The Square of Opposition is a diagram showing the relationships between the four forms of proposition used in the traditional syllogism. These forms and their relationships can be dated back to Aristotle from around 400 BCE. There was a significant resurgence of interest in the Square of Opposition in the mid-19th century and it appears in numerous logic text books, which were prominent around the time Dodgson was developing his ideas on teaching logic.

Dodgson’s views on the precise meaning of propositions, particularly regarding the existence of the subjects of the propositions, differ from the traditional approach, invalidating parts of the Square of Opposition.

Moreover, the current interpretation of propositions removes the validity of all but two of the relationships, making the Square of Opposition of little practical use today.

Four interpretations of the Square of Opposition are presented here:

Traditional: showing how propositions and their relationships have traditionally been interpreted.

Dodgsonian: showing those parts of the traditonal square which remain valid under Dodgson’s interpretation.

Dodgson-Richards: shows how a variant Square of Opposition can be constructed in such a way that it allows for Dodgson’s interpretations whilst retaining the traditional concept of the diagram.

Current: shows what is left of the traditional Square of Opposition if the modern interpretations of propositions are adopted.

All four versions are interactive in that the viewer may click on any of the boxes containing propositions or the links between them to see further explanations. This serves as a useful way to learn about the relationships between propositions but, more importantly, it brings out the fundamental way in which Dodgson’s work on the syllogism differs from both the traditional and the current approaches to the study.