What follows are Newton's "Rules of Reasoning in Philosophy" as they appeared in the 1714-edition of the Mathematical Principles of Natural Knowledge, after which a series of brief comments are made.

It should be noted that by "philosophy" Newton meant "science". The following is quoted from Newton:

"RULES OF REASONING IN PHILOSOPHY

RULE I

We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances.

To this purpose the philosophers say that Nature does nothing in vain, and more is in vain when less will serve; for Nature is pleased with simplicity, and affects not the pomp of superfluous causes.

RULE II

Therefore to the same natural effects we must, as far as possible, assign the same causes.

As to respiration in a man and in a beast; the descent of stones in Europe and in America; the light of our culinary fire and of the sun; the reflection of light in the earth, and in the planets.

RULE III

The qualities of bodies, which admit neither intensification nor remission of degrees, and which are found to belong to all bodies within the reach of our experiments, are to be esteemed the universal qualities of all bodies whatsoever.

For since the qualities of bodies are only known to us by experiments, we are to hold for universal all such as universally agree with experiments; and such as are not liable to diminution can never be quite taken away. We are certainly not to relinquish the evidence of experiments for the sake of dreams and vain fictions of our own devising; nor are we to recede from the analogy of Nature, which is wont to be simple, and always consonant to itself. We no other way know the extension of bodies than by our senses, nor do these reach it in all bodies; but because we perceive extension in all tht are sensible, therefore we ascribe it universally to all others also. That abundance of bodies are hard, we learn by experience; and because the hardness of the whole arises from the hardness of the parts, we therefore justly infer the hardness of the undivided particles not only of the bodies we feel but of all others. That all bodies are impenetrable, we gather not from reason, but from sensation. The bodies which we handle we find impenetrable, and thence conclude impenetrability to be an universal property of all bodies whatsoever. That all bodies are movable, and endowed with certain powers (which we call the inertia) of persevering in their motion, or in their rest, we only infer from the like properties observed in the bodies which we have seen. The extension, hardness, impenetrability, mobility, and inertia of the whole, result from the extension, hardness, impenetrability, mobility, and inertia of the parts; and hence we conclude the least particles of all bodies to be also all extended, and hard and impenetrable, and movable, and endowed with their proper inertia. And this is the foundation of all philosophy. Moreover, that the divided but contiguous particles of bodies may be separated from one another, is matter of observation; and, in the particles that remain undivided, our minds are able to distinguish yet lesser parts, as is mathematically demonstrated. But whether the parts so distinguished, and not yet divided, may, by the powers of Nature, be actually divided and separated from one another, we cannot certainly determine. Yet, had we the proof of but one experiment that any undivided particle, in breaking a hard and solid body, suffered a division, we might by virtue of this rule conclude that the undivided as well as the divided particles may be divided and actually separated to infinity.

Lastly, if it universally appears, by experiments and astronomical observations, that all bodies about the earth gravitate towards the earth, and that in proportion to the quantity of matter which they severally contain; that the moon likewise, according to the quantity of its matter, gravitates towards the earth; that, on the other hand, our sea gravitates towards the moon; and, all the planets one towards another; and the comets in like manner towards- the sun; we must, in consequence of this rule, universally allow that all bodies whatsoever are endowed with a principle of mutual gravitation.

For the argument from the appearances concludes with more force for the universal gravitation of all bodies than for their impenetrability; of which, among those in the celestial regions, we have no experiments, nor any manner of observation. Not that I affirm gravity to be essential to bodies: by their vis insita I mean nothing but their inertia. This is immutable. Their gravity is diminished as they recede from the earth.

RULE IV

In experimental philosophy we are to look, upon propositions inferred by general induction from phenomena as accurately or very nearly true, notwithstanding any contrary hypotheses that may be imagined, till such time as other phenomena occur, by which they may cither be made more accurate, or liable to exceptions.

This rule we must follow, that the argument of induction may not be evaded by hypotheses."

Here are a few brief comments on the above Rules:

Rule I: This clearly is a version of Ockham's Razor, that in turn can be both defended and explained by Probability Theory:

Given that:

pr(P|T)=1 - T implies P so the probability of P given T is 1
pr(T)=t    -  Let t be the probability of T and assume 0<t<1
p(P)=p     -  Let p be the probability of P and assume 0<p<1

it follows that

pr(T|P)=pr(P|T)*p(T):pr(P) - by probability theory
          = t:p                     - by the above assumptions
t:p > t                             - since 0<p<1

Newton's own justification "To this purpose the philosophers say that Nature does nothing in vain, and more is in vain when less will serve; for Nature is pleased with simplicity, and affects not the pomp of superfluous causes." can be similarly defended and explained, but involves an extra hypothesis about nature, namely "Nature is pleased with simplicity, and affects not the pomp of superfluous causes".

This may be true, but it seems more sensible to say that that the human mind is pleased with simplicity, and this predelection for simple hypotheses quite often has been found to be adequate to the facts that nature produces.

Rule II: This is a simple logical consequence from Rule I.

Newton's examples under it are fine applications of it.

Rule III: This is an assumption from which Rule IV follows, that allows induction in Newton's sense.

It is noteworthy that Rule III has two qualification: It concerns "qualities of bodies" which are best taken in the sense of properties, but then Newton adds that these must be such as "admit neither intensification nor remission of degrees" and also such as "are found to belong to all bodies within the reach of our experiments".

The first qualification can be understood as meaning that Newton only wanted to propose the rule for constant i.e. invariant properties, or also, such properties as do not come naturally (we suppose) in terms of more or less. It does not seem to me to be very useful a restriction, but it seems probable that Newton added it to increase the plausibility of Rule III or to avoid possible difficulties with it.

But it seems quite sensible to propose a theory of the form "With this disease, you will get a fever that first goes up to approximately 39 degrees Celsius, after which it will go down, if you live" and the like.

The second qualification is relevant, and what Newton in fact proposes comes down to a rule which may be written in logical notation as follows:

(x)(x is A & x is Experienced --> x is C)
-------------------------------------- ergo by Rule III
(x)(x is A  --> x is C)

In words: If everything in our experience (experiments) that is A also is C, then (by Rule III) everything that is A, also outside our experience, is C.

For the problems related to this rule see Induction and the Problem of Induction.

It should be fairly evident, at least, to any intelligent reader that the rule as stated is not deductively valid. In defense of Newton it should be added that he was quite obviously aware of it, for else he would not have proposed it as a rule, and that if we can learn from experience about nature, then nature must have some properties that satisfy Rule III.

Newton's explanation and defense of Rule III is quite interesting, and he is quite right that something like Rule III "is the foundation of all philosophy" - where "philosophy" means "science".

For more, see Hume's Enquiry concerning Human Understanding, especially section IV (which is on this site with my comments).

Rule IV: This is a logical consequence of Rule III.

It may be restated in terms of abduction, namely as follows - in which case it ceases to be implied by Rule III:

'In experimental philosophy we are to look upon propositions inferred by abduction from phenomena as possible explanations, that may be true, and that can be tested when other relevant phenomena occur, by which they may be either made more accurate, or liable to exception.' 

And in this form it seems to me to be quite true. See section IV of Hume's Enquiry concerning Human Understanding, the Problem of Induction, and Theory. Also, in the sense in which I use terms, that accords with probability theory, it is the last part of the rule - tested when other relevant phenomena occur, by which they may be either made more accurate, or liable to exception - that corresponds with and is properly called induction.

Hence, it may be fairly said that Bacon, Newton and Hume confused induction with abduction, and that they missed the principle of elementary probability theory explained above, that also forms the basis of Bayesian reasoning, whereas my solution of the problem of induction undoes the confusion and adds some new hypotheses for reasoning with probabilities.

And it is interesting to remark on Newton's explanation under Rule IV - "This rule we must follow, that the argument of induction may not be evaded by hypotheses" - although it merely restates what was already affirmed by Newton, may well explain Newton's claim that "Hypotheses non fingo" i.e. "I make no hypotheses", it this is read as he clearly intended it: "I make no ad hoc hypotheses".

However, it should be noted that, unlike Newton, in the restated form of Rule IV it is not to claimed that the propositions inferred by abduction are to be looked upon as accurately and very nearly true, but only as possible explanations, that may be the best we can offer on our present knowledge, but may need further confirmation by induction as I use that term, to come to be regarded as more probably true than not.

And there are several problems here, of which I have treated most in "The Solution of the Problem of Induction", especially the last section.

I here merely remark that I replace there Newton's Rule III by a  postulate or rule that must be added to any empirical theory T in order to test it, and which is to the following effect:

• Everything that is relevant to the predictions that T implies (and thus explains) also is implied as relevant by the theory

where "relevant" is understood in as in probability theory: Q is relevant to T iff the probability of T given Q is different from the probability of T given ~Q.

This rule also implies ad hoc hypotheses are excluded, and implies that what the theory asserts if it holds at all holds unconditionally, which includes time, and thus also solves Goodman's New Riddle of Induction, and also deductively entails what Newton presumed in general, namely that what the theory asserts and is known to hold inside experience also holds outside experience.

Indeed, the postulate enables one to test theories given predictions by enabling one to abstract from irrelevant circumstances, which always exist: Whatever is not entailed as relevant by the theory is irrelevant to it.

It differs from Newton's postulates in six ways:

• It is probabilistic;
• it is not claimed to yield true or nearly true conclusions but merely possibly true conclusions, not known to be false;
• it concerns a postulate for the testing of theories, rather than for the inference of theories;
• it replaces what Newton called "induction" by "abduction", that has another kind of analysis: Newton's inductions are generalizations of experience, whereas abductions are inferences of explanations for given facts
• it requires that an abduction is tested, by "induction" in a new sense, namely confirmation in the probabilistic sense;
• it is a local rather than a universal postulate: It must be supposed to be added to any specific theory, for indeed without the postulate a theory cannot be tested, since anything whatsoever may be relevant to it.  

The last point is to be understood also in a methodological sense, and indeed a good part of experimental methodology consists in trying to make sure that one's experiments are not biased or influenced by unknown factors.

This usually requires considerable care, and in this sense it is local, but since one cannot exclude more factors experimentally than one knows or suspects, and since there always may be factors one does not know or suspect, one must make the postulate.

Accordingly, what follows if such a theory is contradicted by experimental evidence is indeed that the theory is refuted, for it was assumed the theory entails all factors that are relevant.

These matters are more fully analyses in my solution of the problem of induction and in the Rules of Probabilistic Reasoning.