Truth of Reasoning and Truth of Fact

“All that which implies contradiction is impossible, and all that which implies no contradiction is possible.” G.W. Leibniz

“I assume that every judgement (i.e. affirmation or negation) is either true or false and that if the affirmation is true the negation is false, and if the negation is true the affirmation is false; that what is denied to be true-truly, of course- is false, and what is denied to be false is true; that what is denied to be affirmed, or affirmed to be denied, is to be denied; and what is affirmed to be affirmed and denied to be denied is to be affirmed. Similarly, that it is false that what is false should be true or that what is true should be false; that it is true that what is true is true, and what is false, false. All these are usually included in one designation, the principle of contradiction.” G.W. Leibniz

“There are . . . two kinds of truths, those of reasoning and those of fact. Truths of reasoning are necessary and their opposite is impossible; truths of fact are contingent and their opposite is possible. When a truth is necessary, its truth can be found by analysis, resolving it into more simple ideas and truths, until we come to those which are primary. It is thus, that in Mathematics speculative Theorems and practical Canons are reduced by analysis to Definitions, Axioms, and Postulates. In short, there are simple ideas, of which no definition can be given; there are also axioms and postulates, in a word primary principles, which cannot be proved, and indeed have no need of proof, and these are identical propositions, whose opposite involves an express contradiction.” G.W. Leibniz

 So Leibniz obtains all knowable propositions or statements to be divided based on the principle of contradiction. The truth of statements is divided into two realms. This also deals with what people can know, or knowability. It basically says that

“For each statement, if statement is knowable, then statement is either truth of reasoning or truth of fact. For each statement, if statement is truth of reasoning, then statements affirmation is logically possible and statements negation is logically impossible. For each statement, if statement is truth of fact, then statements affirmation is logically possible and statements negation is logically possible.”

A truth of reasoning is always true and not possible it is false. It is logically impossible that it is false. The negation of a truth of reasoning is an impossible statement or impossible proposition. It is self-contradictory. A truth of fact is not always true and possible it is false. It is logically possible that it is true or logically possible it is false. Truth of Reasoning is Logically Necessary and Truth of Fact is Logically Contingent.

“For each statement, if statement is Truth of Fact, then statement is an empirical claim. For each statement, if statement is Truth of Reasoning, then statement is not an empirical claim. For each statement, if statement is Truth of Reasoning, then statement is non-empirical claim. For each statement, if statement is Truth of Fact, then statement is not non-empirical claim.”

What also happens to come from this is that Truth of Facts do not entail or lead to Truth of Reasoning, and Truth of Reasoning do not entail or lead to Truth of Fact. This means that Truth of Facts do not imply or entail non-empirical claims and Truth of Reasoning do not imply or entail empirical claims. This means that statements of experience are not non-empirical claims and means statements of experience are empirical claims.

George Berkeley’s on Immediate Perception & Mediated Perception Pt. II

This is going to work off of a previous blog done on George Berkeley’s point about Immediate Perception and Mediated Perception.

All the human senses (sight, taste, sound, touch, and smell), are distinct from one another, like the sun is distinct from the moon. But these “naturally” distinct things are combined, connected, or associated, with one another. When they are combined, connected, or associated together, they bring forth new phenomena. One example is combining sight and touch together to form depth. We connect these two distinct senses together, and we get depth or three dimensions. This new phenomena is a human creation. It was not an immediate perception but was a mediated perception.

We connect each of the distinct senses together to help to form one “body” or “object”. You have a visual object, and this visual image happens to have color and shape. We also have tactical object, and this tactical feel happens to be hard, cold, and wet. We do this with taste, smell, and sound. We combine these distinct sensual immediate perceptions together to form one “object”, or “body”, and get the mediated perception of “the apple”.

Every sense is distinct from one another. Each distinct sense has their own immediate perceptions. So sight is distinct from touch, but touch has its own immediate perceptions and sight has its own immediate perception. Sight gives the immediate perception of red and touch gives the immediate perception of hot.

Now it appears that each sense would have its own “mediated perceptions” as well, or at least how the human mind has created somethings. Take this example, which is only dealing with sight. [**To speak clearly, these examples are not strictly immediate perceptions as Berkeley uses the term. But they are going to be called “immediate perceptions”, and what comes about from their combination becomes the “mediated perception”.**]

(1) You pick up a rock and look at it.
(2) You look at the rock with a magnifying glass. [(1) is not the same as (2)]
(3) You look at the rock with binoculars. [(2) is not the same as (3)]
(4) You look at the rock with a Light Microscope. [(3) is not the same as (4)]
(5) You look at the rock with an Electron Microscope. [(4) is not the same as (5)]
(6) You look at the rock with Infrared. [(5) is not the same as (6)]
(7) You look at the rock with X-Ray. [(6) is not the same as (7)]

Each of these visual “immediate perceptions” are distinct from one another. But similar to how we connect sight, touch, taste, smell, and sound, to form the “body” or “object” we call “the apple”, so too do we combine the different visual “immediate perceptions” to form the “mediated perception” of the visual “body, or “object”, we call “the rock”. This would hold with the other distinct senses, i.e. touch and taste, as well.

But the different “visual images” are obviously not of the same “body” or “object”. First, the “body” or “object” is a combination of different senses from one another. The rock is a combination of sight, touch, taste, smell, and sound. But each of these are “naturally” distinct from one another, and so the body that is the rock before us, is a creation of the human mind. Each of these senses presents a different aspect of the “same” object, (the object itself being a mediated perception and thus a creation of the human mind).

The senses themselves also have mediated perception, (in an analogical sense and not strictly). The rock has distinct visual aspects. And the human mind combines these distinct visual images and say they are different aspects of the “same” object, (the object itself being a mediated perception in a loose analogical sense, and thus a creation of the human mind). This would go on with the other senses as well, like touch.

Take the example of (1) and (5), or even (4). In these cases, when we look at (5), what we are looking at is something that we cannot touch. It is first of all the sense of sight, and the sense of sight is distinct from touch. Thus, we cannot touch what we see. But we have combined these two distinct senses to say that we can see what we touch. But now we have said that the sense of sight has shown us another “level” of the “body” or “object” that is “the rock” before us. And at the level of (5) or (4), what we see does not match up with what we touch. The rock feels smooth but image (5) or (4) does not feel smooth or even “look” smooth. So the image not only does not match up with the other images, it does not match up with the other senses or the minimum tangibly of touch.

 

 

Pyrrhonian Scepticism Against Time

This comes from Sextus Empiricus, the Pyrrhonian skeptic, speaking out against conceptions of time. This comes from Against the Physicists II.

“Let this, then, serve as our account of the difficulties regarding the real existence of time which arise from the conception of it; but we can also establish our case by means of direct argument. For if time exists it is either limited or unlimited; but neither is it limited, as we shall establish, nor is it unlimited, as we shall show; therefore time is nothing.

For if time is limited, there was once a time when time did not exist, and there will one day be a time when time will not exist. But it is absurd to say either that there was once a time when time did not exist, or that there will one day be a time when time will not exist, for the statements that “there once was” and that “there will be” are (as I said before) indicative of different times. So, then, time is not limited.

-Nor, in fact, is it unlimited. For one part of it is past, the other future. Each of these times, then, either exists or does not exist. And if it does not exist, time is at once limited, and if it is limited the original difficulty remains- that there was once a time when time did not exist and there will one day be a time when time will not exist. But if each exists- I mean both past and future time,- each will be in the present. And as existing in the present, both past and future time will be in present time. But it is absurd to say that past and future are conceived as in present time. So, then, time is not unlimited either.

But if it is neither conceived as limited nor as unlimited, it will not exist at all. – Also, what is composed of non-existents will be non-existent- of the past which exists no longer and of the future which does not as yet exist; time, therefore, is non-existent.

 

Furthermore: if time is anything, it is either indivisible or divisible; but it cannot be either indivisible, as we shall show, or divisible, as we shall establish; no time, therefore, exists.

Now time cannot be indivisible, since it is divided into past, present, and future. And it will not divisible because everything divisible is measured by a part of itself; the cubit, for instance, is measured by the palm, and the palms, is a part of the cubit, and the palm is measured by the finger, and the finger is a part of the palm. So, then, if time too is divisible, it ought to be measured by some part of itself.

But it is not possible for the other times to be measured by the present. For if the present time measures the past, the present time will be in the past, and being in the past it will no longer be present but past. And if the present measures the future, being within this it will be future and not present.

Hence, too, it is not possible to measure the present by the other times; for, as being within it, each of them will be present and not either past or future. But if one must certainly conceive time as either divisible or indivisible, and we have shown that it is neither divisible nor indivisible, it must be declared that time is nothing.

 

Furthermore: time is tripartite; for one part of it is past, one present, and one future. And of these the past no longer exists and the future does not yet exist. It remains to say that one part exists, the present.

The present time, then is either indivisible or divisible. But it cannot be indivisible, for “nothing divisible is of a nature to exist in indivisible time,” as Timon says, – becoming, for example, and perishing, and everything of a similar kind. And if it is indivisible, it will neither have a beginning whereby it is joined on to the past, nor an end whereby it is joined on to the future; for that which has a beginning and an end is not indivisible. But if it has neither a beginning nor an end, it will not have a middle either; for the middle is conceived by way of comparison in its relation to the other two. And as having neither beginning nor middle nor end, it will not exist at all.

And if present time is divisible, it is divided either into existent times or into non-existent. And if it should be divided into non-existent times, it will no longer be time; for that which is divided into non-existent times will not be time. And if it is divided into existent times, it will no longer, as a whole, be present but one part of it will be past, another future. And for this reason it will no longer, as a whole, be [present and] existent, as part of it no longer exists and part of is not as yet existing. But if of the three times- past, future, and present- it has been proved that not one exists, no time will exist.

 

And those who assert that present time is the limit of the past and the beginning of the future,- thus making one out of two non-existent times,- making not only one but every time nonexistent.

– And further : if present time is the limit of past, and the limit of the past has passed away together with that whereof it is the limit, present time will no longer exist, if it really is the limit of the past.

– And again; if present time is the beginning of the future, and the beginning of the future does not yet exist, present time will not yet exist, and thus it will have most opposite properties; for inasmuch as it is present it will exist, but inasmuch as it has passed away together with the past it will exist no longer, and inasmuch as it accompanies the future it will not as yet exist.

But it is absurd to conceive the same time as both existing and not existing, and no longer existing and not yet existing. So, then, in this way too one must deny that any time exists.

 

One may also argue thus: if time is anything, it is either imperishable and ingenerable or perishable and generable; but it is neither imperishable and ingenerable, as shall be proved, nor perishable and generable, as this also shall be established; time, therefore, is not anything.

Now it is not imperishable and ingenerable, seeing that part of it is past, part present, and past future. For the day of yesterday exists no longer, that of to-day exists, and that of to-morrow has not yet come into existence. Hence one part of time (namely, the past) no longer exists antoher (namely, the future) does not yet exist. And for this reason time will be neither ingenerable nor imperishable.

– But if it is perishable and generable, it is hard to say what it will perish into and from what it will come to exist. For neither does the future exist already, nor the past exist any longer. But how can a thing (come into existence) from non-existents, (or how can a thing) perish (into non-existents)? Time, then, is nothing.

 

One may attack it also in this way; if time is anything, it is either generable or ingenerable, or partly generable and partly ingenerable. But time cannot be either generable or ingenerable or partly generable and partly ingenerable; therefore time is not anything.

For if it i were generable, since everything which is generated becomes in time, time too being generated will be generated in time. Either, then, it will be generated as itself in itself or as one time in another.

And if it is generated as itself in itself, it will be a thing which has come to exist before it has come to exist; which is absurd. For since that in which a thing becomes must exist before that which is generated in it, time also, as generated in itself, must have come into existence before itself; just as a statue is wrought in a workshop, but the workshop existed before the statue, and a ship is constructed in a certain place, but the place was existing before the ship, So, then, if time too becomes in itself, it will exist before itself; and thus, inasmuch as it becomes, it will not yet exist, since everything which becomes, while it is becoming, does not exist as yet; but inasmuch as it becomes in itself, it must exist beforehand. Time, then, will be at once both existent and non-existent. Inasmuch as it becomes it will not exist, but inasmuch as it becomes in itself it will exist. But it is absurd that the same thing at the same instant should both exist and not exist; therefore it is also absurd to say that time becomes in itself.

-Nor yet does it become as one time in another,- the future, for instance, in the present, and and the present in the past. For if one time becomes in another, each of the times will necessarily quit its own position and occupy the post of the other. If, for example, the future time becomes in the present time, the future as becoming during the present will be present and not future; and if the present becomes in the past, as becoming during the past it will certainly not be present but past. And the same argument applies if we reverse their order, making the past becoming in the present and the present in the future; for her again the same difficulties follow.

– if, then, time does not become either in itself or as one time in another, time is not generable itself or as one time in another, time is not generable. But if it is neither ingenerable nor generable, and besides these one can conceive no third possibility, one must declare that time is nothing.

Now the fact that it cannot be ingeneralbe is extremely easy to demonstrate. For if it is ingenerable and neither has become nor will become, one time alone, the present, will exist, and neither will the future, and the things therein, be any longer future, nor will the past, and the things done therein, be any longer past. But this is not so; nor, consequently, is time ingenerable.

Nor yet is it partly generable and partly ingenerable, since, if so, the difficulties will be combined. For the generable must become either in itself or in another; but if it becomes in itself it will exist before itself, and if in another it will no longer be that time but , quitting its own post, it will be the time during which it becomes. And the same argument applies also to the ingenerable; for if it is ingenerable, neither will the future time every exist nor the past, but one time only remains, then, to say that as time is neither generable nor ingenerable, nor partly generable and partly ingenerable, time does not exist.”

Pyrrhonian Scepticism Against Learning

This comes from Sextus Empiricus in his book “Against the Ethicists”. I broke up the one long argument to put it in a more readable manner.

“These, then, are the objections of a more general character brought forward by the Sceptics to show the non-existence of learning; and it will be possible also to apply these difficulties in turn to the so-called art of life.

For either the wise man will teach this to the wise, or the unwise to the unwise, or the unwise to the wise, or the wise to the unwise. But neither would the wise man be said to teach it to the wise (for both are perfect in virtue and neither of them needs to learn), nor the unwise to the unwise (for both of them have need of learning and neither of them is wise so as to teach the other). Nor yet will the unwise teach to the wise; for neither is the blind man of instructing the man who sees about colours. It only remains, therefore, that the wise man is capable of teaching the unwise; and this too is a matter of doubt.

For if wisdom is “the science of things good and evil and neither,” the unwise man, when the wise man is teaching him the things good and evil and neither, will merely hear the things since he does not possess any wisdom but is in ignorance of all these things. For if he should comprehend them while he is in a state of unwisdom, unwisdom will be capable of knowing things good and evil and neither. But, according to them, unwsidom is not capable of perceiving these things; therefore the unwise man will not comprehend the things said or done by the wise man in pursuance of the rule of his wisdom. And just as he who is blind from, so long as he is blind, has no conception of colours, and he who is deaf from birth, so long as he is deaf, does not apprehend sounds, so also the unwise man, in so far as he is unwise, does not comprehend things wisely said and done. Neither, therefore, can the wise man guide the unwise in the art of life.

– Moreover, if the wise man teaches the unwise, wisdom must be cognizant of unwisdom, even as art is of lack of art; but wisdom cannot be cognizant of unwisdom; therefore the wise man is not capable of teaching the unwise. For he who has become wise owing to some unwise. For he who has become wise owing to some joint exercise and practice (for no one is such by experience (for no one is such by nature) either has acquired wisdom in addition while his unwisdom still subsists within him, or else has become wise through getting rid of the latter and acquiring the former. But if he has acquired wisdom in addition while his unwisdom still subsists within him, the same man will be at once both wise and unwise, which is impossible. And if has acquired the former by getting rid of the latter, he will not be able to know his pre-existing condition, which is not now naturally so;

for certainly the apprehension of every object, whether sensible or intelligible, comes about either empirically by way of sense-evidence or by way of analogical inference from things which have appeared empirically, this latter being either through resemblance (as when Socrates, not being present, is recognized from the likeness of Socrates), or through composition (as when from a man and a horse we form by compounding them the conception of the non-existent hippocentaur), or by way of analogy (as when from ordinary man there is conceived by magnification the Cyclops who was “Less like a corn-eating man than a forest-clad peak of the mountains,” and by diminution the pygmy). Hence, if unwisdom is perceived by wisdom and also the unwise man by the wise, the perception takes place either by experience or by inference form experience. But perception does not take place by experience (for no one gets to know wisdom in the same way as white and black and sweet and bitter), nor by inference from  experience (for no existing thing resembles unwisdom) [But if the wise man makes the inference from experience this, it is either through resemblance or through composition or through analogy];so that wisdom will never perceive unwisdom.

– Yes, but possibly someone will say that the wise man can discern the unwisdom of another by the wisdom within himself; but this is puerile. For unwisdom is a condition productive of certain works. If, then, the wise man sees and apprehends this in another, either he will apprehend the condition directly by means of itself, or by attention to its works he will also get to know the condition itself, just as one knows the condition of the medical man from works in accordance with the arts of medicine, and that of the painter from works in accordance with the art of painting. But he cannot perceive the condition by means of itself; for it is obscure and invisible, and it is not possible to view it closely through the shape of the body; nor by means of the works which result from it; for all the apparent works are, as we showed above, common to wisdom and unwisdom alike. But if it is necessary that the wise man, in order that he may teach the art of life to the unwsise, should himself be capable of perceiving unwisdom-even as the artist lack of art,- and it has been shown that unwisdom is to him imperceptible, then the wise man will not be able to teach the unwise the art of life.”

George Berkeley’s Immediate Perception and Mediated Perception

George Berkeley makes two distinctions based on things presented to the human senses. There are those things immediately perceived those things that are mediately perceived. But it should be made clear, both things mediately and immediately perceied, only exist in a mind.

“Having of a long time experienced certain ideas, perceivable by touch – as distance, tangible figure, and solidity – to have been connected with certain ideas of sight, I do upon perceiving these ideas of sight forthwith conclude what tangible ideas are, by the wonted ordinary course of nature, like to follow. Looking at an object I perceive a certain visible figure and colour, with some degree of faintness and other circumstances, which from what I have formerly observed, determine me to think that if I advance forward so many paces or miles, I shall be affected with such and such ideas of touch. So that in truth and strictness of speech I neither see distance itself, nor anything that I take to be at a distance. I say, neither distance nor things placed at a distance are themselves, or their ideas, truly perceived by sight. This I am persuaded of, as to what concerns myself; and I believe whoever will look narrowly into his own thoughts and examine what he means by saying he sees this or that thing at a distance, will agree with me that what he sees only suggests to his understanding that, after having passed a certain distance, to be measured by the motion of his body, which is perceivable by touch, he shall come to perceive such and such tangible ideas which have been usually connected with such and such visible ideas. But that one might be deceived by these suggestions of sense, and that there is no necessary connexion between visible and tangible ideas suggested by them, we need go no farther than the next looking-glass or picture to be convinced…when I speak of tangible ideas, I take the word ‘idea’ for any the immediate object of sense or understanding, in which large signification it is commonly used by the moderns.”

“But if we take a close and accurate view of things, it must be acknowledged that we never see and feel one and the same object. That which is seen is one thing, and that which is felt is another. If the visible figure and extension be not the same with the tangible figure and extension, we are not to infer that one and the same thing has divers extensions. The true consequence is that the objects of sight and touch are two distinct things.”

“In order therefore to treat accurately and unconfusedly of vision, we must bear in mind that there are two sorts of objects apprehended by the eye, the one primarily and immediately, the other secondarily and by intervention of the former. Those of the first sort neither are, nor appear to be, without the mind or at any distance off. They may indeed grow greater or smaller, more confused or more clear, or more faint, but they do not, cannot, approach or recede from us. Whenever we say an object is at a distance, whenever we say it draws near or goes farther off, we must always mean it of the latter sort, which properly belong to the touch, and are not so truly perceived as suggested by the eye in like manner as thoughts by the ear.”

“No sooner do we hear the words of a familiar language pronounced in our ears, but the ideas corresponding thereto present themselves to our minds. In the very same instant the sound and the meaning enter the understanding; so closely are they united that it is not in our power to keep out the one, except we exclude the other also. We even act in all respects as if we heard the very thoughts themselves. So likewise the secondary objects, or those which are only suggested by sight, do often more strongly affect us, and are more regarded than the proper objects of that sense, along with which they enter into the mind and with which they have a far more strict connexion than ideas have with words. Hence it is we find it so difficult to discriminate between the immediate and mediate objects of sight, and are so prone to attribute to the former what belongs only to the latter” An Essay Towards a New Theory of Vision (Italics are my own emphasis)

This is George Berkeley showing that each of the senses are distinct from one another, and they have no connection to one another. Sight is not sound, or taste, or tactile, or smell. So what we see by sight has no qualities of sound, taste, tactile, or smell. But we still immediately perceive something by sight, but we can mediate it with sound, taste, tactile, or smell. We can combine these two distinct things, and something new comes about, i.e. sight and touch forms distance or 3d vision.

We interweave one sense with another sense, and confound them together into the same “object”. This object can be a tower. We have a visual idea, or immediate perception of it, and we also have a tactile idea, or immediate perception of it. We connect these two distinct ideas together into the same “object” or “idea”. But each of these things are “naturally” distinct from one another and have no connection expect those that we make by connecting them. From all of these immediate perceptions, we mediate them by connecting them together with former connections we have made, and find that what we are experiencing “suggests” the tower getting closer or moving further away, or its height.

It becomes second nature to take the mediated for the immediate, even though it is not immediately given in by that sense (i.e. visual or tactile). It is something that the human mind creates itself, and takes it for being immediately given or suggested by experience. The immediate things are not done by the human mind, but the mediated things are done by the human mind. They are what the human mind adds on to what is immediately perceived by the senses.

One of Berkeley’s big points is that “sensible things are those only which are immediately perceived by sense.” So those things immediately perceived by the senses are ‘sensible things’, and those things that are mediately perceived by the senses are not sensible things. This would mean that depth or 3d are not sensible things. Like seeing someone’s face flush red, sensible thing, does not mean that it is sensible thing that the person had a passion arise in them. It might be “suggested”, but it is not by vision that the conclusion was reached, but because it was mediated by something else, which was itself not immediately perceived.

The example of written word is a great example. There are certain colors and shapes that are seen, and from these things seen one obtains some thoughts. But the thoughts were not themselves found in the visual experience itself. So one mediated what was immediately perceived through something else, which was thought. Vision was mediated with thoughts, which are two distinct things. This is sometimes closer to what is called theory-ladden observation.

Did Popper Solve The Problem of Induction?

Karl Popper said that he believed he had solved the “Problem of Induction”, or what he called “Hume’s Problem”. But did Karl Popper really solve the Problem of Induction or Hume’s Problem? Maybe we should (1) take a look at what Popper considered to be Hume’s problem, and (2) see what Popper says his solution to the problem is. (Whether or not Popper did correctly identify Hume’s problem, is of no concern here).

Before we do this, I think we should start out with something basic, or part of basic, logic.

(A) Universal Quantifier Affirmative (All S are P): For each x, if x is S, then x is P
(E) Universal Quantifier Negation (No S are P) : For each x, if x is S, then x is not P
(I) Existential Quantifier Affirmative (Some S are P): There exists at least one x, such that x is S and x is P
(O) Existential Quantifier Negation (Some S are not P): There exists at least one x, such that x is S and x is not P

“All of the categorical propositions illustrated above can be expressed by using either the universal quantifier alone or the existential quantifier alone. Actually, what this amounts to is the definition of the universal quantification of propositions in terms of existential quantification and the definition of existential propositions in terms of universal quantification.” p. 349 Formal Logic: An Introductory Textbook by John Arthur Mourant

Now this means that the Universal Quantifier (UQ) can be expressed in a logically equivalent form to an Existential Quantifier (EQ), and the Existential Quantifier can be expressed in a logically equivalent form to Universal Quantifier. For something to be logically equivalent means they mean the same thing in a logical sense. Logically equivalent statements have the exact same truth. One can’t be true and the other false, for this would mean they are both necessarily false.

Universal Quantifiers to Existential Quantifiers

A: For each x, if x is S, then x is P  ↔  There does not exist at least one x, such that x is S and x is not P
E: For each x, if x is S, then x is not P  ↔  There does not exist at least one x, such that x is S and x is P
I: Not for each x, if x is S, then x is not P  ↔  There exists at least one x, such that x is S and x is P
O: Not for each x, if x is S, then x is P  ↔  There exists at least one x, such that x is S and x is not P

A: For each x, if x is Crow, then x is Black  ↔  There does not exist at least one x, such that x is Crow and x is not Black
E: For each x, if x is Crow, then x is not Black  ↔  There does not exist at least one x, such that x is Crow and x is Black
I: Not for each x, if x is Crow, then x is not Black  ↔  There exists at least on x, such that x is Crow and x is Black
O: Not for each x, if x is Crow, then x is Black  ↔  There exists at least on x, such that x is Crow and x is not Black

Existential Quantifiers to Universal Quantifiers

A: There does not exist at least one x, such that x is S and x is not P  ↔   For each x, if x is S, then x is P
E: There does not exist at least one x, such that x is S and x is P  ↔   For each x, if x is S, then x is not P
I: There exists at least one x, such that x is S and x is P  ↔  Not for each x, if x is S, then x is not P
O: There exists at least one x, such that x is S and x is not P  ↔   Not for each x, if x is S, then x is P

A: There does not exist at least one x, such that x is Crow and x is not Black  ↔  For each x, if x is Crow, then x is Black
E: There does not exist at least one x, such that x is S and x is P  ↔  For each x, if x is Crow, then x is not Black 
I: There exists at least one x, such that x is Crow and x is Black  ↔  Not for each x, if x is Crow, then x is not Black
O:There exists at least one x, such that x is Crow and x is not Black  ↔  Not for each x, if x is Crow, then x is Black

It needs to be pointed out first that there are two types of statements.
(1)Necessary Truth: Statement whose denial is self-contradictory.
(2) Contingent Truth: One that logically (that is, without self-contradiction) could have been either true or false.

(1a) “All bachelors are unmarried males”
(2a) “Justin Bieber is an unmarried male”

A necessary truth is said to have no empirical content. A contingent truth is said to have empirical content.

Hume’s problem was that he found that he cannot justify induction by demonstrative argument, since he can always imagine a different conclusion.

What Popper takes to be “Hume’s Problem”

“It is usual to call an inference ‘inductive’ if it passes from singular statements (sometimes called ‘particular’ statements), such as accounts of the results of observations or experiments, to universal statements, such as hypotheses or theories. Now it is far from obvious, from a logical point of view, that we are justified in inferring universal statements from singular ones, no matter how numerous; for any conclusions drawn in this way may always turn out to be false: no matter how many instances of white swans we may have observed, this does not justify the conclusion that all swans are white. The question whether inductive inferences are justified, or under what conditions, is known as the problem of induction.” pg. 3-4 Logic of Scientific Discovery

“The root of this problem [of induction] is the apparent contradiction between what may be called ‘the fundamental thesis of empiricism’- the thesis that experience alone can decide upon the truth or falsity of scientific statements- and Hume’s realization of the inadmissibility of inductive arguments.” pg. 20 Logic of Scientific Discovery

Here’s an Inductive argument

Singular: (P1) There exists at least one x, such that x is Crow and x is Black
Singular: (P2) There exists at least one x, such that x is Crow and x is Black
Universal: (C) For each x, if x is Crow, then x is Black

Popper’s Solution to “Hume’s Problem”

“Consequently it is possible by means of purely deductive inferences (with the help of the modus tollens of classical logic) to argue from the truth of singular statements to the falsity of universal statements. Such an argument to the falsity of universal statements is the only strictly deductive kind of inference that proceeds, as it were, in the ‘inductive direction’ that is, from singular to universal statements.”pg. 21 Logic of Scientific Discovery

Here’s Popper’s solution

Universal: (P1) For each x, if x is Crow, then x is not Black
Singular: (P2) There exists at least one x, such that x is Crow and x is Black
Universal: (C) Not for each x, if x is Crow, then x is not Black

Singular statement leads to a universal statement. From there exists at least one x, such that x is Crow and x is Black, the conclusion is reached that not for each x, if x is Crow, then x is not Black.

Here’s Poppers understanding of Induction: “It…passes from singular statements…to universal statements…”

Here’s Poppers solution to the ‘Problem of Induction: “Such an argument to the falsity of universal statements is… from singular to universal statements.”

So going from singular statement to universal statement can be justified by  going from singular statements to universal statements. This falls for the problem of induction again, because this is a circular argument that is used to defend induction.