[Hamiton 1853, p. 638-639]
Mathematics are not a road of any kind to Logic, whether to Logic speculative, or to Logic practical. A road to logic, did I say ? It is well, if Mathematics, from the inevitability of their process, and the consequent insertion, combined with rashness, which they induce, do not positively ruin the reasoning habits of their votary. Some knowledge of their object-matter and method is requisite to the philosopher; but their study should be followed out temperately and with due caution. A mathematician in contingent matter is like an owl in daylight.
[Mill 1843, p. 13]
The sole object of logic is guidance of one’s own thoughts (. . . ). Logic takes cognizance of our intellectual operations only as they conduce to our own knowledge and to our command over that knowledge for our own uses.
[Mill 1843, p. 23]
Logic, then, is the science of operations of the understanding which are subservient to the estimation of evidence: both the process itself of advancing from known truths to unknown, and all over intellectual operations in so far as auxiliary to this.
[Mill 1843, p. 13]
The view taken in the text, of the definition and purpose of logic, stands in marked opposition to that of the school of philosophy which, in this country, is represented by the writings of Sir William Hamilton and of his numerous pupils. Logic, as this school conceives it, is ‘the Science of the Formal Laws of Thought’, a definition framed for the express purpose of excluding, as irrelevant to logic, whatever relates to Belief or Disbelief, or to the pursuit of truth as such, and restricting the science to that very limited portion of its total province, which has reference to the conditions, not of Truth, but of Consistency.
[Boole 1854, p. 4]
The knowledge of the laws of the mind does not require at his basis any extensive collection of observations. The general truth is seen in the particular instance, and it is not confirmed by the repetition of instances.
[Boole 1854, p. 4]
It is the ability inherent in our nature to appreciate Order, and the concurrent presumption, however founded, that the phenomena of nature are connected by a principle of Order.
[Jevons 1863, p. 3]
My present task, however, is to show that all and more than all the ordinary processes of logic may be combined in a system founded on comparison of quality only, without reference to logical quantity. Before proceeding I have to acknowledge that in a considerable degree this system is founded on that of Prof. Boole, as stated in his admirable and highly original Mathematical Analysis of Logic. The forms of my system may, in fact, be divesting his system of a mathematical dress, which, to say the least, is not essential to it. The system being restored to its proper simplicity, it may be inferred, not that logic is a part of Mathematics, as is almost implied in Prof. Boole’s writings, but that mathematics are rather derivatives of Logic.
[Jevons 1863, p. 6]
we might say that logic is the algebra of kind of quality, as algebra (. . . ) is the calculus of known and unknown quantities.
[Jevons 1863, p. 7]
Terms will be used to mean name, or any combination of names and words describing qualities and circumstances of a thing .
[Jevons 1863, p. 76]
Boole’s symbols are essentially different from the names or symbols of common discourse – his logic is not the logic of common thought.
There are no such operations as addition or subtraction in pure logic. The operations of logic are the combination and separation of terms, or their meanings, corresponding to multiplication and division in mathematics.
[Jevons 1863, p. 79]
My third objection to Professor Boole’s system is that it is inconsistent with the self-evident law of thought, the Law of Unity, ( A +A = A.).
[Jevons 1863, p. 79]
The last objection that I shall at present urge against Professor Boole’s system is, that the symbols 1/1, 0/0, 0/1, 1/0, establish for themselves no logical meaning, and only bear a meaning derived from some method of reasoning not contained in the symbolic system.
[Jevons 1863, p. 74]
Compared with Professor Boole’s system, in its mathematical dress, this system shows the followings advantages.
1- Every process is of self-evident nature and force, and governed by laws as simple and primary as those of Euclids axioms.
2- The process is infallible and gives no uninterpretable or anomalous results.
3- The inferences may be drawn with far less labour than in Professor Boole’s system, which generally requires a separate computation and development for each inference.
[MacColl 1878, p. 27]
My method, however, differs both from Prof. Boole’s and Prof. Jevons in three cardinal points, which a perusal of this paper will show to be important as to necessitate an essentially different treatment if the whole subject. These three points are:
(1) With me every single letter, as well as every combination of letters, always denote a statement.
(2) I use a symbol (the symbol : ) to denote that the statement following it is true provided the statement preceding it be true.
(3) I use a special symbol - namely an accent – to express denial; and this accent, like the minus sign in ordinary algebra, may be made to affect a multinomial statement of any complexity.
On these three points of difference are founded some symmetrical rules on which depends the whole working of the calculus.
[Venn 1881, p. 54]
We cannot 'except' anything from that in which it was not included ; so that x – y certainly means that y is a part of x.
[Venn 1881, p. 75]
if yz = x, then z = x + possibly any portion of that is neither x nor y. If, as before, we put v as indicative of this kind and degree of indeterminateness, we should write it z = x + v.x.y.
[Venn 1881, p. 68]
The symbols are therefore an afterthought to express results and operations to which we are already accustomed, though their introduction is a powerful means of economizing time and thought.
This will set before the eyes, at a glance, the whole import of the propositions collectively.