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Posts Tagged ‘Contingent’

Fallacy of Evidentialism

Posted by allzermalmer on August 18, 2013

There are two philosophers, who are taken to be generally representative of Evidentialism. These two philosophers are David Hume and C.K. Clifford. These two philosophers have two quotes that are examplars of their Evidentialism thesis. They are, respectively, as follows.

“A wise man, therefore, proportions his belief to the evidence…when at last [a wise man] fixes his judgement, the evidence exceeds not what we properly call probability.” – David Hume in “Of Miracles” (Italics are Hume’s)

“We may believe what goes beyond our experience, only when it is inferred from that experience by the assumption that what we do not know is like what we know…It is wrong in all cases to believe on insufficient evidence” – W.K. Clifford in “The Ethics of Belief

Thomas Huxley,

Huxluy Evidence

Those quotes from these three writers are taken as representative of Evidentialism, and thus the Evidentialist Principle. The statements they make might appear to carry some validity & they might even seem to be sound.

However, Karl Popper holds that they are not valid. He also doesn’t hold that they are sound. They even contradict all empirical systems or all empirical propositions. They forbid us from ever believing or holding to any empirical system or empirical proposition, they forbid us from ever believing or holding to any scientific hypothesis or scientific proposition. But the problem of Induction applies to both the truth of this matter of fact assertion and the probability of the truth of this matter of fact assertion.

Both of the propositions contain signs of being based on Induction. Hume points out that a wise man will fix their judgements on a proposition when the evidence indicates that it is probable. Clifford points out that we may infer from experience what goes beyond our experience, but this is based on hypothesis that unknown is similar to the known.

Both of the propositions show that Evidentialism is founded on Induction, or inductive inferences.

Hume, supposedly, showed that it is logically impossible to infer the unknown from the known. It is logically impossible to derive the unknown from the known. Thus, Evidentialism is founded on a logical impossibility.

“The problem of the source of our knowledge has recently been restated as follows. If we make an assertion, we must justify it; but this means that we must be able to answer the following questions.

How do you know? What are the sources of your assertion?’ This, the empiricist holds, amounts in its turn to the question,

‘What observations (or memories of observations) underlie your assertion?’ I find this string of questions quite unsatisfactory.” – Karl Popper in “The Sources of Knowledge and Ignorance

Popper presents the Evidentialist Principle, in that quote, as saying that “If we make an assertion, we must justify it“. If you make an assertion, then you must justify it, or making an assertion implies must justify the assertion. You would have to answer one question, ‘How do you know? What are the sources of your assertion?’, and have to answer another question, ‘What observations (or memories of observations) underlie your assertion?’. 

As Popper points out, the Evidentialist Principle is an answer to The Problem of Source of Knowledge. So we may suppose that Evidentialism and Induction are to be based on the Source of a proposition or an empirical proposition. It seeks that the source of a proposition to be justified.

Criticizing or discrediting a proposition because of the source has some similarity to the Genetic Fallacy: “if the critic attempts to discredit or support a claim or an argument because of its origin (genesis) when such an appeal to origins is irrelevant.”

With the Genetic Fallacy, a proposition is being discredited, or supported, because it is “paying too much attention to the genesis of the idea rather than to the reasons offered for it”. The origin, or source, of the proposition is used to discredit, or support, the proposition.

Evidentialism would discredit a proposition because the source of the proposition is without justification.

We also find that David Hume presents an example of the questions that Popper finds to be unsatisfactory.

“All reasonings concerning matter of fact seem to be founded on the relation of cause and effect. By means of that relation alone we can go beyond the evidence of our memory and senses. If you were to ask a man, why he believes any matter of fact, which is absent; for instance, that his friend is in the country, or in France; he would give you a reason; and this reason would be some other fact; as a letter received from him, or the knowledge of his former resolutions and promises…All our reasonings concerning fact are of the same nature. And here it is constantly supposed that there is a connexion between the present fact and that which is inferred from it. Were there nothing to bind them together, the inference would be entirely precarious.

When it is asked, What is the nature of all our reasonings concerning matter of fact? the proper answer seems to be, that they are founded on the relation of cause and effect. When again it is asked, What is the foundation of all our reasonings and conclusions concerning that relation? it may be replied in one word, Experience. But if we still carry on our sifting humour, and ask, What is the foundation of all conclusions from experience? this implies a new question, which may be of more difficult solution and explication.” – David Hume in “Sceptical doubts concerning the operations of the understanding” (Italics are Hume’s)

David Hume himself goes down the line of questioning that Popper brings up. For example, suppose that some assertion is made like “all ravens are black”. This assertion is what Hume calls a Matter of Fact, i.e. Synthetic proposition or Contingent proposition. It is Possible that it is true that “all ravens are black” and it is possible that it isn’t true that “all ravens are black”. This starts a line of questioning once this assertion is presented.

Question: What is the nature of reasoning concerning that matter of fact?
Evidence: The assertion is founded on the relation of cause and effect.
Question: What is the foundation of reasoning and conclusion concerning that relation of cause and effect?
Evidence: The relation of cause and effect of that assertion is founded on Experience.

These two questions follow a basic form that Popper is bringing up, and the type of basic form that Popper finds unsuitable, or the type of basic form of Evidentialism that is unsuitable. The basic reason for this is because another question follows from the answer to the previous two questions.

Question: What is the foundation of that conclusion drawn from experience?

This new question is where the Problem of Induction arises, or what Popper calls The Logical Problem of Induction.

If all Ravens are Black then justified in the relation of cause and effect. If justified in the relation of cause and effect then justified by experience. If justified by experience then experience is justified by Induction. So if all ravens are black then justified by Induction. But, Induction isn’t justified. So assertion all ravens are black isn’t justified. Therefore, Evidentialism would make it so that the assertion all Ravens are Black isn’t justified. This applies to all matters of fact, and thus all empirical and scientific assertions.

“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.” – Karl Popper in “The Logic of Scientific Discovery” (Italics are Popper’s)

The Problem of Induction comes about because Induction relies on statement that is a matter of fact assertion, but this matter of fact assertion cannot, in principle, be inductively justified. So either all reasonings concerning matter of fact seem to be founded on experience or not all reasonings concerning matter of fact seem to be founded on experience.

This is a logical problem because either Induction relies on a statement that is either a contingent proposition or necessary proposition. We can call this the “Principle of Induction”. But the Principle of Induction can’t be a necessary proposition because the negation of the Principle of Induction is possible to be false. A necessary proposition can’t be possible to be false. So it is possible that Principle of Induction is true and it is possible that isn’t true that Principle of Induction is true. Therefore, the Principle of Induction is a contingent proposition.

Hume points out that matter of facts about dispositions and universal propositions are matters of facts. Thus dispositional propositions and universal propositions are contingent propositions. Dispositional propositions describe law-like behavior and universal propositions describe lawful behavior or law-like behavior. These would both be contingent propositions, and so we wouldn’t be justified, based on Induction, in asserting those dispositional propositions or universal propositions.

We wouldn’t be justified, based on Evidentialism, when it came to assertions about dispositional propositions or universal propositions. Science wouldn’t be justified, based on Evidentialism, when it came to assertions about dispositional propositions or universal propositions. But science is full of assertions about dispositional propositions and universal propositions. Therefore, science wouldn’t be justified in asserting dispositional propositions and universal propositions.

“[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 even 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.” Karl Popper in “Realism and the Aim of Science” (Italics are Popper’s)

The assertion “all ravens are black” isn’t justified as true under Evidentialism and “all ravens are black” isn’t jusified as probably true under Evidentialism. Hume himself points out that the wise man doesn’t fixate his judgement on an assertion in which the evidence exceeds what we properly call probability. In other words, the Evidentialist doesn’t hold to assertions in which the evidence exceeds what we properly call probability. So Evidentialist only hold to assertion in which evidence shows it is true or probably true. So “all ravens are black” is only held by an Evidentialist if evidence shows it is true or at least probably true.

Popper presents a solution to the Problem of Induction, and thus treats assertions differently from Evidentialism. Popper rejects Induction, and thus rejects Evidentialism. The source of an assertion has nothing to do with either discrediting the truth of a proposition or supporting the truth of a proposition.

Matter of fact propositions, or scientific propositions, don’t discredit or support the source of an assertion. Science doesn’t support the truth of a proposition or support the probability of a proposition. It, basically, seeks to discredit the truth of a proposition. Science seeks to show that the proposition is false, not that the proposition is true or probably true. Science always seeks to discredit it’s proposition and not to support it’s propositions. So scientific propositions are, in principle, possible to show they are false and never show they are true or probably true. This includes both dispositional propositions and universal propositions.

In other words, Evidentialism seeks both positive justifications for assertion and negative justifications for assertion. Evidentialism would be based on “full decidability”. Falsifiability, or Falsification, seeks only negative justifications for assertions. Falsifiability would be based on “partial decidability” . These negative justifications, for Falsifiability, basically state that scientific assertion hasn’t been demonstrated false as of yet. This never indicates a positive justification for the assertion being true or probably true.

“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.” – Karl Popper in “The Two Problems of The Theory of Knowledge” (Italics are Popper’s)

For Falsifiability, the source of an assertion is irrelevant when judging whether the assertion is either true or false, and the source of an assertion is irrelevant when judging whether justified in believing that assertion is true or probably true. The source of an assertion is irrelevant for the justification of the assertion. Would have to rely on Induction, and Induction isn’t justified itself. The only justification of an assertion, specifically an empirical assertion, is that it is possible to show that assertion is false. An empirical assertion has the possibility to be shown false, but it doesn’t have the possibility to be shown true (or probably true).

Science, thus, doesn’t care of the source of an assertion. Science is justified in believing, or holding to, an empirical proposition because that empirical proposition allows for the possibility that can be shown that it is false, but hasn’t been shown that it is false yet. For example, science would be justified in believing the empirical proposition that “all ravens are orange” if wasn’t for “some ravens are black”. It would be a negative justification, since don’t have another empirical proposition that contradicts it, or shows that it is false.

One of the basic mechanisms of Falsifiability is that works by deductive inference. Modus Tollens forms an example of deductive inference that Falsifiability uses. Given the conditional claim that the consequent is true if the antecedent is true, and given that the consequent is false, we can infer that the antecedent is also false.

If an empirical assertion is true implies another empirical assertion is true & the other empirical assertion is false, then original empirical assertion is false.

Principle of Modus Tollens:If all ravens are orange implies no ravens are not orange & some ravens are black, then not all ravens are orange. This is how the negative justification of empirical assertions works, which is deductive inference of modus tollens. It wouldn’t be possible for “not all ravens are orange” to be false. So it must be true.

The Principle of Modus Tollens is a necessary truth, which is different from the Principle of Induction. The Principle of Induction isn’t a necessary truth. It is possible that the Principle of Induction is false. So it might be true.

An assertion that is the conclusion of the Principle of Induction, or the assertion of a wise man that reviewed the Evidence, might be true. An assertion that is the conclusion of the Principle of Modus Tollens, or the assertion of a foolish man that never reviewed the Evidence, must be true.

The truth that the Principle of Modus Tollens always produces truth. It is similar to negative theology. It isn’t true that “all ravens are orange” & it isn’t true that “no ravens are not orange”. Each time saying what is true because true isn’t those false statements, since it is true that “not all ravens are black”.

The contradiction between “all ravens are orange” and “not all ravens are orange” are exclusive, they both can’t be true and no intermediary empirical propositions between them. If know that “all ravens are orange” is false then know that “not all ravens are orange” is true. All ravens are orange implied no ravens are not orange & some ravens are black. Therefore, it is necessarily true that not all ravens are orange. If Know that “not all ravens are orange” is true then “not all ravens are orange” is true. “Not all ravens are orange” is true.

Both the Principle of Modus Tollens are dealing with scientific propositions. The scientific propositions are possibly true or possibly false. If combine scientific propositions with the Principle of Induction, then scientific proposition infered might be true. If combine scientific propositions with Principle of Modus Tollens, then scientific proposition infered must be true. The negative justification allows for things that aren’t possibly not true & hold to statements that are only true, while positive justification allows for things that are only possibly true & hold to some statements that aren’t only true.

So Evidentialist like David Hume, or C.K. Clifford, would be justified in holding some scientific propositions that aren’t only true. Evidentialist would hold to both true statements and false statements. While the Non-Evidentialist, which follows Falsifiability or negative justification, would hold only to true statements. The Non-evidentialist wouldn’t be justified in asserting a scientific statement, even though conclusions drawn from it must be true.

Thus, Evidentialism is fallacious because the assertions that it concludes to be justified in holding, based on the evidence, aren’t truth-preserving. It’s conclusions of justified scientific propositions aren’t based on the evidence or derived by positive support it receives from the evidence. However, it is completely opposite with Non-Evidentialism of Falsification, or it isn’t fallacious.

The Evidentialist would be acting irrationally by seeking their justification, while the Falsifiabilist, which is necessarily a Non-Evidentialist, would be acting rationally by not seeking the Evidentialist justification.

Huxley’s assertion, in his examplar of Evidentialism, mentions that “merciless to fallacy in logic.” But we later find out that Evidentialism isn’t “merciless to fallacy in logic”, but is founded on a fallacy in logic itself. David Hume recognized this, even though exemplar of Evidentialism. Instead, he went about acting irrationally by seeking a (positive) justification of proposition by evidence & the rest of Evidentialism followed, like C.K. Clifford and Thomas Huxley. They would all go about by searching for evidence that proposition is true and end right back in the same place.

Finding Evidence

So we finally come full circle with the fallacy of Evidentialism, and find the source of the Evidentialist fallacy.

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Knowability Paradox and Modal Realism

Posted by allzermalmer on August 17, 2013

Equivalency is defined this way: (p–>q)<–>(q–>p)

So if we assume (p–>q) is true and (q–>p) is true, then it necessarily follows that (p–>q) if and only if (q–>p).

(1) p–><>p
(2) <>p–>p [Modal Realism]
(3) (p–><>p)<–>(<>p–>p)

There is one thing that must be made clear.

(1) is necessarily true. It is not possible that it is not true. It is an axiom of modal logic. Now (2) isn’t necessarily true. It is possible that it is not true. It is not an axiom or theorem of modal logic.

(1) can be substituted with variable of x. So x if and only if p–><>p. (1) being necessarily true implies necessarily x. (1) and x, are analytically true.

(2) can be substituted with variable of y. So y if and only if <>p–>p. (2) being not necessarily true implies possibly not y. (2) or y, are contingently true.

(3), based on substitutions of (1) & (2), takes on the form of x<–>y. Or we can say analytically true if and only if contingently true.

x is analytic implies either necessarily x or necessarily not x. [](p–><>p) v []~(p–><>p).
y is contingent implies possibly y and possibly not y. <>(<>p–>p) & <>~(<>p–>p).

(3) shows that we have collapsed any modal distinction between possibility and actuality. There is no modal difference between possibly true and actually true. This specific proposition presents that possible if and only if actual.

(4) p–>Kp [Fitch’s Theorem]
(5) Kp–>p
(6) (p–>Kp)<–>(Kp–>p)

There is one thing that must be made clear.

(5) is necessarily true. It is not possible that it is not true. It is an axiom of epistemic logic. Now (4) is also necessarily. It is not possible that it is not true. It is a theorem of epistemic logic.

(4) can be substituted with variable of x. So x if and only if p–>Kp. (4) being necessarily true implies necessarily x. (4) and x are necessarily true.

(5) can be substituted with variable of y. So y if and only if Kp–>p. (5) being necessarily true implies necessarily y. (4) and y are necessarily true.

(6), based on substitutions of (4) & (5), takes on the form of x<–>y. Or we can say analytically true if and only if analytically true

x is analytic implies either necessarily x or necessarily not x. [](p–>Kp) v []~(p–>Kp).
y is analytic implies either necessarily y or necessarily not y. [](Kp–>p) v []~(Kp–>p).

(6) shows that we have collapsed any epistemic distinction between knowledge and truth. There is no difference between knowing something is true and something is true. What this specific proposition presents is that truth if and only if knowledge

Equivalency: ECpqCqp or ECNqNpCNpNq

(1) CpMp
(2) CMpp
(3) ECpMpCMpp

(4) CpKp
(5) CKpp
(6) ECpKpCKpp

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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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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.

or

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

or

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.

Example:
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.

Example:
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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Truth of Reasoning and Truth of Fact

Posted by allzermalmer on October 26, 2012

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

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

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

 So Leibniz obtains all knowable propositions or statements to be divided based on the principle of contradiction. The truth of statements is divided into two realms. This also deals with what people can know, or knowability. It basically says that
“For each statement, if statement is knowable, then statement is either truth of reasoning or truth of fact. For each statement, if statement is truth of reasoning, then statements affirmation is logically possible and statements negation is logically impossible. For each statement, if statement is truth of fact, then statements affirmation is logically possible and statements negation is logically possible.”

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

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

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

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

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

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

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

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

What Popper takes to be “Hume’s Problem”

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

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

Here’s an Inductive argument

Singular: (P1) There exists at least one x, such that x is Crow and x is Black
Singular: (P2) There exists at least one x, such that x is Crow and x is Black

Universal: (C) For each x, if x is Crow, then x is Black

Popper’s Solution to “Hume’s Problem”

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

Here’s Popper’s solution

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

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

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

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

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

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Averroes, Al-Ghazali, and Causality

Posted by allzermalmer on December 16, 2011

During the time period between 1037 and 1198 in Muslim philosophy were there were two ideas that were prominent on the view of causality. Of that period there were two defenders of the opposing viewpoints of causality.. For the theologian side there was Al-Ghazali. For the philosopher side there was Averroes. Al-Ghazali held that things happened, by causality, contingently. For Averroes, things happened necessarily. But they both agreed that things were caused by God, but for slightly different reasons.

Al-Ghazali read up on the philosophers’ position of causality, and the philosopher that had the most influence, and was the biggest defender of necessary causation, was Avicenna. Al-Ghazali wrote a response to Avicenna’s stance of causality and the relation it shared to that of God. Avicenna talked about the necessitation of causality, and how things are to happen a certain way and could not happen otherwise. Al-Ghazali presented an argument of causality based on things happening contingently. And all these things were done by the will of God.

Being contingent affairs, things can always happen otherwise than they do, and Al-Ghazli is talking about, what it was known as during his times, as efficient causation. This means, that because the sun has risen every day in the east, this doesn’t mean that the sun won’t rise in the west tomorrow. There is nothing logically contradictory in holding to either position separately, but you can’t hold on to both positions at the same time. They are contradictory. So we know that either one of those two things will happen. Which happens, we cannot say that it can’t happen otherwise than one way, like the sun will always rise in the east.

Al-Ghazli likes to use the analogy of cotton and a fire. Imagine, in modern day way, that we have a lighter in our hand and a piece of fresh picked cotton. We can put the flame right under the cotton so that the flame from the lighter is seen to be in “contact” with the cotton. We notice that the cotton starts to catch on fire and turn black as well. We don’t notice anything necessary between them, or for one thing to follow the other. Thus, we don’t observe that the fire has to make the cotton turn black and catch fire. We just notice this correlation between these two different events. And this happens because this is the way that God willed it to happen.

God can always, the next time we put the lighter under the cotton and make contact with the cotton, for the cotton to not catch fire or turn black. God has decided that, for this instance, the cotton should not catch fire. There is nothing that prevents God from doing this, because there is no logical contradiction for it not happening. Thus, when Abraham was surrounded by fire and never caught on fire, that he was not burnt. God did not will it to happen, and broke from the way that God usually willed for things to happen, like they did in past incidents.

We can have two propositions about causality. [1.]At time T^1, a person swallows a date while hungry. [2.]At time T^2, the person’s hunger is gone. Affirming the first proposition doesn’t entail the second proposition. Nor does the non-existence of state of events entail the non-existence of another state of affairs. For God, anything can be done that is possible. Thus God can will a person to remain hunger even after they ate a date, or God can make take aways someone’s hunger when they don’t eat. Things only have those dispositions for which God wills at any instance.

The philosopher Averroes responded to Al-Ghazali’s take on causality, and tried to show the response for which the philosophers have to the theologians take on causality. He wanted to point out that the philosophers do not deny that those things that are possible, but say that there are necessary things as well. Averroes stated that because we can’t observe the cause of certain effects by the senses that we only need to search harder for those causes that brought about these effects. This is because things have certain essences in them that make them comply to do certain things and respond in certain ways. This forms the necessary connection between cause and effect.

Averroes holds that there are four causes, and not just the efficient causation that Al-Ghazali dealt with. Besides efficient causation, there is the cause of form, the cause of matter, and the cause for the end. These other causes play into effects, and the stance from which necessary causation is linked. Averroes points out that when we have two things, that one is active and the other passive. From this, we can draw one relation from infinity of things that could happen between them or come from them causing on one another. But having this one relation limits those things that could come from all the possibilities that aren’t logically contradictory.

One of the reasons for this is that certain things are a certain way. The necessity seems to be in that of the name and definition. For example, it is necessary, for fire to keep its name and definition, that it keeps its “burning power”. Thus, it is similar to “All bachelors are unmarried males”. This means that things are limited unto the meaning of things, and that they can’t deviate from them based on this. Thus, based on the definitions and name, in conjunction with the four causes, things happen necessarily. This is also related to things being one, which is based on its essence. This is what the definitions and names are supposed to capture, which is the essence of things. The very essence of things means that only certain necessary things can happen, and this helps to make cause and effect a necessary connection. There is a necessary connection between cause and effect.

So, for example, let us go back with the example of the date. . [1.]At time T^1, a person swallows a date while hungry. [2.]At time T^2, the person’s hunger is gone. Now there is a necessary connection between these two propositions. The first of them would be from the four causes. There would be the necessary connection between the forms, the matter, the efficient cause, and the final cause. But within this it comes about because of the very essence of the date and the human being. There is an active and passive connection between their essences in this situation, and this means that based on the definition of these essences, one thing follows from necessity because of the other.

One of the major themes involved is that God knows all things, and knowing them, while being omniscient, entails that things necessarily happen because of God knowing them. God knows all the essences of things, which means that God knows the active and passive relation between things at all times. This also entails that God knows the four causes, and what necessarily follows from these things. It is also based on God’s will, because only things that have knowledge can have a will. Thus, when it comes to a causal connection, when we affirm one thing, it necessarily follows the entailment of the second thing. Thus, when we affirm the cause, it necessarily follows that a certain consequent shall follow, and this is the effect.

What seems to be one of the major differences would be involved with the definition of things, or at least their very essence which is trying to be captured by the definition of something. Al-Ghazali doesn’t seem to be affirming the essence of things, while Averroes seems to be affirming. One of the other differences is that Al-Ghazali isn’t using the other three causes, and is only focusing on one of those causes. Averroes is using four causes, which means that it is using efficient cause like Al-Ghazali exclusively focuses on. What they do both agree on, though, is that God is the cause of things, in some way or form.

For Averroes, things happen out of necessity because God knows the essence of all things, and God knowing something entails that it happens out of necessity. While for Al-Ghazali, things happen because God wills them to happen. One can allow for miracles because God decides to break “habit” of usually having the cotton catch fire when it is touched by a flame, like God did with Abraham. For Averroes, there are miracles of this sort as well. God knowing that Abraham will not catch fire means that Abraham will not catch fire. Things happen out of necessity because of this. God’s knowledge of things is the cause of them.

One way to look at this deals with the “Why” question. This can be broken down into two parts. It is that something produces the item, or that something explains the need or function of something. Al-Ghazali deals with what produces the item, while Averroes deals with explaining the need or function of something. The need for an explanation based on the function is the necessity of why it happened, while dealing with the something that produces an item is one that is contingent, and has no real function for something being brought about out of necessity. Both the theological side of Al-Ghazali and the philosophical side of Averroes take different stands on these positions. One only takes the first parts of something that produces an idea, while the second take the first part and also incorporates the second.

For Averroes, the world and God are both co-eternal. And God is the sufficient cause of all things. There is the essential cause and the efficient cause. These are based on teleological cause and efficient cause. When the essential cause exists, it necessitates the efficient cause. But for Al-Ghazali, there is no real essential cause. There is the efficient cause from which it is God’s will from which things happen as they do. There is no necessity, or essential cause, for which things must follow as they do. The only thing that would come close to this would be based on God’s will for things to happen in a certain way that they do. And one of the differences for this is that Al-Ghazali held that the world has not always existed and that God created the world, which is to negate the eternal existence of the world. While for Averroes, the world has always existed alongside the world. This is where the essential cause is affirmed by Averroes and denied by Al-Ghazali.

One of the other differences between Averroes and Al-Ghazali based on their idea of causality is that God can only do things out of necessity for Averroes. While for Al-Ghazali, God can do anything that isn’t self-contradictory and doesn’t have to do things out of necessity. One way to look at this is that they both uphold causality of some sort. But for Al-Ghazali, the efficient causation holds, but it is not because those things that we attribute as the cause is what brought about the effect. There is no link, which we find through the senses, which show this link. Thus, if we hold to the causal relation of A and B, Al-Ghazali doesn’t say that it is because of A that B happened. While for Averroes, it is because of A that B happened. One of the reasons that Al-Ghazali objected to the necessarianism of causality is that it seems to reject miracles, and the miracles are attested to in the Qur’an.

One of the big differences is that one holds that there must be a cause for something, which is what Averroes holds to. While on the other hand, for Al-Ghazali, things don’t need causes in order for them to happen. Now what is understood here is that of a necessary cause, or an essential cause. These would be the four causes. Thus, Al-Ghazali objects to these four causes, and that is because they are based on the Greek philosophers, who were pagans. So Al-Ghazali objects to these ideas, and presents an idea of causality which is devoid of the pagan influences, but also consistent with the view of God in the Qur’an.

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On Causality by Al-Ghazali

Posted by allzermalmer on December 16, 2011

This blog will be based on one chapter of a book done by  the Arabic philosopher Al-Ghazali. This blog will be derived from the 17th discussion/chapter in his book The Incoherence of the Philosophers, which is based on the idea of causation. You can also find his take on causality on page 96 of this PDF.

Al-Ghazali was, what can apply, an Islamic philosopher. He eventually took up Sufism, and was also part of the Ash’arites. He went after some of the influential Islamic Philosophers vies like Al-Farabi and Avicenna (Ibn Sina). These two philosophers held a view of necessary causation. And this is where Al-Ghazali brings up the Ash’arite view, and the one used against the idea of Necessary Causation. And the view espoused by the Ash’arites and Al-Ghazli becomes known as Occasionalism.

I will quote some of the arguments and points made by Al-Ghazli in his book The Incoherence of the Philosophers. These quotes will be taken from the PDF given above, and the chapter starts on page. 96 with the title of “Refutation of their belie in the impossibility of a departure from the natural course of events.”

“In our view, the connection between what are believed to be the cause and the effect is not necessary. Take any two things. This is not That; nor can That be This. The affirmation of one does not imply the affirmation of the other; nor does its denial imply the denial of the other. The existence of one is not necessitated by the existence of the other; nor its non-existence by the non-existence of the other. Take for instance any two things, such as the quenching of thirst and drinking; satisfaction of hunger and eating; burning and contact with fire…or any other set of events observed to be connected together…They are connected as the result of the Decree of God (holy be His name), which preceded their existence. If one follows the other, it is because He has created them in that fashion, not because the connection in itself is necessary and indissoluble. He has the power to create the satisfaction of hunger without eating, or death without the severance of the head, or even the survival of life when the head has been cut off…”

This is the stated view of Al-Ghazli and the Ash’arite. Now some of the philosophers who believe in necessary connection, of those times, might give a certain refutation or questioning of what Al-Ghazli stated. So Al-Ghazali will give an example of what he is talking about with causality, and how certain objections might be presented. Al-Ghazali uses the example of fire and cotton, (in distinction from David Hume’s example with billiard balls.)

“Since the inquiry concerning these things (which are innumerable) may go to an indefinite length, let us consider only one example-viz., the burning of a piece of cotton at the time of its contact with fire. We admit the possibility of a contact between the two which will not result in burning, as also we admit the possibility of the transformation of cotton into ashes without coming into contact with fire. And [the philosophers] reject this possibility.”

Now there are three objections that can be presented. The first thing, possibly, raised by someone who holds into necessary causation, could be along these lines: Fire is, by itself, the agent of burning. It is by it’s very nature to take part of burning, and it cannot refrain from doing it because it does it not by choice. When it comes into contact with a something that is receptive to it, an effect follows necessarily. But Al-Ghazali has a response to something along these lines.

“We say it is God-who through the intermediacy of angels, or directly- is the agent of the creation of blackness in cotton; of the disintegration of its parts, and of their transformation into a smouldering heap of ashes. Fire, which is an inanimate thing, has no action. How can one prove that it is an agent? The only argument is from the observation of the fact of burning at the time of contact with fire. But observation only shows that one is with the other, not that it is by it and has no other causes than it.”

There could be an objection to this, by the philosophers. But they might accept a part of the argument, but have their own twist to it. They will accept that we observe one with the other, not that it is by it and has no other cause. They would say something along these lines: This is not That, but both of them have their own capacities which allow us to distinguish one from the other. We can take the example of how one things softens under the sun and another hardens under the sun. This is because They are different receptive capacities in these objects, and they necessarily respond like This and like That out of the necessity of their capacity when acted upon by another certain capacity.

Al-Ghazali, in a previous section of the Incoherence of the Philosophers, he refutes the idea of the capacity of things that they must necessarily act in a certain, and he references back to that previous refutation. That is a long argument in itself, and shall be skipped over here.

There is another objection the Philosophers could raise: Now that you deny necessary causation, and replace it with contingent causation, this would mean that anything could happen at any moment. For example, you could drop the ball from your hand and it would go shooting up into the sky and outer space. It could turn into an elephant and crush your hand, and other logically possible things like this. This would mean that God has no well-defined course in which He brings things about, and that what God brings together as cause and effect would be arbitrary.And Al-Ghazali responds.

“If you could prove that in regard to things which ‘can exist’ there cannot be created for man a knowledge that they ‘do not exist’, then these absurdities would be inescapable. We have no doubt in regard to the situations described by you. For God has created for us the knowledge that He would not do these things, although they are possible. We never asserted that they are necessary. They are only possible-i.e., they may, or may not, happen. It is only when something possible is repeated over and over again (so as to form the Norm), that its pursuance of a uniform course in accordance with the Norm in the past is indelibly impressed upon our minds.”

“Now, if in extraordinary times, God breaks the Norm by causing such a thing to happen, then our cognitions (that a certain possible thing ‘does not happen’) will slip out of our hearts and will not be recreated by Him. Therefore, there is nothing to prevent us from believing that: (a.) something may be possible, and may be one of those things to which God’s power extends; (b.) in spite of its being possible, it might have been known as a rule in the past that God would not do it; and (c.) God may create for us a knowledge that He would not do it in this particular instance. So the philosophers’ criticism is nothing but obstinate fault-finding.”

Al-Ghazali goes on further to meet the criticism of the philosophers, and this is a partial response to the criticism of the capacities of things, which is suppose to be part of their necessary connection.

“We agree that fire is so created that when it finds two pieces of cotton which are similar, it will burn both of them, as it cannot discriminate between two similar things. At the same time, however, we can believe that when a certain Prophet was thrown into the fire, he was not burnt-either because the attributes of fire had changed, or because the attributes of the prophet’s person had changed. Thus, there might have originated-from God, or from the angels- a new attribute in the fire which confined its heat to itself, so that the heat was not communicated to the prophet. Hence, although the fire retained its heat, its form and its reality, still the effect of its heat did not pass onwards. Or there might have originated a new attribute in the prophet’s body which enabled it to resist the influence of fire, although it had not ceased to be composed of flesh and bones.”

“We see that one who covers himself with asbestos sits down in a blazing furnace, and remains unaffected by it. He who has not observed such a thing will disbelieve it. Therefore, our opponents’ disbelief in God’s power to invest fire or a person’s body with a certain attribute which will prevent it from burning, is like disbelief on the part of a man who has not observed asbestos and its effect. Things to which God’s power extends include mysterious and wonderful facts. We have not observed all those mysteries and wonders. How, then, can it be proper on our part to deny their possibility, or positively to assert their impossibility?”

Now the philosophers might agree that God’s power extends to all that is possible and  that no power extends to that which is impossible. Everything has been divided into three kinds. [1.] an impossibility that is known; [b.] the possibility that is known; and [c.] those things we are hesitant in affirming their possibility or their impossibility. So what does Al-Ghazali mean by “impossibility”? Would it be a combination of affirmation and negation in the same thin, then say those two things don’t presuppose the existence of the other.

For example, it seems that Al-Ghazali is saying that (a.) God has the power to create will without knowledge of the object of will; (b.) God has the power to cause movement of a dead man’s hand to look like they are alive and write a book while holding a conversation with you; (c.) or God could cause a body to move when the person is not alive and in the body and etc. The problem becomes that when these things are possible, all distiniction between voluntary and spasmodic movements are gone. “No controlled action will be an indication of knowledge or power on the part of the agent.” This, indirectly, points out the problem of Other Minds. We could say, with the problem of the “dead man” moving looking like they’re alive, that we can’t tell if their conscious like us or if they’re just machines/zombies without consciousness like us.

Al-Ghazali responds to their point in this manner, but first responds to what he takes to be “impossible”.

“No one has power over the Impossible. What the Impossible means is the affirmation of something together with its denial; or the affirmation of the particular together with the denial of the general; or the affirmation of two together with the denial of one. That which does not fall under these heads is not impossible. And that which is not impossible is within power.”

“The combination of blackness and whiteness is impossible; for by the affirmation of the forms of blackness in a subject we understand the negation of whiteness, and the existence of blackness. Therefore, if the negation of whiteness is understood by the affirmation of blackness, then the affirmation of whiteness together with its (understood) negation will be impossible.”

“It is not possible for one person to be in two places at the same time. For by his being in the house we understand his not-being in the not-house. Therefore, it is impossible to suppose his being in not-house together with his being in house which only means the denial of his being in not-house.”

“Nor is it possible that knowledge should be created in inorganic Matter. For by inorganic Matter we understand something which has no cognition. If cognition is created in it, it will be impossible to call it inorganic Matter in the sense in which we understand it. If in spite of the new-created cognition, the stone does not cognise, then it will be impossible to name as knowledge this new-created thing which does not enable its subject to have any cognition whatsoever. So this is the reason why the creation of knowledge in inorganic Matter is impossible.”

“When we say that blood becomes sperm, we mean that one and the same Matter has put off one form to take on another. So the final outcomes is that one form has passed away, and another has come into existence, while Matter remains unchanged beneath successive forms. Again, when we say that water becomes air because of heat, we mean that Matter which had received the form of water has now discarded that form to receive another. So the Matter is common; it is only its attributes which change. Similarly, therefore, we may speak of the Rob becoming a serpent, or of dust becoming an animal. But between the Substance and the Accident there is no common Matter.”

This is a response to point (b.), which is about the person that we took to be dad looking like they’re alive and writting a book and talking with us.

“…we must say that in itself it is not impossible. For we ascribe all temporal events to the will of One who acts by choice. But it is to be rejected insofar as it is subversive of the usual course of events. Your statement that the possibility of such a thing will destroy the probative value of the adjustment of an action is an indication of knowledge on the agent’s part is not true. For it is God who is the agent; He makes the adjustment, and performs the action-through the dead man.”

Now Al-Ghazli has a response to point (c.), which is based on us not being able to tell the difference between voluntary and spasmodic movements. And this point also deals with the Problem of Other Minds.

“As regards your statement that thee remains no distinction between voluntary and spasmodic movements, we will say that we know such a thing from ourselves. When in our own case, we observe a distinction between the two states, we designate the cause of distinction as power. And then we conclude that what actually happens is only one of the two possible things-i.e. either the state in which movement is produced by power, or the state in which it is produced not-by-power. So when we look at someone else, and see many coherent movements, we acquire the knowledge of his power over the movements. Now, this knowledge is one of those cognitions which are created by God, and which depend upon the continuance of the regular course of events. Knowledge of this kind can only tell us of the existence of one of the two possible things. But, as shown earlier, it does not prove the impossibility of the alternative.”

And such is part of Al-Ghazli’s view on causality, or more specifically, Efficent Causation. His position touched on problem of Other Minds, Things continuing as they have before (Uniformity of Events), Change in things to keep uniformity of Matter, and God is the cause of all events.

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