Month: August 2015

Transitivity of Evidence Revisited

entirelyuseless Probability

It was earlier shown that evidence in general is not transitive. The reason given was that if A is evidence for B, and B for C, A is not necessarily evidence for C, because the cases where A and B are the case can be disjoint from the cases where B and C are the case.

However, this does not prevent evidence from being transitive on particular occasions, and possibly on most occasions, namely whenever situations where A and B are the case are the same or mostly the same situations where B and C are the case.

For example, if someone speaks with me in a loud tone and with a red face, that is evidence that he is angry with me; and if he is angry with me, that is evidence that I have said or done something which offended him; and likewise speaking with me in a loud tone and with a red face is evidence that I have said or done something which offended him. In this case situations where A and B are the case, namely where someone is angry and speaking angrily with me, overlap a good deal with cases where B and C are the case, namely where the person is angry because I have offended him.

This is quite possibly the more common situation — namely that evidence is transitive in practice — and thus the likely reason why people tend mistakenly to conclude that evidence is always transitive.

Sola Me

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Just as the Protestant doctrine that the ultimate authority in matters of revealed truth is Scripture alone is referred to as sola Scriptura, the idea that I am personally the ultimate authority regarding the content of revelation could be called sola me (credit for this name goes to Michael Bolin.)

The idea that faith implies absolute subjective certitude turns out to result in such a claim of absolute personal authority. For example, suppose someone holds this doctrine regarding faith, and also happens to believe that it is a matter of faith that women cannot be ordained to the priesthood in the Catholic Church. Thus in effect he holds that the probability that women will be ordained in the Catholic Church is 0%. According to the rules of evidence, therefore, he can never admit that women have been ordained, no matter what evidence he observes, e.g. even he reads on the Vatican website that the Pope has ruled that Ordinatio Sacerdotalis was mistaken; he reads news reports about bishops all over the world ordaining women; he sees women celebrating Mass at his parish and at other parishes; and so on. If he ever admits that the Catholic Church has ordained women, then he would have to admit that he was mistaken in the first place to claim that he had absolute subjective certitude.

This does not imply that in such a case he would denounce the Catholic Church and lose his faith. On the contrary: someone who claims that he has absolute subjective certitude of such a thing is claiming that it is impossible for him to lose his faith, no matter what happens, not even if he observes such evidence. In practice, in a case like this, this would result in the person adopting sedevacantism or a similar position, namely one implying that his observed evidence is not really relevant to the Catholic Church, but to something else. The result is that the person in effect holds the doctrine of sola me: he claims to possess absolute authority to judge about what is revealed, and his personal authority, being absolute, must take precedence over everyone else in the world. Richard Ibranyi is a good example of the extremes to which this can be taken.

It could be objected that this only happens if the person is mistaken about what is revealed. If it is actually revealed by God that women cannot be ordained in the Catholic Church, then in fact they cannot be, and the evidence described will never be observed by anyone, and thus the bad consequences of this position can be avoided.

But this is the point. Something revealed by God cannot be wrong. This is objective certainty. But this is not the same as subjective certainty precisely because the individual can be mistaken about what is revealed. Thus Muslims are mistaken in believing that Islam is a divine revelation. Nor can it be said, for example,  that Muslims are wrong about what is revealed, but Catholics are not wrong, and consequently it is right for Catholics to claim absolute subjective certitude. For Catholics also disagree about what is revealed. Thus many Catholics claim that it is revealed in Genesis that human beings were not produced by a process of evolution, but were directly created by God, while other Catholics deny this. Thus Catholics (and likewise any other individual, holding any religion whatsoever, or no religion) can be wrong about what is revealed, and consequently cannot have absolute subjective certitude regarding such matters.

Someone could object again, essentially in a way suggesting that even in terms of internal disagreements, his own position is true and the position of others is false. Thus in regard to the creation of man, someone could say that it is a question of fidelity to the Magisterium, which has taught that the idea of human evolution is acceptable, given certain conditions. One problem with this is that it assumes that those who hold the direct creation of man to be divinely revealed are unfaithful to the Magisterium, and if they claim to be faithful to it, they must be lying. For if they are honestly trying to be faithful to the Magisterium, then such fidelity does not exclude being mistaken about what is revealed, and thus one cannot have absolute subjective certainty about the content of revelation based on such fidelity.

And so it goes on. Someone can continually add conditions, reasons why he himself is correct about what is revealed and why others are mistaken. But it is clear what is happening here: even if the person happens to agree with someone else about the entire contents of revelation, there is no guarantee that this will continue to happen. And if a disagreement ever comes up, the person will need to provide a reason why he is right and the other is wrong. And since he is claiming not only that he happens to be right, but that he is absolutely certain to be right, he holds the doctrine of sola me, just as much as Richard Ibranyi: he claims that he alone in the world is the ultimate judge of what is revealed by God.

Edward Feser on Naturalism

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Edward Feser, discussing David Hart on natural law, says, “For Darwinian naturalism, as Hart points out, gives us a view of the mind on which it floats entirely free of truth.  Any belief or argument whatsoever could seem absolutely indubitable even if it were completely wrong, if this were conducive to survival.” He takes this as an argument against Darwinian naturalism, which means that he thinks the claim, “Any belief or argument etc.” is either false or implausible.

It is not entirely clear why he thinks this, given that either he agrees, or at least does not disagree, with the biological theory of evolution. However, it may be that, holding that the intellect is immaterial, he believes that it is not subject to the process of natural selection. But this cannot be true. It is evident that whatever the exact relationship between the mind and the body, there is certainly some relationship, and the null hypothesis is basically always false. Consequently, whether or not the intellect is immaterial, there will be bodily causes that influence a person’s tendency to be certain or uncertain about things, with the result that the claim, “Any belief or argument whatsoever could seem absolutely indubitable etc.”, will surely have at least some truth.

It is also clearly true from experience. For example, in Muslim societies, most of the population are extremely convinced that Islam is true, even though this is completely wrong, but very conducive to survival, since even in the present day the death penalty continues to be used against apostates from Islam.

Obviously Islam has not existed long enough for natural selection to have much effect here, however, so in fact this particular case is probably part of a more general situation where agreeing with the people around is “conducive to survival”, both in the literal sense, and in the sense of producing economic and social advantages.

Nor does this imply that the mind “floats entirely free of truth”, since in most cases wrong beliefs about the world are harmful, and true beliefs helpful. If there is a pit of spikes in front of me and I believe that there is not, this is not conducive to survival at all. It does imply that the mind is not perfect and that there is a need to reflect on its work and frequently correct it. The possibility of self-reflection provides possibilities for progress in truth, even given the existence of such mental flaws.

100%

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Given that probability is a formalization of subjective degree of belief, it would reasonable to consider absolute subjective certainty to correspond to a probability of 100%. Likewise, being absolutely certain that something is false would correspond to assigning it a probability of 0%.

According to Bayes’ theorem, if something has a probability of 100%, that must remain unchanged no matter what evidence is observed, as long as that evidence has a finite probability of being observed. If the probability of the evidence being observed is 0%, then Bayes’ formula results in a division by zero. This happens because a probability of 0% should mean that it is impossible for this evidence to come up, and indicates that one was simply wrong to claim that there was no chance of this, and a different probability should have been assigned.

The fact that logical consistency requires a probability of 100% to remain permanently fixed, no matter what happens, implies that it is generally a bad idea to claim such certainty, even in cases where you have absolute objective certainty such as mathematical demonstration. Thus in the previously cited anecdote about prime numbers, if SquallMage claimed to be absolutely certain that 51 was a prime number, he should never admit that it is not, not even after dividing it by 3 and getting 17. Instead, he should claim that there is a mistake in the derivation showing that it is not prime. Since this is absurd, it follows that in fact he should never have assigned a 100% probability to the claim that the number was prime. And since there was subjectively probably not much difference between 41 and 51 for him at the time, with respect to the claim, neither should he have claimed a 100% probability that 41 was prime.

Belief and Probability

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As was argued in an earlier post, any belief, or at any rate almost any belief, is voluntary insofar as we choose to think, act, and speak as though it were true, and in theory it is always in our power to choose to behave in the opposite way. In practice of course we would not do this unless we had some motive to do it, just as the fact that it is in someone’s power to commit suicide does not make him do so before he has a motive for this.

Such a belief, since it involves affirmation or denial, has a basically binary character — thus I say either that the sun will rise tomorrow, or that it will not. This binary character can of course be somewhat modified by the explicit addition of various qualifiers, as when I say that “I will probably be alive five years from now”, or that “there is a 75% chance that Mike will come to visit next week” or the like. Nonetheless, even these statements have a binary character even if at one remove from the original statement. Thus I can say “Mike will come to visit,” or “Mike will not come to visit”, but also “there is a 75% chance etc” or “there is not a 75% chance etc”.

The interior apprehension of the mind, however, does not have the same binary character, but is more a matter of degree. Thus for example someone may argue that increasing the restrictions on gun ownership in the United States would be a good idea. “More gun control would be good,” he says. This is the affirmation of one side of a contradiction. Then suppose he is involved in a conversation on the matter, with another person arguing against his position. As the conversation goes on, he may continue to assert the same side of the contradiction, but he may grow somewhat doubtful inside. As he walks away from the conversation, he still believes that more gun control would be good. He still chooses to speak, think, and act in that way. But he is less convinced than he was at first. In this sense his interior apprehension has degrees in a way in which the belief considered as an affirmation or denial does not.

Probability theory is a mathematical formalization of such degrees of belief as interior apprehension, and the laws of probability are rules which must be followed in order to maintain logical consistency when working with such degrees. In another post we looked at Hume’s claim about the impossibility of induction. It is likely that one cause of his error was the fact that the formal theory of probability was less developed at the time; in particular, Bayes’ theorem, which we used against Hume, was proved 20 years after Hume’s treatise was written.

Thus, in our speech and behavior, beliefs are basically binary, but we possess various degrees of certainty about our beliefs. And such a degree is reasonably considered to be something like the probability, as far as we are concerned, that our belief is true.

Absolute Subjective Certainty

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Any conclusion at which one arrives after a long process of reasoning will have a possibility of error, just as I said about mathematical arguments. Consequently if we want to find some absolute subjective certainty, it will need to be something either without a process, or with a minimal one; something very basic.

First principles such as the principle of non-contradiction and the like are one possibility.

Another would be immediate apprehensions such as “I exist”, “I am thinking”, “I am awake”, and so on.

Still another would be facts of immediate experience such as “I am currently working on a blog post.”

We might come to a different result depending on whether or not we include the content of the assertion as well as the mode of apprehension. If the mode of apprehension is considered alone, “I exist,” cannot be said to have such absolute certitude if it is possible for someone to be certain in exactly the same way of the claim, “I do not exist,” and it seems to me that this might be possible. This is even more likely to be possible in the case of the claim about being awake; a waking person can believe that he is asleep and dreaming.

If we include the content of the assertion, this kind of certitude is not fully distinct from objective certitude, and there will surely be at least a few cases such as the claim about existence. Thus someone who believes that he exists, cannot fail to exist, not so much because of the certainty of his belief, but because having a belief includes existing.

External facts of immediate experience, such as the fact that I am currently writing this blog post, or that there is a computer on the table in front of me, are not infallible even including both the content and the condition of my certainty, since we are often wrong about such things, e.g. “I know my glasses were there on the table, I just saw them,” but they were not.

Overall the situation is not entirely clear. Absolute subjective certainty, including the content of the assertion as well as the mode, is possible, as in the case of the claim about existence, but it is not clear that it is possible if the content is not included. In any case, whether or not the content is included, such certainty could only refer to immediate apprehensions, and at most only to a few such apprehensions.

One other case which is sometimes proposed is the case of faith in divine revelation. Thus the Catechism of the Catholic Church states, “Faith is certain. It is more certain than all human knowledge because it is founded on the very word of God who cannot lie.” It seems likely that in reality this refers to objective certitude: one who believes something revealed by God cannot be wrong, because God does not reveal things that are false. Such an objective certitude would not imply absolute subjective certitude, for the same reasons that mathematical demonstration does not imply such a subjective certitude. However, some believe that the Catechism refers to an absolute subjective certitude. I will consider this idea in another post.

Against Airport Security

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Having been through airport security screening five times in the last six days, I am officially in favor of abolishing all security procedures and allowing all passengers to go directly to their planes.

Then, after that, we can see whether or not there are any bad consequences to such a policy. It is perfectly possible that there would be none.

If there are, we could give real consideration to the costs imposed on every single passenger by such procedures and determine whether or not it is worth imposing these costs in order to prevent a somewhat greater harm to a few.

Absolute Certainty

entirelyuseless Logic, Metaphysics, Probability ,

If I say that I am certain of something, this can mean that I personally do not have any doubt that it is true. Naturally, this does not ensure that the thing is in fact true. The fact that I do not doubt it, does not prevent it from being false, and people are frequently sure of such things.

But asserting that I am certain can also imply that the thing cannot fail to be true. As discussed in the previous post, this could mean that the thing cannot fail to be true on account of the objective nature of my conviction, or on account of its subjective nature.

As an example of the objective nature of the conviction, someone can say that he has a demonstrative argument for a conclusion, based on first principles. Given this kind of conviction, the thing cannot fail to be true, because something that actually follows from first principles will always be true. Thus I can prove in this way that 13 is a prime number. The objective nature of the conviction here is mathematical knowledge, and given that I have mathematical knowledge of a thing, the thing will always be true.

However, it would be either rare or impossible to have a subjective apprehension of my own knowledge such that I infallibly recognize my own possession of mathematical knowledge, and therefore can judge about the truth of the conclusion infallibly. I consider my knowledge and say, “This is a valid mathematical demonstration,” but my apprehension of this fact is not itself infallible. This was illustrated earlier with the example of a mathematician claiming that there is a flaw in a proof. If my apprehension of the fact that something is a demonstration is infallible, then I will know through this infallible knowledge that his claim is mistaken. But this does not happen in reality, and thus my knowledge is not subjectively infallible, not even when I have a valid mathematical demonstration for some conclusion.

To be continued…

Various Kinds of Necessity

entirelyuseless Logic, Metaphysics ,

(1) It is necessarily true to say that “tomorrow has not yet come.”

(2) Socrates is necessarily a human being.

(3) If something is a human being, it is necessarily rational.

(4) Given the objective nature of my conviction about the matter, it is necessarily true that 13 is prime.

(5) Given the subjective nature of my conviction about the matter, it is necessarily true that I exist.

Type (1) consists in the fact that a statement as formulated cannot fail to be true. Type (2) consists in the fact that given a certain thing, its nature requires certain other things. Type (3) is hypothetical necessity — given one thing, another thing follows of necessity. Types (4) and (5) are claims that I am necessarily right about certain things, either in virtue of the objective nature of my conviction, or on account of its subjective nature, i.e. that which I apprehend concerning my conviction.

These types of necessity are not entirely distinct and it is not surprising that one can often formulate one kind of necessity in terms of another. In particular, types (4) and (5) are particular cases of (3).

It is questionable whether type (5) exists in reality, and if it does, its scope is extremely limited, perhaps to claims such as the one in the example. I will discuss this in the next post.

Voluntary Beliefs

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Belief is a confusing mixture of the voluntary and the non-voluntary. St. Thomas tries to explain this by saying that the knowledge of first principles and of conclusions drawn from them are not voluntary, while other beliefs are voluntary:

Faith implies assent of the intellect to that which is believed. Now the intellect assents to a thing in two ways. First, through being moved to assent by its very object, which is known either by itself (as in the case of first principles, which are held by the habit of understanding), or through something else already known (as in the case of conclusions which are held by the habit of science). Secondly the intellect assents to something, not through being sufficiently moved to this assent by its proper object, but through an act of choice, whereby it turns voluntarily to one side rather than to the other: and if this be accompanied by doubt or fear of the opposite side, there will be opinion, while, if there be certainty and no fear of the other side, there will be faith. (ST. II-II Q. 1, a. 4, corp.

However, in reality it is highly questionable whether the intellect ever assents to anything of necessity in such a way that it is impossible to doubt it. For example, if a skilled mathematician shows you a proof of some new conclusion using premises which are well known to you, you may assent to the conclusion without any doubt. But if he then says, “There is a flaw in this proof,” you are unlikely to respond, “I cannot accept that claim, as I have seen beyond any possibility of doubt that this conclusion is true.” Or if you do respond in this way, suppose he says, “In that case, I will offer to bet you $1 against $100 that there is a flaw in the proof and that I can show it to you. Since you claim that you cannot be wrong, you are bound to win the bet and gain a dollar.” Hopefully in this situation you will come to your senses and doubt the proof, rather than continuing to assert your own infallibility, an act of insane overconfidence which will surely cost you $100.

St. Thomas does point out that the act of making a statement, whether to oneself or to others, is always a voluntary act. However, he does not seem to take note of the fact that this implies that the contrary act is also possible, even if it would violate first principles or conclusions drawn from them. For example, just as I can say or write “2 + 2 = 4” voluntarily, I can equally well say or write “2 + 2 = 5”, should I so choose, and I can say this not only to others but also to myself.

Someone will respond to this that if I do say this, whether to myself or to others, I am lying. Indeed. But how is this lying detected? The main way that we would determine that someone was lying in such a case is that he would refuse to act on this claim. As an obvious example, I might say that I want to change two twenty dollar bills for a fifty (when he has one), and I would expect him to refuse. This would actually be more obvious with statements like “it is raining.” If someone believes it is raining, he will probably carry an umbrella or something of the sort. The problem with the mathematical statement is that it is so basic that it is hardly clear what it would mean to act as though two and two made five. But given that we can determine how a person believing something would act, we can find evidence that someone is lying by showing that he refuses to act in conformity with his claim, at least in certain contexts.

This suggests that there is a sense in which any belief will be entirely voluntary: just as I can voluntarily choose to make any claim, I can voluntarily decide to conform the whole of my life to the claim. In this way I am deciding to act in every way as though the claim were true, both interior and exterior, insofar as these things are in my power.

Choosing to believe, in this sense, would be rather like choosing to love someone. Just as I choose to conform my life to love, I can choose to conform my life to belief. However, there is one significant difference. There is reason to believe that one who chooses to conform his life to love is disposing himself as fully as possible toward love, such that it will always be true that he does indeed love. But there may be reason to think that, at least in some cases, it is different with belief. For example, suppose that someone were to say, “I choose to believe that I can walk out of a fifth story window without getting hurt.” If he does choose this in the manner under discussion, he will be willing to put it into practice. But even if he does actually carry it out, it is possible that he is doing this to prove that he believes, while in fact still not truly giving intellectual assent. In other words, it may be that part of the disposition to believe is in the intellect itself, and that such a disposition must be present there in advance in order for the choice to believe to be efficacious.

If this is the case, we could say that someone who “chooses to believe” when his intellect entirely lacks the appropriate disposition is rather pretending to believe, in a sense, rather than truly believing. However, this is not completely clear and consequently it remains at least possible that such a choice to believe is always efficacious.