Augustus De Morgan

“ Thus, ‘Every A is X, Every X is B,’ which yields ‘Every A is B,’ also yields ‘Some A is B.’ But writers on logic have always considered these syllogisms as useless, conceiving it better to draw from any premises their strongest conclusion. ”

“ Most syllogisms which give an affirmative conclusion are generally meant to imply à fortiori arguments, except only in mathematics. It is seldom, except in the exact sciences, that we meet with a proposition, ‘Every A is B,’ which we cannot immediately couple with ‘Some Bs are not As.’
When an argument is completely established, with the exception of one assertion only, so that the inference may be drawn as soon as that one assertion is established, the result is stated in a form which bears the name of an hypothetical syllogism. ”

“ The principal danger is that of confounding a proposition which is negative with another which is affirmative of something requiring a negative to describe it. Thus ‘he resembles the man who was not in the room,’ is affirmative, and must not be confounded with ‘he does not resemble the man who was in the room.’ Again, ‘if he should come to-morrow, it is probable he will not stay till Monday,’ does not mean the simple denial of the preceding proposition, but the affirmation of the directly opposite proposition. ”

“ Simple identity or non-identity is the ultimate state to which every assertion may be reduced; and we shall, therefore, first ask, from what identities, &c., can other identities, &c., be produced? Again, since we name objects in species, each species consisting of a number of individuals, and since our assertion may include all or only part of a species, it is further necessary to ask, in every instance, to what extent the conclusion drawn is true, whether of all, or only of part? ”

“ The author would not, by any means, in presenting the minimum necessary for a particular purpose, be held to imply that he has given enough of the subject for all the ends of education. He has long regretted the neglect of logic; a science, the study of which would shew many of its opponents that the light esteem in which they hold it arises from those habits of inference which thrive best in its absence. He strongly recommends any student to whom this tract may be the first introduction of the subject, to pursue it to a much greater extent. ”

“ The middle term must enter universally into one or the other premiss. If it were not so, the one premiss might speak of one part of the middle term, and the other of the other; so that there would, in fact, be no middle term. Thus, ‘Every A is X, Every B is X,’ gives no conclusion: it may be thus stated;
All the As make up a part of the Xs
All the Bs make up a part of the Xs And, before we can know that there is any common term of comparison at all, we must have some means of shewing that the two parts are the same; or the preceding premises by themselves are inconclusive. ”

“ Therefore, That race is a race which must have possessed some of the arts of life. A person who makes the preceding assertion either means to imply, antecedently to the conclusion, that all Asiatic races must have possessed arts, or he talks nonsense if he asserts the conclusion positively. ‘A must be B, for it is X,’ can only be true when ‘Every X is B.’ This latter proposition may be called the suppressed premiss; and it is in such suppressed propositions that the greatest danger of error lies. It is also in such propositions that men convey opinions which they would not willingly express. ”

“ Thus, when we say that all men will die, and that all men are rational beings, and thence infer that some rational beings will die, the logical truth of this sentence is the same whether it be true or false that men are mortal and rational. ”

“ Let us take the simple assertion, ‘Every living man respires;’ or, every living man is one of the things (however varied they may be) which respire. If we were to inclose all living men in a large triangle, and all respiring objects in a large circle, the preceding assertion, if true, would require that the whole of the triangle should be contained in the circle. And in the same way we may reduce any assertion to the expression of a coincidence, total or partial, between two figures. ”

“ But it must be remembered, that, in all logical reasoning, the negation is simply negation, and nothing more, never implying affirmation of the contrary.
The common proposition that two negatives make an affirmative, is 6true only upon the supposition that there are but two possible things, one of which is denied. Grant that a man must be either able or unable to do a particular thing, and then not unable and able are the same things. ”

“ From premises both negative no conclusion can be drawn. For it is obvious, that the mere assertion of disagreement between each of two things and a third, can be no reason for inferring either agreement or disagreement between these two things. It will not be difficult to reduce any case which falls under this rule to a breach of the first rule: thus, No A is X, No B is X, gives
Every A is (something which is not X)
Every B is (something which is not X) 17in which the middle term is not spoken of universally in either. ”

“ This contradiction shews that the supposition of particular premises producing a legitimate result is inadmissible.
5. If one premiss be negative, the conclusion, if any, must be negative. If one term agree with a second and disagree with a third, no agreement can be inferred between the second and third.
6. If one premiss be particular, the conclusion must be particular. This is not very obvious, since the middle term may be universally spoken of in a particular proposition, as in Some B is not X. But this requires one negative proposition, whence (Rule 3) the other must be affirmative. ”

“ This is the proposition in proof of which nearly the whole of the demonstration of Euclid is spent.
(2.) It is false that there can be any whole which is not greater than its part (self evident) .
(3.) Therefore it is false that there is any equiangular triangle which is not equilateral; or all equiangular triangles are equilateral.
When a proposition is established by proving the truth of the matters it contains, the demonstration is called direct; when by proving the falsehood of every contradictory proposition, it is called indirect. ”

“ The word hypothesis means nothing but supposition; and the species of syllogism just mentioned first lays down the assertion that a consequence will be true if a certain condition be fulfilled, and then either asserts the fulfilment of the condition, and thence the consequence, or else denies the consequence, and thence denies the fulfilment of the condition. Thus, if we know that
When A is B, it follows that P is Q ”