Proceedings of the 13 ESSLLI Student Session - Multiple Choices ...

Proceedings of the 13 th ESSLLI Student Session
Figure 2: An Euler diagram.
inside A and C but outside B, and so forth. The region outside of all contours is also a
zone. Shading within a zone asserts the emptiness of the set represented by that zone. So,
the shading of the diagram in figure 2 asserts A ∩ B = ∅ and A − C = ∅.
Reasoning is carried out by the application of rules which transform one diagram into
another, such as Add Contour and Remove Shading; a sound and complete set is given in
(Stapleton, Masthoff, Flower, Fish and Southern, 2007). A proof using Euler diagrams is
formed by applying these rules repeatedly to transform an initial diagram (the premise)
into the target diagram (the conclusion); figure 3 shows a short example. The Add Shaded
Zone rule is applied to transform d1 to d2. A new shaded zone can be added at any
time since both a shaded zone and a missing zone assert the emptiness of the represented
set; both d1 and d2 state that A and B are disjoint. The Add Contour rule is applied
to transform d2 to d3. The new contour C intersects all existing zones without changing
their shading. Since this operation introduces no new shading and the way that C is added
ensures that no missing zones are created, d2 and d3 have the same meaning.
Figure 3: An Euler diagram proof.
The diagrams are formalised using an abstract syntax. The abstraction of Euler diagrams
that we present here is obtained from (Stapleton et al., 2007). Each zone is represented
as a tuple of the set of labels of contours that the zone is inside and the set of
labels of contours the zone is outside. For example, in diagram d1, figure 3, the only zone
inside A has the abstraction ({A}, {B}). Diagrams are represented as a tuple of the set
of labels (L), the set of zones (Z) and the set of shaded zones (Z ∗ ). Thus, diagram d2 in
figure 3 has abstraction:
〈L = {A, B}, Z = {({A}, {B}), ({B}, {A}), ({A, B}, ∅), (∅, {A, B})}, Z ∗ = {({A, B}, ∅)}〉
There are a number of logics that extend this system of Euler diagrams, including spider
diagrams (Howse, Stapleton and Taylor, 2005) and the constraint diagrams mentioned
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