The peculiar service which was rendered at this juncture by the ` Cambridge School' was that, instead of opposing a mere dogmatic opposition to the Tubingen critics, they met them frankly on their own ground; and instead of arguing that their conclusions ought not to be and could not be true, they simply proved that their facts and their premisses were wrong.
With these definitions it is now possible to prove the following six premisses applying to finite cardinal numbers, from which Peano 2 has shown that all arithmetic can be deduced i.
The proof of the six premisses requires an elaborate investigation into the general properties of classes and relations which can be deduced by the strictest reasoning from our ultimate logical principles.
They are excellent principles of the highest value, but they are in no sense the necessary premisses which must be proved before any other propositions of cardinal numbers can be established.
But on the assumption that "mathematics" is to denote a science well marked out by its subject matter and its methods from other topics of thought, and that at least it is to include all topics habitually assigned to it, there is now no option but to employ "mathematics" in the general sense' of the "science concerned with the logical deduction of consequences from the general premisses of all reasoning."
But it will be noticed that the second half of the definition in the text - "from the general premisses of all reasoning" - is left unexpressed.
Whatever be the historical worth of this story, it may safely be said that it cannot be disproved by deductive reasoning from the premisses of abstract logic. The most we can do is to assert that a universe in which such things are liable to happen on a large scale is unfitted for the practical application of the theory of cardinal numbers.
We feel its presence in his earliest notable work, The Rationale of Religious Enquiry, 1836; and may there see the rigour with which it applied audacious logic to narrow premisses, the tenacity with which it clung to a limited literal supernaturalism which it had no philosophy to justify, and so could not believe without historical and verbal authority.
4 The echoes of the dying controversy are thus distinct and not very distant in this book, though it also offers in its larger outlook, in the author's evident uneasiness under the burden of inherited beliefs, and his inability to reconcile them with his new standpoint and accepted principles, a curious forecast of his later development, while in its positive premisses it presents a still more instructive contrast to the conclusions of his later dialectic. Nor did the sound of the ancient controversy ever cease to be audible to him.
Secondly, he made no division of logic. In the Categories he distinguished names and propositions for the sake of the classification of names; in the De Interpretatione he distinguished nouns and verbs from sentences with a view to the enunciative sentence: in the Analytics he analysed the syllogism into premisses and premisses into terms and copula, for the purpose of syllogism.
In 1874, when Gladstone published his pamphlet on The Vatican Decrees, Lord Acton wrote during November and December a series of remarkable letters to The Times, illustrating Gladstone's main theme by numerous historical examples of papal inconsistency, in a way which must have been bitter enough to the ultramontane party, but demurring nevertheless to Gladstone's conclusion and insisting that the Church itself was better than its premisses implied.
In the years immediately following he devoted considerable attention to the construction of a logical machine, exhibited before the Royal Society in 1870, by means of which the conclusion derivable from any given set of premisses could be mechanically obtained.
Nor does the process of acquiring the premisses of eristical syllogism, which is fallacious either in its premisses or in its process, differ, except that, when the premisses are fallacious, the dialectical interrogations must be such as to cause this fallacy.