4.5a: The Traditional Square of Opposition (Aristotelian Standpoint)

Traditional Square of Opposition

An arrangement of lines that illustrates logically necessary relations among the four kinds of categorical propositions. - because the ___ standpoint recognizes the additional factor of Existential Import, the ___ square supports more inferences than does the modern square. - for the ___ Square of Opposition to apply, the ___ standpoint must be adopted, and the propositions to which it is applied must assert something about actually existing things.

Contradictory Relation

If two propositions are related in this way, they have opposite truth value. This is the definition of ___ ___. - 2 Important Rules: "assuming that we always use the ___ ___ first, if one of the remaining relations yields a logically undetermined truth value, the others will as well. The other rule is that whenever one statement turns out to have logically undetermined truth value, its ___ will also. Thus, statements having logically undetermined truth value will always occur in pairs, at opposite ends of diagonals on the square." - this definition is for the Traditional Square of Opposition, not the Modern Square of Opposition.

Subcontrary Relation

Like the Contrary Relation, this also expresses a kind of partial opposition. - if a certain I proposition is given as false, the corresponding O proposition is true (because at least one must be true), and if an O proposition is given as false, the corresponding I proposition is true. But if either an I or an O proposition is given as true, then the corresponding proposition could be either true or false without violating the "at least one is true" rule...the I and O propositions cannot both be false, but they can both be true.

Contrary Relation

This differs from the Contradictory Relation in that it expresses only partial opposition. - if a certain A proposition is given as true, the corresponding E proposition is false (because at least onn must be false), and if an E proposition is given as true, the corresponding A proposition is false. But if an A proposition is given as false, the corresponding E proposition could be either true or false without violating the "at least one is false" rule. In this case, the E proposition has logically undetermined truth value. Similarly, if an E proposition is given as false, the corresponding A proposition has logically undetermined truth value...the A and E propositions cannot both be true, however, they can both be false. - important: "whenever one statement turns out to have logically undetermined truth value, its contradictory will also." - is a part of the Traditional Square of Opposition

Subalternate Relation

This relation is represented by two arrows: a downward arrow marked with the letter T (true), and an upward arrow marked with an F (false). These arrows can be thought of as pipelines through which truth values "flow." The downward arrow "transmits" ONLY truth, and the upward arrow ONLY falsity. Thus, if an A proposition i given as true, the corresponding I proposition is true also, and if an I proposition is given as false, the corresponding A proposition is false. But if an A proposition is given as false, this truth value cannot be transmitted downward, so the corresponding I proposition will have logically undetermined truth value...Analogous reasoning prevails for the subalternation relation between the E and O propositions. To remember the direction of the arrows for ___ ___, imagine that truth "trickles down," and falsity "floats" up.