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Generalized quantification as substructural logic
TL;DR: In this article, the Herbrand approach is generalized to the case of generalized quantifiers by turning variables into structured objects, governed by various types of structural rules, interpreted semantically by means of a dependence relation.
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Abstract: We show how sequent calculi for some generalized quantifiers can be obtained by generalizing the Herbrand approach to ordinary first order proof theory. Typical of the Herbrand approach, as compared to plain sequent calculus, is increased control over relations of dependence between variables. In the case of generalized quantifiers, explicit attention to relations of dependence becomes indispensible for setting up proof systems. It is shown that this can be done by turning variables into structured objects, governed by various types of structural rules. These structured variables are interpreted semantically by means of a dependence relation. This relation is an analogue of the accessibility relation in modal logic. We then isolate a class of axioms for generalized quantifiers which correspond to first-order conditions on the dependence relation.
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Citations
Seeking explanations: abduction in logic, philosophy of science and artificial intelligence
Atocha Aliseda-Llera
- 17 Mar 1998
TL;DR: In this paper, a taxonomy for abductive reasoning from an observation to its possible explanations, from a logical point of view, is presented, and a more computational analysis of processes producing abductive inferences is presented in the framework of semantic tableaux.
150
Predicate Logics on Display
TL;DR: The paper provides a uniform Gentzen-style proof-theoretic framework for various subsystems of classical predicate logic and considers predicate logics obtained by adopting van Behthem's modal perspective on first-order logic.
14
The Craig Interpolation Theorem in abstract model theory
Jouko Väänänen,Jouko Väänänen +1 more
TL;DR: It is argued in this paper that the interpolation property is an important litmus test in abstract model theory for identifying “natural,” robust extensions of first order logic.
On a Decidable Generalized Quantifier Logic Corresponding to a Decidable Fragment of
First-Order Logic,Natasha Alechina +1 more
- 01 Jan 1995
TL;DR: In this article, a decidable translation from a language containing a general-ized quantifier Q into a first-order language enriched with a family of predicates is proposed, which is decidable using the method of analytic tableaux.
5
Conditional Quantification, or Poor Man's Probability
TL;DR: The purpose of this note is to take a few steps toward a first order logic of a particular type of incomplete information: incomplete knowledge about the value of a variable, where the ‘degree of incompleteness’ is given by some algebraic structure.
References
Model Theory: Index
Wilfrid Hodges
- 01 Jan 1993
TL;DR: A new structure is produced without the canonical base property, which is interpretable in Baudisch’s group, and is CM-trivial, and thus at the lowest possible level of the ample hierarchy.
1.4K
Completeness and Correspondence in the First and Second Order Semantics for Modal Logic
TL;DR: In this paper, the authors focus on normal modal logics and first order properties of the accessibility relation, and present three main themes, a positive, a negative, and a comparative.
401
Logic with the quantifier “there exist uncountably many”
TL;DR: In this article, it was shown that the completeness theorem for L(Q) holds with a very simple set of axiom schemes, which were later refined in [4] and [15] to give less complex axioms.
274
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