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Wednesday, May 13th, 2009

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Axioms and Illative Reasoning
oscillon writes in and comments: 

"It makes us uncomfortable to need axioms at the bottom of the pile. They are different than everything that lies above them and they look suspiciously like a cheat. We naturally want to break them down and figure out what they're made of, to reason beyond them. Your solution (and Lewis's) is to posit God. Ok, I can't say you're wrong. But that solution is just as far outside the system of logic above it than directly accepting them on faith."


Ah, friend, this is not a problem with God but a problem with axioms. You see, by their nature axioms cannot be deduced, since they are first principles that must be adopted before reasoning on a given topic begins.

However, axioms are open to other forms of reasoning. Axioms can be but are not necessarily a matter of faith. They can be approached via reasoning: it is merely deductive reason that is closed. Inductive reasoning or reductive reasoning will offer us generous room for logical conclusions concerning which axioms are true or must be true.

In addition to deduction, there is (1) inductive reasoning (2) hypothetical or reductive reasoning and (3) illative reasoning. All of these are valid means to achieve wisdom, if not apodictic certainty.

(1)    For example, I can use an inductive argument to show that other minds aside from mine exist in the universe, or, to phrase it another way, to show that solipsism is false. Induction does not lead to perfect, mathematical certainty, but wisdom does not rest on certainty, but on what is sound and sane. I see my inner self from within my own soul, with direct apperception of my own thoughts; and I acknowledge I have an external form that others can see and know. These others have an external form that I can see and know, and one of their forms includes speech and other signs of rationality, which cannot be explained except by recourse to the assumption or axiom that they also have an internal self. This is as valid an induction as the induction that the sun rises tomorrow.  It does not admit of the certainty that mathematicians crave because only those things that cannot be any other way possess that degree of certainty. There are things that are true because any other possibility is impossible, such as twice two is four. There are things that are true because the other possibilities are untrue, such as that the sun will rise in the east tomorrow. For discussions of astronomy, I can take the persistance of the motions of the sun and stars as a given.

(2)       For example, I can use a reductive argument (reductio ad absurdum) to show that “truth can be known” is axiomatic, for if I posit hypothetically that there is no truth, or no truth the human mind can reach, I am left with a paradox. The statement that there is no truth, if true, is false. To avoid that paradox, I must accept that (at least one) truth can be known. I must accept this as an axiom, for, without it, reasoning is not possible. For discussions of any kind, I can and must take the objectivity of truth as a given.

(3)    Illative reasoning is a term coined by Cardinal Newman to describe that act of induction or pattern-recognition that takes place when one idea draws together or satisfies several otherwise desperate strands of reasoning. For example, How do you know your dog loves you? You cannot be certain in a Cartesian sense or by deduction. The dog never says he loves you. But it is only a fool who cannot tell whether his dog loves him, and a bigger fool who says no master can ever tell whether his dog loves him.

The obvious answer is: "If my dog did not love me he would not act as he does." If you think about it, you will realize this is neither Cartesian deduction, nor inductive reasoning, nor an argument from reductio, nor is it an arbitrary assumption nor an axiom. It is a judgment that draws together and accounts for many facts and lines of reasoning that otherwise have no explanation, or none but an awkward and unconvincing explanation.

I personally am astonished and aghast that this type of reasoning has never before been identified, or never identified correctly, since it is the primary form of reasoning and argument used in everyday life to confirm everyday matters of judgment.

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