Logic

"I come now to the capital work of Hilbert which he communicated to the Congress of Mathematicians at Heidelberg... of which...an English translation due to Halsted appeared in The Monist. ...the author's aim is analogous to that of Russell, but on many points he diverges from his predecessor. "But," he says, "on attentive consideration we become aware that in the usual exposition of the laws of logic certain fundamental concepts of arithmetic are already employed; for example, the concept of the aggregate, in part also the concept of number. "We fall thus into a vicious circle and therefore to avoid paradoxes a partly simultaneous development of the laws of logic and arithmetic is requisite." ...what Hilbert says of the principles of logic in the usual exposition applies likewise to the logic of Russell. So for Russell logic is prior to arithmetic; for Hilbert they are 'simultaneous.' We shall find... other differences still greater... I prefer to follow step by step the development of Hubert's thought... "Let us take as the basis of our consideration first of all a thought-thing 1 (one)." Notice that in so doing we in no wise imply the notion of number, because it is understood that 1 is here only a symbol and that we do not at all seek to know its meaning. "The taking of this thing together with itself respectively two, three or more times ..." Ah! this time it is no longer the same; if we introduce the words 'two,' 'three,' and above all 'more,' 'several,' we introduce the notion of number; and then the definition of finite whole number which we shall presently find, will come too late. Our author was too circumspect not to perceive this begging of the question. So at the end of his work he tries to proceed to a truly patching-up process. Hilbert then introduces two simple objects 1 and =, and and considers all the combinations of these two objects, all the combinations of their combinations, etc. It goes without saying that we must forget the ordinary meaning of these two signs and not attribute any to them. Afterwards he separates these combinations into two classes, the class of the existent and the class of the non-existent... entirely arbitrary. Every affirmative statement tells us that a certain combination belongs to the class of the existent; every negative statement tells us that a certain combination belongs to the class of the non-existent. Note now a difference of the highest importance. For Russell any object whatsoever, which he designates by x, is an object absolutely undetermined and about which he supposes nothing; for Hilbert it is one of the combinations formed with the symbols 1 and =; he could not conceive of the introduction of anything other than combinations of objects already defined."

"The very possibility of the science of mathematics seems an insoluble contradiction. If this science is deductive only in appearance, whence does it derive that perfect rigor no one dreams of doubting? If, on the contrary, all the propositions it enunciates can be deduced one from another by the rules of formal logic, why is not mathematics reduced to an immense tautology? The syllogism can teach us nothing essentially new, and, if everything is to spring from the principle of identity, everything should be capable of being reduced to it. Shall we then admit that the enunciations of all those theorems which fill so many volumes are nothing but devious ways of saying A is A! ...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."

"It is in set theory that we encounter the greatest diversity of foundational opinions. This is because even the most devoted advocates of the various new axioms would not argue that these axioms are justified by any basic ‘intuition’ about sets. ... One may vary the rank of sets allowed. Conventional mathematics rarely needs to consider more than four or five iterations of the power set axiom applied to the set of integers. More iterations diminish our sense of the reality of the objects involved."

"In any remembering, thinking or imagining, although the object of my intending is some state of affairs or other, I am also potentially aware as I intend that what I am doing is an act of remembering, thinking, or imagining. My asserting that S is P is not an assertion of mine unless I am implicitly aware as I assert that I am asserting, not entertaining the possibility that, S is P."

"No sincere assertion of fact is essentially unaccompanied by feelings of intellectual satisfaction or of a persuasive desire and a sense of personal responsibility."

"A theory is the more impressive the greater the simplicity of its premises is, the more different kinds of things it relates, and the more extended its area of applicability."

"From the viewpoint of my general purpose, I had come to believe that one way to achieve the education which leads to understanding and compassion is to take some period of the past and to immerse oneself in it so thoroughly that one could think its thoughts and speak its language. The object would be to take this chapter of vanished experience and learn to know it in three if not four dimensions. That would mean coming to understand why certain actions which in the light of retrospect appear madly irrational appeared at that time the indisputable mandate of reason; why things which had been created with pain and care were cast quickly on the gaming table of war; why men who had sat in the senate chamber and debated with syllogism and enthymeme stepped out of it to buckle on the sword against one another. Almost any book of history will give you the form of such a time, but what will give you the pressure of it? That is what I particularly wished to discover."

"I believe that there never was a creator of a philosophical system who did not confess at the end of his life that he had wasted his time. It must be admitted that the inventors of the mechanical arts have been much more useful to men that the inventors of syllogisms. He who imagined a ship towers considerably above him who imagined innate ideas."

"She certainly had a little syllogism in her head as to the Duke ruling the borough, the Duke's wife ruling the Duke, and therefore the Duke's wife ruling the Duke, and therefore the Duke’s wife ruling the borough; but she did not think it prudent to utter this on the present occasion."

"The motion picture does not outlaw the still photograph but combines a series of them according to the laws of motion. Dialectics does not deny the syllogism, but teaches us to combine syllogisms in such a way as to bring our understanding closer to the eternally changing reality."

"In the depth of his heart he knew he was dying, but not only was he not accustomed to the thought, he simply did not and could not grasp it. The syllogism he had learnt from Kiesewetter's Logic: “Caius is a man, men are mortal, therefore Caius is mortal” had always seemed to him correct, but he was Cauis, not an abstract man, but a creature quite, quite separate from all others."

"We now know that w:Limelightlimelight and a brass band do more to persuade than can be done by the most elegant train of syllogisms. It may be hoped that in time anybody will be able to persuade anybody of anything if he can catch the patient young and is provided by the State with money and equipment."

"Tim Russert told George W. Bush, on his military service that several news reporters found that there was no evidence that he had reported to military duty during the time he had claimed. Bush replied, 'Yeah, they're just wrong. There may be no evidence, but I did report. Otherwise, I wouldn't have been honorably discharged.' That's the Bush syllogism: The evidence says one thing; the conclusion says another; therefore, the evidence is false."

"Francis Bacon not only despised the syllogism, but undervalued mathematics, presumably as insufficiently experimental. He was virulently hostile to Aristotle, but though very highly of Democrates."

"In logic, a form of deductive reasoning consisting of a major premise, a minor premise, and a conclusion. Adjective:syllogistic. Here is an example of a valid categorical syllogism: Major premise: All mammals are warm-blooded. Minor premise: All black dogs are mammals. Conclusion: Therefore, all black dogs are warm-blooded."

"I have made every effort to obtain exact information, comparing doctrines, replying to objections, continually constructing equations and reductions from arguments, and weighing thousands of syllogisms in the scales of the most rigorous logic. In this laborious work, I have collected many interesting facts which I shall share with my friends and the public as soon as I have leisure. But I must say that I recognized at once that we had never understood the meaning of these words, so common and yet so sacred: Justice,equity, liberty; that concerning each of these principles our ideas have been utterly obscure; and, in fact, that this ignorance was the sole cause, both of the poverty that devours us, and of all the calamities that have ever afflicted the human race."

"...that the god in the sanctuary was finite in his power, and hence a fraud. One horse-laugh is worth ten thousand syllogisms. It is not only more effective; it is also vastly more intelligent."

"For a thorough understanding of the syllogism we need to understand it not only with respect to its definition but also with respect to its divisions. Some of the divisions that must be presented apply to the syllogism in general –e.g., a syllogism is either perfect or imperfect, either affirmative or negative, and so on. However, other divisions that must be presented apply to the syllogism in such a way as to separate it into distinct types. The following division is of this kind: as syllogism is either demonstrative, or dialectical, or sophistical. A demonstrative syllogism is the one that produces scientific knowledge on the basis of necessary [premises] and the most certain reasons for the conclusion. A dialectical syllogism, however, is the one that produces opinion on the basis of probable [premises]. Finally sophistical syllogism is the one that either syllogizes on the basis of seemingly probable [premises] or seemingly syllogizes on the basis of probable [premises]; in either case it is strictly aimed at glory or victory."

"Aristotle writes that persuasion is based on three things: the ethos, or personal character of the speaker; the pathos, or getting the audience into the right kind of emotional receptivity; and the logos, or the argument itself, carried out by abbreviated syllogisms, or something like deductive syllogisms, and by the use of example."

"Knowledge is ours only if, at the moment of need, it offers itself to the mind without syllogisms or demonstrations for which there is no time."

"Deductive reasoning moves from a generalization believed to be true or self-evident to a more specific conclusion. The process of deduction has traditionally been illustrated with a syllogism, a three-part set of statements or propositions that includes a major premise, a minor premise, and a conclusion. Major premise: All books from that store are new. Minor premise: These books are from that store. The major premise of a syllogism makes a general statement that the writer believes to be true. The minor premise presents a specific example of the belief that is stated in the major premise. If the reasoning is sound, the conclusion should follow from the two premises."

"It appears that reason is not, as sense and memory, born with us; nor gotten by experience only, as prudence is; but attained by industry: first in apt imposing of names; and secondly by getting a good and orderly method in proceeding from the elements, which are names, to assertions made by connexion of one of them to another; and so to syllogisms, which are the connexions of one assertion to another, till we come to a knowledge of all the consequences of names appertaining to the subject in hand; and that is it, men call science."

"A syllogism is valid (or logical) when its conclusion follows from its premises. A syllogism is true when it makes accurate claims--that is, when the information it contains is consistent with the facts. To be sound, a syllogism must be both valid and true. However, a syllogism may be valid without being true or true without being valid."

"The “elements” of the Great Alexandrian remain for all time the first, and one may venture to assert, the only perfect model of logical exactness of principles, and of rigorous development of theorems. If one would see how a science can be constructed and developed to its minutest details from a very small number of intuitively perceived axioms, postulates, and plain definitions, by means of rigorous, one would almost say chaste, syllogism, which nowhere makes use of surreptitious or foreign aids, if one would see how a science may thus be constructed one must turn to the elements of Euclid."

"Lying is not only excusable; it is not only innocent; it is, above all, necessary and unavoidable. Without the ameliorations that it offers, life would become a mere syllogism and hence too metallic to be borne."

"...mathematicians say—the construction propounded will be possible too, and once more the demonstration will correspond, in the reverse order to the analysis; but if we come upon something that is admitted is impossible the problem will also be impossible. It is quite possible to accept that Plato 'discovered' the method of Analysis, in the same sense as Aristotle discovered the syllogism; that is to say, he was the first to reflect upon the process of thought involved and to describe it in contrast with the process of Synthesis."

"They may attack me with an army of six hundred syllogisms; and if I do not recant, they will proclaim me a heretic."

"All important things bear the sign of death: Haven't people learned yet that the time of superficial intellectual games is over, that agony is infinitely more important than syllogism, that a cry of despair is more revealing than the most subtle thought, and that tears always have deeper roots than smiles?"

"In building his theory of rhetoric around the syllogism despite the problems involved in deductive inference Aristotle stresses the fact that rhetorical discourse is discourse directed toward knowing, toward truth not trickery... If rhetoric is so clearly related to dialectic, a discipline whereby we are enabled to examine inferentially generally accepted opinions on any problem whatsoever, then it is the rhetorical syllogism [i.e., the enthymeme which moves the rhetorical process into the domain of reasoned activity, or the kind of rhetoric Plato accepted later in the Phaedrus."

"LOGIC, n. The art of thinking and reasoning in strict accordance with the limitations and incapacities of the human misunderstanding. The basic of logic is the syllogism, consisting of a major and a minor premise and a conclusion --thus: Major Premise: Sixty men can do a piece of work sixty times as quickly as one man. Minor Premise: One man can dig a posthole in sixty seconds; therefore Conclusion: Sixty men can dig a posthole in one second. This may be called the syllogism arithmetical, in which, by combining logic and mathematics, we obtain a double certainty and are twice blessed."

"One horse-laugh is worth ten thousand syllogisms. It is not only more effective; it is also vastly more intelligent."

"SYLLOGISM, n. A logical formula consisting of a major and a minor assumption and an inconsequent."

"[In the introduction to his Middle Commentary on Aristotle's Topics, Averroes said] This art has three parts. The first part sets forth the speeches from which dialectical conversation is composed — i.e., its parts, and the parts of its parts on to its simplest components. This part is found in the first treatise on Aristotle's book. The second part sets forth the topics from which syllogisms are drawn — syllogisms for affirming something or denying it with respect to every kind of problem occurring in this art. This is the next six treatises of Aristotle's book. The third part set forth how the third part sets forth how the questioner ought to question and the answerer answer. It also sets forth how many kinds of questions and answers there are. This is in the eighth treatise of Aristotle's book."

"It is significant that in the greatest religious poem existent, the Book of Job, the argument which convinces the infidel is not (as has been represented by the merely rational religionism of the eighteenth century) a picture of the ordered beneficence of the Creation; but, on the contrary, a picture of the huge and undecipherable unreason of it. ‘Hast Thou sent the rain upon the desert where no man is?’ This simple sense of wonder at the shapes of things, and at their exuberant independence of our intellectual standards and our trivial definitions, is the basis of spirituality as it is the basis of nonsense. Nonsense and faith (strange as the conjunction may seem) are the two supreme symbolic assertions of the truth that to draw out the soul of things with a syllogism is as impossible as to draw out Leviathan with a hook."

"All the sophisticated syllogisms of the ponderous volumes published by Marx, Engels, and hundreds of Marxian authors cannot conceal the fact that the only and ultimate source of Marx’s prophecy is an alleged inspiration by virtue of which Marx claims to have guessed the plans of the mysterious powers determining the course of history. Like Hegel, Marx was a prophet communicating to the people the revelation that an inner voice had imparted to him."

"The peril of the heavy tower, of the restless vault, of the vagrant buttress; the uncertainty of logic, the inequalities of the syllogism, the irregularities of the mental mirror,— all these haunting nightmares of the Church are expressed as strongly by the Gothic cathedral although it had been the cry of human suffering, and as no emotion had ever been expressed before or likely to find expression again."

"St. Thomas Aquinas narrates the life of St. Francis d'Assisi. O MORTAL cares insensate, what small worth, In sooth, doth all those syllogisms fill, Which make you stoop your pinions to the earth!"

"The race of prophets is dead. Europe is becoming set in its ways, slowly embalming itself beneath the wrappings of its borders, its factories, its law courts and its universities. The frozen Mind cracks between the mineral staves which close upon it. The fault lies with your mouldy systems, your logic of 2 + 2 = 4. The fault lies with you, Chancellors, caught in the net of syllogisms. You manufacture engineers, magistrates, doctors, who know nothing of the true mysteries of the body or the cosmic laws of existence."

"Then there is that glorious Epicurean paradox, uttered by my friend, the Historian, in one of his flashing moments: "Give us the luxuries of life, and we will dispense with its necessaries.""

"For thence, — a paradox Which comforts while it mocks, — Shall life succeed in that it seems to fail: What I aspired to be, And was not, comforts me: A brute I might have been, but would not sink i' the scale."

"The mind begins to boggle at unnatural substances as things paradoxical and incomprehensible."

"Paradoxes explain everything. Since they do, they cannot be explained."

"PARADOX: A statement that reduces the matter at hand to complete obscurity while clarifying it. … Paradoxes are sensitive and can be routed by sneering."

"A Paradox is truth spelt with seven letters instead of five"

"You undergo too strict a paradox, Striving to make an ugly deed look fair."

"More than any other Hellenic thinker, Julian insisted on the virtue of paradox and on the importance of the search for religious truth."

"These are old fond paradoxes to make fools laugh i' the alehouse."

"The assumption that anything true is knowable is the grandfather of paradoxes."

"A logical theory may be tested by its capacity for dealing with puzzles, and it is a wholesome plan, in thinking about logic, to stock the mind with as many puzzles as possible, since these serve much the same purpose as is served by experiments in physical science."

"Paradox is thus a much deeper and universal concept than the ancients would have dreamed. Rather than an oddity, it is a mainstay of the philosophy of science."