Observing the Truth: Diagrams, Sets and Free
                       Rides

                                    Gem Stapleton

                                University of Brighton,
                                     Brighton, UK
                             g.e.stapleton@brighton.ac.uk

    There are many different notations that can define relationships between sets,
some of which are diagrammatic and others symbolic. Even when a notation
is selected, there are choices to be made between semantically equivalent, yet
syntactically different, statements. Syntactic choices include variations in both
abstract syntax and concrete syntax where graphical and topological properties
can differ. Whilst it is clearly important to understand the relative benefits of
choices in all senses, the focus here is on the choice of notation and, within that,
abstract syntax choices1 .




                            Fig. 1: Illustrating free rides.


    This importance of notation choice has been explored previously, through
the study of free rides [4]. To illustrate, suppose we are given the set-theory
statements P ∩ Q = ∅, R ∩ Q = ∅ and S ⊆ Q. We can translate these three
statements into the Euler diagram in figure 1. As a consequence, the diagram
automatically expresses P ∩S = ∅ and R∩S = ∅: we can see these two statements
are true ‘for free’. By contrast, one cannot read off P ∩ S = ∅ and R ∩ S = ∅
from P ∩ Q = ∅, R ∩ Q = ∅ and S ⊆ Q; instead, the first two statements must be
inferred. Therefore, the diagram has a perceived advantage over the set-theory
statements, with P ∩ S = ∅ and R ∩ S = ∅ being examples of free rides.
    Free rides are thought to be a major reason why diagrammatic notations
can outperform symbolic notations in reasoning tasks. In particular, free rides
indicate why we may wish to choose Euler diagrams over symbolic set-theory
as a notation for representing relationships between sets. A key requirement
that underpins the theory of free rides is the existence of a translation from one
set of statements, such as symbolic statements about sets, into another set of
1
    Readers interested in concrete syntax choices for Euler diagrams or linear diagrams
    are referred to [1–3].
statements, such as a set of (perhaps a single) Euler diagrams. Such a translation
essentially chooses a representation, at the abstract syntax level, of the originally
defined statements in the other notation. Here, this idea is generalized to the
notion of an observational advantage that does not require the existence of a
translation between notations. Instead, all that is required is for the two sets of
statements to be semantically equivalent.
    To illustrate, suppose we have the two set-theory statements P = Q and
P ∩ R = ∅. The two Euler diagrams d1 and d2 in figure 2 are, between them,
semantically equivalent to P = Q and P ∩ R = ∅:

                      D = {d1 , d2 } ≡ S = {P = Q, Q ∩ R = ∅}.

We can observe P ∩ R = ∅ from S because it is written explicitly in S. However,
we cannot observe P ∩R = ∅ from either d1 and d2 . In this case, from D we must
infer d, which directly expresses P ∩ R = ∅. Therefore, S has an observational
advantage over D. Moreover, D also has an observational advantage over S, since
we can observe Q ∩ R = ∅ from D but not S.




                           Fig. 2: A more complex example.


    Through defining what it means to be able to observe one statement from a
set of statements, we can precisely describe the observational advantages of one
set of statements over another. In particular, in the case of Euler diagrams and
set theory we can prove that Euler diagrams are observationally complete. That is
given a finite set of set-theory statements, S, there exists a single Euler diagram,
d, that is semantically equivalent to S such that any set-theory statement that
can be inferred from S can be observed from d [5]2 . Thus, it is possible to
characterise, formally, the observational advantages of Euler diagrams over set-
theory. Importantly, the theory of observational advantages allows us to explain
why diagrams are advantageous representations of sets in a more general way
than free rides.
Acknowledgement This extended abstract is based on research conducted col-
laboratively with Mateja Jamnik and Atsushi Shimojima [5, 6].

2
    We note that d must explicitly represent all sets occurring in S and that the inferred
    set-theory statements must only represent sets occurring in S; we refer the reader
    to [5, 6] for more details.
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