Sec. 4.  The principle of Identity is usually written symbolically thus:  A is A; not-A is not-A.  It assumes that there is something that may be represented by a term; and it requires that, in any discussion, every relevant term, once used in a definite sense, shall keep that meaning throughout.  Socrates in his father’s workshop, at the battle of Delium, and in prison, is assumed to be the same man denotable by the same name; and similarly, ‘elephant,’ or ‘justice,’ or ‘fairy,’ in the same context, is to be understood of the same thing under the same suppositio.

But, further, it is assumed that of a given term another term may be predicated again and again in the same sense under the same conditions; that is, we may speak of the identity of meaning in a proposition as well as in a term.  To symbolise this we ought to alter the usual formula for Identity and write it thus:  If B is A, B is A; if B is not-A, B is not-A.  If Socrates is wise, he is wise; if fairies frequent the moonlight, they do; if Justice is not of this world, it is not. Whatever affirmation or denial we make concerning any subject, we are bound to adhere to it for the purposes of the current argument or investigation. Of course, if our assertion turns out to be false, we must not adhere to it; but then we must repudiate all that we formerly deduced from it.

Again, whatever is true or false in one form of words is true or false in any other:  this is undeniable, for the important thing is identity of meaning; but in Formal Logic it is not very convenient.  If Socrates is wise, is it an identity to say ’Therefore the master of Plato is wise’; or, further that he ‘takes enlightened views of life’?  If Every man is fallible, is it an identical proposition that Every man is liable to error?  It seems pedantic to demand a separate proposition that Fallible is liable to error.  But, on the other hand, the insidious substitution of one term for another speciously identical, is a chief occasion of fallacy.  How if we go on to argue:  therefore, Every man is apt to blunder, prone to confusion of thought, inured to self-contradiction?  Practically, the substitution of identities must be left to candour and good-sense; and may they increase among us.  Formal Logic is, no doubt, safest with symbols; should, perhaps, content itself with A and B; or, at least, hardly venture beyond Y and Z.

Sec. 5.  The principle of Contradiction is usually written symbolically, thus:  A is not not-A.  But, since this formula seems to be adapted to a single term, whereas we want one that is applicable to propositions, it may be better to write it thus:  B is not both A and not-A.  That is to say:  if any term may be affirmed of a subject, the contradictory term may, in the same relation, be denied of it.  A leaf that is green on one side of it may be not-green on the other; but it is not both green and not-green on the same surface, at the same time, and in the same light.  If a stick is straight, it is false that it is at the same time not-straight:  having granted that two angles are equal, we must deny that they are unequal.