The fact was that the pupil learned to compute correctly irrespective of the deductive explanations.
It may be an illusion, but one seems to sense in the better textbooks a recognition of the futility of the attempt to secure deductive derivations of those manipulations.
At any rate the fact is sure that most pupils do not learn the manipulations by deductive reasoning, or understand them as necessary consequences of abstract principles.
Except for the very gifted pupils, the ordinary preliminarydeductive explanations of what must be done are probably useless as means of teaching the pupils what to do.
What is learned of such deductive theory should rank among the most rather than least permanent of a pupil's stock of arithmetical knowledge and power.
The doctrine that the customary deductive explanations of why we invert and multiply, or place the partial products as we do before adding, may be allowed to be forgotten once the actual habits are in working order, has a suspicious source.
The criticism was valid and should have been met in part by replacing the deductive explanations by inductive verifications, and in part by using the deductive reasoning as a check after the process itself is mastered.
They are protected from adding 3 + 5 and 2 + 3, again, by no deductive reasoning but for the second and third reasons just given.
The first explanation leads to the common preliminarydeductive explanations of the textbooks.
The general deductive theory of arithmetic should not be learned only to be forgotten.
It arose to meet the criticism that so much time and effort were required to keep these deductive explanations in memory.
Post hoc, ergo propter hoc; and the deductive fallacy corresponding to it 364 6.
The deductive science of society will not lay down a theorem, asserting in an universal manner the effect of any cause; but will rather teach us how to frame the proper theorem for the circumstances of any given case.
The experimental process is not here to be regarded as a distinct road to the truth, but as a means (happening accidentally to be the only, or the best, available) for obtaining the necessary data for the deductive science.
Now to such cases we have seen that the Deductive Method, setting out from general laws, and verifying their consequences by specific experience, is alone applicable.
Difficulties of the Direct Deductive Method in the Social Science 489 3.
And as will be shown presently, nothing of a really scientific character is here possible, except by the inverse deductive method.
But, further, this reasoning by counters is only suitable to a very limited portion even of our deductive processes.
The immense part which those laws take in giving a deductive character to the other departments of physical science, is well known; and is not surprising, when we consider that all causes operate according to mathematical laws.
Sciences have become deductive usually through its being shown, either by deduction or by direct experiment, that the varieties of some phenomenon in them uniformly attend upon those of a better known phenomenon, e.
The Deductive Method must be employed to obtain the laws of the formation of character.
Social Science, must use the Concrete Deductive Method, compounding with one another the laws of all the causes on which any one effect depends, and inferring its law from them all.
Observation has its uses in relation to this subject; but only as verification of the results arrived at by the Deductive Method.
Contagious Influence of Chemical Action, and his Theory of Respiration, are among the finest examples, since Newton's exposition of the law of gravitation, of the use of the deductive method for explanation.
The use of a hypothesis is to enable us to apply the Deductive Method before the laws of the causes have been ascertained by Induction.
A transition from the inductive stage to the deductive stage was shown in the answer--the transformation results from the unceasing multiplication of effects.
Eventually the growing generalisation, thus far inductive, might take deductive form: being all at once recognised as a necessary consequence of some physical principle--some established law.
The deductive interpretation of waste is easy, but it is different with repair, for here the analogy between the organism and an inanimate engine breaks down.
When, shortly after, there came the perception that the condition of homogeneity is an unstable condition, yet another step towards the completely deductive stage was made.
Geometry is a whole collection of deductive sciences based on a corresponding collection of sets of axioms.
The succession of phenomena would in such case be altogether arbitrary, and deductive Science impossible.
The seventh chapter of Matthew contains a number of excellent examples of deductive teaching.
If we pursue the opposite order, beginning with some general law or principle and proceed by reasoning to special or particular facts, we use the deductive method in teaching.
By analysis we reduce them to propositions which are as nearly as possible simple and precise, and we arrange them in deductive chains, in which a certain number of initial propositions form a logical guarantee for all the rest.
The discovery of geometry had intoxicated them, and its a priori deductive method appeared capable of universal application.
But an empirical empiricism, in contrast with orthodox deductive empiricism, has no difficulty in establishing its jurisdiction as to deductive functions.
And these are the generations of Thare: Thare begot Abram, Nachor, and Aran.
And Nachor lived after he begot Thare, a hundred and nineteen years, and begot sons and daughters.
And Thare lived seventy years, and begot Abram, and Nachor, and Aran.
And Sarug lived after he begot Nachor, two hundred years, and begot sons and daughters.
This completely certain and safe conclusion, although it is not based upon any direct experience, is a deductive conclusion.
If the latter is the greatest inductive law of biology, then it of necessity follows that the former is its most important deductive law.
The theory of humoral pathology, one of the most important advances in medical science, was based on a conclusion from analogy and arrived at by the deductive method.
Thus astrology may serve as one of the most telling examples of scientific delusions to which the ancient diagnostico-theoretical methods were bound to lead, with their conclusions from analogy and their deductive modes of procedure.
Thus the way was opened for a fundamental separation of the inductive material scientific from the deductive mathematical method.
There is, however, difference of opinion regarding their inner structure and their inner relation to the deductive inferences.
Only the deductive process proceeding syllogistically from the universal (or essential truth) to the particular is scientifically cogent or apodictic.
We are indebted to the Greek, and especially to the Platonic-Aristotelian philosophy for important contributions to the understanding of the deductive method of mathematical thought.
That the science of logic was "completed and perfect" in the time of Kant could only with any degree of accuracy be said of the treatment of syllogistic proof or the deductive logic of Aristotle.
The conceptual determination of the causal relation, and with it in principle the sum total of the methods in the material sciences, becomes a logical, analytical, and deductive one.
Astronomy has already passed through its successive stages: first collections of facts; then inductions from them; and lastly deductive interpretations of these, as corollaries from a universal principle of action among masses in space.
Most of the great religious teachers have had deductive minds: from the basis of certain sublime assumptions they have asserted their commandments.
The minds of men have always moved in two directions, and always will; and as long as men shall write, we shall have, and ought to have, both inductive and deductive fiction.
Every student of philosophy will tell you that the world's thought was prevailingly deductive till the days of Francis Bacon.
So stated, the distinction is as follows: In setting forth his view of life, the realist follows the inductive method of presentment, and the romantic follows the deductive method.
The distinction between inductive and deductive processes of thinking is very simple and is known to all: it is based upon the direction of the train of thought.
He does not conceal his aversion to purely deductive procedures which start from the general and end in the particular.
If this science is deductive only in appearance, whence does it derive that perfect rigor no one dreams of doubting?
We can not therefore even say that in the really analytic and deductive part of mathematical reasoning we proceed from the general to the particular in the ordinary sense of the word.
His Studies in Deductive Logic, consisting mainly of exercises and problems for the use of students, was published in 1880.
Farther, as regards the particular method of investigating that group of phenomena obtained for him by the abstract method, he will adopt, he tells us, neither the purely deductive nor the purely inductive mode but the union of both.
So he is compelled either to use the deductive mode of arguing from general laws or to employ the method of abstraction, which gives a fictitious isolation to phenomena never so isolated in actual existence.
There seemed to be no interstice, no crevice into which he might insert the keen probe of his marvelous deductive power.
I never was good at puzzles when I was little, and I suppose I lack that deductive quality now.
Rolfe, who seemed to his superior officer to be in danger of displaying some admiration for deductive methods.
Rolfe was so interested in Crewe's revelations that he stood beside the deductive expert and studied the paper afresh.
The disdainful pity of thedeductive experts for the rule-of-thumb methods of the police is not to be compared with the vigorous scorn of the official detective for the rival who has not had the benefit of police training.
He was not a deductive expert, and, as he told his wife afterwards, he did not know what the detective was "driving at.
He is taught elementary mathematics, that he may understand all those relations of number and form, upon which the transactions of men, associated in complicated societies, are built, and that he may have some practice in deductive reasoning.
Aristotle criticised the deductive speculations of Plato by means of inductive negative instances, but Polybius will not take the 'Cloud City' of the Republic into account at all.
So he is compelled either to use the deductive mode of arguing from general laws or to employ the method of abstraction which gives a fictitious isolation to phenomena never so isolated in actual existence.
But the consideration of the matter is outside the sphere of formal or deductive Logic.
Inductive inferences are wholly extraneous to the science of formal logic, which deals only with formal, or necessary, inferences, that is to say with deductive inferences, whether immediate or mediate.
On the other hand a deductive inference has all the certainty that can be imparted to it by the laws of thought, or, in other words, by the structure of our mental faculties; but this certainty is purely hypothetical.
Hence in the natural order of treatment inductive logic precedes deductive, since it is induction which supplies us with the general truths, from which we reason down in our deductive inferences.
Now Formal, which is another name for Deductive Logic, is concerned only with the way in which the mind thinks, and has nothing to do with the particular objects thought about.
The fact is that inductive inferences are either wholly instinctive, and so unsusceptible of logical vindication, or else they may be exhibited under the form of deductive inferences.
But the moment this universal proposition is stated, the truth of the proposition in the individual instance flows from it by way of deductive inference.
In deductive inference we never do more than vary the form of the truth from which we started.
There are never more than two propositions in the antecedent of a deductive inference.
As soon as men begin to take foresight for the future and to prepare themselves in advance to meet it effectively and prosperously, the deductive operations and their results gain in importance.
But students of German philosophy, of mathematics, and of psychology, no matter how much they attacked one another, have made common cause in attack upon the orthodox logics both of deductive proof and inductive discovery.
Deductive processes define, delimit, purify and set in order the conceptions through which this enriching and directive operation is carried on, but they cannot, however perfect, guarantee the outcome.
This is the deductive method of ratiocination (wherein lies the difference between the deductive and the inductive method, no one can understand); these are the dogmas of the technological and metaphysical period.
