"The intersection of and is not empty", or . "Some is not " (AoB) is equivalent to " is not a subset of ", or . By definition, the empty set is a subset of all sets. From this fact it follows that, according to this mathematical convention, if there are no 's, then the statements "All is " and "No is " are always true whereas the statements "Some is " and "Some is not " are always false. This also implies that AaB does not entail AiB, and some of the syllogisms mentioned above are not valid when there are no 's (). See also Boole's syllogistic Free logic Logical cube Logical hexagon Octagon of Prophecies Triangle of opposition References External links International Congress on the Square of Opposition Special Issue of Logica Universalis Vol. 2 N. 1 (2008) on the Square of Opposition Catlogic: An open source computer script written in Ruby to construct, investigate, and compute categorical propositions and syllogisms Conceptual models Term logic Inference Syllogism