allzermalmer

Truth suffers from too much analysis

Posts Tagged ‘Necessary’

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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Principles of William of Ockham (Occam)

Posted by allzermalmer on November 18, 2012

These are the basic principles of William of Ockham or William of Occam. This comes from

1. All things are possible for God, save such as involve a contradiction.

In other words, God can do (or make or create) everything which does not involve a contradiction; that which includes a contradiction is absolute non-entity. Ockham expressly bases this principle on an article of faith: ‘I believe in God the Father Almighty’. From this Ockham immediately infers a second principle which is encountered everywhere in his writings:

2. Whatever God produces by means of secondary (i.e. created) causes, God can produce and conserve immediately and without their aid.

Hence any positive reality which is naturally produced by another created being (not of course without the aid of God who is the first cause) can be produced by God alone without the causality of the secondary cause. In other words, God is not dependent on the causality of created causes, but they are absolutely dependent on His causality. This is stated in a more general manner:

3. God can cause, produce and conserve every reality, be it a substance or an accident, apart from any other reality.

Hence God can create or produce or conserve an accident without its substance, matter without form, and vice versa. In order to bring anything under the operation of this principle, it is sufficient to prove that it is reality or entity. These rules or guiding principles are theological in nature, as Ockham does not fail to emphasise. The following is, however, a scientific principle of general application:

4. We are not allowed to affirm a statement to be true or to maintain that a certain thing exists, unless we are forced to do so either by its self-evidence or by revelation or by experience or by a logical deduction from either a revealed truth or a proposition verified by observation. 

That is the real meaning of ‘Ockham’s Razor’ can be gathered from various texts in Ockham’s writings. [Nothing must be affirmed without a reason being assigned for it, except it be something known by itself, known by experience, or it be something proved by authority of holy scripture.’ and ‘We must not affirm that something is necessarily required for the explanation of an effect, if we are not led to this by a reason proceeding either from a truth by itself or from an experience that is certain.’]

It is quite often stated by Ockham in the form: ‘Plurality is not to be posited without necessity’ (Pluralitas non est ponenda sine necessitate), and also, though seldom:  ‘What can be explained by the assumption of fewer things is vainly explained by the assumpition of more things’ (Frustra fit per plura quod potest fieri per pauciora). The form usually given, ‘Entities must not be multiplied without necessity’ (Entia non sunt muliplicanda sine necessitate), does not seem to have been used by Ockham. What Ockham demands in his maxim is that everyone who makes a statement must have a sufficient reason for its truth, ‘sufficient reason’ being defined as either th eobservation of a fact, or an immediate logical insight, or divine revelation, or a deduction from these. This principle of ‘sufficient reason’ is epistemological or methodological, certainly not an ontological axiom.

The scholastics distinguished clearly between a sufficient reason or cause (usually expressed by the verb sufficit) and a necessary reason or cause (usually expressed by requiritur). As a Christian theologian Ockham could not forget that contingent facts do not ultimately have a sufficient reason or cause of their being, inasmuch as God does not act of necessity but freely; but our theological and philosophical, and in general ll our scientific, assertions ought to have a sufficient reason, that is a reason from the affirmation of which the given assertion follows. All created things can be explained ultimately only by a necessary reason, i.e. a cause which is required to account for their existence. For every creature is contingent. The guiding idea of Duns Scotus, to safeguard contingency (servare contingentiam), is present everywhere in the work of Ockham. We can formulate it as follows:

5. Everything that is real, and different from God, is contingent to the core of its being.

If we bear in mind these guiding principles of Ockham, then his philosophical work becomes intelligible as the effort of a theologian who is looking for absolute truth in this contingent world, viz. for truth independent of any of those thoroughly contingent worlds which are equally possible. He is a theologian who views the world from the standpoint of the absolute. Consequently he sees many truths which were called ‘eternal’ dwindling away in the light of eternity, which is God himself. The actual order of creatures remains contingents; the possible order of creatures is above contingency. Hence the tendency of Ockham to go beyond the investigation of the actual order, by asking what is possible regardless of the state of the present universe. What is absolutely possible can never be impossible; and in that sense statements about absolute possibility are always true and free from contradiction, and for that reasons are necessary. Thus the work of Ockham also becomes intelligible- and this is only the converse of the former viewpoint- as the effort of a philosopher who constantly remanded by the theologian in himself that he must not all any truth necessary unless it can be shown that its denial implies a contradiction.

 

 

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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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Possibility and Necessity

Posted by allzermalmer on January 14, 2012

This blog is going to deal with logically necessary and logically possible. This is slightly different from Avicenna, but mostly based on Modal Logic, or some of the basic ideas of Modal Logic.

Because it is a form of logic, it also deals with one of the foundations of logic. Logic is concerned with statements, and the inferences that we draw from these statements. It is about having a couple of statements, and seeing if we can draw another statement with those statements that we accept. Logic helps give us some rules to follow in order to say that we drew a statement from the other statements that we held to, in a correct manner.

Law of Identity is “every individual thing is identical to itself”. Law of Excluded Middle is “every statement is either true or false”. Law of Non-Contradiction is “given any statement and its opposite, one is true and the other false”. And with possibility and necessary, they are mostly based on the Law of Non-Contradiction.

Possible means not self-contradictory. For example, “The sun won’t rise tomorrow” or “I ran a 2 minute mile” are possible. There’s nothing logically self-contradictory there. Necessary means self-contradictory to deny, which are based on logic, meaning of concepts, or necessary connection between properties. For example, “2+2=4” or “a bachelor is an unmarried male”.

A possible world is “a consistent and complete description of how things might have been or might in fact be.” A possible world is a consistent world, and this means that the statements that describe a possible world don’t entail self-contradiction. We can’t have statement X of possible world N and statement ~X of possible world N, being both affirmed at the same time like “statement X of possible world N and statement ~X of possible world N”. To do this would be to affirm a contradiction, or show that that possible world couldn’t exist because it’s a contradictory world. But the actual world is a description of how things in fact are. Yet, it would seem, that the actual world has to be consistent as well, which means there are no self-contradictions in the world. So the actual world is a possible world itself.

One of the difference between necessary and possible is that necessary statements are known to be true or false without experience. This means that information obtained from observation or sense-perception play no rule in determining if the necessary statement is true. This means we can know that a statement is true without recourse to evidence supplied by observation. But possible statements are known to be true or false with experience. This means that information from observation or sense-perception plays a part in determining if the possible statement is true. This means we know that a statement is true with recourse to evidence supplied by observation.

(A side note is that a necessary statement is a possible statement as well. This is because a necessary statement is a statement that it would be self-contradictory to deny, but the statement itself shouldn’t be self-contradictory and that’s what a possible statement is as well.)

From this idea of possible and necessary, we can say there are three types of statements. There are necessary statements, impossible statements, and contingent statements. A necessary statement is a statement that couldn’t be false. A impossible statement is a statement that couldn’t be true. A contingent statement is a statement that could be true or could be false, or could have been true or could have been false, or could be true in the future or could be false in the future.

So take “a bachelor is an unmarried male”. This is a necessary statement and means it’s necessarily true. But now say that I say “a bachelor isn’t an unmarried male”. This is a impossible statement, and means it’s necessarily false. To actually affirm the second statement is to affirm something that is false. Now say that I affirm “Justin Bieber is a bachelor”. That statement is a contingent statement. This means that Justin Bieber is a married male or isn’t a married male. The only way we could tell which of the two propositions is true is through experience.

So take the statement “Justin Bieber is a bachelor“, and accept it’s a contingent statement. So when we say “‘Justin Bieber is a bachelor‘ is a contingent statement”, we are also saying “‘Justin Bieber is a bachelor‘ is possible and not ‘Justin Bieber is a bachelor‘ is possible.”But take the same statement, and accept it’s a contingent truth. So when we say “‘Justin Bieber is a bachelor‘ is a contingent truth”, we are also saying “‘Justin Bieber is a bachelor’ is true but could have been false.”

As Raymond D. Bradley said, “Our own world – the real world, the actual one – is just one of many possible worlds. Indeed, it is just one of infinitely many, since for any possible world containing say n atoms there is another logically possible world containing n+1 atoms, and so on ad infinitum.” So there are an infinity of possible worlds, or possible ways of describing what might have been or what might be. Say we have the possible world of N, and within this world it contains M and M contains 140 particular things that are M. That is one possible world, but anther possible world, which is logically contradictory from N, and we can call it N*, contains 141 particulars in M. But 140 is logically contradictory from 141. So N is one possible world and N* is a different possible world, but they are logically contradictory from one another.We can continue on doing this infinitely, and so there’s an infinity of possible worlds.

Now science is concerned with contingent statements. This is because science is said to be empirical. The quotes, in order of the authors, are from Richard Feynman, Pierre Duhem, Stephen Hawking, and Henir Poincare: “The principle of science, the definition, almost, is the following: The test of all knowledge is experiment. Experiment is the sole judge of scientific “truth.””; “Agreement with experiment is the sole criterion of truth for a physical theory.”; “[the] scientific method…w[as]…developed with goal of experimental verification.”; “Experiment is the sole source of truth. It alone can teach us something new; it alone can give us certainty. These are two points that cannot be questioned.” Now experiment is based on coming to human sense-perception. This can be anything from looking at the squirrel climbing a tree to reading the numbers off of a volt-meter.

The reason that science wants to deal with contingent matters, besides dealing with sense-perception, is that science likes to try to have the ability to be shown that the theories are false. But if science only dealt with necessary statements, then scientific statements could never be shown to be false. For if a statement was presented that said a necessary statement was false, that statement would be an impossible statement and necessarily be false. But a contingent statement can be shown to be false.

But science creates models of how the actual world could possibly be. Take this example of what Richard Dawkins says of science and what science does: “There is a less familiar way in which a scientist can work out what is real when our five senses cannot detect it directly. This is through the use of a ‘model’ of what might be going on, which can be tested. We imagine- you might say we guess- what might be there. That is called the model. We then work out (often by doing a mathematical calculation) if the model were true. We then check whether that is what we see…We look carefully at the model and predict what we ought to see (hear, etc.) with our sense (with the aid of instruments, perhaps) if the model were correct. Then we look to see whether the predictions are right or wrong.  ”

So science creates a model of what is possible, and what happens in this possible world. And we can deduce what should be observed if this possible world is the actual world. We have the possible world of N. But N is our model. So, “If N then O. O. Therefore N.” This is the fallacy of affirming the consequent. Just because the model has a true prediction in the actual world, that doesn’t mean that the model is how the actual world is. In other words, just because this possible world (model) had one right prediction of how the actual world is (which is also another possible world), doesn’t mean that this actual world is that possible world. It also wouldn’t matter if the model has made nothing but correct predictions up till now, because the problem is still there.

If I robbed a bank, then I have 100 million dollars. I have 100 million dollars. Therefore, I robbed a bank. But me having 100 million dollars is also consistent with me winning the lottery, it is also consistent with me investing my money in a way where I got a lot of return in my investments, me starting up a business and my business got me 100 million dollars, or me getting the money from the death of a family member as a part of inheritance. In other words, there are many other possibilities that are consistent with the actual results or observations. As W.V. Quine once said, “Whatever observation would be counted for or against the one theory counts equally for or against another.”

This seems to raise a skeptical problem.

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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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