A Modern Introduction to Logic, by L. Susan Stebbing (London: Methuen & Co. Ltd. (1930) 5th ed., 1946)
[Neither Bradley, nor Bosanquet, nor any of this school of Idealist logicians, has ever succeeded in making clear what exactly is meant by the principle of identity-in-difference upon which the metaphysical logic of the Idealists is based.] : Invariance under transformation? This cannot be made clear to a mathematical logician, since he does not know what it is to exist.
[Mr. I. A. Richards has suggested the convenient terminology 'The scientific use of language' and 'the emotive use of language'. When language is used simply in order to refer to a referend its use is scientific.] : Nonsense! This suggests that any exact statement is scientific, i.e. impersonal or objective. A subjective statement can be no less precise.
[As Prof. Whitehead puts it: 'Operations of thought are like cavalry charges in a battle—they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.'] : No wonder science is such an unintelligent pastime!
[a Chinese philosopher...is reported to have said that if there is a dun cow and a bay horse, then there are three things; for the dun cow is one thing, and the bay horse is another thing, and the two together are a third. We must inquire wherein precisely lies the absurdity of this statement.] : Far from being absurd, this statement is of fundamental ontological importance. (If there is a bowl and a stem, is the pipe that comes of taking the two together simply a class? If so, then the bowl and stem are also classes, since different parts of them can be distinguished.)
: How can you say there certainly are no unicorns? How can you tell? The whole point is that if they can can be thought of they might exist—their existence would not involve contradiction. In other words to be thought of is to be possible. But it does not follow that unicorns exist as possibilities apart from the act of thinking about them. And it does not follow from the fact that to be seen is to be certain that horses exist as certainties apart from the act of seeing them.
[As Professor Moore points out, 'unreal' does not stand for any conception at all. We use the expression 'are unreal' to express the denial of existence, not to assert a special mode of existence.] : Ass! Unreal = Imaginary.
[There is no doubt that properties may be present to mind in a way analogous to, but different in important respects from, the way in which an individual object may be present to mind. But properties are not individual objects, and can be thought of even if there are no objects which possess these properties.] : A property is a thing. I.e. it is distinct from other properties. What distinction is there between a thing and an individual object?
: 'This lion exists' = 'this lion is certain' (not to be confused with 'this is certainly a lion'). 'Lions exist' = 'this lion is possible' (not to be confused with 'this is possibly a lion'). (Cf. Russell: '"it is a lion" is sometimes true.') In other words, 'Lions exist' = 'I am thinking of a lion'.
[...Mr. Russell puts this point by saying that 'it is of propositional functions that you can assert or deny existence'...] : Ergo; I am a propositional function.
[he says that 'Lions exist' means '"x is a lion" is sometimes true'.] : Therefore 'we exist' means '"x is I" is sometimes true'.
[From the common sense point of view we may say that if it is true that A is red, then A can be regarded as possessing the quality of being red independently of any reference to any other object.] : This is a mistake. A red thing implies not-red things. If red is present, it is so absolutely and alone: not-red is absent and plural—i.e. not-reds. Red is related to—i.e. is not—each not-red individually; and when all these relations or negatives are taken together we have singular (or present) red related to plural (or absent) not-red. This is simply the Principle of Identity—A must be given before not-A can appear. But this does not make A independent of not-A, it simply states that 'an irreducible multiple relation' cannot be given (or exist) without a point of view (or orientation). This is an ontological necessity, hidden from the eye of the non-existing logician. (A is A = A or not-A exclusively).
[Self-evidence is a relative notion. What we are able to doubt depends upon our previous knowledge and our mental capacity.] : Rubbish!
[The necessity of logical principles is nothing but the necessity of constructing systems. The construction of such systems may be the expression of the thinking of rational beings. But this would not establish the necessity.] : This assumes that 'the thinking of rational beings' is a fortuitous and arbitrary quality of certain beings, like having blue eyes, or red hair: man + blue eyes = blue-eyed man; being + thinking = thinking (or rational) being. But cogito ergo sum: thinking implies being.
[We...deny that any significance can be attributed to the notion of absolutely necessary principles and absolutely indemonstrable propositions.] : Wrong. Absolute principles are those that cannot be conceived without being tacitly assumed in the act of conception.
[There is a class of all possible individuals, called the 'universe'.] : This assumption is unjustified.
[If by 'the world' we mean 'everything that is the case', then it may be doubted whether the world is a system.] : The expression 'is the case' applies only to propositions (we cannot, for example, say 'a lion is the case'), and 'everything that is the case' means 'all true propositions'. A world of propositions is truly a logician's world.
[A mathematical proposition is independent of what happens to exist.] : Even of the existing mathematician?
[Owing to its independence of empirical facts mathematics is a wholly deductive science; hence it employs a method of exact demonstration.] : If mathematical propositions are, as you say, independent of what exists, then they are not concerned with matters of fact—they do not depend upon inductive verification. And if this is so, mathematics is neither deductive nor demonstrative. I rather fancy that the contradictory statements on pages 192 and 193 are due to a wish to avoid this conclusion. (From p. 415 it is clear that mathematical propositions—e.g. 2 + 2 = 4—cannot be asserted as true in the same way as propositions about matters of fact (e.g. 'all crows are black'). The Principle of Deduction does not therefore apply to mathematics: this seems to be as good as admitted at the top of p. 489, and footnote 1.)
[In order that a proposition should be scientific it must relate to something other than the immediate experience of an individual.] noted
[there seems not the slightest justification for the view that, for example, the causal law Sugar dissolves in water must hold in all possible worlds, in the sense in which 'must' means 'could not be otherwise'.] : If it did not hold, could we still speak of sugar and water? It must hold in every world where there is sugar and water. The question is: will this still be sugar in all possible worlds?
p. 287/2-3 [the property being on this table is an external relational property of this book.] : It is not a property of this book. It is a property of the situation book-on-a-table.
[This is equivalent to the assertion that every property of A is an internal property. There is no reason to suppose that this assertion is true.] : On the contrary.
[Professor Whitehead goes so far as to say that 'the incredible labours of the scientist would be without hope' were it not for 'the inexpungable belief that every detailed occurrence can be correlated with its antecedents in a perfectly definite manner exemplifying general principles'. This is perhaps an overstatement. The scientist is quite ready to leave out of account a number of details which do not fit into his scheme.] : Whitehead is justified: the scientist does not merely 'leave out of account' what does not fit his scheme, he denies its existence. Thus only what fits his scheme is an occurrence.
[But if nature exhibited a type of order of unimaginable complexity, finite minds could not discover it.] : What, pray, is a 'finite mind'?
[The demand for continuity is closely bound up with the demand for persistence and identity. There must be no sudden breaks, no arbitrary discontinuities. The appearance of such discontinuities presents a problem; the discovery that, in spite of appearances, something identical or conserved is felt to be an acceptable solution.] : Confusion! There cannot be identities without discontinuity. It is continuity that presents a problem.
['the paradox is now fully established that the utmost abstractions are the true weapon with which to control our thought of concrete fact.'—A. N. Whitehead.] : This simply boils down to the observation that things endure, which is only a paradox if one starts from the scientific assumption that there are no such things as things. The paradox is in the presupposition. Whitehead mistakenly assumes that the concrete is instantaneous, and then proclaims a paradox when he discovers it is not. Concrete things are transcendent, but this does not make them abstract.
['To be abstract', says Professor Whitehead, 'is to transcend particular concrete occasions of actual happenings'.] 'transcend' u/l: No, it is to ignore them. the full sentence noted: A rock is therefore abstract if, by its remaining unchanged (as the same rock), it transcends the successive particular concrete states of the advance of the tide.
[...Thus the man watching the sea-gull may notice a second sea-gull, and it is possible that he should be sensibly aware of the same specific shade of whiteness in the throat of each of them, although he cannot name this shade. The particular occasion, then, is irrelevant to what is meant by the 'absolute specific shade of white', since it can be within more than one particular occasion. It is in this sense that the absolutely specific shade of white is abstract.] : From this argument it follows that my cousin Bill is abstract, since he is independent of the particular occasions in which he is present. Cousin Bill is transcendent in that he is the invariant of a number of different situations, but he only becomes abstract if he is thought of apart from any situation. But as a concrete existing transcendent he is always in some situation.
[But as the plain man would admit, there are an infinite number of points in a line.] : Only if adjacent points are separated by a line. If not, there is no way of getting beyond one point.