I think that your first point is related to Gem’s comment, and my reply is similar. In a formal sense, you are right, no real diagrams conform to our idealized specification. However, in practice a teacher would assess a diagram as correct if it were “close enough” to the desired diagram, incorrect if it is outside of tolerance.

The main point of our paper is that it is possible to imagine a collection of representations systems all being used with a single diagram. Once concerning the actual lengths of all of the bars of a bar chart, another concerning the relative lengths of the bars, a third concerning only the first bar, and so on. We show how to formalize this idea, in terms of representation systems that are “derived” from the basic underlying point-by-point semantics. So as you say, reader approach the diagram with different issues in mind, and the representation system that they choose to use is determined by those issues. If they can choose a representation system which makes the diagram true for their purposes, then they can interpret the diagram appropriately.

Thanks again for your comment.

I have a question about the definition and continuity of “falseness”. For example, we often take data about students’ self-constructed diagrams (e.g., hand-drawn diagrams). In such cases, almost all diagrams might be categorized as “false diagrams”. In such situations, “correct” diagrams might not be generated. Is this ok as far as your definition is concerned? Even among the self-constructed diagrams, some diagrams are more erroneous as they don’t convey important information (e.g., trends) correctly. If considering this, the falseness might be conceptualized as a continuum.

Also how much the diagrams are “erroneous” might be affected by people’s purpose in reading the diagrams. For example, if a person tries to understand the trends, then a diagram might be a bit false, but if a person tries to understand the exact values, then the diagram might be much more of a false diagram. Falseness is not simply decided based on the features of diagrams, but it will be decided according to the purpose of readers.

I am sorry that I did not read all of your paper carefully, so if I am commenting on something you already thought about, I apologize. But I am happy to know whether those issues I raise above are considered in your theory or not. Again, thank you very much for your very interesting presentation and ideas.

In our conception, a bar chart whose bars should be of lengths 2,3,4, but which are in fact of lengths 2.01, 3.001, 4.05, is false in every bar (but still true in trend data). It is likely then that there are few or no accurate representations in our idealized system, since they all have to be realized on imperfect printing equipment, displays etc. In practice however, people will consider these as accurate diagrams. If we were modeling this, we would do as you suggest and adopt a semantics that considers a bar as accurate if its actual length approximates to some tolerance the intended length.

Our paper, however, concerns discrepancies between the assigned semantics and the realized diagram. Even with a formal semantics that is tolerant of small discrepancies, there is the issue of bars whose lengths are outside the tolerance that is built into the semantics. These are the diagrams that we consider inaccurate in this paper.

Your question concerning the accuracy with which the assigned semantics captures the represented object is an interesting one. With the notion of “tolerance” you just suggested, it might be possible to compare two representation systems with different tolerances as being more or less “accurate”. I think that this is a cognitive question as much as a formal one. How much tolerance do people have to highly tolerant semantic mappings (how useful can such systems be)?

Dave (not speaking for Atsushi)

That is
6:15am Wednesday (Japan)
10:15pm (London)
5:15pm (New York)
2:15pm (California)

Here’s the zoom link
https://stanford.zoom.us/my/dave.barkerplummer?pwd=MTBSeHA2bUpLYnRHdFN3TDJhekVyQT09

Please drop by to chat about the paper, or just to hang out.

Is the notion of whether a diagram is “false” determined by (a) its assigned semantics or (b) by the degree to which the assigned semantics correspond to an object that the diagram is intended to represent? I appreciate in some cases the assigned semantics may be defined by directly appealing to the object being represented, but need not be in general. Could one measure the strength of a diagram in terms of how accurately it’s assigned semantics corresponds to a particular object that it may represent?