Trilateral Diagrams

With the applet below you drag counters to one of the eligible positions on the trilateral diagram. This is where you formalize your premises. The conclusion is read off from the bilateral diagram on the right.

Some actions have an immediate effect. For example, a single red counter in the inner North-West square which corresponds to Some xym exist is immediately interpreted as Some xy exist. No interpretation for the trilateral diagram is provided in this case. The same happens in other cases too. For example, gray counters in the two NW cells (meaning No xym exist and No xym' exist) are not interpreted in the trilateral diagram while still leading to a meaningful configuration on the right - No xy exist.

There are two reasons for such an implementation. Firstly, many (actually most) configurations of the trilateral diagram have no meaning as premises of syllogisms (with m or m' as eliminands.) Secondly, the amount of interpretation on the left may overwhelming without having bearing on the conclusion; thus actually obscuring a valid syllogism when such one is present.

Configurations of the trilateral diagram meaningful in this context are collected in a special table.

Remark

Much as Venn diagrams, trilateral diagrams rely on our geometric intuition. When you place a red counter on the diagram you mean to make an assertion of existence. For who would doubt existence of a counter right in front of one's nose? When you say Some m are x the intention is usually to Some m exist, Some x exist, and, in addition, Some m are x. Strangely, this view point may not be logically viable. K.Devlin gives this example:

1. All green pigs are pigs. (m = "green pig", x = "pigs")
2. All green pigs are green. (m = "green pig", y = "green animal")

Therefore, Some x are y meaning that green pigs exist which may no be true. (Change the color to magenta to make the statement even less plausible.)

Formalism of Boolean Algebra easily detects this pitfall.

• Introduction
• Controversial Venn Diagrams
  • From K.Devlin, MATHEMATICS: The Science of Patterns
  • From W.Dunham, The Mathematical Universe
  • Venn Diagrams applet I
  • Venn Diagrams applet II
• Bilateral Diagrams
  • Interpretation of Bilateral diagrams
  • Bilateral diagrams applet
• Syllogisms
• Trilateral Diagrams
  • Interpretation of Trilateral diagrams
  • Trilateral diagrams applet
• Sample soriteses
• The Game