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Truth suffers from too much analysis

Posts Tagged ‘Falsifiability’

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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Max Tegmark and Multiverse Hypothesis

Posted by allzermalmer on May 26, 2013

Max Tegmark, a theoretical physicist that teaches at the Massachusetts Institute of Technology, has proposed that hypothesis that “all logically acceptable worlds exist“. Not only has Max Tegmark proposed this hypothesis itself, he believes that it is an empirical hypothesis or scientific hypothesis.

Possibly and Necessarily: Modal Logic

Before I go into some of the ideas proposed by Tegmark, I will first go into a rough sketch of a form of logic known as Modal logic. More specifically, this form of modal logic is known as the S-5 system of modal logic and was formally created by Clarence Irving Lewis, C.I. Lewis. This system of logic plays off of the ideas of possible and necessary discussed about by Gottfried Wihelm von Leibniz, G.W. Leibniz.

Possible and Necessary are interchangeable, or we may define one based on the other. We may define them as so:

(1) Necessarily so if and only if Not possibly not so
(2) Possibly so iff Not necessarily not x so
(3) Not possibly so iff Necessarily not x so
(4) Possibly not so iff Not necessarily so

Truth is defined based on Necessary and Possible, which is done by Possible Worlds. A statement is Necessary if it is true in every possible world. A statement is Possible if it is true in some possible world.

There are some axioms in Modal Logic, and one of them is what I shall call NP: Whatever is necessarily so is actually so. It is necessarily so implies it is actually so. If it is necessarily so then it is actually so.

NA, in conjunction with some other axioms of modal logic and some rules of inference, is a theorem derived in modal logic. This theorem I shall call AP: Whatever is actually so is possibly so. It is actually so implies that it is possibly so. If it is actually so then it is possibly so.

One inference of Modal Logic is what I shall call GR: Whatever is provably so is necessarily so. It is provably so implies it is necessarily so. If it is provably so then it is necessarily so.

One comment is required of GR. Pythagorean Theorem is provably so, and in fact has been proved to be so, so it is necessarily so. It was proved based on a formal system known as Euclidean Geometry, which has its own definitions, axioms, and rules of inference. From these we are able to prove some statements. These proved statements show that it’s negation is not possible, and so the processes of elimination leads to that proved statement necessarily being so.

(GR) Whatever is provably so is necessarily so; (NP)Whatever is necessarily so is actually so; Thus Whatever is provably so is actually so. This in turn means that AP is actually so since it was proved like the Pythagorean Theorem was proved. Since AP being provably so implies AP is necessarily so. And since AP is necessarily so, AP is actually so.

All that is logically possible to be the case is actually the case

Max Tegmarks hypothesis is the converse of AP. We may call this MH: Whatever is possibly so is actually so. It is possibly so implies it is actually so. If it is possibly so then it is actually so.

We may thus assume MH is true and assume that AP is true. Since both of these are assumed true, they form a logical equivalence. We may call this *MH*: Whatever is actually so is possibly so if and only if Whatever is possibly so is actually so. If it is actually so implies it is possibly so then  it is possibly so implies it is actually so.

Max Tegmark presents his hypothesis, similar to how Albert Einstein presented Special Relativity, by his hypothesis being based on two assumptions. One of these assumptions, as already previously stated is MH. The second hypothesis of Max Tegmark is what we may call EW: There exists an external physical reality and it is independent of human observers.

So Tegmark’s two assumptions are as follows:

EW: There exists an external physical reality and it is independent of human observers.
MH: Whatever is possibly so is actually so.

EW is an existential statement and MH is a universal statement. This is very important to keep in mind, as shall be shown later on.

Mr. Tegmark prefers to talk about MH being something like this, “Our external physical reality is a mathematical structure”. A mathematical structure, or mathematical existence, is “merely freedom from contradiction.” I use MH as I do because the definition of mathematical existence is the same as possible. For something to be possible it must not contain a contradiction. For something to be impossible it must contain a contradiction.

Euclid’s geometry, for example, is a mathematical structure, and also has a mathematical existence. This means that Euclid’s geometry is “free from contradiction”. One cannot derive a contradiction within Euclid’s geometry.

We may say that there are two categories. There is what is possible and there is what is impossible. What is possible contains two sub-categories. These are Necessary and Contingent. Something is necessary because it not being actual is impossible. Something is contingent because it not being actual is possible and it being actual is possible. For example, it is necessary that all bachelors are unmarried males and it is contingent that all like charges repel.

Mathematics and Logic, at least, deal with what is Necessary. Metaphysics and Science deal with what is Contingent. The Criterion of Demarcation, or Line of Demarcation, between Metaphysics and Science, or Metaphysical Arguments and Empirical Arguments, is Falsifiability. Falsifiability was first laid out by Karl Popper in his book The Logic of Scientific Discovery, and throughout his other writings.

Some Criterion of Falsifiability for Empirical Hypothesis

There is one thing that all hypothesis must conform to, which is that of consistency, i.e. don’t allow contradictions. Necessary statements obviously conform to this, and Contingent statements are also suppose to follow consistency.

“The requirement of consistency plays a special role among the various requirements which a theoretical system, or an axiomatic system, must satisfy. It can be regarded as the first of the requirements to be satisfied by every theoretical system, be it empirical or non-empirical…Besides being consistent, an empirical system should satisfy a further condition: it must be falsifiable. The two conditions are to a large extent analogous. Statements which do not satisfy the condition of consistency fail to differentiate between any two statements within the totality of all possible statements. Statements which do not satisfy the condition of falsifiability fail to differentiate between any two statements within the totality of all possible empirical basic statements.” Karl Popper

Karl Popper points out, basically, that both metaphysics and science must adhere to consistency. One of the ways to refute a hypothesis is to show that it leads to a contradiction, which is known as a Reductio Ad Absurdum. You assume the opposite of a statement, and from this assumption you deduce a contradiction from the assumption. This proves the statement derived to be necessarily true, since its negation is impossible.

One tests of Scientific hypothesis is to make sure it is consistent with all other scientific hypothesis (generally, unless a new hypothesis that alters the edifice of science like Galileo and Einstein did). Another test is to show that the hypothesis is internally consistent.

Max Tegmark’s hypothesis, which contains both EW and MH are contradictory to one another. This is because MH allows for, what I shall call IW: There exists world and it is not independent of human observers. IW does not state how many human observers there are. There could be only one human observer, which is solipsism, or there can be infinitely many human observers, i.e. Human observer + 1 or N+1. MH allows for these possibilities, since there is no contradiction in such a situation. This implies that there exists a possible world where I am the only human observer, and it also implies that you,the reader, exists in a possible world where you are the only human observer. This also implies there exists a possible world in which only you the reader and I are the only inhabitants of a possible world where we are only human observers, and etc and etc.

Instead of accepting MH itself, which means both accepting EW and IW, Max Tegmark accepts only a part of MH by accepting only EW and denying IW. MH is both being affirmed and denied since denying a part of MH and accepting a part of MH. This would also follow by a simple example of Modus Tollens.

(1) All logically possible worlds exist implies there exists an external physical reality and it is independent of human observers and there exists a world and it is not independent of human observers.
(2) There doesn’t exist a world and it is not independent of human observers. (Because of EW)
(3) Thus, not all logically possible worlds exist. (Thus, Not MH)

The general point is that it is logically possible that there exists a world and it is dependent on human observers. But it is also possible that there exists a world and it is not dependent on human observers. Both of these are contained in MH, and Tegmark denies one but accepts the other, while also accepting MH. This would be similar to holding to the Theory of Special Relativity (which would be MH here) as a whole and accepting the first postulate (which would be EW here) and denying the second postulate (which would be IW). This is impossible since the Theory of Special Relativity is defined by both postulates together.

“A theoretical system may be said to be axiomatized if a set of statements, the axioms, has been formulated which satisfies the following four fundamental requirements. (a) The system of axioms must be free from contradiction (whether self-contradiction or mutual contradiction). This is equivalent to the demand that not every arbitrarily chosen statement is deducible from it. (b) The system must be independent, i.e. it must not contain any axiom deducible from the remaining axioms. (In other words, a statement is to be called an axiom only if it is not deducible within the rest of the system.) These two conditions concern the axiom system as such;” Karl Popper (Bold is my own emphasis and Italics are Popper’s own emphasis.)

It has already been shown that Tegmark’s hypothesis already violates (a). But Tegmark’s hypothesis also violates (b). This means that the two axioms of Tegmark’s hypothesis (MH & EW) are not independent of each other. We may deduce EW from MH, which means that EW is not independent of MH. It would be charitable to believe that Tegmark doesn’t hold that EW is not possible, which means that Tegmark doesn’t believe that EW is impossible.  But MH deals with everything that is possible. And so EW would be possible and thus be part of MH.

These two “proofs” don’t assume that Max Tegmark’s hypothesis aren’t an empirical hypothesis, but they are consistent with Max Tegmark’s hypothesis not being an empirical hypothesis, i.e. consistent with Max Tegmark’s hypothesis being a metaphysical hypothesis. These are also theoretical proofs, not practical or “empirical proofs” themselves.

There are two steps at falsifiability. One of them is that we show that the theoretical structure of the hypothesis is not itself contradictory. If the theoretical structure is not found to be contradictory, then we try to show that the theoretical structure is contradictory with empirical observations. If the theoretical structure is contradictory with the empirical observations, then the theoretical structure is falsified. First we try to show that the theoretical structure is contradictory or we try to show that the theoretical structure is contradicted by the empirical observations.

There will always be partial descriptions

The paper “A Logical Analysis of Some Value Concepts” was written by the logican Frederic B. Fitch, and appeared in the peer-review journal called The Journal of Symbolic Logic, Vol. 28, No. 2 (Jun., 1963), pp. 135-142.In this paper, a formal system was created for dealing with some “Value Concepts” like “Truth”, “Provability”, “Knowledge”, “Capability”, and “Doing”, to name a few. This deals with an abstract relationship, one as usually described by formally consistent systems like S-5 Modal logic.

What Frederic Fitch presents in the paper is what Tegmark would call a “Mathematical Structure”. This “Mathematical Structure” also has some Theorems that are proved within it. Like AP was a Theorem in a “Mathematical Structure” known as S-5 Modal Logic and the Pythagorean Theorem is a “Mathematical Structure” in Euclidean Geometry, so too are there two specific Theorems that are counter-intuitive, and can both respectively be called the “Knowability Paradox” and “Provability Paradox”. These are, respectively, Theorem 5 and Theorem 6 in Fitch’s paper.

Being Theorems, by the rule of inference GR, they are proved to be the case then they are necessarily the case. Whatever is provably so  is necessarily so. By MP, whatever is necessarily so is actually so. So Theorem 5 and Theorem 6 are actually so, which is also consistent with the hypothesis of Tegmark with MH, i.e. whatever is possibly so is actually so. Which in turn means that Tegmark would have to accept that Theorem 5 and Theorem 6 are true if they accept that their hypothesis MH is true.

Theorem 5, the “Knowability Paradox”, states that “If there is some true proposition which nobody knows (or has known or will know) to be true, then there is a true proposition which nobody can know to be.”

Some equivalent ways of stating Theorem 5 is such as: It is necessary that it isn’t known that both “P is true” & it isn’t known that “P is true”. It isn’t possible that it is known that both “P is true” & it isn’t known that “P is true”. The existence of a truth in fact unknown implies the existences of a truth that necessarily cannot be known. There exists such a true statement that both statement is true & for every agent no agent knows that statement is true implies there exists a true statement that both statement is true and for every agent it isn’t possible agent knows that statement is true.

Theorem 6, the “Provability Paradox”, states that “If there is some true proposition about proving that nobody has ever proved or ever will prove, then there is some true proposition about proving that nobody can prove.”

Some equivalent ways of stating Theorem 6 is such as: It is necessary that it isn’t provable that both “P is true” & it isn’t provable that “P is true”. It isn’t possible that it is provable that both “P is true” & it isn’t provable that “P is true”. The existence of truth in fact unproven implies the existence of a truth that necessarily cannot be proven.There exists such a true statement that both statement is true & for every agent no agent proves that statement is true implies there exists a true statement that both statement is true and for every agent it isn’t possible agent proves that statement is true.

These two Theorems show that it is necessary that agents, like human observers, know everything that can be known by those agents and proved everything that can be proven by those agents. This implies that Goldbach’s Conjecture, which hasn’t been proven to be true by human observers or proven false, cannot possibly be proven true or proven false. It will forever remain unprovable to human observers. It also implies that MH, or  cannot possibly be known and will forever remain unknown. This would also hold for all agents, which are not omniscient agents. These is necessarily so and means it is actually so, especially by MH and GR.

This is interesting because MH is presented as a hypothesis that is possibly true and it is not known that it is true or false. But since it is not known to be true and it is not known to be false, it cannot known to be true or false. MH, in conjunction with GR and Fitch’s Theorems, tells us that it cannot be known to be true or false and that it also isn’t provable that it is true or false, i.e. unprovable that it is true or false.

The Knowability Paradox and Provability Paradox also attack one of the aspects of Tegmark’s hypothesis, which is that of EW. EW implies that other agents that are not human observers, which can be supercomputers or aliens, would also fall for these paradoxes as well. This shows that we can never have a complete description of the world, but can only have a partial description of the world. This means that human observers, supercomputer observers, or alien observers, all cannot have a complete description of the world. We, the agents of EW, will never have a complete description.

What is interesting is that both paradoxes are very closely aligned with IW, or lead one to accept IW as true. Sometimes pointed out that the Knowability Paradox leads to Naive Idealism, which is part of IW and is thus not part of EW. This, in some sense would appear to imply that MH again implies another contradiction.

Must a Mathematical Structure be Free from Contradiction? 

“Mathematical existence is merely freedom from contradiction…In other words, if the set of axioms that define a mathematical structure cannot be used to prove both a statement and its negation, then the mathematical structure is said to have [Mathematical Existence].” Max Tegmark

Does mathematical existence really have to be freedom from contradiction? It is possible to develop formal systems that allow for both violations of non-contradiction and violations of excluded middle. A formal system of such a sort was developed by Polish logical Jan Lukasiewicz. This logic was created by using three values for logical matrices than two values.

Lukasiewicz three value logic has been axiomatized, so that there axioms, definitions, and logical relationships between propositions. And from this three value logic one may obtain violations of non-contradiction and violations of excluded middle. If there is a violation of non-contradiction then there is a violation of mathematical existence.

As Tegmark points out, A formal system consists of (1) a collection of symbols (like “~”, “–>”, and “X”) which can be strung together into strings (like “~~X–>X” and “XXXXX”), (2) A set of rules for determining which such strings are well-formed formulas, (3) A set of rules for determining which Well-Formed Fomrulas are Theorems.

Lukasiewicz three value logic satisfy all three of these criterion for a formal system.

The primitives of Lukasiewicz’s three valued calculus is negation “~”, implication “–>”, and three logical values “1, 1/2, and 0”. 1 stands for Truth, 1/2 stands for Indeterminate, and 0 stands for False. From negation and implication, with the three values, we can form a logical matrices of both negation and implication. And from these primitive terms we may define biconditional, conjunction, and disjunction as follows:

Disjunction “V” : (P–>Q)–>Q ; Conjunction “&” : ~(~P–>~Q) ; Biconditional “<—>” : (P–>Q) & (Q–>P)

“&” is symbol for Conjunction, “V” is symbol for Disjunction, “<—>” is symbol for Biconditional. Lukasiewicz’s Three-value calculus have the following truth tables:
Luk Truth TLukasiewicz’s axioms are as follows:
[Axiom 1] P –>(Q –>P)
[Axiom 2] (P –>Q ) –>(( Q–>R) –>(P –>R))
[Axiom 3](~Q –>~P ) –> (P –>Q)
[Axiom 4] ((P –>~P) –>P) –>P

Lukasiewicz’s rule of inference was Modus Ponens, i.e. Rule of Detachment:
(Premise 1) P –> Q
(Premise 2) P
(Conclusion) Q

From this it becomes obvious that formal systems do not need to be free from contradictions. This formal system allows for both (P & ~P) to have a truth value of neither True nor False. This is because, as the Conjunction Truth table shows, P= 1/2 or Indeterminate and ~P= 1/2 or Indeterminate is a well formed formula that is itself Indeterminate.

Does this mean that mathematical structures must be free from contradiction? It appears that Lukasiewicz’s formal system, and there are some others that can be created, show that mathematical structures and thus mathematical existence, do not need to follow the being free from contradiction. Lukasiewicz’s formal system can be expanded to allow for infinite number of truth values.

One important part of Tegmark’s idea of MH, which implies EW, is that it prohibits Randomness. He states that “the only way that randomness and probabilities can appear in physics (by MH) is via the presence of ensembles, as a way for observers to quantify their ignorance about which element(s) of the ensemble they are in.” Now Lukasiewicz’s logic can be the way our actual world is. This would mean that the world is random or indeterminate. Lukasiewicz’s even himself says that his three value logic is based on the position of indeterminacy, which is contradictory to determinacy.

[This post will be updated at sometime in the future….with more to come on this subject.]

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Hume and The Impossibility of Falsification

Posted by allzermalmer on May 5, 2013

Hume’s logical problem of induction as Hume presents it and Popper presents it, deals with contingent statements. The affirmation or the negation of the same contingent statement is possible. Take the contingent statement that “All Swans are White”: It is both possible that “All Swans are White” and it is also possible that  not “All Swans are White”. Logic alone cannot decide if “All Swans are White” is either true or false. So it would be decided by some other way as to wither its affirmation or negation to be true. Hume, and Popper, say that experience cannot show the truth of the contingent statement “All Swans are White”.

“Hume’s argument does not establish that we may not draw any inference from observation to theory: it merely establishes that we may not draw verifying inferences from observations to theories, leaving open the possibility that we may draw falsifying inferences: an inference from the truth of an observation statement (‘This is a black swan’) to the falsity of a theory (‘All swans are white’) can be deductively perfectly valid.” Realism and The Aim of Science

(H) Hypothesis: All Swans are White
(E) Evidence: This is a Black Swan

Hume, as Popper takes him in his problem of induction, showed that we cannot show that (H) is true, no matter how many individual swans that are white we have observed. To show that (H) is true, we must verify every case of (H). (H) is a Universal statement, its scope is that of all times and all places. The universal statement is both omnipresent and omnitemporal in its scope. It makes no restriction on temporal location and spatial location. (E) makes a Singular statement, its scope is of a particular time and a particular place. It makes a restriction on temporal location and spatial location. Popper held that we can know (E) is true, ‘This is a Black Swan’. Thus, we cannot know (H) All Swans are White but we can know (E) This is a Black Swan.

Hume’s logical problem of induction, as Popper takes it, goes something like this:

(i) Science proposes and uses laws everywhere and all the time; (ii) Only observation and experiment may decide upon the truth or falsity of scientific statements; (iii) It is impossible to justify the truth of a law by observation or experiment.

Or

(i*) Science proposes and uses the universal statement “all swans are white”; (ii*) Only singular observational statements may decide upon the truth or falsity of ‘all swans are white’; (iii*) It is impossible to justify the truth of the universal statement ‘all swans are white’ by singular observational statements.

It is taken as a fact that (i) or (i*) is true. So there is no question about either (i) or (i*). So the conflict of Hume’s logical contradiction arises between (ii) and (iii) or (ii*) and (iii*). Popper accepts (iii) or (iii*). So the only way out of Hume’s logical problem of induction is to modify or reject (ii) or (ii*) to solve the contradiction.

Popper thus solves Hume’s logical problem of induction by rejecting (ii) or (ii*) and replacing it with a new premise. This new premise is (~ii).

(~ii) Only observation and experiment may decide upon the falsity of scientific statements
Or
(~ii*) Only singular observation statements may decide upon the falsity of ‘all swans are white’.

Popper rejects (ii) or (ii*), which basically said that only singular observation statements can show that either universal statements are true or false. Popper rejects this because of (iii), and says that Singular observation statements can only show that universal statements are false. Popper believes, as the quote at the beginning of the blog says, that Hume’s logical problem of induction doesn’t show that we can’t show that a universal statement is false by a singular observational statements. But is this what Hume showed to be true?

It does not appear that Hume’s logical problem of induction even allows Popper to escape with the modification of (ii) to (~ii). It appears that Hume’s logical problem of induction does not allow Popper to escape from “fully decidable” to “partially decidable”, i.e.  decide both truth or falsity to cannot decide truth but only falsity.

Take the singular observational statement that Popper gives in the quote, i.e. ‘This is a black swan’. It is a singular statement, but the statement contains a universal within it, it contains “swan”. “Swan” are defined by their law-like behavior, which are their dispositional characteristics, and is a universal concept. These dispositions are law-like, and thus universal in scope as well. And by (iii) we cannot determine if something is a “swan” because of that. The concept “swan” is in the same position as “all swans are white”. They are both universal, and because of (iii) cannot be shown to be true.

“Alcohol” has the law-like behavior, or disposition, or being flammable. So if we were to say that ‘This is alcohol’. We would have to check all the alcohol that existed in the past, present, future, and all places in the universe in which it was located. We would have to light them to see if they catch fire, and thus flammable. Only than could we say that “This is alcohol”, and know that it is alcohol. But to do so would be to verify a universal through singulars, which is impossible by (iii).

In fact, Hume even talks about dispositions and law-like behavior in his talks about the problem of induction. For example, Hume says that “we always presume, when we see like sensible qualities, that they have like secret powers, and expect that effects, similar to those which we have experienced, will follow from them.” Hume is specifically attacking dispositions as well, which means he is attacking universal concepts and universal statements.

“Our senses inform us of the colour, weight, and consistence of bread; but neither sense nor reason can ever inform us of those qualities which fit it for the nourishment and support of a human body…The bread, which I formerly eat, nourished me; that is, a body of such sensible qualities was, at that time, endued with such secret powers: but does it follow, that other bread must also nourish me at another time, and that like sensible qualities must always be attended with like secret powers?” Enquiry’s Concerning Human Knowledge

From Popper’s point of view, science can only show the falsity of a universal statement through the truth of a singular statement. The singular statement would have to contradict the universal statement and the singular statement would have to be true.

(h) If it rained then wet ground.
(e) Not a wet ground
(c)Thus, it didn’t rain.

If we assume that both (h) and (e) are true, then we accept a contradiction. Contradictions can’t possibly be true. So we know that at least one of these two must be false. But which one is false and which one is true, (h) or (e).

But how can we show the truth of a singular observational statement when it relies on a universal concept, and universal concepts fall for (iii) just as much as universal statements? Hume’s position of the logical invalidity of of induction, i.e. (iii), also holds not only with universal statements but also universal concepts, i.e. law-like behavior/ dispositional characteristics. How does Popper respond to this?

Popper accepts the invalidity of reaching universal statements through experience, but takes it that we accept singular observational statements based on conventions. We conventionally accept the singular observation statement as true.

Hume’s logical problem of induction shows this:

(H) All Swans are White
(E) This swan is black

Now we may either accept (H) as a convention or accept (E) as a convention, or both as conventions. Popper rejects accept (H) as a convention, because you cannot show that a convention is false. Showing something false is what (~ii) was used to solve the original problem of induction. He wants to show that (H) is false, which is consistent with (~ii), but the only way to do that is if (E) can be shown true. But (E) contains a universal concept and (iii) prevents us from experiencing dispositions or law-like behaviors, i.e. Swan or Alcohol. (iii) applies just as much to universal statements as it does to universal concepts. (E) is based on universal concepts and so has to be accepted as a convention, to escape (iii), in order to show that (H) is false and be consistent with (i) and (~ii). (H) has to have the ability to be shown false to be falsifiable, and not being a convention means it has the ability to be shown false.

Contrary to what Popper thinks, Hume’s logical problem of induction doesn’t even allow you to show a falsifying instance. Thus, following full implications of Hume’s logical problem of induction, we can neither show the truth of a universal statement or show the falsify of a universal statement.

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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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Popper, Hume, Induction, Falsifiability, and Science

Posted by allzermalmer on September 30, 2012

Here are some interesting things from Karl Popper on Falsification and Induction, or Hume on Induction.

“we merely have to realize that our ‘adoption’ of scientific theories can only be tentative; that they always are and will remain guesses or conjectures or hypotheses. They are put forward, of course, in the hope of hitting upon the truth, even though they miss it more often than not. They may be true or false. They may be tested by observation (it is the main task of science to make these tests more and more severe), and rejected if they do not pass…Indeed, we can do no more with a proposed law than test it: it is no use pretending that we have established universal theories, or justified them, or made them probably, by observation. We just have not done so, and cannot do so. We cannot give any positive reasons for them. They remain guesses or conjectures- though well tested ones.” Realism and the Aim of Science

Now someone might wonder how we cannot give any positive reasons for establishing the universal theories, or justified them, or made them probable, by all the observations that confirm its predictions on tests. This comes from what Popper takes to be Hume’s problem of induction.

“[Hume] tried to show that any inductive inference- any reasoning from singular and observable cases (and their repeated occurrence) to anything like regularities or laws- must be invalid. Any such inference, he tried to show, could not een be approximately or partially valid. It could not even be a probable inference: it must, rather, be completely baseless, and must always remain so, however great the number of the observed instances might be. Thus he tried to show that we cannot validly reason from the known to the unknown, or from what has been experienced to what has not been experienced (and thus, for example, from the past to the future): no matter how often the sun has been observed regularly to rise and set, even the greatest number of observed instances does not constitute what I have called a positive reason for the regularity, or the law, of the sun’s rising and setting. Thus it can neither establish this law nor make it probable.” Realism and the Aim of Science

I think it should be pointed out, Hume did bring up that the basic idea of induction was that “we suppose, but are never able to prove, that there must be a resemblance betwixt those objects, of which we have had experience, and those which lie beyond the reach of our discovery.” Induction is also done in other ways besides going from particular statements to universal statements.

[I.] Move form particular statement to particular statement.
In 1997 the Chicago Bulls beat the Utah Jazz in the NBA Finals. In 1998 the Chicago Bulls beat the Utah Jazz in the NBA Finals. Thus, the Chicago Bulls will win against the Utah Jazz the next time they play in the NBA Finals.

[II.] Move from general statement to general statement.
All NFL teams made tons of money this year. Thus, all NFL teams will make tons of money next year.

[III.] Move from general statement to particular statement.
All NFL teams made tons of money this year. Thus, the Ravens will make tons of money next year.

[IV.] Move from particular statement to general statement.
This crow is black. Thus, all crows are black.

Each of these, though, follow what Hume points out for Induction. They are going from the known to the unknown, which does not have to include the future or past.Hume also says that the only thing that can take us from the known to the unknown is causality, or a necessary connection between two events to form a necessary causal relation. But Hume already pointed out that this relation is not found by experience. So Hume comes to the conclusion that since the necessary relation between cause and effect or continuation of that relationship, is not shown by experience nor demonstrative,  or that the principle of induction is not known by experience or demonstrative, but that they are creations of the human imagination that cannot be shown to be true based on experience or reason, and any justification of them will either rely on an infinite regress or circular reasoning. So they cannot be proven to be true.

This would mean that when science proposes either a causal connection, or what will happen in the future, or what happens beneath sensible qualities, cannot be proved by experience to be true , or by reason to be true, or even held to be probably true. IOW, we are not justified in proposing things beyond what is known, since they cannot be shown to be true or probably true. So scientific hypotheses are unjustified and cannot be shown to be true or probably true, or natural laws cannot be shown to be true or probably true or justified.

Popper comes along and tries to save science, in some way. But you notice where his position eventually leads as well. He admits with Hume that we cannot demonstrate the truth of a scientific hypothesis or explanation; we cannot show by experiment the truth of a scientific hypothesis or explanation; we cannot show that a scientific hypothesis or explanation is probably true. All we can do is show if they are false. We can give negative reasons to a scientific hypothesis or explanation by it failing its severe experimental/observational tests. This is because it follows the demonstrative inference of modus tollens and disjunctive syllogism, so we can demonstrate that a scientific hypothesis or explanation is false.

So falsifiability, or refutabilty, can show you only that a scientific hypothesis or explanation is false. Refutability cannot demonstrate that the hypothesis or explanation is true, or has been shown by experience to be true, or is probably true.  It can only tell you that it may be true, and it has not failed any of its tests so far. It doesn’t even appears to care if something is true, only that it can be shown to be false.

And here are Hume on what Induction is, or relies on.

“that which we have had no experience, must resemble those which we have had experience, and nature continues uniformly the same.” Treatise of Human Nature:  Book I (Of the Understanding), Part III (Of Knowledge & Probability), Sect.VI.Of the Inference from the Impression to the Idea

“probability is founded on the presumpition of a resemblances betweixt those objects, of which we have had experience, and those, of which we have had none…” Treatise of Human Nature:  Book I (Of the Understanding), Part III (Of Knowledge & Probability), Sect.VI.Of the Inference from the Impression to the Idea

“Thus not only our reason fails us in the discovery of the ultimate connexion of causes and effects, but even after experience has informed us of their constant conjunction, it is impossible for us to satisfy ourselves by our reason, why we should extend that experience beyond those particular instances, which have fallen under our observation. We suppose, but are never able to prove, that there must be a resemblance betwixt those objects, of which we have had experience, and those which lie beyond the reach of our discovery.” Treatise of Human Nature:  Book I (Of the Understanding), Part III (Of Knowledge & Probability), Sect.VI.Of the Inference from the Impression to the Idea

“we always presume, when we see like sensible qualities, that they have like secret powers, and expect that effects, similar to those which we have experienced, will follow from them.” An Enquiry Concerning Human Understanding: Section IV. Sceptical Doubts Concerning the Operations of the Understanding, Part II

“all arguments from experience are founded on the similarity which we discover among natural objects, and by which we are induced to expect effects similar to those which we have found to follow from such objects.” An Enquiry Concerning Human Understanding: Section IV. Sceptical Doubts Concerning the Operations of the Understanding, Part II

“From causes which appear similar we expect similar effects. This is the sum of all our experimental conclusions.” An Enquiry Concerning Human Understanding: Section IV. Sceptical Doubts Concerning the Operations of the Understanding, Part II

 

 

 

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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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What does Solipsism tell us?

Posted by allzermalmer on May 3, 2011

There is this idea within philosophy known as Solipsism. Solipsism says one of two things, which is either based on epistemology or metaphysics. Within metaphysics, it says that only my perceptions and thoughts exist. The epistemological position says that I can only know my experiences. The purposes of this post is only to deal with the epistemological position, and what it tells us. However, it seems impossible to deal with the epistemological without having ones toe in the water of the metaphysical issue.

Epistemological solipsism comes from the 1st person perspective, like what one experiences with their senses of sight, touch, taste, sound, and smell. It also deals with the thoughts that you have. So say that I am walking in the street, I see trees, and so I know there are trees. I am walking and I hear someone shouting at another, and I know this because of sound. I pick up this object and eat it, and I find it tastes good. I also felt this object, and know how it feels. I know that I had a thought where I was like, ‘This tastes very good.’

Now say that I come to meet Albert Einstein. He tells me all sorts of things, like how space and time are interconnected, and how space curves with things of mass, that light is both a wave and a particle. He tells me all of these things. Do I know that what he says is true? Of course I do not. I have never experienced anything that he tells me, and so I do not know. All I can do is believe that what they tell me is true, but I cannot know what he tells me is true.This goes with anything that someone tells me that I have no experienced myself.

So say that someone says that they performed an experiment, and that such-and-such happened. I do not know that it happened, and I can only believe him or not believe him. Now, I can come to have knowledge if I perform the experiment that they did. So say that the person set up certain conditions, like that of a recipe for making a cake. I get all the ingredients that are on the recipe, and I put it all together. I can find out if I get the same result, and then I know what happened.All I can say is this, the recipe that they gave me lead to me having this particular experience. I cannot say that they had the same experience as me, and all I can say is that they said they had such-and-such experiment.

Now this brings up some interesting issues. One of them is this, I cannot know what happened in the past, except for my memory of what happened in my past. So say that I was born in 1990. I would not know what happened in 1989, 1988, 1987 and backwards. This is not knowledge I can have. I can read books that talk about things that happened before 1990, but I cannot know that they happened. All I can know is that I read such-and-such that said such-and-such happened. I can believe it or not. I never know it happened, but I can believe that it happened.

The second issue is this, I cannot know what is going to happen in the future. I can only experience what happened in the past from my memory, or what is going on now in the present. I never experience the future, and so I can never know the future. I can only know the present, and the present becomes the past, which is my past. I can make an inductive inference that because such-and-such happened in the past, that it will happen in the future. This I cannot know.

The third issue is this, I cannot know that other people are having experiences like me, or similar to experiences like me. I can never experience what someone is experiencing, or even know that they are having experiences. All I can know is what I see and hear. I can see their body doing certain things, like bending over to pick up an object. I can know that they are saying “I am going to pick up this box on the floor to clean up the room.” I cannot know what they are thinking, or that they thought to do it. I cannot know if they are having an emotional reaction, or if they feel sad. I can only see their facial expression or the sound they make.

The fourth issue is this, I cannot know that things are happening, or going on, when I am not experiencing them myself.  So say that I am in my study, surrounded by books and a computer. I decide to leave this room to go to the kitchen to make a peanut-butter sandwich. As soon as I am out of sight of the study, I know longer know that it exists. I am not experiencing it, and have no first person experience of it to say that it exists. It is something that I just do not know. So as I watch the news, I see that there are things being brought up that are supposedly going on at Washington D.C. What I do know is that I am watching TV and there are these images on the TV, and these images are saying certain things. I know this. But now say that I turn off the TV. I n longer know that those things exist anymore, let alone that the city of Washington D.C. exists, or that the Library of Congress in D.C. exists. These are not things that are being experienced.I cannot know that a fish are swimming in the Atlantic Ocean, since I have no first hand experience of it.

Now there is this idea that is called falsifiability. Falsifiability states, a statement must be able to be refuted by experience through inter-subjective observation or experiment. So say that I make the statement, All Ravens are Black. Now the statement is a categorical statement, and is logically equivalent to No Raves are not Black, or All non Black are non Ravens. It also can carry this logical equivalent in propositional logic: If Raven, then Black. This carries the same meaning as, If not black, then not Raven.

So let us go with the typical conditional statement of propositional logic; If Raven then Black. Now falsifiability says that this statements has to have the possibility of being refuted by an observation by others, besides myself. It carries the logical inference of modus tollens. So say I come across a White Raven. This means that I have come across a Raven that is not Black. So If A then B; not B; thus not A.

Now what makes solipsism untestable is that it is not inter-subjective. You only have your first person experience, and cannot experience what someone else experiences or know that they have experiences to begin with, since you are assuming that this other person exists without being able to test if they do have experiences. Like I brought up before, I do not know that they experienced what they said they experienced, but I do know that they said they had a certain experience.

The second part of it is that one cannot refute that they are not the only thing that exists. One cannot know this, and so they can only believe it. From this belief, they make all sorts of other beliefs, and none of them can be known. So you cannot know that things exist when you are not experiencing them, know that other people are having experiences that they say they are or are thinking, cannot know that things happened in the past before you existed. From all of this, we find that a great deal of our ‘knowledge’ of things is just belief. It is not really knowledge. We cannot refute that things do not exist when they are not being experienced, since all we can know is what we experience. In order to refute solipsism, you must assume that it is false to show that it is false. This would just beg the question, and would not show that you refuted it.

The short man’s version is this, you cannot know that things are a certain way without you experiencing it, and all that you know are your experiences, which is  all that exists. So when it comes to the statement, “Julius Cesar was killed by Brutus”, is something that cannot be falsified. We cannot refute it because we can have no experience of it through observation, let alone an observation that is in principle able to be experienced by another person. Not to mention, one is assuming that this happened when you have no experience of it happening.

Now we can make the statement, “the Universe started from the Big Bang around 14 Billion years ago.” Now once we ask, What does ‘Universe’ mean, we come to realize it means ‘all that exists’. Now this is strange, since all that we know to exist is our experiences, and our experiences do not seem to have been 14 Billion years ago. Thus, if we really held to this idea, we have found that our experience refutes it, or at least our memory. But this is not inter-subjective. But this aside, we are stating that something happened when we did not experience it and that this happened in the past.

Now, supposedly, everyone is supposed to be subjective, and have their own point of view that no one else can logically experience. Thus, all of us only know what we experience. So the question could be, “what does this mean about our knowledge of things before humans came about?” The answer seems to be that they are just theoretical background view/belief. We carry this theoretical background view/belief that things exist when we are not observing them. We carry this theoretical background view/belief that the past exists. We carry this theoretical background view that there are other minds.

From these general theoretical background views, we can falsify statements. The statements themselves are not falsifiable, for the most part, if at all, without these theoretical background views. Without them, statements like, “the universe started 14 billion years ago”, would not be falsifiable. First, we would re-define universe to be those things that exist beyond our own 1st person experiences, and those of humans, since are giving them both a theoretical existence, which is not knowledge, but it is belief. From these beliefs we are going to build up other beliefs that we would call knowledge. They are not really knowledge, but for short hand we call them knowledge. They are just belief.

However, we come up with this other distinction. There is a difference between these two statements, even though they are both be theoretical background views/beliefs:
(1.) There is a penguin in Antarctica
(2.) The Universe began 14 Billion years ago

The difference is that the first statement is one that we can observe with our senses. The second is something that we would not be able to observe in principle, and this also holds for other people’s thoughts an experiences. Much of what we are told is based on things that go beyond our senses, and so they are theoretical background beliefs/views as well.

So in the end, we find that epistemological solipsism, with a hint of metaphysical solipsism, shows us that most of what we call knowledge is not really knowledge. It shows us that most of what we call knowledge is just a belief. Thus, we find that falsifiable statements, or beliefs, are built off of unfalsifiable beliefs. Our falsifiable statements are based off of theoretical background views/beliefs, and from these views we build up statements that we can show are false. This means, though, that they are not really falsifiable. What is more interesting is this, 1st hand knowledge is what we know and 2nd hand knowledge is just belief. We would also have to differentiate between observable and unobservable, since only the observable is falsifiable.

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