Philosophical Studies · G. E. Moore
°B = A.°
We have now got everything which is required for expressing, in a short symbolic form, the proposition, with regard to a given thing A and a given relational property P, which A in fact possesses, that P is _internal_ to A. The required expression is _xP_ entails (°_x_ = A°) which is to mean the same as "Every proposition which asserts of any given thing that it has not got P _entails_ the proposition, with regard to the thing in question, that it is other than A." And this proposition is, of course, logically equivalent to (_x_ = A) entails _x_ P where we are using "logically equivalent," in such a sense that to say of any proposition _p_ that it is logically equivalent to another proposition _q_ is to say that both _p_ entails _q_ and _q_ entails _p._ This last proposition again, is, so far as I can see, either identical with or logically equivalent to the propositions expressed by "anything which were identical with A would, in any conceivable universe, necessarily have P" or by "A could not have existed in any possible world without having P"; just as the proposition expressed by "In any possible world a right angle must be an angle" is, I take it, either identical with or logically equivalent to the proposition "(_x_ is a right angle) entails (r is an angle)."
We have now, therefore, got a short means of symbolising, with regard to any particular thing A and any particular property P, the proposition that P is _internal_ to A in the second of the two senses distinguished on p. 286. But we still require a means of symbolising the general proposition that _every_ relational property is internal to any term which possesses it--the proposition, namely, which was referred to on p. 287, as the most important consequence of the dogma of internal relations, and which was called (2) on p. 289. In order to get this, let us first get a means of expressing with regard to some one particular relational property P, the proposition that P is internal to _any_ term which possesses it. This is a proposition which takes the form of asserting with regard to one particular property, namely P, that any term which possesses that property also possesses another--namely the one expressed by saying that P is internal to it. It is, that is to say, an ordinary universal proposition, like "All men are mortal." But such a form of words is, as has often been pointed out, ambiguous. It may stand for either of two different propositions.
It may stand merely for the proposition "There is nothing, which both is a man, and is not mortal"--a proposition which may also be expressed by "If anything is a man, that thing is mortal," and which is distinguished by the fact that it makes no assertion as to whether there are any men or not; or it may stand for the conjunctive proposition "If anything is a man, that thing is mortal, _and there are men."_ It will be sufficient for our purposes to deal with propositions of the first kind--those namely, which assert with regard to some two properties, say Q and R, that there is nothing which both does possess Q and does not possess R, without asserting that anything does possess Q. Such a proposition is obviously equivalent to the assertion that _any_ pair of propositions which resembles the pair "AQ" and "AR," in respect of the fact that one of them asserts of some particular thing that it has Q and the other, of the same thing, that it has R, stand to one another in a certain relation: the relation, namely, which, in the case of "AQ" and "AR," can be expressed by saying that "It is not the case both that A has Q and that A has not got R." When we say "There is nothing which does possess Q and does not possess R" we are obviously saying something which is either identical with or logically equivalent to the proposition "In the case of every such pair of propositions it is not the case both that the one which asserts a particular thing to have Q is true, and that the one which asserts it to have R is false."
We require, therefore, a short way of expressing the relation between two propositions _p_ and _q,_ which can be expressed by "It is not the case that _p_ is true and _q_ false." And I am going, quite arbitrarily to express this relation by writing _p_ * _q_ for "It is not the case that _p_ is true and _q_ false." The relation in question is one which logicians have sometimes expressed by "_p_ implies _q_." It is, for instance, the one which Mr. Russell in the _'Principles of Mathematics_ calls "material implication," and which he and Dr. Whitehead in _Principia Mathematica_ call simply "implication." And if we do use "implication" to stand for this relation, we, of course, got the apparently paradoxical results that every false proposition implies every other proposition, both true and false, and that every true proposition implies every other true proposition: since it is quite clear that if _p_ is false then, whatever _q_ may be, "it is not the case that _p_ is true and _q_ false," and quite clear also, that if _p_ and _q_ are both true, then also "it is not the case that _p_ is true and _q_ false." And these results, it seems to me, appear to be paradoxical, solely because, if we use "implies" in any ordinary sense, they are quite certainly false. Why logicians should have thus chosen to use the word "implies" as a name for a relation, for which it never is used by any one else, I do not know.
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