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Journal of Logic, Language and Information

Erschienen in:

07.04.2017

What Makes an Effective Representation of Information: A Formal Account of Observational Advantages

verfasst von: Gem Stapleton, Mateja Jamnik, Atsushi Shimojima

Erschienen in: Journal of Logic, Language and Information | Ausgabe 2/2017

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Abstract

In order to effectively communicate information, the choice of representation is important. Ideally, a chosen representation will aid readers in making desired inferences. In this paper, we develop the theory of observation: what it means for one statement to be observable from another. Using observability, we give a formal characterization of the observational advantages of one representation of information over another. By considering observational advantages, people will be able to make better informed choices of representations of information. To demonstrate the benefit of observation and observational advantages, we apply these concepts to set theory and Euler diagrams. In particular, we can show that Euler diagrams have significant observational advantages over set theory. This formally justifies Larkin and Simon’s claim that “a diagram is (sometimes) worth ten thousand words”.

Vorheriger Artikel In All But Finitely Many Possible Worlds: Model-Theoretic Investigations on ‘Overwhelming Majority’ Default Conditionals

Nächster Artikel Language-Theoretic and Finite Relation Models for the (Full) Lambek Calculus

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By statement, we mean a syntactic entity (in any representation system) that represents some information. For example, a set-theoretic sentence is a single statement, and so is an Euler diagram.

By the term ‘set theory’ we mean a symbolic representation of sets, such as \(P\cap Q\), and the assertion of relationships between sets, such as \(P\cap Q\subseteq R\).

We use the term semantically entailed in the standard, model theoretic way: one set of statements, \(\varSigma \), semantically entails a single statement, \(\sigma \), if all the models for \(\varSigma \) are models for \(\sigma \).

From this point forward, we will not adopt the standard convention of omitting brackets where no semantic ambiguity arises, in order to avoid potential confusion as to the structures in set-theoretic sentences that are meaning carriers.

We use the term semantically equivalent in the standard, model theoretic way: two (sets of) statements are semantically equivalent if they have the same models.

Such situations can arise in application areas like ontology engineering, where teams of people devise statements in order to define a domain of interest. Team members may have their own preferred notation, such as description logic (Baader et al. 2003), whereas others may prefer visual approaches, such as VOWL (Lohmann et al. 2014) or concept diagrams (Stapleton et al. 2013). But if we can determine what is observable for each representation, and consequently what the potential observational advantages of each representation are, then we can better choose our appropriate representation.

Note that \(d_1\) and \(d_2\) are semantically equivalent because they are both true under the same circumstances. Formally, these diagrams have the same models, a concept made precise in the next section.

One may wonder if the partial overlap of the curves labelled P and Q in \(d_1\) is really ‘necessary’, since even if these curves stood in a different relationship (e.g., separated), the diagram could support \(\gamma \) and express the information that P union Q is a subset of R. However, any of such alternative relationships would make the diagram express something stronger than this information (e.g., that P and Q are disjoint). Having the partial overlap of the curves labelled P and Q is necessary in this sense—it is the only way to translate s to an Euler diagram without expressing anything stronger.

Another type of cost would arise if the diagram from which observation is made was not just given, but had to be constructed. Imagine how you would construct a diagram given the premises \((P \cup Q) \subseteq R\) and \(R \subseteq P\). Once a diagram is successfully constructed, it lets you observe interesting consequences such as \(P = Q\) and \(Q = R\), but the construction of the diagram must have required a significant effort on your part in figuring out how to enclose regions on a plane with curves to satisfy a set of containment conditions. Such a ‘construction cost’ would have to be weighed if we were to consider not just a given Euler diagram and a given set of set-theoretic sentences, but also their construction processes out of a set of premises. We thank one of our reviewers for the suggestion of this example.

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Titel: What Makes an Effective Representation of Information: A Formal Account of Observational Advantages
verfasst von: Gem Stapleton
Mateja Jamnik
Atsushi Shimojima
Publikationsdatum: 07.04.2017
Verlag: Springer Netherlands
Erschienen in: Journal of Logic, Language and Information / Ausgabe 2/2017
Print ISSN: 0925-8531
Elektronische ISSN: 1572-9583
DOI: https://doi.org/10.1007/s10849-017-9250-6

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