# Posts Tagged ‘Impossible’

## 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:
Lukasiewicz’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.]

## When You Have Eliminated the Impossible…

Posted by allzermalmer on March 20, 2013

Sherlock Holmes is written to have said “when you have eliminated the impossible, whatever remains, however improbable, must be the truth.”

The full quote is found in the book The Sign of Four: “You will not apply my precept,” he said, shaking his head. “How often have I said to you that when you have eliminated the impossible, whatever remains, however improbable, must be the truth? We know that he did not come through the door, the window, or the chimney. We also know that he could not have been concealed in the room, as there is no concealment possible. When, then, did he come?”

The method that Holmes is using is an Eliminative Method, or a form of Eliminative Inference. This form of reasoning has at least two forms of inference. They are the Disjunctive Syllogism and Modus Tollens.

Disjunctive Syllogism: Because at least one disjunct must be true, by knowing one is false we can infer that the other is true.
Premise 1: Either A or B
Premise 2: ~A
Conclusion: Therefore, B

Premise 1: Either they Came through the Door, or Came through the Window, or Came through the Chimney, or Came through the hole in the roof.
Premise 2: Didn’t come through the door and Didn’t come through the window and didn’t come through the chimney.
Conclusion:Therefore, they Came through the hole in the roof.

This is an eliminative inference. We have some possible solutions to the mystery that Holmes is investigating. Holmes has also made some observations that are inconsistent with some of those possible solutions. These observations contradict the possible solutions, and this makes them impossible. Holmes rejects some of the conjectured solutions and has now limited possible solutions that are still consistent with his observations.

Suppose Holmes were to list alternative hypothesis. Suppose that Holmes ordered them in preference, and those that he preferred the most were considered to be the most probable. For example: Come through the Window (65%) > Come through the Door (20%) > come through the Chimney (10%) > Came through the hole in the roof (5%).

Here are four alternative hypothesis, and each one of them is graded on their probability of being true and his preference going from most probable to improbable. Holmes collects some observations. Some of the hypothesis are inconsistent with the observations. The more probable hypothesis are eliminated by the observations collected. To hold to the observations collected and hypothesis that are inconsistent with those observations, would be equivalent to holding to what is impossible. The hypothesis are contradicted by the observations. No matter how probable the impossible is, it still isn’t possibly true. And once we have eliminated the impossible, whatever remains, however improbable, must be the truth.

## Whatever Is Conceivable Is Possible

Posted by allzermalmer on September 27, 2012

I am going to quote one little section in a book called Hume’s First Principles by Robert Fendel Anderson. This first part of the book is on Perceptions, and the first principle gone over on Perceptions is “Whatever is Conceivable is Possible”.

“The principle of the possible existence of whatever is conceivable is one which Hume finds both an evident principle and already an established maxim in metaphysics[1]. The application of the principle is frequently restricted to that which is clearly and distinctly conceivable: “…nothing of which we can form a clear and distinct idea is absurd and impossible.”[2] Again: “To form a clear idea of anything, is an undeniable argument for its possibility…”[3]. The possibility of existence, therefore, is of the essence of whatever is clearly and distinctly conceived; that is, its possibility is included or implied within it: “ ‘Tis an establish’d maxim in metaphysics, That whatever the mind clearly conceives includes the idea of possible existence…”[4] and: “Whatever can be conceiv’d by a clear and distinct idea necessarily implies the possibility of existence….”[5]

A clear and distinct idea, according to Hume’s doctrine, is one which neither contains nor implies a contradiction: “Now whatever is intelligible, and can be distinctly conceived, implies no contradiction…”[6] Again: “How any clear, and distinct idea can contain circumstances, contradictory to itself, or to any other clear, distinct idea, is absolutely incomprehensible….”[7] In saying that whatever is clearly and distinctly conceived is possible, therefore, it appears to be Hume’s intention also that whatever is self-consistent and noncontradictory is possible:

“Whatever can be conceiv’d by a clear and distinct idea necessarily implies the possibility of existence; and he who pretends to prove the impossibility of its existence by any argument deriv’d from the clear idea, in reality asserts, that we have no clear idea of it, because we have a clear idea. ‘Tis in vain to search for a contradiction in any thing that is distinctly conceiv’d by the mind.”[8]

The expression employed in the remarks thus far examined may lead the reader to suppose that there are some things clearly and distinctly conceived and some not- that some of our ideas are clear and distinct and some of them unclear and indistinct. Were this true, then it would follow that we have ideas of things the existence of which we must regard as impossible. There is evidence, however, that Hume considers all our ideas to be clear and distinct. He offers an argument to this conclusion, based on his doctrine that ideas are derived from impressions:

“…we need but reflect on that principle so oft insisted on, that all our ideas are copy’d from our impressions. For from thence we may immediately conclude, that since all impressions are clear and precise, the ideas, which are copy’d from them, must be of the same nature…”[9]

Since all perceptions are either impressions or ideas[10], we must conclude that there are no perceptions of any kind that are not clear and precise.

From the clarity and preciseness of all ideas, we may infer, moreover, that we possess no ideas of those things whose existence we must regard as impossible, but that any idea we may have is the idea of something the existence of which is possible. We find, indeed, that Hume does not always restrict the possibility of existence to that which is clearly and distinctly conceived, but extends it as well to everything that is conceived or imagined at all: “…whatever we conceive is possible.”[11] And: “…whatever we can imagine, is possible.”[12]Hume appears, indeed, to make no firm distinction between what is clearly and distinctly conceived and what is conceived or imagined merely, as is evidenced in his full statement of the metaphysical maxim: “ ‘Tis an establish’d maxim in metaphysics, That whatever the mind clearly conceives includes the idea of possible existence, or in other words, that nothing we imagine is absolutely impossible.”[13] We are thus again justified, apparently, in supposing that all our ideas are equally clear and distinct, and that all things conceived are possible. Things which are contradictory and therefore impossible, on the other hand, cannot be conceived or imagined at all: “We can form the idea of a golden mountain, and from thence conclude that such a mountain may actually exist. We can form no idea of a mountain without a valley, and therefore regard it as impossible.”[14] Again: “ ‘Tis in vain to search for a contradiction in any thing that is distinctly conceiv’d by the mind. Did it imply any contradiction, ‘tis impossible it cou’d ever be coneiv’d.”[15]

Knowing then that self-contradictory things are neither conceivable nor possible, and knowing that whatever is conceived or imagined is possible, we may next inquire what things are in fact conceived or imagined and hence possible. From certain of Hume’s remarks one might infer that we conceive only perceptions; for it is only perceptions that are “present to” the mind: “…nothing is ever really present with the mind but its perceptions or impressions and ideas…”[16] If this be true, then it is reasonable to suppose that we have clear and distinct ideas only of perceptions, as Hume sometimes appears to agree: “We have no perfect idea of any thing but of a perception.”[17] Now if we can conceive only of perceptions, then according to Hume’s principle it is only perceptions whose existence we may regard as possible. We may observe, moreover, that the remarks we have thus far examined do not imply that perceptions, as such, exist, but only that their existence is possible. Were there no further texts available to us from among Hume’s writings, we might justifiably conclude that what he calls “perceptions” are to be understood as a realm of mere essences which, taken together, comprehend all possibility, but which are not, of themselves, existence.”

[1] David Hume, A Treatise of Human Nature, ed. by L.A. Selby-Bigge (Oxford: Clarendon Press, 1888), pp. 32, 250, Hereafter cited as Treatise.

[2] Treatise, pp.19-20. Cf> David Hume, Dialogues Concerning Natural Religion, ed. and with an introduction by Henry D. Aiken(New York: Hafner Library of Classic, Hafner Publishing Company, 1948), p. 19, Philo speaking. Hereafter cited as Dialogues.

[3] Treatise, p. 89

[4] Treatise, p.32

[5] Treatise, p. 43

[6] David Hume, “An Enquiry Concerning the Human Understanding,” in An Enquiry Concerning the Human Understanding and an Enquiry Concerning the Principles of Morals, ed. and with an introduction by L.A. Selby-Bigge (2d ed.; Oxford: Clarendon Press, 1902), p. 35. Hereafter cited as “Understanding.” Cf. Dialogues, p. 58, Cleanthes speaking.

[7] “Understanding.” P. 157

[8] Treatise p. 43.

[9] Treatise, p. 72; cf. p. 366.

[10] Treatise, pp. 1, 96.

[11] Treatise, p. 236.

[12] Treatise, p. 250

[13] Treatise, p. 32.

[14] Treatise, p. 32.

[15] Treatise, p.43. Cf. “Understanding,” p. 164.

[16] Treatise, p. 67; cf. pp.197,212.

[17] Treatise, p. 234.