et al, 1984) (if predicate variables are considered as distinguished predicate symbols): The structure in the conventional sense corresponds to I, as described above, except that mappings of predicate variables are omitted. The assignment is β, extended such that all predicate variables p are mapped to { t 1 , ..., t n | +p(t 1 , ..., t n ) ∈ I}.…”

Section: Relation To Conventional Model Theorymentioning

confidence: 99%

Projection and scope-determined circumscription

Wernhard

2012

Journal of Symbolic Computation

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We develop a semantic framework that extends first-order logic by literal projection and a novel second semantically defined operator, raising, which is only slightly different from literal projection and can be used to define a generalization of parallel circumscription with varied predicates in a straightforward and compact way. We call this variant of circumscription scopedetermined, since like literal projection and raising its effects are controlled by a so-called scope, that is, a set of literals, as parameter. We work out formally a toolkit of propositions about projection, raising and circumscription and their interaction. It reveals some refinements of and new views on previously known properties. In particular, we apply it to show that wellfoundedness with respect to circumscription can be expressed in terms of projection, and that a characterization of the consequences of circumscribed propositional formulas in terms of literal projection can be generalized to first-order logic and expressed compactly in terms of new variants of the strongest necessary and weakest sufficient condition.

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Mathematical Logic

Cited by 274 publications

References 0 publications

Probabilistic modelling, inference and learning using logical theories

Probabilistic modelling, inference and learning using logical theories

A Survey of Nonstandard Sequent Calculi

Projection and scope-determined circumscription

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