December 13, 2010


me+ME: Introduction to probability


"Probability is the very guide to life."
   (Bishop Butler)

"Every year, if not every day, we have to wager our salvation upon some prophecy based upon imperfect knowledge."
   (O.W. Holmes, Jr.)

"Life is the art of drawing sufficient conclusions from insufficient premises."
   (Samuel Butler II)

"The actual science of logic is conversant at present only with things either certain, impossible, or entirely doubtful, none of which (fortunately) we have to reason on. Therefore the true logic for this world is the calculus of Probabilities, which takes account of the magnitude of the probability which is, or ought to be, in a reasonable man's mind."
   (James Clerk Maxwell, quoted by E.T. Jaynes)

"Probability theory is nothing but common sense reduced to calculation."
   (Laplace, quoted by E.T. Jaynes

I continue being not well, and otherwise also as before, so I cannot do much.

Today I reproduce part of something I wrote on induction, since it seems to me a good introduction to the very basis of the mathematics of probability, at least for those who are willing to think for themselves and fairly naive about both mathematics and probability, since I explain most and added notes for clarity and background.

And the reason for my wanting to give some reasonable explanations of the very basis of the mathematics of probability is given by the opening quotations, and by my last note, that shows why the concept of irrelevance is so important.

Introduction to probability: There are a number of different axiomatizations of probability[1], and also at least three different interpretations of any given axiomatization[2]. Before considering the different interpretations I shall give what has become in this century the standard mathematical axiomatization of probability theory, namely Kolmogorov's.[3] For the finite case[4] Kolmogorov proposed three axioms, which may be stated as follows:

Suppose that $ is a set of propositions P, Q, R etc. and that this set is closed for negation, conjunction and disjunction, which is to say that whenever (P e $) and (Q e $), so are ~P, (P&Q) and (PVQ). [5] Now we introduce pr(.) as a function that maps the propositions in $ into the real numbers [6] in the following way, that is, satisfying the following three axioms:

A1. For all P e $ the probability of P, written as pr(P), is some non-negative real number.
A2. If P is logically valid, pr(P)=1.
A3. If ~(P&Q) is logically valid, pr(PVQ)=pr(P)+pr(Q).

In the present formulation I have chosen to attribute probabilities to propositions. This is not necessary, for probabilities may be attributed to sets as well. Since the mathematics is the same[8], and since the present reading is more convenient, I have chosen to say that probabilities apply to propositions. There is more to be said on this choice but as this properly belongs to the interpretations of probability[9], I shall return there to this issue.

Basic theorems: Irrespective of the axiomatization or interpretation of probability, there are a number of important theorems which we shall need - just as we need laws like (a+b)=(b+a) for counting, irrespective of axioms used to prove them or of what we choose to count. The advantage and use of axioms is that one can use them to prove the theorems one needs - and having given a valid proof one knows that any objection against the theorem must be directed against the axioms, for the theorem was proved to follow from them.[10] So what we shall do first is to derive some useful theorems. First, then, there is

T1. pr(~P)=1-pr(P).

This shows how we can find the probability of ~P from pr(P). T1 is proved by noting that pr(PV~P)=pr(P)+pr(~P) by A3, since ~(P&~P) is logically valid, and also pr(PV~P)=1 by A2, since (PV~P) is logically valid. It should be noted that here and in the rest of the chapter I merely indicate the proofs, so that the reader can do the rest.[11] Next there is

T2. 0 <= pr(P) <= 1.

which says that all probabilities are in the interval of 0 to 1 inclusive. That pr(~P) is not less than 0 follows from A1. Now if pr(P) would exceed 1, pr(~P) would be less than 0 by T1, which is a contradiction. So it follows pr(P) does not exceed 1, and T2 now follows by A1.

T3. If P |= Q, then pr(P) <= pr (Q).

This says that if P logically entails Q, then pr(P) is not larger than pr(Q). It can be proved by noting that if P indeed does logically entail Q, that then ~(P&~Q), and so A3 entails pr(PV~Q) = pr(P)+pr(~Q). By A1 and T2 the LS[12] is <= 1, and so pr(P)+pr(~Q) <= 1, from which follows the theorem on transposing 1-pr(Q). T3 immediately entails

T4. If P is logically equivalent to Q, then pr(P)=pr(Q),

as logical equivalence amounts to: P |= Q and Q |= P. So logical equivalents have the same probability. This is a very important theorem, and is used all the time. [13] Thus we can use it to prove the following expansion of any proposition P:

T5. pr(P)=pr(P&Q) + pr(P&~Q), for arbitrary P and Q in $.

It is noteworthy that this expansion can be repeated: pr(P&Q) + pr(P&~Q) = pr(P&Q&R)+ pr(P&Q&~R)+pr(P&~Q&R)+pr(P&~Q&~R), and repeated ad lib.

T5 is proved by noting that T4 entails that pr(P)=pr((P&Q)V(P&~Q)), since P iff P&Q V P&~Q, which turns into T5 upon noting that ~((P&Q)&(P&~Q)) is logically valid, and applying A3. From T5 we can conveniently prove

T6. pr(PVQ)=pr(P)+pr(Q)-pr(P&Q)

which extends A3, and shows how we may calculate the probability of any disjunction. To prove this, we first note that by T4 pr(PVQ) = pr((P&Q) V (P&~Q) V (~P&Q)) = pr(P&Q) + pr(P&~Q) + pr(~P&Q) by A3. Since by T5 pr(Q) = pr(P&Q) + pr(~P&Q), it follows that adding pr(P)+pr(Q) and subtracting once their common term pr(P&Q) yields pr(PVQ) i.e. T6. If we now combine T5 and T6 we get

T7. (pr(P&Q) <= pr(P))  &  (pr(P) <= pr(PVQ))

that is: The probability of a conjunction is not larger than the probability of any of its conjuncts, and the probability of a disjunction is not smaller than the probability of any of its disjuncts.

Basic conditional theorems: Most probabilities are not, as they were in this chapter so far, absolute, but are conditional: Rather than saying "the probability of Q = x" we usually introduce a condition and say, "the probability of Q, if P is true, = y". This idea, that of the probability of a proposition Q given that one or more propositions P1, P2 etc. are true is formalised by the following important definition:

Definition 1 : pr(Q/P) = pr(P&Q):pr(P)

That is: The conditional probability of Q, given or assumed that P is true, equals the probability that (P&Q) is true, divided by the probability that (P) is true.[14] NB, as this fact has important implications for the interpretation and application of probability theory: A conditional probability is defined in terms of absolute probabilities, so therefore we need absolute probabilities to establish conditional ones. [15]

Definition 1 has many applications, and many of these turn on the fact that it also provides an implicit definition of pr(P&Q), namely as pr(P)pr(Q/P) (simply by multiplying both sides of Def 1 by pr(P)). Consequently, we have as a theorem (if pr(P)>0 and pr(Q)>0), since P&Q and Q&P are logically equivalent

T8. pr(P&Q)=pr(P)pr(Q/P)=pr(Q)pr(P/Q)

The second equality is, of course, also an application of Def 1, and T8 accordingly says that the probability of a conjunction equals the probability of one conjunct time the probability of the other given that the one is true.

Another consequence of Def 1 is

T9. pr(Q/P)+pr(~Q/P)=1

which results from T5 and Def 1 upon division by pr(P), and says that the probability of Q if P plus the probability of ~Q if P equals 1. Of course, this admits of a statement like T1:

T10. pr(Q/P)=1-pr(~Q/P)

which shows that conditional probabilities are like unconditional ones. A theorem to the same effect, that parallels T3 is

T11. 0 <= pr(Q/P) <= 1.

That 0 <= pr(Q/P) follows from D1, because the components of a conditional are both >=0 by A1; and that pr(Q/P)<=1 is equivalent to pr(P&Q) <= pr(P), which holds by T7. A theorem in the vein of T4 is

T12. If P |= Q, then pr(P&~Q)=0

This is proved by noting that if P |= Q holds, then so does ~(P&~Q), which, by A3, entails that pr(PV~Q)=pr(P)+pr(~Q). As by T6 pr(PV~Q)=pr(P)+pr(~Q)-pr(P&~Q), it follows pr(P&~Q)=0 if P |= Q. From this it easily follows that

T13. If P |= Q, then pr (Q/P)=1 provided pr(P)>0

which is to say that if Q is a logical consequence of P, the probability of Q is P is true is 1. The proviso is interesting, for it denies the possibility of inferring Q from a logical contradiction or known falsehood. This means that the def: P |= Q =df pr(Q/P)=1 strengthens the logical "|=" by adding that proviso. [16] T13 immediately follows from T5, T12 and Def 1.

Def 1 may, of course, list any finite number of premises, as in pr(Q/P1&....&Pn) = pr(Q&P1&....&Pn):pr(P1&....&Pn). Such long conjunctions admit of a theorem like T8:

T14. pr(P1&.....&Pn)=

This says that the probability that n propositions are true equals the probability that the first (in any convenient order) is true times the probability that the second is true if the first is true times the probability that the third is true if the first and the second are true etc. The pattern of proof can be seen by noting that for n=3 pr(P1)pr(P2/P1)pr(P3/P1&P2) = pr(P1&P2)pr(P3/P1&P2) = pr(P3&P2&P1) because the denominators successively drop out by Def 1. That the premises can be taken in any order is a consequence from T4: Conjuncts taken in any order are equivalent to the same conjuncts in any other order.

T11 and T13, together with T9 and T10, show that conditional probabilities are probabilities. We need just one further theorem:

T15. If R |= ~(P&Q), then pr(PVQ/R) = pr(P/R)+pr(Q/R)

which parallels A3. It is easily proved by noting that pr(PVQ/R) = (pr(P&R)+pr(Q&R)-pr(P&Q&R)):pr(R) by Def 1, T4 and T6, and that pr(P&Q&R)=0 by T12 and T4 on the hypothesis. The conclusion then follows by Def 1.

Irrelevance: A second important concept which now can be defined is that of irrelevance. Two propositions P and Q are said to be - probabilistically - irrelevant, abbreviated PirrQ if the following is true:

Def 2: PirrQ iff pr(P&Q)=pr(P)pr(Q)

Evidently, irrelevance is symmetric (by T4):

T16. PirrQ iff QirrP

But there are more interesting results. Let's call a logically valid statement a tautology and a logically false statement a contradiction. Then we can say:

T17. Any proposition is irrelevant to any tautology and to any contradiction.

Note that this entails that tautologies are also mutually irrelevant. To prove T17, first suppose that P is tautology. By A2 pr(P)=1. Since tautologies are logically entailed by any proposition, Q |= P, and so pr(Q&~P)=0 by T12. Consequently, it follows pr(Q)=pr(Q&P) by T5, and so pr(P).pr(Q) pr(P&Q) and we have irrelevance. Next, suppose (P) is a contradiction. If so, ~(P) is a tautology, and so pr(P)=0 by T1. By T7 pr(P&Q) <= pr(P) and as by A1 all probabilities are >= 0, it follows pr(P&Q)=0. But then pr(P)pr(Q) 0=pr(P&Q), and again we have irrelevance.

Def 2 is often stated in two other forms, which are both slightly less general, as they require respectively that pr(P)>0 or that pr(P)>0 and pr(~P)>0, in both cases to prevent division by 0. Both alternative definitions depend on Def 1, and the first is given by

T18. If pr(P)>0, then PirrQ iff pr(Q/P)=pr(Q).

This is an immediate consequence of Defs 1 and 2. It states clearly the important property that irrelevance signifies: If P is irrelevant of Q, the fact that P is true does not alter anything about the probability that Q is true - and conversely, by T16, supposing that Q is not also a contradiction. So irrelevance of one proposition to another is always mutual, and means that the truth of the one makes no difference to the probability of the truth of the other.

This can again be stated in yet another form, with once again a slightly strengthened premise, for now it is required that both pr(P) and pr(~P) are > 0:

T19. If 0 < pr(P) < 1, then PirrQ iff pr(Q/P)=pr(Q/~P)

Suppose the hypothesis, which may be taken as meaning that P is an empirical proposition, is true. T19 may be now proved by noting the following: pr(Q/P) = pr(Q/~P) iff pr(Q&P):pr(P) = pr(Q&~P): (1-pr(P)) iff pr(Q&P) - pr(P)pr(Q&P) = pr(P)pr(Q&~P) iff pr(Q&P) = pr(P)(pr(Q&P)+pr(Q&~P)) iff pr(Q&P) = pr(P)pr(Q).

Another important property of irrelevance is that if P and Q are irrelevant, then so are their denials:

T20. PirrQ iff (~P)irrQ iff Pirr(~Q) iff (~P)irr(~Q).

This too can be proved by noting some series of equivalences that yield irrelevance. First consider pr(P&~Q), assuming PirrQ. Then pr(P&~Q) = pr(P)-pr(P&Q) = pr(P)-pr(P)pr(Q) = pr(P)(1-pr(Q))= pr(P)pr(~Q) (using T1). So Pirr(~Q) if PirrQ. The converse can be proved by running the argument in reverse order, and so Pirr Q iff Pirr(~Q). The other equivalences are proved similarly.

Finally, the concept of irrelevance, which so far has been used in an unconditional form, may be given a conditional form, when we want to say that P and Q are irrelevant if T is true:

Def 3: PirrQ/T iff pr(Q/T&P) = pr(Q/T)

This says that the probability that Q is true if T is true is just the same as when T and P are both true - i.e. P's truth makes no difference to Q's probability, if T is true. It should be noted that Def 3 requires that pr(T&P) > 0 (which makes pr(T) > 0), but that on this condition T19 shows that Def 3 is just a simple extension of Def 2. And as with Def 2 there is symmetry:

T21. PirrQ/T iff QirrP/T.

For suppose PirrQ/T. By Def 3 pr(Q/T&P)=pr(Q/T) iff pr(Q&T&P):pr(T&P)=pr(Q&T):pr(T) by Def 1. This is so iff pr(Q&T&P):pr(Q&T) = pr(T&P):pr(T) iff pr(P/Q&T)=pr(P/T) iff QirrP/T by Def 3.

And this conditional irrelevance of Q from P if T does not only hold in case P is true, but also in case P is false. That is:

T22. PirrQ/T iff (~P)irrQ/T.

For suppose PirrQ/T, i.e. pr(Q/T&P) = pr(Q/T). By def 1 this is equivalent to pr(Q&T&P):pr(T&P) = pr(Q&T):pr(T) iff pr(Q&T&P) = pr(T&P)pr(Q&T):pr(T).
Now pr(Q&T&P) = pr(Q&T)-pr(Q&T&~P), and so we obtain the equivalent pr(Q&T&~P) = pr(Q&T)-(pr(T&P)pr(Q&T):pr(T)) = pr(Q&T)(1-(pr(T&P):pr(T)) = pr(Q&T)((pr(T)-pr(T&P)) : pr(T)) = pr(Q&T)(pr(T&~P):pr(T)) from which we finally obtain as equivalent to PirrQ/T pr(Q&T&~P):pr(T&~P) = pr(Q&T) : pr(T), which is by Def 3 the same as (~P)irrQ/T. Qed.

And finally T21 and T22 yield the same result for conditional irrelevance as for irrelevance:

T23. PirrQ/T  iff QirrP/T
                  iff (~P)irrQ/T
                  iff Pirr(~Q)/T
                  iff (~P)irr(~Q)/T

The proof is: The first line is T21, the second T22. The third results thus: By both theorems, QirrP/T iff (~P)irrQ/T whence PirrQ/T iff (~P)irrQ/T by T21. The fourth results from this by substituting (~Q) for Q as in the third line. Qed. [17]

So far as regards the mathematics of probability for the moment.


[1] It is interesting to note that while the ancient Greeks and Romans already gambled, the mathematical theory of probability only started in the 17th Century, when Pascal and Fermat laid some of its foundations, mainly related to games of chance or combinatorics, followed by Bernoulli's Ars Conjectandi, published a little after 1700 and Laplace's Theorie of Probabilities, published a little after 1800. Even so, a sound and widely accepted set of axioms for probability theory, that is, a set of propositions that, if true, logically imply many other propositions of the theory, only can about in the 1930ies.

[2] The "three interpretations" I have in mind are (1) ontological: probabilities are somehow real and relate to real facts, such as the chance that this bit of radioactive material emits a particle the next minute or the chance that 90% of the gas in a container is in the containers' left part or the chance that one's teeth are bad given one has a low income; (2) epistemological: probabilities are somehow related to knowledge one has about something, that may be far from complete or certain, and thus gets expressed as a fraction of certainty; (3) subjective: probabilities are related to persons' willingness to bet on events, and are not so much related to what the real facts are or may be, nor to the knowledge there is about the real facts or that the person has, but to the odds a given person is willing to accept that a certain event will be found to be true.

Here I have been sketching, and within each interpretation there are sub-interpretations, while many entertain several interpretations, e.g. real probabilities for quantum mechanics and insurance; epistemic probabilities or historical events and testimonies of witnesses; and subjective probabilities when one is betting on the horses.

The interesting fact is that the mathematics of probability is generally supposed to be the same whatever interpretation, the underlying reason being that probabilities are in any case much like proportions. Also, one can see the three interpretations combined, when considering e.g. the probability that a given bit of radioactive material will emit a particle the next minute: Presumably, there is a real chance, about which there is some but not full scientific knowledge, that some person may know some fraction of, while the person may be willing to accept certain odds on the outcome for subjective reasons of his own.

[3] As I mentioned in Note [1], Kolmogorov's axiomatization, given in the text, dates from the 1930ies. It was rapidly and widely accepted by mathematicians, physicists and statisticians, because it is simple, elegant, true for the intended interpretation - which is to say, in other words, what one takes probabilities to be intuitively (something like proportions, with an interpretation as in Note [2]) intuitively satisfies Kolmogorov's axioms - and powerful in that it implies logically many theorems about probability that should be true of probability in any intuitive sense (and that often had been proved before, from more specific asssumptions). See also the next note.

[4] Kolmogorov also provided an axiom for infinite sums, that is much needed for the more intricate applications of probabilities that involve the calculus, since integrals are infinite sums (sums of infinitely many, perhaps infinitely small, quantities). This will be left out here, but it should be mentioned that since Kolmogorov proposed his axiomatization that axiomatization, and indeed the purely mathematical theory of probability have become part of measure theory, which is a yet more general theory of things having some measure, in a fairly intuitive and mathematically precise sense. (See Halmos: "Measure Theory".)

As in Note [2], the problems relating to probability these days are not so much mathematical (standard probability theories are all part and parcel of measure theory, mathematically speaking) but philosophical: How is this all to be interpreted? What is probability? (But those much inclined to skepticism should also realize that standard probability works, as in applied technology, and physics, and also for insurance and many kinds of statistics.)

[5] Notational note: "e" is read as "is" or "is an element of (the set of)"; "~" is read as "not" or "it is not true that"; "&" is read as "and" or "both"; "V" is read as "or" in the inclusive sense, i.e. when the one or the other or both are true; and "$" as said is an arbitrary set of propositions (e.g. about a given subject) that is closed for the just mentioned logical operators (which accordingly means that the propositions that one can form with "~", "&" and "V" from propositions in the set are also in the set, as indeed seems intuitively desirable).

Finally, "iff" is "if and only if" (and [(P iff Q) iff ((P&Q) V (~P&~Q))]).

[6] A function is a rule that assigns precisely one value to some expression. Many ordinary terms, such as "your age", "your height", "your gender" are functional in this sense, also depending on time, as one has - at any given moment - just one age, height and gender. The rule refers to the procedure by which the value for the expression is to be found.

Sums, products, quotients etc. are also functions and one learns the rules for these at school.

The real numbers are numbers which may have infinitely long fractional parts that do not recur, and are needed as soon as one wants to calculate roots from arbitrary numbers, such as the square root of 2, which is a real number in the sense defined.

[7] The reader should note that the three axioms are intuitively true for proportions, and that "logically valid" is as in standard bivalent propositional logic, where it means "is true in each and any possible circumstance", such as "it is true it rains or it is not true it rains", which is so regardless from the weather. Finally, the hypothesis in A3 means that P and Q are not both true.
With these provisos, it will - probably (!) - be clear to the reader that the three axioms are true of proportions, as when "P" and "Q" are interpreted as areas or regions, possibly overlapping, as in Venn diagrams.

[8] The reason it is said "the mathematics is the same" for propositions and sets in this case is that sets generally are taken to be given by statements that are equivalent to statements involving "&", "V" and "~". Thus, the complement of a set consists of all elements that are not - "~" - in the set; the intersection of two sets is the set that consists of all elements that are in both - "&" - sets a.s.o.

[9] See Note [2] for the interpretations of probability. Another reason to prefer propositions and propositional logic is that it is more simple than alternatives like set theory.

[10] This is one reason why a mathematical or logical theory and proofs from axioms or assumptions are so important for coming to know things: Everything can be shifted back to one's assumptions, for in a good mathematical or logical theory everything else follows logically from these. (So one also knows that if a provable consequence is not true in fact, at least one of the assumptions used to deduce it cannot be true in fact - for this follows logically.)

See axiomatic method.

[11] And "the rest" is here mostly seeing that as pr(PV~P)=1 it follows pr(P)+pr(~P)=1, subsituting into pr(PV~P)=pr(P)+pr(~P)), from which it follows, subtracting pr(P) from both sides that pr(~P)=1-pr(P), which is the theorem.

[12] Notational note: "LS" is "left side". (Mathematicians also use LHS: left hand side).

[13] Here "logical equivalent" is meant: "by standard propositional logic". (Generally, the logical equivalences are also intuitively equivalent, but it is nice to have a procedure to appeal to when it is not intuitively obvious or needs proof.)

[14] Here it should be added for clarity that the usual arithmetical rule for fractions is followed: If in a fraction n/d it is the case that d=0 the fraction is undefined and does not exist. Other rules might be adopted, conceivably, but then there is the problem that the usual arithmetical rule is as stated.

Also, for conditional probabilities one may intuitively appeal to Venn diagrams, as in Note [7], and say that in these terms pr(Q|P) means that in the area corresponding to P, the proportion of Q is the conditional probability of Q if P.

[15] Here it should be added for clarity that what is claimed holds in the present context, but that there are axiomatizations of probability - Renyi's, for example - which are based on conditional probabilities. Some prefer such axiomatizations for philosophical reasons, which generally amount to the - true - claim that most statements of probability are somehow conditional rather than absolute.

[16] "Def" and "df" are short for "definition", and the claim in the text is conditional: If one defines entailment in probability theory in terms of conditional probability, it follows that the ex falso rule - "from a false assumption anything follows" no longer holds, in case of that definition.

[17] The concept of irrelevance (aka independence) is of very great cognitive and theoretical importance, for various reasons such as the following:

A. In any empirical test, for any theory whatsoever, we must make some assumption to the effect that almost all circumstances that also are the case, besides the prediction and theory we are testing, of which there generally is almost a whole universe full of other facts, are irrelevant for the outcome.

We must make such an assumption, because otherwise the test, whatever the outcome, is not conclusive in any way, since then, apart from the assumption (which generally is supported by methodological designs, randomizations etc.), any of the facts that obtain while test is done may be relevant to the outcome of the test - and while we can exclude possibly relevant conditions we know of by methodological design, we cannot methodologically design away all of the also occuring surrounding facts, and thus we must make the assumption. (This is widely missed, and I lectured on it - and related logical matters - some 21 years ago.)

B. In anything we do - also apart from testing theories - we generally assume that very much that also is happening at the time, close by or far away, is practically speaking quite irrelevant for what we do.

Indeed, we may be mistaken, but this is what we do do - disregard most of the things we don't already know to be relevant as irrelevant - and what seems to be necessary to want to do most things one does at all, for there is little point of trying to do things if one assumes the outcome may depend as much on oneself as on anything whatsoever, apart from one's efforts and knowledge.

C. It is very hard to see how a human intellect may come to know and understand much of the universe it finds itself in, if very many processes in that universe are not, once certain conditions are met, independent of very many other ongoing or preceding processes: Copper conducts electricity, regardless of the political situation; water remains H20 also if the earth gets blown up a.s.o.

D. See my Newton's Rules of Reasoning in Philosophy for some brief fairly systematic explanations, including probability theory and Ockham's Razor.

So far for the short exposition of basic probability theory, that was originally written in the early eighties, essentially because I hadn't found anything like it - an elementary, simple, fast, mathematically correct explanation, with proofs of many fundamental basic theorems - that suited my own purposes. The 17 notes were written today.

Finally, another reason - apart from its fundamental importance for human reasoning - to compile and write this Nederlog is that I love mathematics, and the above is a good basic piece of mathematics, and besides I try to find relief from the miseries of everyday by  reconnoitering the beauties of real science and of mathematics - see: Real science & real psychology = joy.

P.S. Corrections must wait till later.

-- Dec 14, 2010: Some small corrections and precisifications have been added.

P.P.S. It may be I have to stop Nederlog for a while. The reason is that I am physically not well at all. I don't know yet, but if there is no Nederlog, now you know the reason.

As to ME/CFS (that I prefer to call ME):

1. Anthony Komaroff

Ten discoveries about the biology of CFS (pdf)

3. Hillary Johnson

The Why

4. Consensus (many M.D.s) Canadian Consensus Government Report on ME (pdf)
5. Eleanor Stein

Clinical Guidelines for Psychiatrists (pdf)

6. William Clifford The Ethics of Belief
7. Paul Lutus

Is Psychology a Science?

8. Malcolm Hooper Magical Medicine (pdf)

Short descriptions:

1. Ten reasons why ME/CFS is a real disease by a professor of medicine of Harvard.
2. Long essay by a professor emeritus of medical chemistry about maltreatment of ME.
3. Explanation of what's happening around ME by an investigative journalist.
4. Report to Canadian Government on ME, by many medical experts.
5. Advice to psychiatrist by a psychiatrist who understands ME is an organic disease
6. English mathematical genius on one's responsibilities in the matter of one's beliefs:
   "it is wrong always, everywhere, and for anyone, to believe anything upon
     insufficient evidence
7. A space- and computer-scientist takes a look at psychology.
8. Malcolm Hooper puts things together status 2010.

    "Ah me! alas, pain, pain ever, forever!

No change, no pause, no hope! Yet I endure.
I ask the Earth, have not the mountains felt?
I ask yon Heaven, the all-beholding Sun,
Has it not seen? The Sea, in storm or calm,
Heaven's ever-changing Shadow, spread below,
Have its deaf waves not heard my agony?
Ah me! alas, pain, pain ever, forever!
     - (Shelley, "Prometheus Unbound") 

    "It was from this time that I developed my way of judging the Chinese by dividing them into two kinds: one humane and one not. "
     - (Jung Chang)


See also: ME -Documentation and ME - Resources

Maarten Maartensz

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