The Square of Opposition

The Square of Opposition

We place the four types of categorical sentences at the four corners of a square and we connected opposite corners with diagonal lines. We call the resultant figure the "Square of Opposition". There are several things we can note about the categorical sentences as they are placed in the square. Notice first that the categorical sentence on the left side of the square are affirmative, while those on the right are negative. The categorical sentence at the top of the square are universal while those at the bottom are particular:

Using the Venn diagrams, we can also determine the semantical relations among the various categorical sentences. In particular, we can show that categorical sentences which are at opposite ends of the diagonal lines are contradictories. They always have "opposite" truth values. For example, if "All S are P" is true, then "Some S are not P" must be false. The Venn diagrams show us that this is the case. If the A-categorical sentence is true, then, according to the Venn diagram, the area of the S circle which is outside of the P circle must be empty. If this is so, then the O categorical sentence cannot be true, since it is only true when the area of the S circle which is outside the P circle is not empty. 

Similar reasoning shows that when an E categorical sentence is true, the corresponding I categorical sentence must be false, and vice versa.