Only classical logic, as opposed to Hegelian dialectic synthesis, is used in this book: (a) ß is ß, (b) ß is-not not-ß and (c) α is either ß or not-ß, where α and ß are nouns (persons, places or things). This can be stated another way by writing a sentence called the thesis (α is ß) and another sentence called the antithesis (α is not-ß).
Given any thesis and its antithesis, one is true, the other is false and the decision must be based on evidence.
The concept of thesis/antithesis can be illuminated using the language of probability theory. Consider a sample space comprising the set of all possible choices or outcomes. Each individual choice is called a sample point. If the sample space is discrete -- the sample points can be counted -- then ß is any one, but not more than one, of the sample points. not-ß is the complement of ß which contains all sample points except ß. The thesis (α is ß) means α corresponds to the sample point ß. The antithesis (α is not-ß) means α corresponds to one of the sample points in not-ß.
If the sample space is nondiscrete -- noncountably infinite -- then it has as many points as there are real numbers corresponding to the points on a line interval such as 0 <= x <= 1 denoted by γ. ß is any sub-interval of γ and not-ß corresponds to all the points on γ not contained in ß. The thesis (α is ß) means α corresponds to one of the points in ß. The antithesis (α is not-ß) means α corresponds to one of the points in not-ß.
This is the logic of absolutes and does not permit α to be a synthesis, which is neither, ß nor not-ß. Consider, for example, the thesis (my height is 1.70 to 1.71 meters) and its antithesis (my height is not-1.70 to 1.71 meters). One is true; the other is false. No third option is rational such as (my height is 1.70 to 1.71 meters for some but not-1.70 to 1.71 meters for others because "truth" is personal).
A thesis may be stated in variant formats but can always be converted to the form (α Ω ß) where Ω represents any tense of the verb "to be." For example, (Jesus did exist) can be converted to (Jesus was a man).
The antithetical expression (α is not-ß) means α corresponds to a member of a set that does not include ß. If not-ß contains more than one member, the one member that is equivalent to α is not specified. For example, if α = 1 and ß = 2, the thesis and antithesis become (1 is 2) and (1 is not-2). The antithesis states that 1 is a member of the set of all integers not including 2. This is a true statement but lacks the operational specificity of the analogous statements (1 is-not 2) and not-(1 is 2). From an operational standpoint, the expression (α Ω not-ß) can be more usefully rephrased as (α Ω-not ß) or not-(α Ω ß).
This can be illustrated, by analogy, using a programming language called C. In this language, the symbols ==, != and ! represent the "equal to" and "not equal to" relational operators and the "not" logical operator respectively. The expressions (α != ß) and !(α == ß) always evaluate to the same integer: one if true and zero if false. However, the expression (α == !ß) may not evaluate to that same integer because !ß is always assigned the value one if ß = 0 and zero otherwise. For example, if α = 1 and ß = 0 then (α != ß), !(α == ß) and (α == !ß) evaluate to 1, 1 and 1 respectively. However, if α = 1 and ß = 2, the three expressions evaluate to 1, 1 and 0. To eliminate ambiguity, the antithesis will be expressed as (α Ω-not ß) or not-(α Ω b) in the remainder of this book.
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