Abstract

This paper argues that questions have an important role to to play in logic, both semantically and proof-theoretically. Semantically, we show that by generalizing the classical notion of entailment to questions, we can capture not only the standard relation of logical consequence, which holds between pieces of information, but also the relation of logical dependency, which holds between information types. Proof-theoretically, we show that questions may be used in inferences as placeholders for arbitrary information of a given type; by manipulating such placeholders, we may construct formal proofs of dependencies. Finally, we show that such proofs have a specific kind of constructive content: they do not just witness the existence of a certain dependency, but actually encode a method for transforming information of the types described by the assumptions into information of the type described by the conclusion.

Highlights

  • 1.1 A motivating exampleSuppose a certain disease may give rise to two symptoms, S1 and S2, the latter much more distressing than the former

  • The third purpose of the paper is to show that such proofs admit a constructive interpretation, similar to the proofs-as-programs interpretation of intuitionistic logic: they do not just witness the existence of a dependency, but encode a method for computing the dependency, i.e., a method for turning information of the type described by the assumptions into information of the type described by the conclusion

  • We have seen that classical logic can be given an alternative, informational semantics in terms of support conditions, which determines when a sentence is settled by a body of information, rather than when it is true at a world

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Summary

A motivating example

Suppose a certain disease may give rise to two symptoms, S1 and S2, the latter much more distressing than the former. Given the hospital’s protocol, whether or not the treatment is prescribed for a patient is determined by two things: (i) which symptoms the patient presents and (ii) whether the patient is in good physical condition This means that, in the given context, a certain relation holds between the following questions: μ1. We will say that the question ν is determined by the questions μ1 and μ2 in the given context, and we will refer to this relation as a dependency.1 This relation may be viewed as connecting three different types of information: given the hospital’s protocol, complete information about a patient’s symptoms, combined with information about whether the patient is in good condition, yields information about whether the treatment is prescribed. Using the questions as labels, we may say that information of type μ1, together with information of type μ2, yields information of type ν

Natural sciences
Linguistics
Databases
Aim and structure of the paper
Questions enter the stage
Pieces and types of information
Logical entailment
Entailment in context
From contextual to logical entailment
Internalizing entailment
Summing up
Questions in propositional logic
Propositional information states
Support semantics for classical propositional logic
Adding questions to propositional logic
Resolutions and inquisitive normal form
Entailment and propositional dependencies
Reasoning with questions
A natural deduction system for InqB
Conjunction
Implication
Inquisitive disjunction
Double negation elimination
On the role of questions in inference
Non entailment-directed approaches to questions
The Logic of Interrogation
Nelken and Shan’s modal approach
The modal translation of InqB
Parsimony
Insight
Logical operations
Inferences and computational interpretation of proofs
Dependence logic
Previous work on inquisitive semantics
Conclusion and further work
Full Text
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