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 isimpossible, 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.
It is, indeed, the type of reasoning used by lawyers to demonstrate the innocence or guilt of their cases to a jury. No lawyer ever performs what a mathematician or a scientist would call a proof or an experiment before a jury. Lawyers do not rely on empirical evidence presented to the sense impressions of the jury: they never show the crime or the tort to the juror’s eyes. All physical evidence is merely an ‘exhibit’, something shown to the jury to establish one of several strands of reasoning that the jurors will draw together into a coherent picture of the crime scene. All testimony is an exhibition of another kind, one where the honesty and the powers of observation of the testifiers are also open to question.
I do not know if you have seen the movie TWELVE ANGRY MEN, but it is an engaging dramatization of jury deliberation. The whole drama revolves around illative reasoning: that is, each juror imagines a picture or ‘narrative’ of the crime, and tries to fit the available evidence, and his own sense of how human nature and the world works, into their narrative. Where the evidence cannot be fit into the narrative, it causes a jar or shock of doubt.
Let me use an example from that movie for those of you who have seen it. A switchblade knife is found at the crime scene, apparently the murder weapon. The defendant testified that he had such a blade of that same make and model, but that it fell from a hole in his pocket. One juror says it is “impossible” that the murderer could find such a knife that looked so much the same. Henry Fonda dramatically stands up and throws a knife of the exact same make and model into the table top. He says he bought it over his lunch hour while he went for a walk.
Theforensic evidence is that the murder blow was an overhand blow (the blade below the pinky). One of the jurors, played by Jack Klugman, is a kid from a low-class and violent neighborhood, and he is familiar with such knives. He says that switchblades are designed to be held underhand (the blade above the thumb and forefinger), and opened with a thumb-switch. If you were striking a foe, you would have to click open the knife, and then turn the knife 180 degrees in your grip in order to deliver an overhand blow, which no one (Klugman says) who knows how to use a switchblade would do.
Now, dear reader, contemplate what goes on in the jurors’ minds in these scenes when they weigh and balance the evidence. It is often called “probabilistic” reasoning, but no name could be more misleading. There is no statistical enumeration of probabilities going on. No juror says “There is a 45.778% chance that the knife is question resembles the murder weapon” and then when Henry Fonda produces a matching knife, circuits in the juror’s brain click over, and he says, “Revised estimate: there is a 76.433% chance that the knife in question resembles the murder weapon” – that is completely ridiculous.
When Henry Fonda produces a matching knife, the “probability” that the defendant’s knife is not the murder weapon does not go up or down. Probability has to do with how many times out of a hundred repeated tests under the same circumstances a given result obtains. Here, there are no repeated tests, and hence no probabilities. We, in English, use the word “probability” not only to refer to probability in the scientific sense, we also use the word as a metaphor to refer to sound judgment. My judgment that one knife looks exactly like the murder weapon depends, among other things, on how often I have seen such a knife. When Henry Fonda throws a knife of the exact same make and model into the tabletop and testifies that he found one easily, my judgment about how many such knives there might be must adapt to the new information.
Likewise, Jack Klugman, when he says switchblades are designed to be used underhand says nothing about probability. He does not say that normal knives are held overhand 50% of the time and switchblades held overhand 12.5% of the time. Even if he did, this evidence would be of no use to the jurors if they did not apply illative reasoning powers, for they could not know that this one case was not one of those rare 12.5% of the time cases. Probability has nothing to do with anything here. What is being judged is how many threads of reasoning can be accounted for with a given narrative, and how weighty each thread is.
What the jurors are actually doing is using their illative reasoning powers to see which narrative can account in sound judgment for all the known facts. They draw together many separate chains of reasoning, and the jurors do not rest until one narrative that matches with their experience can account for them all.
To use another example from the movie, one juror asks Henry Fonda why the defense attorney did not raise the questions he is raising. Fonda says perhaps the defense attorney was young and overworked. Now, this is neither empirical reasoning nor deductive reasoning on the part of Henry Fonda’s character. This type of reasoning is often dismissed and scoffed at by philosophers as being uncertain and worthless—and it may be worthless for the goals sought by philosophers. Nonetheless, this is the most common form of reasoning, used by you and me in everyday life, and used for everything except philosophical deductions. It is illative reasoning. Fonda is offering a model of the universe, a narrative, that would explain what is otherwise be a discontinuity or a jar or shock. The jurors are shocked if the defense attorney did not raise points to defend his client that seemed obvious in hindsight; and that shock must be explained in one of two ways: either Fonda’s ideas are outrageous and the defense attorney did not raise them because they are unsound; or Fonda’s ideas are sound, and the defense attorney did not raise them due to incompetence.
This is illative reasoning because this thread of reasoning (which deals with the soundness of the judgment of the defense attorney’s case) cannot be resolved merely on its own merits. The jurors have no information. They are trapped in the jury room the way you are trapped in your skull. They have no information about the outside world aside from the exhibits offered by the court, and their own native wisdom and experience, in the same way you, trapped in your skull, have only your sense impressions and innate human nature to use to deduce a narrative of what is happening outside.
To use an example from the movie of incorrect illative reasoning, one juror whose own son betrayed his love thinks of young kids as ungrateful punks. His experience is that kids are punks and his narrative is that punks should be punished. Likewise, one juror is a race bigot, and he reasons that since the defendant is a minority, and that all minorities are scoundrels, the defendant must likewise be a scoundrel.
A philosopher would dismiss both these arguments by calling them overbroad generalizations. Merely because one kid is a punk does not necessarily mean all kids are punks. Merely because one minority is a crook does not necessarily mean all minorities are crooks. However, that same philosopher would also have to dismiss Fonda’s argument and Klugman’s and on the same basis. Merely because Fonda found a matching knife does not necessarily mean that the defendant’s knife was or was not the murder weapon. Merely because Klugman would use a switchblade underhand does not mean the defendant did or did not use the switchblade overhand. The philosopher must dismiss both the bigot’s argument and the sound arguments because neither is apodictically certain.
You see? The philosopher is on the wrong track. Nay, the philosopher is useless. He is attempting to apply to a law case the type of reasoning that can only rightly be applies to abstract concepts and mathematical objects.
The reason why the argument that the kid is a punk or the wogs are scoundrels is a bad argument is NOT because this argument fails to achieve apodictic certainty: the reason why it is a bad argument is that the arguments do not explain or draw together in correct illative reasoning that other pieces of evidence or chains of reasoning that need to be explained. There is nothing wrong with the type of inductive reasoning that says, for example, blacks are more likely to be criminals than white. From a statistical point of view, that is correct. Anyone who has ever decided to walk around rather than through a “bad section” of Chicago, where whites are not welcome, performs, whether he admits it or not, this kind of illative reasoning. The problem here is that the defendant’s age and race do not weigh heavily against the other lines or threads of reasoning. The problem here is that bigotry is at work: the jurors in question are exaggerating the importance of a minor factor. It is a distortion of judgment, not a abrogation of logic.
Having said all this, we now can see, dear readers, that a scientist or mathematician or Cartesian skeptic who applied to the deliberations of a jury their means and method would not merely be useless, they would be comically counterproductive.
The moment the jury room door was closed, the Cartesian skeptic would say: “Come now! The door is closed, and I do not see the prisoner. How do I know the defendant actually exists? Perhaps the entire universe outside this room winked out of existence the moment the door shut. If so, weare under no duty to try the defendant, nor can we come to any firm conclusions!”
The empirical scientists is even more worthless. His art is to measure physical events and physical objects and to deduce possible first principles to account for physical events. Anything that cannot be reduced to a measurement of duration, extension, mass and energy is outside his field. When asked to determine the guilt or innocence of the accused, all he can do is blink like an owl and say that guilt and innocence are undefined terms. Guilt has no mass nor length. Innocence has no vector, no spin value, no inertia. It cannot be measured with a stopwatch and a yardstick.
For some reason, the modern age has fallen into the sick delusion that only Cartesian reasoning and empirical reasoning exist, and that no other form of reasoning is valid: all other forms of reasoning are not reasoning at all, but merely arbitrary prejudice. This delusion is so complete, that Cardinal Newman had to invent a new term to describe something we do everyday, to distinguish it from the type of formal reasoning that only scholars do only once or twice in their lives (If that. How often have you been convinced by a formally logical syllogism of a proposition you did not on other grounds accept as wise and wholesome?)
If you notice the action of your conscience, you will notice that all moral reasoning is illative reasoning. You look at what you have done and you see whether or not it fits logically with a narrative or model of the universe. If you think the rules that apply to you apply to other people, it fits. If you think that a special exception exists just for you, this causes a jar or a shock. No principle of logic is offended. No principle of empirical reasoning is abridged. Nonetheless, you find yourself accepting the Golden Rule (or some form of it) because you cannot perform any act of moral reasoning at all without affirming that the moral universe must be consistent with itself, which means, a narrative with no shocks or jars to explain away.
No one bothers to argue with a solipsist. If the solipsist were sincere, he would not bother to argue with any of the automatons he believes people his universe. How can you talk to someone who thinks you do not exist? How can you listen to someone you think is an automaton? But, from a purely Cartesian or Empirical point of view, solipsism is just as logically valid as sanity. We could, after all, be living in The Matrix, with our brains being used as batteries to power evil machines, could we not? It is not strictly impossible in the logical sense. However, it jars against the narrative or model of the universe our illative sense obtains for us. Every other sense impression and line of reasoning has to be explained away with some sort of special pleading or contrived excuse.
A moral solipsist is one who believes other people exist, but that he and he alone just so happen to be immune from the Golden Rule and the laws of morality. Now, out of all possible universes, a universe where one and only one being was immune from moral law is no more impossible than a universe where one solipsist is real and everyone else is an automaton. But it shocks our sense that various chains of reasoning must fit together harmoniously to explain why a universe could exist where one and only one being is immune from moral law, why this happens to be that universe, and why I happen to be that man. If I cannot offer a sound and sane explanation, my illative sense, and not my empirical nor my Cartesian deductions, must assume axiomatically that moral law is one and the same for me as for others.
Please note that moral relativists are merely collectivists of moral solipsism. Their argument is that there is no moral law, or, rather, that so-called moral law is a tissue of assumptions and arbitrary judgments into which certain tribes and races stumble blindly, and that ergo no one tribe has any ground to criticize the folk ways of any other tribe. This stupid argument would not and could not be taken seriously except by a people who fall into the trap of assuming that every form of reasoning outside empirical reasoning or Cartesian deduction is arbitrary or false. Naturally, the same illative reasoning that dismisses solipsism dismisses moral relativism. Naturally, moral relativism also cannot pass muster with reductive reasoning: if there is no shared moral law between tribes, then it is not morally wrong for me and my tribe to criticize, judge, condemn, conquer, or exterminate another tribe. If you say that I “ought not” to condemn the folk ways of aliens, I ask on what ground you make that moral judgment? If you make it on the ground of the moral rules of my tribe, of Christendom, then by the same authority I condemn polygamy and pederasty and genocide and misogyny of the heathens. If you make it on the moral ground of some other tribe, of Marxist Progressivism or Socialist Utopianism, I need only announce that your tribe and its rules do not apply to me, and you must retract the condemnation. Your tribal rules dictate that you must tolerate my diverse cultural differences, including my intolerance; whereas my tribal rules dictate that I condemn the devil and all his works, including the works of darkness of you and your tribe.
God is not, in the Christian scheme, introduced merely as some sort of make-shift or jury-rig to lend more credit to otherwise shaky axioms. Knowledge of God in general comes because of a revelation from God. However, the way we distinguish true revelation from false, or true revelation from mere opinion, the way we select which of competing authorities to trust for those things we must accept on authority: and this is by means of illative reasoning.
In a conversation with a friendly Objectivist, the disciple of that noble if incomplete version of Capitalist Aristotelianism could agree with me on many moral and aesthetic judgments. We agree human nature is sacred. However, certain judgments which in his system of thinking were arbitrary assumptions, in my system are deductions from an organic whole, a vision of glory, as well as affirmations from reason, from authority, from tradition, and from divine revelation. I have multiple reasons for believing the human nature to be divine and sacred, whereas my Objectivist ally, limited as he is to certain fixed axioms of Cartesian deduction, can only assert this arbitrarily.
To him, the axiom of human sanctity is like Playfair’s axiom. If you grant the axiom, you can deduce the Objectivist system in the same way that Playfair can deduce the Euclidean system. If you deny it, you deduce something more like Libertarianism, in the same way denying Playfair’s axiom allows you to deduce Lobachevskian or Riemannian geometry.
His assumption is correct, but (in his system) is arbitrary: it stands or falls by itself. It can be backed up with certain arguments, to be sure, but those supporting arguments rest on axioms less self-evident than the self-evident statement that human nature necessitates human dignity.
I am not merely making the less self-evident argument that human nature is sacred because God is real. I am claiming that belief in God makes more sense of more disparate strands of reasoning than the contrary assumption, and that this “narrative” is more coherent than the atheist or agnostic narrative, which cannot even explain the obvious, much less the subtle.
You can argue for objective moral law without God, but the argument is not obvious. You can argue for the axiom that the universe is rational without God, but the argument is arbitrary. You can argue for the meaningfulness of human life and human history without God, but the argument is awkward, and veers alarmingly toward being merely an apologetic for a make-shift or pseudo-God, as with Marxist personifications oftheir goddess called Historical Dialectic. You can argue for the reconciliation of empirical cause and effect and human free will without God, but the arguments are so complex and subtle that even hardened philosophers blench. You can argue for the innate dignity of women without God, but that argument is risible. Without God, you can argue that slavery is economically not feasible, but you cannot argue that it is an abomination without making an arbitrary assumption.
You can argue that life consists of nothing but a mechanical process, that life is a bitch and then you die, and when you die you are dust and worm food, and all the tears of your loved ones are merely saltwater of no particular significance; and you can argue at the same time that human life and human freedom is sacred and precious, so that those who kill, maim, despoil and enslave others are an abomination not to be endured. But, without God, you cannot reconciled those two conclusions without a jar or a shock. It is shocking to call a man a meat machine built by blind nature for no purpose, but then say his life is sacred. Perhaps there are ways around this jarring note, but they are awkward.
To use another movie example, when Neo, that puny Jesus of the third Matrix movie, is asked by Agent Smith why life is worth living and truth and justice and the American Way is worth fighting and dying to preserve, Neo (since he is a modern new-Age new type of post modern fellow—hence his name) cannot answer that life is sacred since Man is made in the image of God, and endowed with certain inalienable rights. Neo, being modern, cannot answer that we hold these truths to be self-evident. Modern men are too cool and hep to believe in truth or to consider evidence. Truth and evidence is for squares! (We space hippies call such creatures of reason and natural reason “Herbert”).
No, indeed, the only thing neo enough for way cool hepcat Neo to announce as his reason for living and dying is “Because I chose to!”
This is the manifesto of nihilism:
Nothing means anything except by force of the personal willpower of my wonderful all-important wonderful self-actualized self-expressing selfhood. Hurrah, me! A meaningless meat machine whose death means nothing can be meaningful to me because, as the God of my personal cosmos inside my own tiny skull, I let is be so decreed by fiat. Life is sacred because ME SAID SO!
The line is delivered with stirring crescendo is background music, and I assume the movie director wanted the audience to cheer. I giggled in embarrassment and immediately wrote a check to Agent Smith for President political action committee.
Nihilist defiance is one way of smoothing out the jar between thinking human life is nothing but a collection of atoms blown together like a sand-dune by the blind winds for a brief moment, long enough to admire the stars in their elfin beauty, before the wind changes quarter, and you are dispersed like you never were; and thinking human life is sacrosanct.
But it is a way that my judgment (my illative sense) condemns as absurd and illogical.
My judgment condemns nihilist defiance of The Abyss because it is meaning to defy something that cannot notice you. It does nothing and changes nothing. Neo cannot make life, truth and justice sacred in an indifferent cosmos merely by an act of will. If he could, Agent Smith could, with an equal act of will, make death, lies and injustice sacred instead.
You cannot make the meaningless have ultimate and sacred meaning merely because you say so. Reality is not plastic. You do not have a magic ring, and even if you grit your teeth and are unafraid, you cannot change reality. Neverneverland is not an option.
Oddly enough, this is the exact same argument the agnosticsand Nihilists use to dismiss religion. The difference is, what they call my wishful thinking posits that I am recognizing a reality that exists outside myself, and that these properties are true whether I wish it or not. What I call their wishful thinking posits that no reality exists outside themselves, and their reality would wink out like a snuffed candle if they were run over by a streetcar. But if the word “reality” does not mean that which wishful thinking cannot change, then the word has no meaning.
To cleave to God is an act of faith, or, if you prefer, a gift of divine grace. It is like the sun, at which we cannot stare lest our eyes be dazzled: an inexplicable mystery. However, this mystery casts a light that shows all other things in sharp relief, so that all lesser mysteries are resolved. We know whence we come and where we are going. We know the meaning of life and why things are as they are and not some other way.
It may be a hard thing to accept. Let us not mince words: to believe God became Man and suffered and died for our sins and rose again drawing all men to Him is ridiculous, scandalous, and foolishness. And yet, if we accept this foolishness, everything else in life becomes not foolish at all, not absurd, not meaningless. Life makes sense, suffering can be endured, the grave not move us to terror, the poor and weak not move us to contempt. We will have something to say to Agent Smith, or to the Accuser who accused Job, if he demands of us to account for ourselves and our life: and what we say will be clear and piercing as a trumpet blast, as quiet and significant as a whispered promise from a lover, not some mere garble of postmodern nonsense, the caw of a proud and ugly crow. We will speak the truth because it is true, not because mere wishing makes it so.
About John C Wright
John C. Wright is a practicing philosopher, a retired attorney, newspaperman, and newspaper editor, and a published author of science fiction. Once a Houyhnhnm, he was expelled from the august ranks of purely rational beings when he fell in love; but retains an honorary title.
