The passage below is from A System of Logic by John Stuart Mill, Book II, Chapter 3, where he argues (following Thomas Browne, and George Campbell) that syllogistic argument involves a petitio principii: the premisses assume or assert the conclusion. He says in that the universal syllogism
All men are mortal
Socrates is a man
Ergo, Socrates is mortal
the major premiss 'All men are mortal' asserts of every man whatsoever, i.e. of everyone that there has been, or is or will be, that he or she is mortal. Since Socrates is a man, the major premiss therefore asserts of Socrates that he is mortal. But that is the very proposition that was to be proved. Therefore the syllogism contains a petitio.
Whoever pronounces the words, All men are mortal, has affirmed that Socrates is mortal, though he may never have heard of Socrates; for since Socrates, whether known to be so or not, really is a man, he is included in the words, All men, and in every assertion of which they are the subject (System of Logic, Book II, Ch. 3, § 8, note)
To the objection (made by a writer in the 'British Quarterly Review', August, 1846) that in asserting the major premiss we do not assert the conclusion, since we may allow that all men are mortal, without having examined the particular cases of Socrates, Aristotle &c, Mill replies that if we are certain that that all men are mortal, we are certain that Socrates is mortal, even if we have never heard of Socrates. We cannot be sure about the mortality of all men unless we are already sure of the mortality of every individual man. E converso, if it is doubtful that Socrates is mortal (which was what the syllogism was meant to prove), it is doubtful that all men are mortal. Thus, we cannot prove that Socrates is mortal, i.e. remove the doubt in question, by appealing to a proposition that is no less doubtful than the conclusion which it is meant to prove. The general principle, rather than being evidence of the particular instance it was to prove, cannot be taken as true, unless every shadow of doubt that could hang over any particular instance, is dispelled. But what then is there left for the syllogism to prove?
This is, of course, an astonishingly bad argument. It may be perfectly certain that every man is mortal, but if it is uncertain whether Socrates is a man, it is also uncertain whether Socrates is mortal. I may be certain, having marked the exam, that all the candidates passed. But I may not know, if the papers were anonymous, that Susan was a candidate. Certainty about the universal proposition does not imply certainty about any of the propositions which are true because of it. Now Mill assumes that the universal judgment (all men or mortal) is plural or enumerative: that it consists or contains a set of singular judgments about individuals (Socrates is mortal, Plato is mortal, Aristotle is mortal, &c) 'collected into one'. For example, the enumerative judgment 'Seneca and Cicero were philosophers' consists of the individual judgments 'Seneca was a philosopher and Cicero was a philosopher'. Yet even if that assumption were true, certainty about the truth of the enumerative judgment does not strictly imply certainty about any of the individual judgments included under it. For we may be uncertain about whether Cicero is the same person as Tully. Thus, being certain of the truth of 'Seneca was a philosopher and Cicero was a philosopher' does not imply certainty about the truth of 'Tully was a philosopher' even though, as it happens, the truth of Cicero being a philosopher is the same truth as Tully being a philosopher, since they are one and the same person.
Mill's argument is now forgotten, although it underlies his theory of induction: that all reasoning is from particulars to particulars, and that no reasoning from generals to particulars can, as such, prove anything, since from a general principle we cannot infer any particulars, but those which the principle itself assumes as known. But is is of more than just historical interest, as it is connected with a development later in the century that had a lasting impact on logic. Frege replies to a very similar argument in the Grundlagen (and in a number of other places) objecting that the proposition 'All whales are mammals' is not really about whales, but about concepts. It says that the concept 'whale' is subordinate to the concept 'mammal'. It appears to be about whales, but it cannot be, for if we are asked which whale is being spoken of, there none that can be picked out. We cannot infer from the proposition 'all whales are mammals' that the animal present in front of us is a mammal, without the additional proposition that it is a whale. But the proposition in question says nothing about this animal.
In general it is impossible to speak of an object without in some way designating or naming it. But the word 'whale' does not name any individual creature.
Frege thus accepts Mill's assumption if universal judgments are enumerative, the syllogism involves a petitio. This leads him (wrongly, it would seem) to reject plural or enumerative judgments. Judgments are either singular, consisting of proper name and predicate, or they are what traditional logicians called 'hypothetical', asserting that if any X is an A, X is a B, without asserting or presupposing that any X is in fact an A. Despite our intution that 'all whales are animals' is about whales, Frege claims that its real form is 'if any X is a whale, X is an animal'. It cannot be about whales, otherwise it would imply the proposition that this animal here is a mammal, which it does not.
This of course was written in 1874, before Frege discovered the distinction between sense and reference, i.e. that a singular proposition does not logically imply another proposition predicating the same thing of the same individual, but differently named. We cannot infer from 'Cicero is a philosopher' that Tully is a philosopher, without the additional proposition that Cicero is Tully. But he failed to see how this undermines both his explanation of why 'All whales are mammals' does not imply that this animal is a mammal, and his whole view of the universal judgment. The rest is of course history. There is an excellent paper by Hanoch Ben-Yami here, pointing out this fallacy in Frege's argument.
Edward Buckner. London, June 2007
A System of Logic (18..)
[120] § 2. It must be granted that in every syllogism, considered as an argument to prove the conclusion, there is a petitio principii. 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, is presupposed in the more general assumption, All men are mortal: that we cannot be assured of the mortality of all men unless we are already certain of the mortality of every individual man: that if it be still doubtful whether Socrates, or any other individual we choose to name, be mortal or not, the same degree of uncertainty must hang over the assertion, All men are mortal: that the general principle, instead of being given as evidence of the particular case cannot itself be taken for true without exception, until every shadow of doubt which could affect any case comprised with it, is dispelled by evidence aliunde; and then what remains for the syllogism to prove? That, in short, no reasoning from generals to particulars can, as such, prove anything, since from a general principle we cannot infer any particulars, but those which the principle itself assumes as known.
This doctrine appears to me irrefragable; and if logicians, though unable to dispute it, have usually exhibited a strong disposition to explain it away, this was not because they could discover any flaw in the argument itself, but because the contrary opinion seemed to rest on arguments equally indisputable. In the syllogism last referred to, for example, [121] or in any of those which we previously constructed, is it not evident that the conclusion may, to the person to whom the syllogism is presented, be actually and bona fide a new truth? Is it not a matter of daily experience that truths previously unthought of, facts which have not been, and cannot be, directly observed, are arrived at by way of general reasoning? We believe that the Duke of Wellington is mortal. We do not know that by direct observation, so long as he is not yet dead. If we were asked how, this being the case, we know the Duke to be mortal, we should probably answer, Because all men are so. Here, therefore, we arrive at the knowledge of a truth not (as yet) susceptible of observation, by a reasoning which admits of being exhibited in the following syllogism:-
All men are mortal,
The Duke of Wellington is a man,
therefore
The Duke of Wellington is mortal;
And since a large portion of our knowledge is thus acquired, logicians have persisted in representing the syllogism as a process of inference or proof, though none of them has cleared up the difficulty which arises from the inconsistency between that assertion and the principle that if there be anything in the conclusion which was not already asserted in the premises, the argument is vicious. For it is impossible to attach any serious scientific value to such a mere salvo as the distinction drawn between being involved by implication in the premises, and being directly asserted in them. When Archbishop Whately says [N1] that the object of reasoning is 'merely to expand and unfold the assertions wrapt up, as it were, and implied in those with which we set out, and to bring a person to perceive and acknowledge the full force of that which he has admitted', he does not, I think, meet the real difficulty requiring to be explained, namely, how it happens that a science, like geometry, can be all 'wrapt up' in a few definitions and axioms. Nor does this defence of the syllogism differ much from what its assailants urge against it as an accusation, when they charge it with being of no use except to those who seek to press the consequences of an admission into which a person has been entrapped without having considered and understood its full force. When you admitted the major premise, you asserted the conclusion; but, says Archbishop Whately, you asserted it by implication merely: this, however, can here only mean that you asserted it unconsciously; that you did not know you were asserting it; but, if so, the difficulty revives in this shape – Ought you not to have known? Were you warranted in asserting the general proposition without having satisfied yourself of the truth of everything which it fairly includes? And if not, is not the syllogistic art prima facie what its assailants affirm it to be, a contrivance for catching you in a trap, and holding you fast in it? [N2]
§ 3. From this difficulty there appears to be but one issue. The proposition that the Duke of Wellington is mortal is evidently in inference; [122] it is got at as a conclusion from something else; but do we, in reality, conclude it from the proposition, All men are mortal? I answer, No.
The error committed is, I conceive, that of overlooking the distinction between two parts of the process of philosophising, the inferrring part, and the registering part, and ascribing to the latter the functions of the former. The mistake is that of referring a person to his own notes for the origin of his knowledge. If a person is asked a question, and is at the moment unable to answer it, he may refresh his memory by turning to a memorandum which he carries about with him. But if he were asked, how the fact came to his knowledge, he would scarcely answer, because it was set down in his notebook: unless the book was written, like the Koran, with a quill from the wing of the angel Gabriel.
Assuming that the proposition, the Duke of Wellington is mortal, is immediately an inference from the proposition, All men are mortal; whence do we derive our knowledge of that general truth? Of course from observation. Now, all which man can observe are individual cases. From these all general truths must be drawn, and into these they may be again resolved; for a general truth is but an aggregate of particular truths; a comprehensive expression, by which an indefinite number of individual facts are affirmed or denied at once. But a general proposition is not merely a compendious form for recording and preserving in the memory a number of particular facts, all of which have been observed. Generalisation is not a process of mere naming, it is also a process of inference. From instances which we have observed, we feel warranted in concluding that what we found true in those instances, holds in all similar ones, past, present, and future, however numerous they may be. We then, by that valuable contrivance of language which enables us to speak of many as if they were one, record all that we have observed, together with all the we infer from our observations, in one concise expression; and have thus only one proposition, instead of an endless number to remember or to communicate. The results of many observations and inferences, and instructions for making innumerable inferences in unforeseen cases, are compressed into one short sentence.
When, therefore, we conclude from the death of John and Thomas, and every other person we ever heard of in whose case the experiment had been fairly tried, that the Duke of Wellington is mortal like the rest, we may, indeed, pass through the generalisation, All men are mortal, as an intermediate stage; but it is not in the latter hald of the process, the descent from all men to the Duke of Wellington, that the inference resides. The inference is finished when we have asserted that all men are mortal. What remains to be performed afterwards is merely deciphering our own notes. Archbishop Whately has contended that syllogising, or reasoning from generals to particulars, is not, agreeably to the vulgar idea, a peculiar mode of reasoning, but the philosophical analysis of the mode in which all men reason, and must do so if they reason at all. With the deference due to so high an authority, I cannot help thinking that the vulgar notion is, in this case, the more correct. If, from our experience of John, Thomas, &c., who were once living, but are now dead, we are entitled to conclude that all human beings are mortal, we might surely without any logical inconsequence have concluded at once from those instances that the Duke of Wellington is mortal. The mortality of John, Thomas, and others is, after all, the whole evidence we have for the mortality of the Duke of Wellington. Not one iota is added to the proof by interpolating a general proposition. Since the individual cases are all the evidence we can possess, evidence which no logical [123] form into which we choose to throw it can make greater than it is; and since that evidence is either sufficient in itself, or, if insufficient for the one purpose, cannot be sufficient for the other; I am unable to see why we should be forbidden to take the shortest cut from these sufficient premises to the conclusion, and constrained to travel the 'high priori road' by the arbitrary fiat of logicians. I cannot perceive why it should be impossible to journey from one place to another unless we 'march up a hill, and then march down again'. It may be the safest road, and there may be a resting-place at the top of the hill, affording a commanding view of the surrounding country; but for the mere purpose of arriving at our journey's end, our taking that road is perfectly optional; it is a question of time, trouble, and danger.
Not only may we reason from particulars to particulars without passing through generals, but we perpetually do so reason. All our earliest inferences are of this nature. From the first dawn of intelligence we draw inferences, but years elapse before we learn the use of general language. The child who, having burnt his fingers, avoids to thrust them again into the first, has reasoned or inferred, though he has never thought of the general maxim, Fire burns. He knows from memory that he has been burnt, and on this evidence believes, when he sees a candle, that if he puts his finger into the flame of it, he will be burnt again. He believes this in every case which happens to arise; but without looking, in each instance, beyond the present case. He is not generalising; he is inferring a particular from particulars. In the same way, also, brutes reason. There is no ground for attributing to any of the lower animals the use of signs of such a nature as to render general propositions possible. But those animals profit by experience, and avoid what they have found to cause them pain, in the same manner, though not always with the same skill, as a human creature. Not only the burnt child, but the burnt dog, dreads the fire.
I believe that, in point of fact, when drawing inferences from our personal experience, and not from maxims handed down to us by books or tradition, we much oftener conclude from particulars to particulars directly, than through the intermediate agency of any general proposition. We are constantly reasoning from ourselves to other people, or from one person to another, without giving ourselves the trouble to erect our observations into general maxims of human or external nature. When we conclude that some person will, on some given occasion, feel or act so and so, we sometimes jude from an enlarged consideration of the manner in which human beings in general, or persons of some particular character, are accustomed to feel and act; but much oftener from merely recollecing the feelings and conduct of the same person in some previous instance, or from considering how we should feel or act ourselves. It is not only the village matron, who when called to a consultation upon the case of a neighbour's child, pronounces on the evil and its remedy simply on the recollecton and authority of what she accounts the similar case of her Lucy. We all, where we have no definite maxims to steer by, guide ourselves in the same way; and if we have an extensive experience, and retain its impressions strongly, we may acquire in this manner a very considerable power of accurate judgment, which we may be utterly incapable of justifying or of communicating to others. Among the higher order of practical intellects there have been many of whom it was remarked how admirably they suited their means to their ends, without being able to give any sufficient reasons for what they did; and applied, or seemed to apply, recondite principles which they were wholly unable to state. This is a natural consequence of having amind stored with appropriate particulars, [124] and having been long accustomed to reason at once from these to fresh particulars, without practising the habit of stating to oneself or to others the corresponding general propositions. An old warrior, on a rapid glance at the outlines of the ground, is able at once to give the necessary orders for a skilful arrangement of his troops; though if he has received little theoretical instruction, and has seldom been called upon to answer to other people for his conduct, he may never have had in his mind a single general theorem respecting the relation between ground and array. But his experience of encampments, in circumstances more or less similar, has left a number of vivid, unexpressed, ungeneralised analogies in his mind, the most appropriate of which, instantly suggesting itself, determines him to a judicious arrangement.
The skill of an uneducated person in the use of weapons or of tools is of a precisely similar nature. The savage who executes unerringly the exact throw which brings down his game, or his enemy, in the manner most suited to his purpose, under the operation of all the conditions necessarily involved, the weight and form of the weapon, the direction and distance of the object, the action of the wind, &c., owes this power to a long series of previous experiments, the results of which he certainly never framed into any verbal theorems or rules. The same thing may generally be said of any other extraordinary manual dexterity. Not long ago a Scotch manufacturer procured from England, at a high rate of wages, a working dyer, famous for producing very fine colours, with the view of teaching to his other workmen the same skill. The workman came; but his mode of proportioning the ingredients, in which lay the secret of the effects he produced, was by taking them up in handfuls, while the common method was to weigh them. The manufacturer sought to make him turn his handling system into an equivalent weighing system, that the general principle of his peculiar mode of proceeding might be ascertained. This, however, the man found himself quite unable to do, and therefore could impart his skill to nobody. He had, from the individual cases of his own experience, established a connection in his mind between fine effects of colour, and tactual perceptions in handling his dyeing materials; and from these perceptions he could, in any particular case, infer the means to be employed, and the effects which would be produced, but could not put others in possession of the grounds on which he proceeded, from having never generalised them in his own mind, or expressed them in language.
Almost every one knows Lord Mansfield's advice to a man of practical good sense, who, being appointed governor of a colony, had to preside in its courts of justice, without previous judicial practice or legal education. The advice was to give his decision boldly, for it would probably be right; but never to venture on assigning reasons, for they would almost always infallibly be wrong. In cases like this, which are of no uncommon occurrence, it would be absurd to suppose that the bad reason was the source of the good decision. Lord Mansfield knew that if any reason were assigned it would be necessarily an afterthought, the judge being in fact guided by impressions from past experience, without the circuitous process of framing general principles from them, and that if he attempted to frame any such he would assuredly fail. Lord Mansfield, however, would not have doubted that a man of equal experience who had also a mind stored with general propositions derived by legitimate induction from that experience, would have been greatly preferable as a judge to one, however sagacious, who could not be trusted with the explanation of his own judgments. The cases of men of talent performing wonderful things they know not how, are ex[125]amples of the rudest and most spontaneous form of the operations of superior minds. It is a defect in them, and often a source of errors, not to have generalised as they went on; but generalisation, though a help, the most important indeed of all helps, is not an essential.
[…]
[126] § 4. From the considerations now adduced the following conclusions seem to be established. All inference is from particulars to particulars: General propositions are merely registers of such inferences already made, and short formulae for making more. The major premise of a syllogism, consequently, is a formula of this description; and the conclusion is not an inference drawn from the formula, but an inference drawn according to the formula; the real logical antecedent or premise being the particular facts from which the general proposition was collected by induction. Those facts, and the individual instances which supplied them, may have been for[127]gotten; but a record remains, not indeed descriptive of the facts themselves, but showing how those cases may be distinguished, respecting which, the facts, when known, were considered to warrant a given inference. According to the indications of this record we draw our conclusion; which is, to all intents and purposes, a conclusion from the forgotten facts. For this it is essential that we should read the record correctly; and the rules of the syllogism are a set of precautions to ensure our doing so.
[…]
§ 8 [footnote]. A writer in the 'British Quarterly Review' (August, 1846), in a review of this treatise, endeavours to show that there is no petitio principii in the syllogism, by denying that the proposition, All men are mortal, asserts or assumes that Socrates is mortal. In support of this denial, he argues that we may, and in fact do, admit the general proposition that all men are mortal, without having particularly examined the case of Socrates, and even without knowing whether the individual so named is a man or something else. But this of course was never denied. That we can and do draw conclusions concerning cases specifically unknown to us is the datum from which all who discuss this subject must set out. The question is, in what terms the evidence or ground on which we draw these conclusions may best be designated – whether it is most correct to say, that the unknown case is proved by known cases, or that it is proved by a general proposition including both sets of cases, the unknown and the known? I contend for the former mode of expression. I hold it an abuse of language to say that the proof that Socrates is mortal is that all men are mortal. Turn it in what way that we will, this seems to me to be asserting that a thing is the proof of itself. Whoever pronounces the words, All men are mortal, has affirmed that Socrates is mortal, though he may never have heard of Socrates; for since Socrates, whether known to be so or not, really is a man, he is included in the words, All men, and in every assertion of which they are the subject.
[N1]Logic, p. 239 (9th edition)
[N2] It is hardly necessary to say, that I am not contending for any such absurdity as that we actually 'ought to have known' and considered the case of every individual man, past, present and future, before affirming that all men are mortal: although this interpretation has been, strangely enough, put upon the preceding observations. There is no difference between me and Archbishop Whately, or any other defender of the syllogism, on the practical part of the matter; I am only pointing out an inconsistency in the logical theory of it, as conceived by almost all writers. I do not say that a person who affirmed, before the Duke of Wellington was born, that all men are mortal, knew that Duke of Wellington was mortal; but I do say that he asserted it; and I ask for an explanation of the apparent logical fallacy of adducing in proof of Duke of Wellington's mortality a general statement which presupposes it. Finding no sufficient resolution of this difficulty in any of the writers on Logic, I have attempted to supply one.
THE LOGIC MUSEUM Copyright (c)
E.D.Buckner 2007.