In this essay I will attempt to convey my findings about the use of Null sets within Quantificational Logic. By using notions set down by Aristotle, I will convince those of sound mind that this is the definitive logical method for the use of Null Sets in relation to Quantificational Logic. I will systematically explain this philosopher’s concepts and relate them all together and then display the only logical result from their combined wisdom. Firstly let’s talk of Aristotle’s system. Aristotle invented a system of using only 4 sentences to describe categorical propositions. In this system he uses the letter S to stand for the subject term, and the letter P stands for the predicate term. With that in mind the only propositions you can make in his logical system are: All S are P. Example: All humans are Mortal. No S are P. Example: No humans are Immortal. Some S are P. Example: Some humans are males. Some S are not P. Examples: Some humans are not males. These four propositional templates will be the foundation for understanding how subjects will be described in relation to Predicates. Insofar I have not mentioned Null Sets, and I would like to remedy this now. A “set” will be defined as: A Logical grouping of subjects by a definable parameter. It’s a fancy way of saying a group of things with a descriptor that links them all together. S is a T because X. Example: A hairless man is hairless because he has Alopecia Areata. The hairless man belongs to a set of hairless people because he has Alopecia Areata. In this case there is only 1 subject that belongs to our set, this would be “A hairless man”. If we wanted to have then one subject in our set would could say: Hairless men are hairless because they have Alopecia Areata. Then each hairless man found would considered a single subject, and then be added to our set of “hairless”. A null set is simply a set that has no possible subjects. One way to have a Null Set is to have to have a contradiction in the statement. What I mean by contradiction is a situation where the premises are true and the conclusion is false. In relevance to Sets, this means that of all the possible subjects none of them can fit into the set you have defined. Example: Hairy men are hairless because they have Alopecia Areata. Since hairy men are not hairless by definition they cannot belong to the set of “hairless”. Your set of subjects for hairless derived from Hairy men will be Null. The only other way to have a null set is to acquire your subjects from an empty (null) list of subjects to fill your set with. Example: Unicorns are horses with horns. There are no known Unicorns in the world, therefore when you list all the subject that would fit into this propositional statement, you would be pulling your results from an empty list and would therefore result in a Null set. In this example you are making an implied assertion that there are Unicorns. Or in more general terms that the Subject exists. Since it does not, the logical statement is once again a contradiction. You have the Subject existing, and not existing at the same time. Example: Let U = Existence of Unicorns. The statement implies you have Unicorns, there for you have U. But the empty Subject list implies you don’t have Unicorns. Premise 1 implied from statement: U Premise 2 implied from empty Subject list: ~U Therefore you have a contradiction: (U & ~U). In more general terms: Subject are predicate Let E= Existence of Subject. Premise 1 implied from statement: E Premise 2 implied from zero possible subjects: ~E (E & ~E) Contradiction. Therefore I assert that Null Sets in relation to Quantificational Logic is that Null sets cannot be used. As they might be logically atomic, they are not logically sound. Contradictions are not possible in the known Universe, and therefore should not be allowed into any system of Logic designed to understand that very Universe.
Posted on: Mon, 21 Oct 2013 21:00:31 +0000