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Posts Tagged ‘Rudolph Carnap’

Positivism

Posted by allzermalmer on January 18, 2012

This blog will be based on a paper done by W.T. Stace. It was published in the philosophical journal Mind, New Series, Vol. 53, No. 211 (Jul., 1944), pp. 215-237. It was called Positivism.

During the time of writing this paper, there was a big movement in parts of Europe and eventually came to America, and it was Logical Positivism or Logical Empiricism. This group was known for the principle for which they became infamous and later collapsed. Stace says what he takes the principle to be, and calls it the Positivism Principle.

“A set of words purporting to express a factual proposition P is significant only if it is impossible to deduce or infer from it, in combination if necessary with other premises, some proposition or propositions (at least one) Q^1, Q^2, Q^3…etc., the truth or falsity of which it would be logically possible to verify by direct observation. If no such directly verifiable deductions from P are possible, then the set of words purporting to express P is non-significant, and P is not really a proposition at all, but a pseudo-proposition.”

Now there are some terms in that principle that would need to be clarified. These terms are “significant” and “meaning”. We, in our common speaking, talk of meaning of a sentence and meaning of a word. This will also deal with the difference between significant and meaning. The meaning of a sentence is called significance, because only a sentence can be true or false. The meaning of a word is called meaning, but words can’t be either true or false. This is because meaning of sentences is where the predicate of true or false apply, but those predicates don’t apply to single words like “red”. Sentences have significance, and words have meaning. This forms a big distinction from which the rest of the paper follows.

The difference between significance and meaning are based on a distinction within the genus of semantical meanings. So Stace shall be dealing with the semantical significance of sentences and the semantical meaning of words. And Stace will deal with “deduce or infer” as being a deduction or being a causal inference. So when we make the statement of “P”, it can hold the form of “P→Q”. This will help spell out some of the deduction and causal inference that is going on.

But Stace also quotes the Logical Positivist A.J. Ayer, as he states what is meant by the Positivist Principle.

“Let us call a proposition which records an actual or possible observation and experiential proposition. Then we may say that it is the mark of a genuine factual proposition…that some experiential proposition can be deduced from it in conjunction with certain other premises without being deducible from those other premises alone.”

There seems to be no clear distinction between A.J. Ayer and what W.T. Stace has said as well. And so what Stace would say would also seem to hold for the Positivism Principle.

The Positivism Principle seems to be of the same of the Verification principle, but there is some slight difference from when the original Verification principle was proposed. Schlick was one of the first to propose the Verification principle. Verification was meant to be direct and complete verification, which was the significance of a statement was the method of its verification. It had to be direct and complete verification. But such a principle makes universal statements to be insignificant. For to directly and completely verify a universal statement, you wold have to observe an infinite number of facts, which would have been of the past, present, and future, and all locations. And even singular statements about material objects would be insignificant sine the complete verification would also involve an infinite number of observations. Also, talk of the past wouldn’t be allowed for to be significant.

In order to escape some of these downfalls of what Schlick presented as the Verification principle, it was mean to soften it some, but keep some of the same points. Now, instead of complete and direct verification, the new principle allowed for indirect and partial verification. So we can verify the past occurrence of something by checking the present effect. And this is part of Carnap seems to mean with “testing” the proposition. Stace points out how we come to indirectly and partially verify something.

“What is now required in order to make a statement about the past significant is, not that the facts asserted in the statement should be themselves now observable, but that some of their effects should be observable (indirect verification). And what is required to make a universal proposition significant is not that all the facts which it asserts (an infinite number) should be observable, but that some of them should be observable (partial verification). These are the requirements which are embodied in the positivist principle as formulated in the first paragraph of this article.”

Now with the Positivist principle, and what was originally in the Verification principle which was later modified, seemed to be another principle. In other words, the Positivist Principle seems to be based on another principle. And now see what this other principle is would seem to help cast light on the Positivist Principle. This principle would be called The Principle of Observable Kinds:

“A sentence, in order to be significant, must assert or deny facts which are of a kind or class such that it is logically possible directly to observe some facts which are instances of that class or kind. And to observe some facts which are instances of that class or kind. And if a sentence purports to assert or deny facts which are a class or kind such that it would be logically impossible directly to observe any instance of that class or kinds, then the sentence is non-significant.”

Now let us use an example to see what this principle is trying to get across. Take “Napoleon crossed the Alps”. It is logically impossible for us to now directly observe this particular fact asserted in the sentence. This fact no longer exists, and so we can’t directly observe it. But this particular fact of Napoleon crossing the Alps is also part of the class of “men crossing mountains”, which we can experience. So the point becomes that we might not be able to logically observe all particular members of the class, but it is logically possible for us to observe some of the particular members of that class. Thus, the fact itself might not be observable but it is of the observable kind. This means that if the kind of thing said is unobservable, then we definitely can’t observe the particulars.

So the point becomes, “A sentence is significant if what it asserts or denies is the sort of thing which it is logically possible to observe, even if the particular instance in the sentence is such that it would not be logically possible to observe it.”

Now Stace will maintain that the Positivist Principle implies the Principle of Observable Kinds. But the Positivist Principle might not directly say it. But the Principle of Observable Kinds doesn’t seem to be what the Positivist Principle is saying, or what the Logical Positivist themselves maintain. There seem to be two differences between the two principles: 1. The principle of observable kinds introduces the notion of classes, while, on the other hand, the positivist principle says nothing about classes. 2. The Positivist principle makes use of the concept of indirect verification while the principle of observable kinds contains only concepts of direct verification or observation.

Now the Principle of Observable Kind is based on direct verification, and would seem to go back to that of the Verification principle that was given by Shclick. But this wouldn’t be correct to think that it goes back to the Verification principle. But the Principle of Observable Kinds does make a distinction from that of the Verifiability principle, which is based on that of classes.

“Suppose the proposition to be examined for its significance is P. Then according to the original principle of verifiability the facts asserted or denied in P must themselves be capable of being observed. What the new positivist principle says is that the facts asserted or denied in Q (the proposition or propositions deduced from P), must themselves be capable of being observed. It does not say anything at all about the observability or non-observability of the facts asserted or denied in P. It entirely ignores that question.

Now as was pointed out earlier, P can be replaced with P→Q. So P of the P→Q might not be directly verifiable by experience, or direct experience. But the consequence of it, which is Q, would be of the observable kinds. But now there is a divide between both principles, the original formulation and the later formulation that deals with indirect verification. And this divide is filled with the Principle of Observable Kinds. This is because the original formulation doesn’t state whether the facts stated in P must be observable. Nothing is said on this.

So what is the Positivist to say with this divide? “He ought to say, not that the particular facts asserted in P must be observable (this is what he wished quite rightly to avoid), nor yet that they may be of a wholly unobservable kind; but rather that the particular facts asserted in P, although they cannot be themselves be observed, must yet be of a kind of facts which other instances can be observed. And this is what the principle of observable kinds does say.” Thus, the Principle of Observable Kinds is implied by the new Positivist Principle.

The Principle of Observable Kinds seems to carry some of the same meaning of the Positivist Principle, but would seem to give more gain than that of the Positivist Principle. One of the reasons is that the Positivist Principle is based on the way of the propositions logical consequences, while the Principle of Observable kinds don’t seem to be worried too much about that.

Now A.J. Ayer makes a point in his book The Foundation of Empirical Knowledge, that we can have two different philosophers disagree over a certain point. So take that Philosopher A said that “We do not perceive the table or the chair, we only perceive sense-data which we believe to belong to a  table and a chair.” But Philosopher B says, “No, what we perceive is the actual table and the actual chair”. What Ayer says is that both philosophers disagree with the language, but they agree with the observations. They agree about the color, shape, weight, and about every fat which could possibly be observed. They agree about all possible observable facts, and there is no disagreement between these two philosophers on the facts. Thus, they both disagree over the language and not the facts.

But what do we say, in this position, when they disagree over unobservable facts? This seems to be something that Ayer might have overlooked, and is overlooked by the Positivist Principle.

“For instance, Philosopher A may hold the view that there is a “physical object”, X, which possesses intrinsic qualities which correspond to the perceived qualities of shape, size, color, smell, etc, but which are forever hidden fro us, so that we can never know anything about these intrinsic qualities except the fact of their correspondence to perceived qualities; and that this physical object X stands in a causal relation to our sense-data…Philosopher B may deny that any such object as X exists. He may say that the table or the chair simply is the collection of all the sense-data which (according to A) are caused by it. Thus both of them may admit the existence of the sense-data, and may entirely agree about all their characteristics, which means that they will agree about all the observable facts. But they will assert that they are in disagreement whether x exists or not. X, if it is a fat, is a fact forever unobservable These philosophers will therefore say that they are in disagreement about whether or not there exists an alleged unobservable fact. Mr. Ayer appears to have overlooked this in his argument.”

This seems to be one reason why the Positivist Principle would have many things listed as insignificant. It is more of a difference over language than it is over some actual fact. And thus, for the Positivist, it is only about a way of speaking than some actual facts. For they both contain the same observable kinds.

But now maybe the Positivist principle doesn’t rely on the principle of observable kinds. What might be the case if this isn’t so? Than although the facts stated in Q of P→Q are logically capable of being directly verified, then P itself might states facts which would be logically impossible to observe. And the Positivist principle seems to tell us that this might be the case. For Q tells us that it is observable, but it doesn’t say anything about the observability of P. And there seem to be two cases: (1.) Where P→Q is a deductive argument, and (2.) where P→Q is a cause and effect inference.

Case 1: If P→Q is a deductive argument, then either (A.) Q states some facts as P, either in whole or in part or (B.) Q states some facts or elements of fact which are not asserted in P. (A.) is the view that logical rules are rules of linguistic transformations. (B.) is the view that in the conclusion of a deductive argument there may be some element of fact that may be “new”, i.e. not “contained” in the premises.

If we accept (A.), then it is clear that if Q is of the observable kind then P will be of the observable kind as well. This is because Q states the same thing as P, but in different words. Thus, if Q is observable, then P would also be observable since P is just another way of saying Q. This means P is another way of saying observable kind, which is what Q did as well. Thus, the Positivist principle implies the Principle of Observable Kind.

If we accept (B.) then it doesn’t seem that we can rigorously prove that if Q states observables P must state observables. For p might conceivably be of a different kind from the facts stated in Q. But, what seems to be enough, is that the Logical Positivist themselves held to the linguistic transformations.

Case 2: If P→Q is a cause and effect inference, then it’s certain that facts stated in P cannot be unobservable if the facts stated in Q are observables. This is because inference must rest on a causal law. “The cause C will be the fact stated in P, while the effect E will be the fact stated in Q. For instance, P any state the fact that it rained five minutes ago, while Q states in effect of this rain, namely, that the ground will be wet now. But it is impossible that the causal connection between C and E can have been established except on the basis that E has been observed to follow C. Therefore C, the fact stated in P, must be an observable.”

Thus, from these considerations, it appears obvious that the Positivist Principle does rely on the Principle of Observable Kinds. And the Positivist Principle, if adopted because that is the definition for significance that they choose, it carries no real force. This is because they just freely choose this criterion while someone else can pick whatever else criterion they wish to use for significance. But if they wish for it to carry some meaning about it based on experience, it would have to be shown through some sort of experience. This would be be based on inductive generalizations.

Now we might wonder how did we arrive at this Positivist Principle. Stace has an answer on how he thinks that the Positivist derived their principle.

“I think it is almost certain that positivists believe that their principles are a development of the general principle of empiricism. They call themselves “empiricists”. And thus by implications they claim that, whatever evidence there is to support the general principle of empiricism is also evidence which supports them. They think that,although their position is in some way different from that of the other empiricists (such as Hume)- more “advanced” no doubt- yet it grows out of the same root as does the tree of empiricism., and that therefore the sap which nourishes that tree will also nourish them. This is a very interesting and also a very important claim. And I propose to examine it. the question is: Is positivism a legitimate development of empiricism, and are the grounds which support empiricism also grounds which support positivism?”

Now the Logical Positivist claim that they’re “empiricists”, but we might wonder what type of empiricist they are. They’re seems, in history, to be different types of empiricist, but they haven’t, as Stace says, made it clear what type of empiricists they are. But there does seem to be two different types of Empiricism. (1.) the doctrine that all knowledge is “based upon ” or “derived from” experience, and (2.) the doctrine that all “ideas” are “based upon” or “derived from” experience.

With the first kind, the meaning of the phrase “based upon” or “derived from” seem to be different in (1.) and (2.). With (1.) is that if any proposition is known to be true, it can only be so known because there is empirical evidence for it, or must be empirical grounds for it. John Stuart Mill brought this up and tried to use it to support that 2+2=4 is an empirical generalization and illustrates this kind of empiricism. The (2.) kind is based upon ideas, like that of a “centaur”. The idea is neither true or false. What happens is that we can break down our ides into some basic parts, or that our experiences are built up off of some basic parts like “blue”, “horse”, “human head”, and etc. This is like Hume saying that “complex ideas” are based on “simple ideas”.

It appears that the Principle of Observable Kinds isn’t based on the first type of empiricism. As Stace says, “For the principle of observable kinds professes to be a criterion, not of the truth of propositions, nor of ways of knowing them to be true, but of whether they have significance. But the first kind of empiricism has nothing to do with significance at all, and cannot so far as I can see have any bearing on that subject.”

For whatever the relaations between “being known to be true” and “being significant” may be, they are certainly not the same thing, since a proposition may be significant and yet not known to be true. A significant proposition may in fact be known to be false. The long and short of it is that the first kind of empiricism is a theory about the truth of propositions (more correctly about how their truth can be known) while the principle of observable kinds is a theory about the significance of propositions. And since the two theories are “about” different subjects, one cannot possibly follow from, or be legitimately developed out of, the other.”

Of the first kind of empiricism, which is about the truth of propositions, the Logical Positivist would have seemed to hold this position as well. And the position that they state is that a priori statements are analytic statements, which means they’re not “derived from” experience. This stance follows from the first kind of empiricism, but we’ve also noticed that the significance of a proposition isn’t based on the first kind of empiricism. And all a priori propositions being analytic would follow from the first principle.

Now take the second kind of empiricism. We might wonder if the Principle of Observable Kinds comes from the second kind of empiricism. David Hume, after all, does bring up something that would be similar to that of the second kind of empiricism, “from what impression is that supposed idea derived.” But the Principle of Observable Kinds doesn’t follow from the second kind of empiricism.

The second kind of empiricism dealt with “ideas” being derived from experience. But as was pointed out earlier, the Principle of Observable Kinds is based on sentence significance, and not word significance. The second kind of empiricism is worried about the ideas, like that of “red”. But it is only the sentences that make significance and not the words itself. The second kind of empiricism is strictly concerned from what those experiences come from, like “red”, “sweater”, “blue-jeans”, “tennis shoes”, and etc, but isn’t concerned with whole sentences. The Principle of Observable Kinds is concerned with only whole sentences, like “James wore a red sweater while also having some blue-jeans to match their tennis shoes.” That sentence carries significance.

So it doesn’t look like the Principle of Observable Kinds follows from empiricism as well, in either kind. Thus, since the Positivist Principle seems to be implied by the Principle of Observable Kinds, the Positivist Principle seems to carry no weight when it comes to empiricism. Thus, those who call themselves empiricists and support the Positivist Principle don’t seem to have such a right.

The reason is that the principle of empiricism was stated by David Hume. The idea can be listed as the mind cannot spontaneously generate “simple ideas”, nor create them out of nothing, but has to derive them from “impressions”. These simple ideas would be those things that you can’t break down any further from your experience. For example, you have “red, “hot”, “cold”, “round”, “soft”, “sweet”, “loud”. From these unanalyzable, simple, building blocks, we can create different things from them by combining them in different ways. Giving some basic material, you can combine it in many different ways. But the basic idea is that you can’t create these simple ideas out of nothing, which means that you needed some impression of them.

The principle of empiricism implies nothing on how we form these simple ideas are to be combined into complex ideas. It provides no rules for combination. This means that we are free to combine the simple ideas in any way we would like, at least we have no rules on how to combine them, or at least according to the principle of empiricism. But there could be some laws, like the law of non-contradiction, or incompatible characters cannot be combined in the mode of spatio-temporal coincidence, though they can in the mode of spatial juxtaposition. This second idea is the Principle of Incompatibles. It basically states that the surface of a ball may be red and blue simultaneously if juxtaposed over one another. Also, we might have the laws of syntax to deal with how to combine our ideas. But the point is that none of these ideas follow from the principle of empiricism.

“The principle of empiricism concerns only the origination of simple ideas, nothing else. It tells us: no impressions,then no simple ideas. We may add as part of the principle, if we wish, the fact that certain of our ideas are not simple but are compounded out of simple ideas.”

But sentences are based on some ideas being placed in relation to one another in a certain way, or whatever way since the principle of empiricism does not care. Sentences, it seems, deals with complex ideas. So take a word to be symbolized like “F”. This word is composed of different simple ideas, like a certain color, shape, smell, taste, or sound, to go along with it. So all these different, simple ideas, can be symbolized as m,n,o,p,q. Thus, F=m,n,o,p,q. And sentences are composed of complex ideas which talk about the relations between something like A being B or A being related to B.

Thus, when empiricism isn’t concerned on how we form complex sentences, which is how we form our simple ideas together, it doesn’t imply the Principle of Observable Kinds because that principle relies on sentences or how complex ideas are to be put together. And this would also mean that the Positivist Principle isn’t implied by the principle of empiricism.

“What [the principle of empiricism] tells us is that if a sentence asserts or denies a fact F, which is a complex of a,b,c,d…, then each of these simples, a,b,c,d,…, must be an observable. But what the principle of observable kinds does is to assert that the total complex fact F, or abcd, must as a whole, be an observable. But for this there is not the slightest warrant in the principle of empiricism.”

The main point is that the Positivist have no right to claim to their principle of significance follows from empiricism. And that they’re constriction on propositions is arbitrary on it’s own, and has no standing in the principle of empiricism.

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Are All Empirical Statements Merely Hypotheses?

Posted by allzermalmer on December 19, 2011

This blog will be based on an article done by W.T. Stace. It is called, Are All Empirical Statements Merely Hypotheses? It appeared in the philosophical journal known as The Journal of Philosophy Vol. 44, No. 2 (Jan. 16, 1947), pp. 29-38.

It is sometimes stated that all empirical statements are only probable. This was stated by those like, and especially by, Rudolph Carnap. One philosopher who disagreed, and said that some empirical statements are certain, was G.E. Moore. Stace shall agree with Moore, but with some qualifications. The statement that will be the exemplar of what is being talked about will be the statement of “This key is made of iron”. Now this statement is a singular statement like x is Y.

“To say that this proposition can never be more than probable means, I assume, that there must always be some doubt as to its truth. The question we have to get clear about is: what is the doubt, or what are the doubts, which those philosophers who say that such a statement can never be more than probable, have in mind?”

Some of the doubts could be as follows for what makes this empirical statement probable: the laws of nature are statistical, we could be deceived by some sort of demons or might be dreaming, or statements that we make rely on memory and our memory could be wrong. None of these things seems to be what has lead some to think that all empirical statements are probable. That is because these doubts are arising from practical doubt because of the frailty of human faculties.

The philosophers, like Carnap, seem to be relying on theoretical/logical doubt. This seems to be based on the logic at which we arrive at empirical truths, regardless of the frailties of particular human beings. They seem to be saying that we arrive at these empirical statements, like “this key is made of iron”, are arrived at by means of induction. And, through the means of induction, we never arrive at certainty by by means of probability.

Stace quotes Carnap on the basic idea of which is to lead to all empirical statements are merely probable. Take the statement that “This key is made of iron”. This proposition will be known as P1. We can test P1 by seeing if it is attracted by a magnet, if it is then we have partial verification of P1. So here is what Rudolph Carnap says, which leads him to state that all empirical statements are merely probable in his book Philosophy and Logical Syntax:

“After that, or instead of that, we may make an examination by electrical tests, or by mechanical, chemical, or optical tests, etc. If in these further investigations all instances turn out to be positive, the certainty of the proposition P1 gradually grows…but absolute certainty we can never attain. the number of instances deducible from P1 is infinite. Therefore there is always the possibility of finding in the future a negative instance.”

Now this is the logical problem that we face. Anytime we perform a new test, and the test is passed, it only adds a degree of probability to the statement that “this key is made of iron”. And the problem, further, is that we can’t completely verify the statement, or be certain of it, because we would have to complete an infinite number of observations. But this is not only practically impossible, it is also logically impossible.

But there is some ambiguity of what Carnap means, because there are two ways that this can be taken. The first thing could be about the different kinds of tests. For we noticed that he brought up the tests that could be done, like magnetic, electrical, chemical, and etc. So the it could be meant that the number of different kinds of test is infinite, which means we would have to make an infinite number of kinds of tests in order to achieve complete verification of the statements truth. But Stace has an objection to this position.

“If an infinite number of kinds of tests of the key were possible, this would imply that the key must have an infinite number of different characteristics or properties to be tested for. But even if an object can have an infinite number of characteristics, it would not be necessary to test for them all in order to identify the object as iron. All we need is to verify the defining characteristics of iron, which are certainly finite in number. and there is, of course, no logical difficulty about doing that.”

Now there is a second possible meaning for which Carnap has in mind. We could do a single test of a defining characteristic like “being attracted by a magnet”, or what other defining characteristics there might be. These tests only make the statement probable because we may find that the key is attracted one time and perform many of the same tests a thousand times in succession and find the same results as the first test. But we can never be sure that an instance will not turn up in the future in which the object will not be attracted by a magnet (problem of induction). “If the same thing happens in the same circumstances in a vast number of times, each time it happens makes it a little more probable that it will happen again, but it can never be quite certain.”

It is true that scientists perform the same experiments, this is the repeatably of the scientific tests. What one scientist is able to do in a test, it has to be reproducible by other scientists around the world. The same experiment can be repeated by the same experimenter over and over, or can be done by other experimenters around the world. But why are experiments repeated? Is it because each fresh instance of a positive result of the same test adds to the probability of the conclusion? It seems not.

Let us assume that we have an object that is to be tested. We want to test whether it is composed of a certain substance, which we can call X. Now let us suppose that there is only one defining characteristic of X which we call A. The scientist is testing for Y. If Y is found it is a sign that the substance is X. Now, is it true that A may be repeated many times. But why?

“It is not because he supposes that a barren repetition of instances of A makes it more probable that the substance is X. It is always, on the contrary, because he has doubts whether he has satisfactorily established by his observations of the presence of A. It is not the validity of the inductive inference from A to X that he is doubting, but whether A is really present…the doubt which the experimenter is trying to exclude is not any logical doubt about induction, but practical doubts arising from difficulties of observation, possible deficiencies in apparatus, difficulty in ensuring that the experiment is made in the exact conditions required, and so on. He is not doubting that the inductive premises will lead to an absolutely certain conclusion. He is doubting whether he has satisfactorily established the inductive premises.”

What is going on is that the scientist procedure is that a single observation is sufficient to establish an inductive conclusion with certainty. But this is only the case provided that the premises have been established. So it is not the inductive conclusion that is being questioned, but it is the premises that are being questioned. As Stace says, “What is implied by the scientist’s procedure is that a single observation or experiment is sufficient to establish an inductive conclusion with certainty, provided the premises have been established. I hold that the scientist is right.”

Stace locates the problem at three points. And this is the problem of how some philosophers have reached the conclusion that all empirical statements are merely probable.

(1.) One of the problems was how philosophers thought that scientists were repeating experiments to try to dispel logical doubts about the validity of induction. What the scientists were doing, in fact, was trying to dispel practical errors in observing or establishing the premises on which an induction rests. The question of probability doesn’t fall within the inductive argument, but outside of the inductive argument.

“That is to say, what is only probable is not that, if A is once associated with B, it will always be associated with B, but that A has actually been found associated with B; not that if a substance has a certain specific gravity it is gold, but that the substance now before me actually has that specific gravity…a natural mistake located the question of probability within the inductive argument instead of outside of it; have extrapolated it from the practical sphere of observation, measurement, and so on, where it actually belongs, to the logical sphere of the inductive inference in which in reality it has no place.”

So the problem is not in the inductive argument itself, but outside of the argument. What is outside of the argument is making sure that you have made an observation that meets with the premises of the argument. This is what constant testing is about, to make sure that the observations are in line with the premises. It is not the argument being questioned, but something outside of the argument that is being questioned.

(2.) Another reason that it seems that it is brought up that empirical statements are probable deals with the view of induction where an application of the inductive principle to a type of cases different from that of the Iron key. This other application is based on generalizing from observations. For example, we generalize from observations of a number from a certain class to the whole class. This means, from observing some white swans, we go on to generalize to the class of swans. From seeing a certain number of swans being white, and not observing any black swans, we go on to say that All swans are white. This will be dealt with a little later on.

(3.) This view seems to follow, as some philosophers think, from what David Hume had to say on the problem of Induction. Hume showed that we can’t “prove” a conclusion in an inductive argument. Because of this, some seem to have imagine that because we can’t prove it, we can at least make it probable. But it doesn’t seem that this follows from what Hume said on the problem of Induction. But Stace does think that something follows from what Hume said on this problem.

Imagine that we have a single instance of A being associated with B, and we’ve ruled out all practical doubts from possible errors of observation or experiment. We now have, logically, two positions that we can take up.

The first is that we can assume the validity of the principle of Induction. So, in this single instance, we can conclude that A is always associated with B, and our conclusion follows with absolute certainer from our two premises of single observed association of A with B and the principle of induction. With these two premises, the conclusion is certain to start with, and so there is no increasing probability or probability at all.

The second is that you may not assume that validity of the inductive principle. Now this means that we follow Hume, which means that there’s no logical connection between the premises and the conclusion of induction. This means, nothing follows from induction, neither certainty nor probability. No matter how many single instances that support our inductive conclusion, the probability never arises above zero. (Karl Popper would agree with this point). There is no connection to say that because the conclusion obtained, that we can say that the probability of the premises rises some more. They are disconnected. It is like having three dots on a sheet of paper. They are disconnected from each other. So when we affirm one, we can’t affirm any of the others because they’re not connected with one another.

“I have affirmed that, given the inductive principle, a single case will prove the inductive conclusion with certainty, I ought to give a formulation to the inductive principle which embodies this…”If in even a single instance, we have observed that a thing of the sort A is associated with a thing of the sort B, then on any other appearance of A, provided the other factors present along with A are the same on both occasions, it is certain that A will be associated with B.””

There is the clause of “provided the other factors present along with A are the same on both occasions.” This forms part of the principle, which comes down to “Same cause, same effect”. There is an example to help make this point clear. If the bell is struck in air then it produces sound. But it doesn’t follow that a bell struck in a vacuum will produce sound. This is because of the clause that was inserted into the principle. The factors aren’t the same, and so they’re not the same type of thing. But it does introduce a new inductive discovery.

There is one obvious objection that one could make to this principle. It could be said that this new interpretation is merely an assumption that is incapable of proof. So if this is a matter of being arbitrary choice of how to formulate it in terms of certainty and probability, then we ought not to assume more than is necessary to justify our sciences and our practice. So someone could say, “it will be quite sufficient for these purposes to assume that, if A is associated with B now, it will probably be associated with B at other times and places. On this ground the probability formulation should be preferred.”

But putting the term certainty in there is not meant to be arbitrary, but it is mean to represent a formulation of the assumption which has been the basis of science and practice. But maybe Stace should be more clear, which is what he tries to do like as follows:

“If you have one case of a set of circumstances A associated with B, and you are quite sure you have correctly established this one association, then, assuming the uniformity of nature, or the reign of law, or the principle of induction-call it what you will- a repetition of identically the same set of circumstances A is bound to be associated with B. For if not, you would have a capricious world, a world in which A sometimes produces B, and sometimes it does not, a world in which the kettle put on the fire may boil today, but freeze tomorrow. And this would clearly be a violation of the principle of induction which you have assumed.”

Now, if you assume the principle of induction, then a single case validates an induction. But now Stace will try to prove his second contention that if you don’t assume the principle of induction, your inductive conclusion aren’t probable at all and there’s no repetition of instances, so no matter how great the number, then the probability is never raised above zero.

To establish this position, Stace will assume that Hume is right. This means, between the premises and the conclusion of an inductive argument there is absolutely no logical connection at all. This means that there is nothing to establish the slightest probability because they’re is no connection between them. So if we affirm one part, it has no connection to another to raise the probability of this part that is connected to what we affirmed. They are so completely disconnected that there’s no logical connection to even bring up probability.

For example, here is what Al-Ghazali said about causality, which is the same position that David Hume took up, and this is based in some ways on the principle of induction. “The affirmation of one does not imply the affirmation of the other; nor does its denial imply the denial of the other. The existence of one is not necessitated by the existence of the other; nor its non-existence by the non-existence of the other.” So when we affirm one thing with induction, like a correct experiment, this in no way can increase any probability when the affirmation of one doesn’t imply the affirmation of the other. How can you raise the probability when what you affirm has no connection to anything else to raise the probability of this other thing? You can’t.

Stace goes on to try to examine the types of cases in which generalize a whole class from a number of instances that are smaller than the whole class. Try to generalize about a whole class of swans from observing a few of the swans that are suppose to make up the whole class. If we observe one swan and it is white,nto conclude that all swans are white, we might be accused of generalizing from one instance. But if we make 10,000 observations, we might think we have a degree of probability to support the generalization. We go on to make observe 1 billion swans and they were white. This might lead us to go on to admit that the hypothesis has become even more probable. So, someone might say to defend the probability view, that how can we deny that we probability and use the probability view of induction?

“But the inductive principle only holds with the proviso, “if the factors present along with A are the same” in subsequent repitition of A. And this case of the swans is simply a case in which it is extremely difficult to be sure that this is so. A in this case means the defining characteristics of the class swan, and B means whiteness. Now different swans will have, along with the defining characteristics A, a number of other characteristics. and these will differ with different individual swans, not to mention circumambient differences of environment. Thus the first case of A you observed was really ACDE, and this was associated with B. The second case was APQR, the third AXYZ. Now, of course, it does not follow from the principle of induction that because ACDE was associated with B, therefore APQR and AXYZ must be associated with B. For we do not have there that exact repetition of the same sets of circumstances which the inductive principle requires.”

To try to remedy the situation that we are in, we constantly repeat observations of this class of swans. Now if we keep making these observations of A, and they’re found to have B, then we think it becomes more and more likely that we have eliminated other certain possibilities, and raise the probability. We want to eliminate some of the accidental characteristics of certain swans. This would be something like they’re size. food they eat, and the climates that they live in. When we rule out sets of circumstances as irrelevant, they become more probable.

The fundamental reason why there is constant repetition of observation on new members of class is that although in theory the association of A with B, once it is observed must always hold, is because in practice we never get our cases of pure A. “We can not isolate the system. It is always mixed up with extraneous circumstances. Thus the doubt which we are trying to dispel by repeated observations has nothing at all to do with Hume’s doubt about the validity of induction…” That doubt can’t be dispelled, no matter now many numerous observations we make. But the doubt that we are trying to get rid of isn’t the logical doubt. The doubt we are trying to get rid of is the practical doubt from the enormous complexity of nature, our frailty of our intellects which are unequal with the task to disentangle the complexities, or the inadequacy of the instruments that we have at our disposal to isolate the system present.

Some, like Carnap, have divided knowledge into empirical knowledge and necessary propositions. Necessary propositions would be those like mathematics and logic. Now the empirical propositions could be considered doubtful because the practical doubts that arise from our human infirmities. But this means that we ought to have the same doubts in concern with mathematics. This is why we have people that check our work in mathematics, to make sure that we made no practical doubts in the process that we followed.

“There is one sense in which mathematical, or, in general, deductive conclusions are certain this may be called the logical or theoretical sense. And there is another sense, which may be called the practical sense, in which they are only probable, since the mathematician or the syllogizer may err in his reasoning. The mathematician may miscalculate, and the syllogizer may make any one of a hundred mistakes. And if practical doubts are not a ground for denying that, in an appropriate sense, mathematics is certain, then practical doubts can not be a ground for denying that, in an appropriate sense, empirical conclusions are uncertain.”

“As it is with mathematical truths, so precisely it is with empirical truths. There is one sense in which an inductive conclusion is certain, namely, the theoretical sense that it follows with certainity from a single observation plus the inductive principle. And there is another sense, the practical one, in which it is probable only, because there may be errors in observation, experimentation, and the like.”

“The statement that empiricial knowledge may be theoretically certain is, of course, subject to the proviso that we accept the inductive principle. If we don’t accept it, then, of course, empirical knowledge is not even probable. It has no validity at all. In no case does any question of probability enter into the matter.”

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