On Syllogism II

"For all who argue per impossibile infer by syllogism a false conclusion, and prove the original conclusion hypothetically when something impossible follows from a contradictory assumption, as, for example, that the diagonal [of a square] is incommensurable [with the side] because odd numbers are equal to even if it is assumed to be commensurate. It is inferred by syllogism that odd numbers are equal to even, and proved hypothetically that the diagonal is incommensurate, since a false conclusion follows from the contradictory assumption." (Aristotle, "Laws")

"It must be granted that in every syllogism, considered as an argument to prove the conclusion, there is a petitio principii [assumption of premises]. When we say, All men are mortal Socrates is a man therefore Socrates is mortal; it is unanswerably urged by the adversaries of the syllogistic theory, that the proposition, Socrates is mortal." (John S Mill, "A System of Logic, Ratiocinative and Inductive", 1843)

"The maxim is, that whatever can be affirmed (or denied) of a class, may be affirmed (or denied) of everything included in the class. This axiom, supposed to be the basis of the syllogistic theory, is termed by logicians the dictum de omni et nullo [the maxim of all and none]." (John S Mill, "A System of Logic, Ratiocinative and Inductive", 1858)

"Does the mathematical method proceed from particular to the general, and, if so, how can it be called deductive? […] If we refuse to admit these consequences, it must be conceded that mathematical reasoning has of itself a sort of creative virtue and consequently differs from a syllogism." (Henri Poincaré, "Science and Hypothesis", 1901)

"Syllogistic reasoning remains incapable of adding anything to the data that are given it; the data are reduced to axioms, and that is all we should find in the conclusions." (Henri Poincaré, "Science and Hypothesis", 1901)

"The essential characteristic of reasoning by recurrence is that it contains, condensed, so to speak, in a single formula, an infinite number of syllogisms. […] We see, then, that in reasoning by recurrence we confine ourselves to the enunciation of the minor of the first syllogism, and the general formula which contains as particular cases all the majors. This unending series of syllogisms is thus reduced to a phrase of a few lines." (Henri Poincaré, "Science and Hypothesis", 1901)

"The very possibility of mathematical science seems an insoluble contradiction. If this science is only deductive in appearance, from whence is derived that perfect rigour which is challenged by none? If, on the contrary, all the propositions which it enunciates may be derived in order by the rules of formal logic, how is it that mathematics is not reduced to a gigantic tautology? The syllogism can teach us nothing essentially new, and if everything must spring from the principle of identity, then everything should be capable of being reduced to that principle." (Henri Poincaré, "Science and Hypothesis", 1901)

"Trust is the core of human relationships, of gregariousness among men. Friendship, a puzzle to the syllogistic and critical mentality, is not based on experiments or tests of another person's qualities but on trust. It is not critical knowledge but a risk of the heart which initiates affection and preserves loyalty in our fellow men." (Abraham J Heschel, "Moral Grandeur and Spiritual Audacity: Essays", 1997)