Logic

One and Many

entirelyuseless Logic, Metaphysics

“Many” has two meanings:

  1. That which is divided, namely something and something else such that the something is not the something else. Taken in this way, the idea of many comes before the idea of one, because many in this sense is simply defined by distinction.
  2. A whole composed of ones as parts. In this sense many comes after one.

Using the second definition, we can define numbers according to what sort of parts they have. Thus for example two is something many in the second way, such that it does not have any part which is itself many. Similarly, three is something many such that it has a part which is two, but does not have any part which has a part which is two. One can define other numbers in a similar way. Of course such definitions will quickly become nearly unintelligible as one increases the value of the number. This is not so much a problem with this kind of definition, as a sign of the fact that numbers are not very intelligible to us in themselves, and that we grasp them in practice mainly by the use of the imagination.

Whole and Part

entirelyuseless Logic, Metaphysics

To have a whole made of parts requires at least three things: the whole, one part, and another part.

The whole must be distinct from each of the parts, i.e. it must not be one of the parts, since if the whole were the part, the part would not be a part, but the whole. Likewise each of the parts must be distinct from one another.

On other other hand, if there was no other relationship between the parts and the whole, we would simply be talking about three unrelated things. In order for the parts to be parts, they must be something of the whole. The part thus expresses something of the existence of the whole, a mode of its existence, but not its existence overall. Thus for example body and soul are aspects of a man, but neither is the man overall.

In order to be a whole, therefore, a thing must exist in three ways: as itself, as something which it is not, and as something else which it is not. Each of these three must be distinct from the other two.

The whole is greater than the part insofar as the whole exists not only as that part (which of itself would cause equality), but also as itself and as the other part.

Being and Unity

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The unity of a being is simply a certain negation of distinction or division. As was said in the last post, distinction consists in the fact that this thing is not that thing. To say that a thing is one is to say that it is “this thing” rather than “this thing which is not that thing, and that thing which is not this thing”. Thus saying that the thing is one does not deny all distinction, since “this thing” remains “not that thing.” But it denies the distinction within “this thing and that thing,” since this is not one thing.

Or with a concrete example, if I am talking about an apple and an orange, the apple is not the orange, and the orange is not the apple. By reason of this mutual distinction, “the apple and the orange” does not constitute something one. But the apple is one precisely because it is not something like this; “apple” does not name a distinct something and something else. Likewise the orange is one, for the same reason, despite the fact that the apple is not the orange.

St. Thomas explains that it follows that one and being are in some way the same:

“One” does not add any reality to “being”; but is only a negation of division; for “one” means undivided “being.” This is the very reason why “one” is the same as “being.” Now every being is either simple or compound. But what is simple is undivided, both actually and potentially. Whereas what is compound, has not being whilst its parts are divided, but after they make up and compose it. Hence it is manifest that the being of anything consists in undivision; and hence it is that everything guards its unity as it guards its being.

In other words, if you cut the apple into two halves, there is no longer an apple, but one half and another half. And just as you no longer have an apple, you no longer have the being that you had, since that being was an apple. The apple is always one apple; and one apple is always an apple. In this way being and unity are convertible.

On the other hand, just as there are many ways of being, there are many ways to be one. Thus although the two halves are not an apple, and consequently not one apple, they are a pair of apple halves. And being a pair of something is being something at least in some way; and consequently they are also one pair.

Real Distinction

entirelyuseless Logic, Metaphysics

To say that two things are really distinct simply means that one thing is not the other. Thus a chair and a desk are really distinct simply because the chair is not the desk.

This argues against the formal distinction of Duns Scotus, because if we take two things which are said to be formally distinct in his sense, either the first is not the second, or the first is the second. If the first is not the second, they are really distinct, and there is no need for his formal distinction. If the first is the second, then they are really the same, and it is enough to speak of a distinction in concept.

However, it should be noted that in another sense, no distinction is real. The distinction consists in the fact that one thing is not another, and “not being another” is not a being, but a lack of being. Thus distinction is always something generated by the mind when understanding reality, rather than something that exists in itself.

Nonetheless, “this concept differs from that one” differs from “this is not that,” and consequently we distinguish between conceptual distinctions and real distinctions in this way, even though in themselves all distinctions are conceptual.

The Unexpected Hanging

entirelyuseless Logic

Wikipedia tells the tale of the unexpected hanging:

A judge tells a condemned prisoner that he will be hanged at noon on one weekday in the following week but that the execution will be a surprise to the prisoner. He will not know the day of the hanging until the executioner knocks on his cell door at noon that day.

Having reflected on his sentence, the prisoner draws the conclusion that he will escape from the hanging. His reasoning is in several parts. He begins by concluding that the “surprise hanging” can’t be on Friday, as if he hasn’t been hanged by Thursday, there is only one day left – and so it won’t be a surprise if he’s hanged on Friday. Since the judge’s sentence stipulated that the hanging would be a surprise to him, he concludes it cannot occur on Friday.

He then reasons that the surprise hanging cannot be on Thursday either, because Friday has already been eliminated and if he hasn’t been hanged by Wednesday night, the hanging must occur on Thursday, making a Thursday hanging not a surprise either. By similar reasoning he concludes that the hanging can also not occur on Wednesday, Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur at all.

The next week, the executioner knocks on the prisoner’s door at noon on Wednesday — which, despite all the above, was an utter surprise to him. Everything the judge said came true.

Doubtless there are various ways to explain what is going on here. But the moral of the story is simply that no matter how solid your reasoning seems to you, no matter how absolutely conclusive, reality does not have to care. You can be wrong anyway.

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

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(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.

More on Induction

entirelyuseless Logic, Probability, Science ,

Using the argument in the previous post, we could argue that the probability that “every human being is less than 10 feet tall” must increase every time we see another human being less than 10 feet tall, since the probability of this evidence (“the next human being I see will be less than 10 feet tall”), given the hypothesis, is 100%.

On the other hand, if tomorrow we come upon a human being 9 feet 11 inches tall, in reality our subjective probability that there is a 10 foot tall human being will increase, not decrease. So is there something wrong with the math here? Or with our intuitions?

In fact, the problem is neither with the math nor with the intuitions. Given that every human being is less than 10 feet tall, the probability that “the next human being I see will be less than 10 feet tall” is indeed 100%, but the probability that “there is a human being 9 feet 11 inches tall” is definitely not 100%, but much lower. So the math here updates on a single aspect of our evidence, while our intuition is taking more of the evidence into account.

But this math seems to work because we are trying to induce a universal which includes the evidence: if every human being is less than 10 feet tall, so is each individual. Suppose instead we try to go from one particular to another: I see a black crow today. Does it become more probable that a crow I see tomorrow will also be black? We know from the above reasoning that it becomes more probable that all crows are black, and one might suppose that it therefore follows that it becomes more probable that the next crow I see will be black. But this does not follow, since this would be attempting to apply transitivity to evidence. The probability of “I see a black crow today”, given that “I see a black crow tomorrow,” is certainly not 100%, and so the probability of seeing a black crow tomorrow, given that I see one today, may increase or decrease depending on our prior probability distribution – no necessary conclusion can be drawn.

On the other hand, we would not want in any case to draw such a necessary conclusion: even in practice we don’t always update our estimate in the same direction in such cases. If we know there is only one white marble in a bucket, and many black ones, then when we draw the white marble, we become very sure the next draw will not be white. Note however that this depends on knowing something about the contents of the bucket, namely that there is only one white marble. If we are completely ignorant about the contents of the bucket, then we form universal hypotheses about the contents based on the draws we have seen. And such hypotheses do indeed increase in probability when they are confirmed, as was shown in the previous post.

Hume’s Error on Induction

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David Hume is well known for having argued that it is impossible to find reasonable grounds for induction:

Our foregoing method of reasoning will easily convince us, that there can be no demonstrative arguments to prove, that those instances, of which we have had no experience, resemble those, of which we have had experience. We can at least conceive a change in the course of nature; which sufficiently proves, that such a change is not absolutely impossible. To form a clear idea of any thing, is an undeniable argument for its possibility, and is alone a refutation of any pretended demonstration against it.

Probability, as it discovers not the relations of ideas, considered as such, but only those of objects, must in some respects be founded on the impressions of our memory and senses, and in some respects on our ideas. Were there no mixture of any impression in our probable reasonings, the conclusion would be entirely chimerical: And were there no mixture of ideas, the action of the mind, in observing the relation, would, properly speaking, be sensation, not reasoning. ‘Tis therefore necessary, that in all probable reasonings there be something present to the mind, either seen or remembered; and that from this we infer something connected with it, which is not seen nor remembered.

The only connection or relation of objects, which can lead us beyond the immediate impressions of our memory and senses, is that of cause and effect; and that because ’tis the only one, on which we can found a just inference from one object to another. The idea of cause and effect is derived from experience, which informs us, that such particular objects, in all past instances, have been constantly conjoined with each other: And as an object similar to one of these is supposed to be immediately present in its impression, we thence presume on the existence of one similar to its usual attendant. According to this account of things, which is, I think, in every point unquestionable, probability is founded on the presumption of a resemblance betwixt those objects, of which we have had experience, and those, of which we have had none; and therefore ’tis impossible this presumption can arise from probability. The same principle cannot be both the cause and effect of another; and this is, perhaps, the only proposition concerning that relation, which is either intuitively or demonstratively certain.

Should any one think to elude this argument; and without determining whether our reasoning on this subject be derived from demonstration or probability, pretend that all conclusions from causes and effects are built on solid reasoning: I can only desire, that this reasoning may be produced, in order to be exposed to our examination.

You cannot prove that the sun will rise tomorrow, Hume says; nor can you prove that it is probable. Either way, you cannot prove it without assuming that the future will necessarily be like the past, or that the future will probably be like the past, and since you have not yet experienced the future, you have no reason to believe these things.

Hume is mistaken, and this can be demonstrated mathematically with the theory of probability, unless Hume asserts that he is absolutely certain that future will definitely not be like the past; that he is absolutely certain that the world is about to explode into static, or something of the kind.

Suppose we consider the statement S, “The sun will rise every day for at least the next 10,000 days,” assigning it a probability p of 1%. Then suppose we are given evidence E, namely that the sun rises tomorrow. Let us suppose the prior probability of E is 50% — we did not know if the future was going to be like the past, so in order not to be biased we assigned each possibility a 50% chance. It might rise or it might not. Now let’s suppose that it rises the next morning. We now have some new evidence for S. What is our updated probability? According to Bayes’ theorem, our new probability will be:

P(S|E) = P(E|S)P(S)/P(E) = p/P(E) = 2%, because given that the sun will rise every day for the next 10,000 days, it will certainly rise tomorrow. So our new probability is greater than the original p. It is easy enough to show that if the sun continues to rise for many more days, the probability of S will soon rise to 99% and higher. This is left as an exercise to the reader. Note that none of this process depends upon assuming that the future will be like the past, or that the future will probably be like the past. The only way out for Hume is to say that the probability of S is either 0 or infinitesimal; in order to reject this argument, he must assert that he is absolutely certain that the sun will not continue to rise for a long time, and in general that he is absolutely certain that the future will resemble the past in no way.

Evidence and Implication

entirelyuseless Logic, Probability

Evidence and logical implication can be compared; we can say that logical implication is conclusive evidence, or that evidence is a sort of weak implication.

Evidence is commutative. If A is evidence for B, B is evidence for A. But logical implication is not; if A implies B, B does not necessarily imply A. However, even in the case of implication we can say that if A implies B, B is evidence for A.

Implication is transitive. If A implies B, and B implies C, then A implies C. We might be tempted to think that evidence will be transitive as well, so that if A is evidence for B, and B is evidence for C, A will be evidence for C. But this is not necessarily the case; this sort of thinking can lead to believing that the evidence can change sides. Attempting to make evidence transitive is like trying to draw a conclusion from a syllogism without any universal terms; if A is evidence for B, then some B cases are A cases, but not necessarily all of them; if every B case were an A case, then A would imply B, not merely be evidence for it. So if A is evidence for B, and B for C, then some C cases are B cases, and some B cases are A cases; but we cannot conclude that any C cases are A cases. A and C may very well be entirely disjoint. Thus the theory of evolution, taken as given, is evidence that transitional fossils between man and ape can be found; and the finding of such a transitional fossil is evidence for the (completely implausible) theory that some fossils have been preserved from every kind of organism that has ever inhabited the earth. But the theory of evolution taken as given does not provide evidence for the completely implausible theory; rather, the theory of evolution and the completely implausible theory would refute one another, at least if we are also given a little bit of background knowledge.