Apologetics
The Logical Minimum
By David Wiseman
In Euclidian geometry, there are numerous ways to prove the associations of angles. There are proofs for “Angle A is congruent to angle B,” “Angle A is supplementary to angle B,” and “Angle A is complimentary to angle B.” However, there is no proof for “Angle A exists.”
In any proof involving an angle, the assumption made to begin with is that angles exist. The question must at some point arise, why is it an angle? The natural answer that springs to the mind is, “Because two lines intersect,” which is the definition of an angle. How can the definition of a term be the proof that the term applies to a given figure? It may be true that the figure is, indeed, an angle, or the intersection of two lines, but how can it be proven?
The only valid proof is found in common sense, or instinct. It is the intuition that, “This figure is an angle.”
This idea applies to more than simply proving that a figure is an angle. How can it be proven that a point exists? How can it be proven that there is only one line that runs parallel to a given line through a given point? By this same intuition, naturally imbedded in the person. How can it be proven that one exists? It can only be proven by this intuition. How can it be proven that anything is there? By this same intuition that allows every person to function.
There is a given amount of information that must be assumed in order for logic to function. Without this given information, logic is only an illogical, existential leap that contradicts itself. It contradicts itself in that it does not reason everything into existence before it takes them for granted. This is because logic cannot prove everything.
In order to understand logic, it must be realized that there is a point where logic begins, and a point where it ends. The foundation on which logic is laid is this natural intuition of what things are. The givens of an angle begin as, “Given: Angle ABC…” Because how else is the one doing the proof to know that this is an angle, except that it is already given?
The human problem is that mankind is conscious. He is able to question this intuition. While a monkey will simply accept that the banana in front of him, and the tree that he is hanging from are really there, mankind insists on questioning these self-evident facts.
So there are two ways of looking at logic. Either logic is the basis of everything, and only those things exist which can be reasoned, as the philosophers say, or there are certain givens, and definite bounds within which logic must operate. Either one or the other must be right. Geometrically, “Either a statement is true, or it is not true.” Either logic is bounded, or logic is boundless. If logic is boundless, it ought to be able to justify itself. What proof is there of logic? This is where logic defeats itself. If logic is boundless, then assuming that logic works is a complete falsehood, because it cannot be reasoned.
What, then, establishes logic? Even given bounds within which it works, how can it be certain that logic is true? Logic must be one of these assumptions, one of these pieces of intuition. This means that logic is on the same level as the things that are assumed, and actually beneath, inferior to, intuition.
What, then, is assumed? Obviously, logic can’t prove what is assumed, since logic must be one of these assumptions. All that remains is to rely on intuition in order to reveal what must be assumed. Until reasoning enters in, is it assumed that the physical things are real?
However, there must be an amendment to the rule regarding illusions. Optical illusions are assumed to be real until investigation and logic prove them otherwise. There are other types of illusions, aswell. So what, exactly, is the place of logic in disproving illusions? Quite obviously, something is there. Quite obviously, some things, logic, and physical things at least, are factual because of intuition. Equally obviously, intuition cannot always be trusted. However, intuition is not on equal footing with logic, but superior to it, because intuition can prove itself. Intuition is intuited, thus establishing itself.
What remains in logic is a structure like a building, intuitive logic is the foundation, but there is a point at which intuition fails, and linear reasoning takes precedence. There is also a place for block reasoning, that is, knowing or proving one thing, and taking it as proof of related things.
Along with the distinction between the realms of reasoning and intuiting, there is overlap. Going back to Euclidian Geometry, in order to prove that an angle is obtuse, a proof must be used. The proof will vary based on the situation, but no matter what proof is used, it is a type of conscious, linear, reasoning. However, by simply looking at the angle, it becomes apparent that the angle is greater than ninety degrees. The logicians, mathematicians, and philosophers, will generally say that the latter proof is not as valid as the former. But if intuitive logic is actually more basic than linear logic, why is linear logic more valid in a given situation? Certainly, there are situations, such as illusions, where linear logic is generally stronger. But even then, with close examination, intuitive logic can prove illusions wrong.
In conclusion, the mere action of giving linear logic precedence over intuitive logic is contrary to linear, and intuitive, logic. Linear logic submits to intuitive logic, and to certain bounds within which it is set. Reasoning is not an independent fact, apart from all rules, except its own.
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