Logical Preliminaries


Logical Preliminaries

Only classical logic, as opposed to Hegelian dialectic synthesis, is used in this book: (a) ß is ß, (b) ß is-not not and (c) α is either ß or not, 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 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 γß 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.

Theses and antitheses can be joined by two other logical operators called "and" and "or" to form conjunctions and disjunctions respectively.  If (x, y) represent two theses, two antitheses or a thesis and an antithesis, then the truth table for these logical operators is given by Table 1.

Table 1. Truth table for "and" and "or" logical operators

x y x and y x or y
1 1 1 1
1 0 0 1
0 1 0 1
0 0 0 0

Based on Table 1, not-(x and y) = not-x or not-y and, similarly, not-(x or y) = not-x and not-y.  The equal sign indicates equivalence; both sides have the same truth table.

If A and B represent logically combined theses and antitheses [e.g. A = x or (not-y and z); B = u and not-v], then A can be connected to B to form the sentence "if A then B" (conditional statement).  In such a statement, A and B may be different ways of stating exactly the same idea.  If so, the sentence "if A then B" is a type of tautology -- it is always true.  Conversely, if A and B express different ideas then the statement "if A then B" is regarded as true unless A is true and B is false.

"If A then B" can also be expressed as "B if A" and "A only if B."  The expression "A if and only if B" (biconditional statement) is true when both "if A then B" and its converse "if B then A" are true.  Finally, the statement "if A then B" and its contrapositive "if not-B then not-A" are equivalent as shown in Table 2.

Table 2.  Truth table for conditional, biconditional and contrapositive statements

A B if A then B A if and only if B if not-B then not-A
1 1 1 1 1
1 0 0 0 0
0 1 1 0 1
0 0 1 1 1

Although the formal concepts of negation, conjunction, disjunction, conditional and biconditional were not developed until the early 19th century, none of these concepts is counterintuitive and reasoning, based on them, is characteristic of historic documents such as the Christian Bible.