Does every claim assert that its subject class has members? If so, what is a claim’s truth value when its subject class is empty? For instance, are all claims about my daughters false if I have no daughters? If so, then the rules of contradictories and subcontraries fail, and any attempt to maintain the rule of contradictories succeeds only if we abandon contraries and subalternation along with subcontraries. Modern syllogistic logic, following George Boole, adopts a different interpretation of A and E statements to deal with this. Graphically represented in Venn diagrams, this interpretation provides a convenient way to determine the validity of three-term syllogistic arguments, but is still severely limited in its scope of application.
Some classes have no members. For instance, the class of unicorns, the class of round squares, and the class of my daughters are all null. This creates problems because we don’t always know whether the classes we are discussing are populated or not. We need a logical apparatus that can be relied on, either way.
Since any particular (I or O) claim about a null class clearly asserts that its subject class is populated, then they must all be false—for example, “Some of my daughters are blonde” and “Some of my daughters are not blonde.” But then, if we insist that the law of contradictions holds, their contradictory universals (A and E) must both be true—for example, “All of my daughters are blonde” and “None of my daughters is blonde.”
Modern syllogistic logic, following the 19th-century mathematician/logician George Boole, recognizes that particular (I and O) claims assert that their subject classes are populated but reads universal (A and E) claims differently, so as to preserve the law of contradiction.
· All S are P is read in obverse—No S are ~P, or S outside of P is null—which is clearly true when Some S are not P is false.
· No S are P is read straightforwardly as asserting that the intersection of S and P is null, which is clearly true when Some S are P is false.
· In this analysis, both of the “contraries” are true of a null class because they truly assert that certain sets are empty, and both of the “subcontraries” are false because they falsely assert something to exist that does not.
· Consequently, the rules of contraries, subcontraries, and subalternation disappear from the modern square of opposition, and a further syllogistic rule is established: No valid categorical syllogism can have a particular conclusion (I or O) unless it has at least one particular premise.
· A convenient way to represent categorical propositions, so interpreted, is in terms of null forms. Here, we represent the intersection or overlap of two classes by placing the class names side by side. Every class S has a complement, written ~S, and read “curl S” or “tilde S.” For example, the intersection of S and P is written SP, and the intersection of S and ~P is written S~P, and whether that intersection is populated or null is indicated by saying it is, or is not, equal to zero. Thus, “All S are P” can be represented with “S~P = 0,” which is read as “The intersection of S and non-P is empty” or “S outside of P is empty.”
Null forms help us work with Venn diagrams.
Venn diagrams provide a graphic way to test the validity of three-term syllogistic arguments, by shading out empty areas and placing an X in populated areas. We know that a syllogism is valid if, upon inspection, it is evident that diagramming its premises is all it takes to provide a complete diagram for its conclusion.
A valid AAA-1 syllogism.
Anything meritorious (M) is praiseworthy (P).
All scholarship winners (S) are meritorious (M).
Therefore, all scholarship winners (S) are praiseworthy (P).
A valid AII-1 syllogism.
Everyone who is meek (M) is polite (P).
Some sophomores (S) are meek (M).
Therefore, some sophomores (S) are polite (P).
An invalid OOO-1 syllogism.
Some moderates (M) are not politically savvy (P).
Some Senators (S) are not moderates (M).
Therefore, some Senators (S) are not politically savvy (P).
Using diagrams to show graphically that a categorical syllogism is valid—or that it’s not—was a wonderful advance over the more traditional ways of handling syllogisms but perfectly consistent with that ancient system. (Note to readers: Even with these embellishments, however, the logic of categorical syllogisms is still severely limited in its scope of application. It will not comfortably handle categorical arguments with more than three terms, and it does not reveal the relationship between syllogistic logic—which is part of a larger realm known as predicate logic—and sentential logic. Those gaps will be partly filled in Lectures Twenty through Twentytwo.)
[Courtesy: Professor James Hall]
[Courtesy: Professor James Hall]