Truth suffers from too much analysis

Posts Tagged ‘Falsification’

Science Aims for the Improbable

Posted by allzermalmer on July 27, 2013

Karl Popper helped to present the principle that separates scientific statements from non-scientific statements. This separation was based on principle of falsifiability.

One of the things that follows from the principle of falsifiability is that those scientific statements that are highly improbable have more scientific content. Those statements that are improbable say more about world.

The content of an empirical statement is based on simple logical observation.

Suppose that we have the statement (1) “Ravens won Superbowl 35 and Ravens won Superbowl 47”.

This statement is a conjunction, and joins two individual statements. These individual statements, respectively, are (i) “Ravens won Superbowl 35” and (ii) “Ravens won Superbowl 47”.

The content of statement (1) is greater than each part. For example, (i) says more than (ii), and vice versa. The Ravens winning Superbowl 35 doesn’t say anything about winning Superbowl 47, or vice versa.

Here is another example, but a more general example.

(2) All ravens in North America are black.

This statement is a conjunction, in some sense. Because we can say it has three parts. (i) All ravens in US are black, (ii) All ravens in Canada are black, and (iii) All ravens in Mexico are black.

So the content of (1) is greater than “The Ravens won Superbowl 47”, and the content of (2) is greater than “All ravens in US are black”.

Law of Content:

Content of (i) ≤ Content of (1) ≥ Content (ii)

Content of “the Ravens won Superbowl 35” ≤ Content of “the Ravens won Superbowl 35 & the Ravens won Superbowl 47” ≥ Content of “the Ravens won Superbowl 47”.

Content of (1) is greater than or equal to the Content of (i) and the Content of (ii).

Law of probability:

P(i) ≥ P(1) ≤ P(ii)

Probability of “the Ravens won Superbowl 35” ≥ Probability of “The Ravens won Superbowl 35 and the Ravens won Superbowl 47” ≤ Probability of “the Ravens won Superbowl 47”.

Probability of (1) is less than or equal to the Probability of (i) and the Probability of (ii).

So we immediately notice something. When we combine statements, the content increases and the probability decreases. So an increase in probability means a decrease in content, and increase in content means decrease in probability.

Let us work with the second example, i.e. (2) All ravens in North America are black.

This statement would only be true if all the individual parts of it are true. This means that (2) can only be true if all ravens in Canada are black, and all ravens in US are black, and all ravens in Mexico are black. Supposing that all ravens in Canada aren’t black because there exists a raven in Canada that is white, shows that (2) is false. It shows that (2) as a conjunction is false, and shows that (ii) is false. But this doesn’t show that (i) and (iii) are false. All ravens in US or Mexico are black, hasn’t been falsified yet.

(2) would be false because for a conjunction to be true both of it’s parts or conjuncts must also be true. If one of them is false, then the whole conjunction is false.

We immediately find that those empirical statements that have more content are going to have lower probability. And those empirical statements that have lower probability are also easier to falsify. It is easier to find out if they are false, and help us to make progresses.

For example, we find that hypothesis (2) has a lower probability than each of its parts. But finding out that this hypothesis is false by observation will eliminate either one of its conjunctions, like eliminate (ii) and not eliminating (i) and (iii).

These hypothesis not being eliminated means it opens progresses of science. It shows what is false, which informs us of a modification need to make. In making this modification we learn that (i) and (iii) haven’t been falsified. So our new hypothesis would have to contain both (i) and (iii), and also the falsification of (ii).

This new empirical statement would also have more content, it contains (i), ~(ii), and (iii), as part of its content. It contains both what hasn’t been shown false yet, and also shows what has been false.


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A Solipsist Can’t Falisfy their Falsifiable Hypothesis

Posted by allzermalmer on July 27, 2013

Karl Popper’s methodological system of Falsifiability, which is to demarcate between empirical statements or systems of statements from non-empirical statements or systems of statements, relies on empirical statements being “public” or “inter-subjectively criticizable”.

Popper goes on to say that “Only when certain events recur in accordance with rules or regularities, as is the case with repeatable experiments, can our observations be tested- in principle- by anyone. We do not take even our own observations quite seriously, or accept them as scientific observations, until we have repeated and tested them. Only by such repetitions can we convince ourselves that we are not dealing with mere isolated ‘coincidence’, but with events which, on account of their regularity and reproducibility, are in principle inter-subjectively testable.” The Logic of Scientific Discovery pg. 23

How would a Solipsist fit into methodological falsification, as an individual trying to take part in empirical statements? 

One simple answer would be that Solipsist can’t take part in science or produce empirical statements. The solipsist cannot take part in science because their is no discussion to be had. Discussions involve more than one individual, and the Solipsist would be the only individual. One sock short of warm toes.

A solipsist, however, could produce in a weaker version of what Popper presents.

To do this the Solipsist weaker version of methodological falsification would have many things in common, but at least one difference.  The one difference is about the empirical statements for a Solipsist aren’t necessarily “public” or “inter-subjectively testable”.

Empirical statements would have to be contingent statements. A contingent statement is possible true and possibly false. It is possibly true the Ravens won the Superbowl and it is possibly false the Ravens won the Superbowl. So “the Ravens won the Superbowl” is a contingent statement.

Popper’s point about “public” appears to have one thing in common with a solipsist. Popper points out that “We do not take even our own observations quite seriously, or accept them as scientific observations, until we have repeated and tested them.” So a Solipsist would appear to meet this level that is mentioned.

So a Solipsist could make statements that are public, and check to see if the statements end up being shown false by future observations. But the only individual to check for observations that show it is false is the Solipsist. In principle, only the Solipsist could show their own statements are false.

From the obvious principle, it would mean that the Solipsist could not meet the second condition of being “public”, as laid out by Popper. “Only when certain events recur in accordance with rules or regularities, as is the case with repeatable experiments, can our observations be tested- in principle- by anyone.”

The Solipsist may produce a hypothetical system, and check the internal consistency of that system. The Solipsist makes sure that no contradictions may be derived from it, and may also check to see what statements may be derived that can be tested against observations. It finds that no contradictions are derived and may move on to check the system against some observations.

In the processes of looking for some observations, it is guided by the system being of a reproducible nature, and forbidden certain events from happening. So the Solipsist could hold to a statement that says “All x are y”, and goes looking for a single “x and ~y”. Such an observation would show the hypothesis is false.

For all practical purposes, the Solipsist would be going through the same mechanism without being “public” in the full sense of what Popper mentions.



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Superbowl and Falsifiability

Posted by allzermalmer on March 20, 2013

There was a Superbowl a couple of months ago. Now keep in mind that with the rules of NFL and Superbowl, no Superbowl can end in a tie. One of the two teams must win the game. But let us take a look at falsifiability and unfalsifiability.

Suppose that the Superbowl is only between the Baltimore Ravens and the San Fransisco 49ers. You must predict who will win the game.

We can make the claim “The team with the most points will win”. Now this statement is true and cannot be false. This statement is necessarily true. “The team with the fewest points will lose”. This statement is also true and cannot be false, so it is necessarily true. We cannot show that this statement is false, since it is true, so it is unfalsifiabile.

We may now take a further step and predict that either the Ravens will win. This is falsifiable, since we can watch the game and see if the Ravens won or lost. Further, we make take the opposite position and predict that the 49ers will win. We know that either the Ravens win or the 49ers win. Only one of these options can be right.

But we may take a further step and predict who will win the game and what the score will be. We can predict that the Ravens will win and that they will win 14 to 7. We may also do the same with the 49ers. But now we are getting more specific with our predictions. It makes it easier and easier to show that our prediction is wrong. The score could be that the 49er won 14-7, which shows the prediction of Ravens winning 14-7.

Let us further assume that the total amount of points that can be scored in the game is 40 points, and there are a total of 2 teams. We may use the formula of computational probability, nCr = ( n! ) / (r! (n-r)! ). n= total number of subjects, r= number of objects in arrangement, != factorial. Through computation of (40!)/(2!(40-2)!)=780 different possible combination of the =score. This means we can make 780 different predictions. For example, Ravens win 14-7 or 49ers win 14-7, and etc. Maybe the Ravens win 21-14 or the 49ers win 21-14. There are 780 different specific predictions we can make which contain (1) the team that wins of the two, and (2) the score.

Now if we pay attention, we start from the most general and further move down from less general to more and more precise. We can start with a general prediction of “The team with the most points will win”. It is true but it is not falsifiable. We may further move down with a more precise prediction of “The Ravens will win”, which is falsifiable. We may further move down with a more precise prediction of “The Ravens will win 14-7”. Each stage down we go, the claim is more and more falsifiable. It eliminates other possible outcomes, and can be shown if one of these other possible outcome were to be obtained.

We have two general claims that we know are necessarily true and not informative about who will win the game between the Ravens and 49ers, which is 100% you will get the answer correct. We have two further general claims which are not necessarily true and are informative. This claim is general and can be shown to be right or wrong, but it 50% you will get the answer correct. When we further move down to what the score will be, then we have 0.00128% of getting the answer correct. We have 780 possible correct predictions, and only one of them can be correct, so we only have a 0.00128% of getting the correct prediction.

We may make a further specification that you must pick the winner, the points, and how they obtained those points. For example, the only ways to obtain points would be (1) 6 for a touchdown, (2) 3 for field goal, (3) 2 for safety, (4) 1 for extra point. This now means one may say, “The Ravens will win 28-14, with Ravens getting 3 touchdowns, 3 field goals, 1 extra point.” Things are becoming more and more specific, which makes it easier and easier to show it is false.

Suppose we have these claims: (1) All Ravens on Earth are Black, (2) All Ravens on the Northern Hemisphere are Black, (3) All Ravens in North America are Black, (4) All Ravens in the United States are black, (5) All Ravens in California are Black, (6) All Ravens in Los Angeles are black.

These claims are based on the most general of them all and is moving down to more and more precise general claims. We know that If all ravens on earth are Black and all Ravens in Los Angeles are not black, then we know that all ravens on earth are not black. But if all ravens on earth are not black, that does not mean that all ravens in Los Angeles are not black. All the Ravens in Los Angeles can still be black.

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Why Science Doesn’t Invoke Metaphysics

Posted by allzermalmer on November 1, 2012

All those things in italics come from Popper, and those that are in bold & italics  are my own personal emphasis and not Popper’s.

But before I get to that, I want to start out by making one big distinction. There is the distinction between statements that are logically necessary and those that are logically contingent.

Logically Necessary: For each x, if x is logically necessary, then x’s affirmation is logically possible and x’s negation is not logically possible.
Logically Contingent: For each x, if x is logically contingent, then x’s affirmation is logically possible and x’s negation is logically possible.

Popper thinks that things that are Logically Necessary are not in the domain of empirical science. Logically Necessary statements make no claim about reality or what exists, while those things that are Logically Contingent do make claims about reality or what exists. Logically Contingent statements are what empirical science deals with. But from within this domain of Logically Contingent statements, Popper is going to make a distinction.

His distinction is basically this: Not for every statement, if statement is logically contingent, then logically possible for humans to verify that statement is actually true instead of possibly true.

This is because it relies logical distinction between singular statements and universal statements.  “The raven is black in color” or “There exists at least one x, such that x is raven and x is black in color”, are examples of “Singular statements”. They are a proposition that asserts that a particular individual has (or has not) some specified attribute. “All ravens are black in color” or “For every x, if x is raven, then x is black in color”, are examples of “Universal statements”. They are a proposition that refers to all the members of a class. The members of class could have all sorts of particular individual things contained in them, like all ravens that have existed, are existing, or will exist. This can be logically infinite domain in time and space. Singular statements are at specific times and specific places, not all times and all places. So these are logically distinct from one another.

One of the basic points is that sense experience, or observation, is of particular things or individuals. We do not have sense experience, or observation, of all times and places, or all things that have existed, are existing, or will exist. In other words, observation only gives singular statements but science, or empirical science, seeks universal statements that apply to all particular things, for all times and all places. Empirical science is seeking universal statements that apply to singular statements, like universal statements that apply to all particular ravens.

“The fact that theories are not verifiable has often been overlooked. People often say of a theory that it is verified when some of the predictions derived from it have been verified. They may perhaps admit that the verification is not completely impeccable from a logical point of view, or that a statement can never be finally established by establishing some of its consequences. But they are apt to look upon such objections as due to somewhat unnecessary scruples. It is quite true, they say, and even trivial, that we cannot know for certain whether the sun will rise tomorrow; but this uncertainty may be neglected: the fact that theories may not only be improved but that they can also be falsified by new experiments presents to the scientist a serious possibility which may at any moment become actual; but never yet has a theory had to be regarded as falsified owing to the sudden breakdown of a well confirmed law. It never happens that old experiments one day yield new results. What happens is only that new experiments decide against an old theory. The old theory, even when it is superseded, often retains its validity as a kind of limiting case of the new theory; it still applies, at least with a high degree of approximation, in those cases in which it was successful before. In short, regularities which are directly testable by experiment do not change. Admittedly it is conceivable, or logically possible, that they might change; but this possibility is disregarded by empirical science and does not affect its methods. On the contrary, scientific method presupposes the immutability of natural processes, or the ‘principle of the uniformity of nature’.

There is something to be said for the above argument, but it does not affect my thesis. It expresses the metaphysical faith in the existence of regularities in our world (a faith which I share, and without which practical action is hardly conceivable).*1 Yet the question before us— the question which makes the non-verifiability of theories significant in the present context—is on an altogether different plane. Consistently with my attitude towards other metaphysical questions, I abstain from arguing for or against faith in the existence of regularities in our world. But I shall try to show that the non-verifiability of theories is methodologically important. It is on this plane that I oppose the argument just advanced.

I shall therefore take up as relevant only one of the points of this argument—the reference to the so-called ‘principle of the uniformity of nature’. This principle, it seems to me, expresses in a very superficial way an important methodological rule, and one which might be derived, with advantage, precisely from a consideration of the non-verifiability of theories.*2 (I mean the rule that any new system of hypotheses should yield, or explain, the old, corroborated, regularities. See also section *3 (third paragraph) of my Postscript.

Let us suppose that the sun will not rise tomorrow (and that we shall nevertheless continue to live, and also to pursue our scientific interests). Should such a thing occur, science would have to try to explain it, i.e. to derive it from laws. Existing theories would presumably require to be drastically revised. But the revised theories would not merely have to account for the new state of affairs: our older experiences would also have to be derivable from them. From the methodological point of view one sees that the principle of the uniformity of nature is here replaced by the postulate of the invariance of natural laws, with respect to both space and time.  I think, therefore, that it would be a mistake to assert that natural regularities do not change. (This would be a kind of statement that can neither be argued against nor argued for.) What we should say is, rather, that it is part of our definition of natural laws if we postulate that they are to be invariant with respect to space and time; and also if we postulate that they are to have no exceptions. Thus from a methodological point of view, the possibility of falsifying a corroborated law is by no means without significance. It helps us to find out what we demand and expect from natural laws. And the ‘principle of the uniformity of nature’ can again be regarded as a metaphysical interpretation of a methodological rule—like its near relative, the ‘law of causality’.

One attempt to replace metaphysical statements of this kind by principles of method leads to the ‘principle of induction’, supposed to govern the method of induction, and hence that of the verification of theories. But this attempt fails, for the principle of induction is itself metaphysical in character. As I have pointed out in section 1, the assumption that the principle of induction is empirical leads to an infinite regress. It could therefore only be introduced as a primitive proposition (or a postulate, or an axiom). This would perhaps not matter so much, were it not that the principle of induction would have in any case to be treated as a non-falsifiable statement. For if this principle— which is supposed to validate the inference of theories—were itself falsifiable, then it would be falsified with the first falsified theory, because this theory would then be a conclusion, derived with the help of the principle of induction; and this principle, as a premise, will of course be falsified by the modus tollens whenever a theory is falsified which was derived from it. *3 (The premises of the derivation of the theory would (according to the inductivist view here discussed) consist of the principle of induction and of observation statements. But the latter are here tacitly assumed to be unshaken and reproducible, so that they cannot be made responsible for the failure of the theory.) But this means that a falsifiable principle of induction would be falsified anew with every advance made by science. It would be necessary, therefore, to introduce a principle of induction assumed not to be falsifiable. But this would amount to the misconceived notion of a synthetic statement which is a priori valid, i.e. an irrefutable statement about reality. Thus if we try to turn our metaphysical faith in the uniformity of nature and in the verifiability of theories into a theory of knowledge based on inductive logic, we are left only with the choice between an infinite regress and apriorism.” The Logic of Scientific Discovery pg. 249-252

Popper is trying to make the distinction between a metaphysical principle and a methodological principle. He is trying to point out that science is a methodology without metaphysical principles. The line of demarcation between science and metaphysics is falsifiability or refutability.  He holds that “we must choose a criterion which allows us to admit to the domain of empirical science even statements which cannot be verified.” (pg. 18) Popper’s line of demarcation for statements that are allowed into science, or more specifically universal statements allowed into empirical science. “But I shall certainly admit a system as empirical or scientific only if it is capable of being tested by experience. These considerations suggest that not the verifiability but the falsifiability of a system is to be taken as a criterion of demarcation.*3 In other words: I shall not require of a scientific system that it shall be capable of being singled out, once and for all, in a positive sense; but I shall require that its logical form shall be such that it can be singled out, by means of empirical tests, in a negative sense: it must be possible for an empirical scientific system to be refuted by experience.” (pg. 18)

We can verify singular statements, it is logically possible for us to find out if that statement is true. If we have not verified that it is actually true, we cannot infer that it is actually false. It is still logically possible that it is true. So we find out that we can, at least in principle, verify the truth of a singular statement. However, it is not logically possible for us to affirm a universal statement, like empirical claims of science. However, we can show that they are false. We cannot verify them but we can falsify them. We falsify these universal statements with one singular statement, or one observation, which the universal statement does not logically allow for, i.e. says is not logically possible to be true if the universal statement is true. This can be shown by simple modus tollens.

Universal Statement: All ravens are black.
Singular Statement: This raven is white.
Conclusion: Some ravens are not white.


Universal Statement: No ravens are not black.
Singular Statement: This raven is not black.
Conclusion: Some ravens are not black.


Universal Statement: For each x, if x is a raven, then x is black.
Singular Statement: There exists at least one x, such that x is a raven and x is not black.
Conclusion: Not each x, if x is raven, then x is black.

What needs to be kept in mind that the Universal statement has a logical equivalent as “No ravens are not black.” So it logically excludes a raven that is white, since white is the logical opposite of black, so it is not black.

Popper shows that if we do accept a metaphysical principle (i.e. a universal statement) which is logically contingent, then it means it is possibly true or possibly false. And if we choose to invoke a metaphysical principle in our science, and we derive another universal statement from it, then when that derived universal statement is refuted by observation, then the universal statement and the one it was derived from are shown to be false. For example, assume that “All ravens on Earthare black” is a metaphysical principle. We may derive that “All ravens on Earth in  in the United States are black”. When we observe that one particular raven on Earth in the United States is not black, which means that “All ravens on Earth in the United States are black” and “All ravens on Earth are black” are false.

Metaphysical Statement: All ravens on Earth are black.
Scientific Statement: All ravens on Earth in the United States are black.
Observation: This raven on Earth in the United States is not black.
Conclusion: Not all ravens on Earth in the United States are black & Not all ravens on Earth are black.

This means that if someone believes that science holds to the metaphysical principle of induction, then it was shown to be false by scientific theories that are false. Now as a methodology there is nothing wrong with holding to it, because methodology makes no truth claim itself. Also, the example of causality is an example, if we take it as a metaphysical principle that science is based on. So this would mean that science would hold to this metaphysical principle and derive other statements from this principle and test them with experience or observation. From this we find that one of our theories made a false prediction, which means that the metaphysical principle of causality has been shown to be false by experience as well, and all other theories that were derived from the metaphysical principle, but have not been shown false yet, would also by logical implication be false. The same thing would hold with naturalism, physicalism, materialism, dualism, or the world is parsimonious or simple, or determinism, or indeterminism, or presentism and eternalism, and etc.

Now science, or experience, would have never been able to verify these metaphysical principles in the first place. There would be no support for them to be derived from experience. It would still be logically possible for them to be true, but we cannot find out if they are actually true. Experience cannot help us to figure out if they are actually true or possibly true, no matter the amount of observations we make that are consistent with them. But science may use methodological principles in its activities, but holding to those methodological principles does not mean that one is logically obliged to hold to the metaphysical principles.

What is even more interesting is that if we do try to make some sort of inductive argument, we could argue that since science has used metaphysical principle x, and science continually comes up with false theories, or refuted theories, it will continue to derive false theories from that metaphysical principle. But of course, once something was refuted we have shown that it is logically impossible to be true. However, we can still use it and we may derive “true” theories, or theories that have not been shown to be false by observation, yet. This is because anything follows from a logical contradiction. This means you can derive both true statements and false statements. So it would not be surprising if the metaphysical principle also helped you to derive theories that have not been shown false by observation as of yet (even though still logically possible to be shown false with next observation).

Here is an example from basic logic which will rely on two basic rules of logical inference. These two rules are Disjunctive Addition and Disjunctive Syllogism.

Rule 1 – Disjunctive Addition: Given that a statement is true, we can infer that a disjunction comprising it and any other statement is true, because only one disjunct needs to be true for the disjunctive compound to be true.

Premise: It is snowing
Conclusion: Either it is snowing or it is raining

Rule 2 – Disjunctive Syllogism: Because at least one disjunct must be true, by knowing one is false we can infer that the other is true.

Premise: Either the New York Yankees will win the pennant or the Baltimore Orioles will.
Premise: The Yankees will not win the pennant.
Conclusion: Therefore, the Orioles will win the pennant.

For it can easily be shown that these rules permit us to deduce from a pair of contradictory sentences, for instance, from the two sentences,  ”  The sun is shining ” and “The sun is not shining “, any sentence whatsoever.  Let us take these two premisses (a) “The sun is shining”  (b) “The sun is not shining “.  We can deduce with the help of rule (1) from the first of these premisses, the following sentence:”The sun is shining or Caesar was a traitor “. But from this sentence, together with the second premiss (b), we can deduce, following rule (2), that,Caesar was a traitor. And by the same method we can deduce any other sentence. This is extremely important, for if we can deduce any sentence whatsoever, then, clearly, we can always deduce any negation of any sentence whatsoever: It is clear that instead of the sentence “Caesar was a traitor ” we can, if we wish, deduce “Caesar was not a traitor “. In other words, from two contradictory premisses, we can logically deduce anything, and its negation as well. We therefore convey with such a contradictory theory-nothing. A theory which involves a contradiction is entirely useless, because it does not convey any sort of information.”

Logically possible Affirmation: The sun is shining.
Logically possible Negation: The sun is not shining.

The sun is shining. Therefore, by rule 1, The sun is shining or Ceasar was a traitor. But now the sun is not shining. Therefore, by rule 2, Ceasar was a traitor; The sun is not shinning. Therefore, by rule 1, The sun is not shinning or Ceasar was not a traitor. But now the sun is shinning. Therefore, by rule 2, Ceasar was not a traitor. Rule 1 allows you to pull up any premise you want, and be able to affirms this premise and also negate this premise by using Rule 2. So if you affirm a logical impossibility, anything and everything you want follows. They contain no “content” or “information” for empirical science. This is because empirical science wants to eliminate theories because they said something cannot happen and it was found that it did happen. Since there is a contradiction, we know it is logically impossible for the theory to be true.

This process of elimination, though, does not tell you which theories are true. It just says what is not true. There are still many other logically possible universal statements that have not been eliminated by singular statements, or observations, as of yet.

(This will be updated at least 24 hours after posting or publication). Edits need to be done.

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Did Popper Solve The Problem of Induction?

Posted by allzermalmer on October 3, 2012

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
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 
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
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.

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Difference Between Verification and Falsification

Posted by allzermalmer on September 30, 2012

Karl Popper developed the idea that the demarcation between empirical statements, which was mostly taken to be scientific statements, and metaphysical statements was based on the idea of falsification. Popper was speaking out, or presenting, a different criterion to differentiate between empirical statements and metaphysical statements.

“The problem of induction arises from an apparent contradiction between the basic empiricist requirement (only experience can decide the truth or falsity of a scientific statement) and Hume’s insight into the logical impermissibility of inductive decision (there is no empirical justification of universal statements). This contradiction exists only if we assume that empirical statements must be empirically “fully decidable”, that is, that experience must be able to decide not only their falsity, but also their truth. The contradiction is resolved once “partially decidable” empirical statements are admitted: Universal empirical statements are empirically falsifiable, they can be defeated by experience.” The Two Fundamental Problems of the Theory of Knowledge

Verification meant that empirical statements, or scientific statements, are those that it is possible be decided to be true or false by experience. You can fully decide that the statement is true because experience has shown the statement is true. Like experience can show that “this apple is red in color”, so too can experience show the statement that “all apples in the refrigerator are red in color”. The refrigerator is in a specific place, at a specific time, and logically possible to see if all the apples in the refrigeration  are red in color. It can be opened and found that all the apples are red in color, or that all but one of the apples in the refrigerator are red in color, like one can be yellow. Thus, it is both logically possible to empirically verify the statement or empirically falsify the statement. It is logically possible to either show it is true or show it is false.

However, the statement that “all apples in refrigerators are red in color” is logically impossible to empirically verify. This is because this universal statement applies to all times and all places, while the previous universal statement applies to a specific time and specific place.Thus, this universal statement cannot be verified, but it can still be empirically falsified. You might not be able to check all the refrigerators that will, or have, existed in all places or all times, but those that you have observed have the empirical possibility of showing the statement to be false. You might not be able to check all refrigerators in all places and times, but finding a specific refrigerator that has a yellow apple, shows that all refrigerators, in all times and place, do not have all red apples in them. One case has been found to run counter to the universal claim. Thus, we learn that some refrigerators have only red apples in color and some refrigerators have yellow apples in color.

The point becomes that science can introduce whatever universal statement it wants, so long as it is logically possible to make one empirical observation to show it is false. We do not have to show that what it introduces is true by experience, just that it can make predictions that are logically possible to show false by experience.

Let us imagine that there is a person who walks amongst us, and this person knows all the laws of nature. Let us also assume that it is a trickster like Loki. It mixes some truth with some falsity, knowingly. It decides to come up with a falsifiable statement, which means that it is not fully decidable, i.e. it is partially decidable. It knows that this universal statement is false, but it still makes predictions that are possible to be shown false by experience. This being that is like Loki knows that all attempted experiments to show that statement is false by experience will fail, which means it passes every single experimental test that can be presented. You would be justified in accepting a false statement because you cannot show it is true but you can show it is false.




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Strong Inference: The Way of Science

Posted by allzermalmer on September 27, 2012

This is a copy of an article from the journal The American Biology Teacher;  Vol. 65, No. 6 (Aug., 2003), pp. 419-424. The article is called Strong Inference: The Way of Science, by Thomas B. Kinraideand R. Ford Denison. You can read the article here.

“Valentine: It may all prove to be true.
Hannah: It can’t prove to be true, it can only not prove to be false yet.
Valentine: (Pleased) Just like science.
– From “Arcadia,” a play by Tom Stoppard

Science teachers and science textbooks commonly introduce students to the scientific method in elementary and junior high school, but the study of scientific method and philosophy can be a life-long endeavor. Our essay concentrates on a particular aspect of the scientific method -the testing of hypotheses. Concepts of hypothesis testing have changed even within the relatively short period of modern science. Specifically, the concept of proof has been abandoned for reasons we shall describe. Although we can not prove hypotheses, we can almost certainly disprove some hypotheses, if they are false.

To describe the modern method of hypothesis testing, we borrow the term “strong inference” from John R. Platt’s Science (1964) essay by the same name. In brief, strong inference is the method of testing a hypothesis by deliberately attempting to demonstrate the falsity of the hypothesis. A hypothesis that repeatedly withstands attempts to demonstrate its falsity gains credibility, but remains unproven. We are confident that our essay reflects the thinking of most scientists that hypotheses are potentially disprovable but not provable. Nevertheless, we qualify these views somewhat, arguing that neither proof nor disproof is certain.

Strong inference is an avenue to knowledge that is systematically applied in cience, but some practice of strong inference has occurred in human endeavors for thousands of years. For example, courts of law in ancient civilizations occasionally used elements of strong inference – facts were assembled from physical evidence and the testimony of witnesses; hypotheses  were developed (only the grand vizier could have stolen the documents); and impossible or illogical consequences of the hypotheses were grounds for rejecting  the hypotheses (an alibi would establish the grand vizier’s innocence) Nevertheless, former and present methods of inference sometimes differ significantly- an ancient magistrate may have awaited a ghostly visitation during which the truth of a case would be revealed; the body of an accused witch may have been examined for incriminating marks; and confessions may have been extracted by torture. [This mixture of strong inference and alternative methods is described in tales of the historical Chinese magistrate, Judge Dee, by the Dutch diplomat and scholar Robert Van Gulik (1976).]

Even today, people rely upon alternative avenues to knowledge that may include intuition, revelation, and adherence to authority. We are reluctant still to use strong inference outside of enterprises that are recognizably scientific, and the application of strong inference to some beliefs may be impossible. Even when strong inference is possible, its application may be uncomfortable, and its application to the beliefs of others may be considered hostile. Challenges to authority and received wisdom may seem disloyal or arrogant. This reluctance to use strong inference follows understandably from the requirement that belief (or hypotheses) be subjected to deliberate attempts to demonstrate the falsity of the beliefs and by formulating and testing competing beliefs. Nevertheless, strong inference can be practiced with civility and can do much to offset our prejudices and natural gullibility.

A Definition of Hypothesis

Because the formulation and testing of hypotheses are at the heart of strong inference, we will present a definition of hypothesis here, however, a detailed discussion of hypotheses will be delayed until some other terms, incorporated in the definition, are considered. For the definition of hypothesis, and most other terms, we have consulted Webster’s Third New International Dictionary, Unabridged (Gove, 1976)

Hypothesis: [An explanatory] proposition tentatively assumed in order to draw out its logical or empirical consequences and so test its accord with facts that are known or may be determined.

Inevitably, the burden of definition is shifted to other words. In the present case, “fact” is one of those words. Strong inference ultimately rests upon facts, and facts and hypotheses are sometimes confused with each other. Therefore, we shall consider first the concept of fact.

The Concept of Fact

Fact: An occurrence, quality, or relation the reality of which is manifest in experience or may be inferred with certainty.

Here, too, the burden of definition is shifted to other words, among them, “experience” and “reality”. To deal with these terms we must concede that science rests upon a few basic assumptions. Science assumes that nature has a reality independent of the human mind, and science assumes that the human mind can grasp the reality of nature. These epistemological issues are rarely considered in the ordinary practice of science.

Manifest Fact & Inferential Fact

The definition of fact indicates the existence of two kinds of fact- manifest fact and inferential fact. Again, some definitions may be helpful.

Manifest: Capable of being easily understood or recognized at once by the mind: not obscure: obvious.

Inference: The act of passing from one or more propositions…considered as true to another the truth of which is believed to follow from that of the former.

Manifest facts are not highly dependent upon inference. We will call a fact that is highly dependent upon inference an inferential fact. To illustrate inferential and manifest facts, consider the case of a forest fire. If the fire occurred recently, then its occurrence is likely to be a manifest fact. It may have been observed by hundreds of people, and newspaper readers and television viewers are certainly being reasonable in accepting the occurrence of the fire as a manifest fact.

What if the fire had occurred 200 years ago? Most scientist would accept as fact (inferential fact) that a fire had occurred in an area if several observations pointed, convergently, toward a fire. These observations might include the absence of any trees in the area older than 200 years (despite the presence of older trees in surrounding areas), the scarcity or absence of old wood on the forest floor, and the presence of an ash layer beneath the recent leaf and twig litter. Perhaps none of these observations was convincing by itself (the ash may have been blown in from another fire some distance away). Convergence of evidence is the clincher.

In some cases, facts and hypotheses may be confused, but confusion may be avoided by remembering that a hypothesis is a candidate explanation, not a candidate fact. The statement “The Earth is spherical” in ancient times was a candidate fact, and in the present age of satellite photographs, and other evidence, the statement may be regarded as a manifest fact. The statement was also a hypothesis in ancient times, but only when used as an explanation for some other observation. Thus the statement “vertical objects cast shadows of different length at different latitudes because the Earth is spherical” is a hypothesis (a candidate explanation) and not merely a candidate fact. If we confuse a candidate fact for a hypothesis, then we may conclude mistakenly that hypotheses are provable.

Scientific Facts are Public

Another feature of scientific facts is that they are public; that is, a fact (especially a manifest fact) is accessible to all competent observers. The issue of competence is sometimes problematical. In science, public accessibility to facts is crucial even though comprehension of the facts is not always easy. The devotees of mystery cults may be entitle to both their own private opinions and their own private facts, but science disallows private facts.

The Concept of Hypothesis

“Science” and “strong inference” are not synonymous. Science is both a method and a body of knowledge. Facts can be compiled and many questions can be answered without the formulation and testing of hypotheses. Natural history inventories (lists of birds, plants, minerals, and other items) play a role in science and in society. The answer to some questions (What is the speed of light?) may require high technical skill but can be answered without the formulation of hypotheses. In some cases, laws of nature may be formulated without the explicit testing of hypotheses. (Laws are descriptive, often quantitative, but not explanatory, statements having a value intermediate between fact and hypothesis. Examples are Ohm’s law [I=V/R], Newton’s law of motion [e.g. F=ma], and the law of conservation of charge.)

Despite the possibility of some success in science without the testing of hypotheses, science attempts to do more than just compile and describe. Science attempts to explain. This requires the formulation of hypotheses in a creative process that may require the investigator to think beyond readily available explanations. A good hypothesis must be explanatory, but it must have another feature too: It must be testable by strong inference. If it is false, it must be possible to show that it is false.

A Case Study of Hypothesis Testing

A textbook that one of us (T.B.K.) assigned years ago as a college professor was The Study of Biology, 3rd Edition (Baker & Allen, 1977). The first two chapters of that book, The Nature and Logic of Science and Testing Hypotheses and Predictions, are excellent.The following case study was taken from that book.

The Pacific salmon Oncorhyncus kisutch hatches in streams in the Northwest, swims to the sea, then eventually, returns to streams to spawn. We may ask, and answer, the question “Do individual fish return to the stream of their birth?” without formulating an explanatory hypothesis. Tagging experiments have confirmed the fact that the fish predominantly do return to their natal streams. In order to determine how the fish do this, we can proceed in one of two ways. We can continue to study the fish, compiling facts in the hope that an answer may emerge. Sometimes “fishing expeditions” such as these can lead to serendipitous results, but eventually strong inference (hypothesis formulation and testing) is usually needed.

Platt, in the Science article cited above, makes an important suggestion: Formulate more than a single hypothesis. With more than one hypothesis, the investigator is less likely to adopt a “pet” hypothesis to which he/she becomes emotionally attached, and the necessary attempt to demonstrate the falsity of the hypotheses is less worrying- perhaps one will survive. Incidentally, the negation of a significant hypothesis is a significant contribution to science.

In our case study, two hypotheses as to how salmon find their way back to their natal streams might be these:

1. Salmon find their way back by using their sense of sight.
2. Salmon find their way back using their sense of smell (detecting dissolved substances from their birth streams).

Hypotheses are formulated on the basis of prior knowledge, and we know that fish both see and smell. The hypotheses just stated were rather obvious possibilities, but the formulation of hypotheses may be very difficult. The observations for which an explanation is sought may be a very strange (divorced from ordinary experience). Sometimes a hypothesis may be formulated that seems very good because it is compatible with almost all of existing knowledge, but not all of it. In that case, we must consider that the hypothesis, however attractive, may be wrong or that some of the accepted knowledge is wrong.

The next step in strong inference is to test the hypotheses. That is done by deliberately subjecting them to jeopardy, that is, by attempting to demonstrate their falsity. In our fish story, each of the two hypotheses has logical consequences that give rise to predictions as to the outcome of certain experiments. The hypotheses and the predictions are often stated together in if…then… statements. It is very important to make these statements explicit. Such a formulation applied to our example may be “if salmon find their way back using their sense of sight, then salmon with shielded eyes (black plastic discs were used in an actual experiment) will predominantly fail to find their birth streams.” The salmon did, in fact, find their way back in experiment, and the hypothesis was thus considered to be false. The alternative was tested after formulating the statement “If salmon find their way back using their sense of smell, then salmon with a blocked sense of smell (benzocaine ointment was used) will predominately fail to find their birth streams.” This prediction came true, and the second hypothesis was regarded as supported, but not proved.

The Impossibility of Proof

The problem is that even false hypotheses may sometimes give rise to correct predictions. For example, consider the false hypothesis that salmon find their way back to their birth streams by the sense of sight. This gave rise to the prediction that sightless salmon will predominantly fail to find their birth streams. This prediction turned out to be incorrect in the experiment cited earlier, but conceivably the prediction could have been correct. Suppose blindfolded salmon were so traumatized by the blindfolding operation that they did not try to return or that they became so confused without their sight that they ignored their sense of smell and swam off randomly from their release site. In such cases the prediction would have been correctly fulfilled. Is the hypothesis in that case “proved?” Certainly not, though the investigators may claim support for their sight hypothesis if they failed to observe the trauma or the confusion.

A logical truth table presented by Baker and Allen, and others, shows the relationship.

According to the table, an incorrect prediction always corresponds to a false hypothesis, but a correct prediction can come from either true or a false hypothesis. Because of these relationships, hypotheses are often regarded as potentially disprovable (falsifiable) but rarely proveable. How then do some hypotheses come to be regarded as true?

A hypothesis is supported, but not proved, when repeated attempts to negate the hypothesis fail, when competing hypotheses are discredited, and when additional facts (not used in the initial development of the hypothesis) are successfully embraced by the hypothesis.

In the case of the fish, the smell hypothesis withstood an opportunity for disproof, and the competing sight hypothesis was disproved. Still, the smell hypothesis is not proved. Perhaps smell plays no role, and a third sense is the key. Perhaps the benzocaine treatment so traumatized the fish that they could not function properly, or perhaps the benzocaine knocked out the third sense. These worried lead to additional hypotheses, predictions, experiments, and facts.

Another way considering the general unprovability of hypotheses is that no hypothesis can be considered proved if an alternative hypothesis, that excludes the possibility of the first hypothesis and is equally compatible with the facts, is possible. Since we can never be sure that we have considered all possible hypotheses, proof remains unattainable.

Earlier, we stated that a hypothesis is a candidate explanation, not a candidate fact. The case of the salmon provides an illustration of the difference. Early on, people may have observed that the salmon in a particular stream were physically similar to each other and different from salmon in another, distant stream. A couple of hypotheses may be stated:

1. Only salmon of a particular body type are able to navigate a particular stream and that is why they look alike.
2. Salmon return to their natal streams to spawn and look alike because they are genetically similar.

The “fact” that salmon do return to their natal streams establishes the truth of the statement “Salmon return to their natal streams,” but this statement was a candidate fact, not a hypothesis, and the second hypothesis remains unproved.

The Uncertainty of Disproof

Although scientists often refer to the disprovability of hypotheses (as we have), we contend that disproof is uncertain also. The reason for this requirement for the prediction of logical consequences in the testing process, but we can never be certain that our predicted consequences are logical. As an example let’s return to one of our if…then… statements. “If salmon find their way back using their sense of smell, then the Red Sox will win the World Series.” If the Red sox failed to win, we should have concluded falsely, that the hypothesis was false.

The Red Sox example used a preposterously illogical prediction, but some illogical predictions are not so obviously illogical, and the problem is not trivial in some cases. Sometimes scientists disagree over the cogency of a predicted outcome, especially in complex situations where variables are hard to control (see The Triumph of Sociobiology by John Alcock [2001] for interesting discussions of some uncertainties and controversies). An outcome that constitutes adequate grounds for the rejection of a hypothesis for one investigator may be viewed as inadequate by another investigator. The problem of the illogical prediction can be illeviated by testing additional predictions and by the public critique of the methods and conclusions. (The initial stage of public critique is the expert “peer review” of scientific manuscripts prior to publication. See the Acknowledgement in this essay.) Despite the uncertainty of disproof, scientists accept the qualified use of terms such as “disproof”, “falsification,” and “negation,” but not the term “proof”.

The Concept of Theory

When a hypothesis has undergone very extensive testing, especially if the testing attacked the hypothesis from many different angels using independent lines of evidence, then the hypothesis may graduate to the status of theory or, together with other hypotheses and principles, become incorporated into a theory. A dictionary definition of theory is this:

Theory: The coherent set of hypothetical, conceptual, and pragmatic principles forming the general frame of reference for a field of inquiry.

The term theory implies that the component hypotheses are very likely to be true and that together are important and comprehensive. Theories, like well-supported hypotheses, give rise to predictions that are consistently correct, but in the case of theories the range of predictions is often wider than the range of predictions for hypotheses. Theories come to provide a conceptual framework for scientific thought. Some examples include The Atomic Theory, The Theory of Evolution, The Germ Theory of Disease, The Theory of Relativity, and The Quantum Theory. Despite their high status, theories are still hypothesis-like (perhaps we could call them metahypotheses), and as such they are necessarily vulnerable. That is, they must be testable, and potentially falsifiable.

Will Strong Inference Always Work?

Some issues that would seem to be accessible by strong inference remain controversial because of emotional involvement, inadequacy of definitions, or a variety of technical difficulties. For example, a few scientists and public policy makers refute to acknowledge that HIV is the causative agent of AIDS, and the causes, and even the occurrence, of global warming remain controversial.

For many people, science is not the only pathway to knowledge. For them, propositions may rest upon personal revelation or upon religious authority, to cite just two additional pathways to knowledge. For the faithful, faith propositions are considered to be truths, not hypotheses. With regard to the term hypothesis, believers and scientists are in agreement. In most cases, neither scientists (many of whom are religious) nor religious believers (some of whom are scientists) consider religious beliefs to be hypotheses; believers because they consider applying the term to religious teachings to be belittling, and scientists because the term hypothesis can be applied only to statements that their adherents are willing to subject to possible disproof.

Although not scientific, faith propositions are not necessarily in conflict with science, but they may be. A tenet of faith that cannot be accessed by strong inference because it is beyond the technical or epistemological scope of science is not in conflict with science. Examples include doctrines that claim consciousness in inanimate objects, a purpose to life, or rewards or punishments after death. Science cannot now address these propositions, although it may be able to do so in the future (formerly, only faith, not science, could address such issues as the cause of disease, the change of seasons, and the formation of stars).

Some faith propositions are clearly in conflict with science. A tenet of faith that can be accessed by strong inference may be, but is not necessarily, in conflict with science. The indigenous religion of Hawaii provides a fascinating case study. At the time of European discovery, Hawaiian society was encumbered by hundreds of taboos whose violation was though to ensure calamity for individuals and society (Malo, 1959). This religion disintegrated quickly as Hawaiians observed that Europeans (and Hawaiians influenced by Europeans) could violate the taboos and live to tell about it. The Hawaiian nobility quickly embraced the religion of the Europeans and ordered the destruction of idols and the abandonment of many taboos. The causes of this religious transition are complex, but the obvious conflict between reality and some of the faith propositions surely played a role.

A Summary of Strong Inference

1. Observed and inferred facts inspire a question.

2. The question inspires one (or preferably more) hypotheses. This is a creative process. Several hypotheses may be proposed, and they need not have a high likelihood of being supported, but a good hypothesis must be an explanatory statement that is testable.

3. The hypotheses are deliberately subjected to jeopardy (falsification) by, first, stating the logical consequences of the hypotheses. Statements in the form “if (the hypothesis), then (the consequences)” are useful.

4. Next, the accuracy of the predicted consequences are tested by the acquisition of new facts from experimentation, or observation, or from the body of known facts not already used to formulate the hypotheses.

5. Incompatibility between prediction and outcome leads to the rejection of hypotheses, and compatibility leads to tentative acceptance. In all cases, repeated incompatibility or compatibility from separate lines of testing is desirable.

6. The hypotheses, together with the facts and the record of the inferential process, are submitted to public scrutiny and may become accepted into the body of public knowledge.

7. An accepted hypothesis typically spawns the acquisition of more facts and the formulation of new hypotheses (perhaps by the critics of the old hypothesis). These ongoing exercises in strong inference may cause the revision or rejection of the accepted hypothesis.”

8. A hypothesis, or more often a collection of complementary hypotheses, may become incorporated into a theory.

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