The elements of logic

Front matter

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Preface

Logic, as a science, is simple and limited. Most modern treatises upon the subject have erred, by extending it beyond its proper department, on the one hand, and by excluding it from its legitimate province on the other. By some it has been made to occupy the ground of mental science or of rhetoric ; by others it has been denominated, " cm art of reasoning," as if there were any sound reasoning which is not logical, while others have claimed for it the unlimited sphere of teaching "the right use of reason." These errors have been exposed by Archbishop

Whately, and the true nature and appropriate office of logic have been explained and vindicated. His learned and able treatise has obtained favor in the universities of Great Britain and the United States, and will go far, undoubtedly, to revive and extend a neglected, but invaluable science. The principles of that work, which are none other than those of Aristotle, have beeji adopted as the basis of the present volume.

The treatise now presented to the public is designed for a department hitherto unoccupied.

A science, so rudimentary in its principles, and so extensive in its applications, ought to be studied with the common elements of learning. Nor, when properly explained, will it be found any more difficult to the younger student than grammar or arithmetic. It will not task

PEEFACE. 5

the powers beyond what is desirable in salutary discipline, while its tendency to promote a habit of thinking will be greater than that of any other science. Indeed, logic must be studied early, and rendered perfectly familiar, in' order to be of much practical utility in the business of life. It is so long postponed in existing systems of education, and, after all, so superficially studied, that there is scarcely one educated man in a thousand w r ho professes to be master of logic.

This work, though simple in its arrangement, embraces all that is essential to logic, while everything which does not strictly and necessarily come within the appropriate province of the science has been excluded. Collateral matter and discursive explanations have been avoided, as rather calculated to embarrass and confuse the youthful mind. The principles

6 PKEFACE.

and rules of the science have been stated distinctly, and illustrated by a variety of examples. If any further explanation is necessary the enlightened teacher can easily supply it. The great points will thus stand out prominently to view, and all that is added by way of comment will be associated with them in the mind. The attention of the learner will not be distracted by many particulars, nor the memory encumbered with unessential matter.

Boston, August 14, 1840.

Preface To The Revised Edition

The first edition of this book has been used chiefly in academies of a high grade and as a manual in some colleges. The improvements in this edition will make it still more acceptable to advanced students, while it will be no less adapted to the object for which it was originally designed, namely, to follow immediately after grammar in all schools where the higher branches are commenced. Will teachers of grammar schools now give this book a trial, and make known to the public the results of the experiment? Consider: Will it not be of the greatest

8 PREFACE TO THE REVISED EDITION.

advantage to form the habit in early life of analyzing one's process of thought in reasoning, and especially observing whether the reason given for an opinion is a general reason. For example: It is of itself no proof, that you are not at the scene of a murder, because you are here. Indeed ! what other proof is necessary? Clearly this general principle — that no being but God can be in two places at the same time. Without this as a major premise, your alibi as a minor premise answers no purpose. To be sure this principle is implied in the minor premise, and it happens to be a sound one ; but how often in practical life do we imply in our statements a general principle which is not sound ; but its unsoundness escapes us, because we are not in the habit of considering it distinctively. It would be easy enough

PEEFACE TO THE REVISED EDITION. 9

to give illustration of this from any political or religious newspaper that comes to hand. I insist, therefore, that a science so pertinent to every-day practical life should not be excluded from elementary studies, but should be taught in every grammar school.

The most important improvements in this edition are the Analytical Outline, the Chapter on Distinctions of Reasoning, and the Essay on the Philosophy of Induction.

New Yoke, October, 1860.

I. Dissertation on Induction. 121

II. Miscellaneous Examples for Practice : 152

Supposed Exceptions to Rules 168

ELEMENTS OF LOGIC.

Introduction

ANALYTICAL OUTLINE.

1. Logic is the science of inference; it teaches how one judgment may be inferred from other judgments. To reason is to infer, hence it is usually called the science of reasoning.

2. It assumes that every mind conceives intuitively some ideas or judgments which are at once primary and certain ; otherwise we could have no foundation for inference ; and to infer one idea or judgment from others would give no certainty.

These ideas are called first truths. They are given by the senses, the consciousness, and the reason, and are innumerable. I exist. There is an external vjorld. This body is solid, extended, round, red, warm, or cold, are first truths.

3. At first these ideas are particular, but afterward the mind unites those which are the same in some respect into classes by simple addition. This is called generalization. To express this we no longer say, This tody, and that tody, and yonder body, etc., but body. Med body would be a lower class made up of this red thing, that red thing, etc.

4. It is evident, furthermore, that in order to reason the mind must have some general ideas, or judgments, that are conceived intuitively, and not formed by mere addition or generalization ; for if you make a class by adding all the individuals, you gain nothing by drawing one or more out again. These general ideas are called first principles, or axioms, and are the offspring of the reason. Some of the earliest are these : Every body is in space. JYo event happens without a cause. Like material causes produce like effects.

5. It is the province of psychology to explain under what circumstances these primary ideas are given by the senses, the consciousness, and the reason; but logic assumes their existence as the indispensable basis of inference, and its appropriate office is to explain in what way we infer one judgment from another.

6. The process of reasoning when completed is found to be simply this : Something is predicated, that is, affirmed or denied of a class ; an individual is affirmed to belong to this class, and then, of course, the same thing can be affirmed or denied of that individual.

In whatever form a sound argument is expressed, it may always be shown to involve this process, and every unsound argument deviates from it. In reasoning we always proceed from generals to particulars, and never from particulars to generals, for this is impossible, as it would be to draw out what was never put in. A general principle, to be sure, may be inferred from one still more general ; but in relation to that more general principle it is only a particular ; it is a class in a class of classes — a species under a genus. For example, the general law of terrestrial gravity is an inference, not from particular instances of bodies falling to the earth, but from a more general law, which these particulars indicate. This is an instance of induction.

7. As induction has been supposed to be a mode of reasoning opposite to deduction, let us take a simple and common case. I infer that heat to such a degree as will cause the mercury in the thermometer to rise to the point marked two hundred and twelve degrees on Fahrenheit's thermometer will always cause water to boil; in other words, it is proved by induction to be a law of nature that two hundred and twelve degrees Fahrenheit will cause water to boil.

Now this conclusion is not drawn from any number of instances of the boiling of water, but from a few instances combined with the principle that like cause will produce like effects ; for if this principle were not true, then forty thousand instances of water boiling by such a degree of heat would not prove that another case would happen, no more than finding forty thousand clovers bearing three leaves only would prove that clover always has only three leaves; or finding forty different varieties of cloven-footed animals marked with horns would prove that swine must have horns. But now I know that like causes will produce like effects, and I know also by observation that two hundred and twelve degrees Fahrenheit did once, or twice, or thrice cause water to boil, and I therefore infer that it will always cause water to boil. Admit the premises and the conclusion is unavoidable ; and to do this is simply to affirm something of a class, then to refer an individual to that class, and then to affirm the same thing of that individual.

Now the first premise is a general principle, which is intuitively true. The only question is about the second premise, namely, whether two hundred and twelve degrees Fahrenheit was the cause of boiling in the instances observed.

You may now prove this by another argument made up of another intuitive principle, and an observed fact, or perhaps by two arguments, thus : no event happens without a cause ; the boiling of water is an event, therefore it happened not without cause ; in other words, it happened by some cause. What cause?

It is a presumption of reason that those things which immediately and invariably precede a certain event are its cause, or include its cause.

Now observation and experiment have shown when two hundred and twelve degrees Fahrenheit was present water invariably and immediately boiled; and when it was absent, the other circumstance remaining just as before, it did not boil. Hence it is concluded that it was the cause of the boiling in these cases.

Thus the second premise of the main argument being proved, the first conclusion is established, namely, that two hundred and twelve degrees Fahrenheit is always the cause of water boiling.

Every case of induction proper proceeds upon the same grounds and in the same way. It is, therefore, evident induction is no exception to the rule that inference is always from generals to particulars, and not from particulars to generals.

8. Seasoning by analogy proceeds in the same way ; the difference is only in the character of the first premise, which is, that similar causes are likely to produce similar effects, or that things which agree in certain attributes or relations are likely to agree in certain other attributes or relations. Thus we reason by analogy that Jupiter is likely to be inhabited as well as the earth, and that retribution may be expected in a future life as well as in this.

9. Reasoning a priori and a posteriori are not different modes of reasoning, but arguments differing in the character of one of the premises : in the one we reason from antecedents, and in the other from consequents. From the idea of perfection, as an antecedent in the human mind, Des Cartes argued a priori for the existence of a Perfect Being ; and Paley, from the marks of contrivance in the world as an effect, proved a posteriori an intelligent Creator.

10. It will be seen that the value of any conclusion depends upon the degree of certainty which belongs to the premises. If they are certain the conclusion is certain ; if they are probable the conclusion is only probable. This is the only distinction between mathematical and moral or practical reasoning ; nor does this always exist, for some moral arguments may claim premises that are absolutely certain.

11. It remains only to observe that the syllogism is merely a certain convenient mode of stating an argument ; and that is the most perfect syllogism which is framed so as to make the true process of inference the most apparent. The above instance of induction would be stated as a syllogism, thus :

Like causes will ever produce like effects. Two hundred and twelve degrees Fahrenheit is like the cause that produced boiling of water; Therefore, it will ever produce the like effect.

Whatever caused the boiling of water once will always cause it.

Heat 212° Fahrenheit caused the boiling of water once;

Therefore, it will always cause it.

12. Analyzing a syllogism, for example :

All men are mortal.

Mohammed was a man ; Therefore, Mohammed was mortal :

we find that it is made up of three propositions ; that each proposition contains two terms and a copula, expressing an agreement or disagreement of the terms. Each term denotes an idea, as all men, mortal, Mohammed; each proposition expresses a judgment as to the relation of two terms to each other, and the last of these propositions is an inference from the other two judgments. The three operations of mind concerned in reasoiling are therefore simple apprehension, judgment, and inference.

13. Language serves to express these mental operations, but the mind has ideas, judgments, and inferences before it has language, inasmuch as a sign must of necessity be subsequent to the thing signified.

14. From this analysis it appears that logic enters into the vital processes of the mind, and conducts it from the known to the unknown. A demonstration is essentially a discovery : the propositions in a book of geometry are involved in the axioms and definitions on the first page, but to draw them out is as much a discovery to the mathematician as the continent of America was to Columbus.

SYNTHESIS OF LOGIC*

Part I. Oist Teems

1. The first part of Logic treats of Terms.

A Term is one or more words expressing a thing, or what is thought of a thing ; as,

Grass is green. Grass is a terra, and green is a term.

The sun shines brightly. The sun is a term, and shines brightly is a term.

Section I. Different Kinds Of Terms

2. A Singular Term expresses only one individual; as,

Boston, Connecticut River, this rock, the discoverer of America.

* Younger students commence here.

3. A Common Term is one which is not confined to a single object, as City, which may apply to Boston and to all other cities.

4. A Relative Term expresses an object that is related to another object ; as. Husband, which is related to wife. Parent, which implies offspring.

5. An Absolute Term expresses a thing considered by itself, without reference to any other thing; as,

Eiver, Mountain, Power, Wisdom.

6. A Positive Term expresses a thing as actually existing; as,

Sight, Seeing, Speech, A man speaking.

7. A Privative Term is one which expresses the absence of an attribute from a thing capable of it; as, a blind man, which denotes the absence of the power to see.

DIFFERENT KINDS OF TERMS. 25

8. A Negative Teem denotes the absence of an attribute from a subject, which is not capable of it at all ; as,

9. An Abstract Term is one which expresses a quality, without reference to any subject in which it may be found ; as,

10. A Concrete Term expresses both the attribute and the object to which it belongs ; as, wrong, which expresses both an action and its quality ; ruler, which indicates an agent and his office.

11. An Indefinite Term is one which does not define or mark out an object, and has the particle- not attached to it, expressed or understood ; as, not a man, which may imply any other being.

12. A Definite Term is one which does define or mark out an object and has not the particle not attached to it ; as,

Man, Brutus, Finished, Complete, Established. EXAMPLES.

1. Man is a rational being.

2. Cicero was a great orator.

3. This town is pleasantly situated.

4. Mothers have much solicitude.

6. Far-sightedness is peculiar to seamen.

7. A dumb man is a pitiful object.

9. Wisdom is more precious than rubies.

10. Just and good are the laws of God.

11. The society is organized.

12. The plan is incomplete.

Section II. Opposition Of Terms

13. Consistent Teems are those which may at the same time be affirmed of the same thing ; as, dry and cold.

OPPOSITION OF TEEMS. 27

14. Opposite Terms cannot at the same time be affirmed of the same thing; as, black and white.

The opposition of terms is fourfold.

15. Relative Opposition is that which is between relative terms, that cannot at the same time be applied to the same subject; as, father and son.

16. Contrary Opposition is that between absolute terms, that expel one another from a subject capable of either ; as, wise and foolish.

IT. Privative Opposition is that between a positive and privative term ; as, seeing and Hind.

18. Contradictory Opposition is that between a definite and indefinite term ; as Cesar and not Cesar.

EXAMPLES.

1. Good and small.

2. Master and servant.

3. Ruler and subject.

4. Material and immaterial.

5. Lovely and hateful.

6. Hearing and deaf.

Section III. Species And Genus

This subject involves no mystery, as formerly under a false philosophy.

19. Species is a term that denotes a class, including several individuals ; as, Beast, which includes the horse, cow, lion, deer, etc.

20. Genus is a term that denotes a class, that includes several species ; as Animal, which includes beast, bird, fish, man, insect.

A genus which cannot be comprehended under a higher genus is called highest genus; and a species which includes no lower species is called lowest species.

21. The Essential Difference denotes an essential part of a species that distinguishes it from other species ; as rational, which is the essential difference of the species man, because it is the essential part of man that distinguishes him from beast, bird, etc.

22. A Property is something necessarily joined to the essential difference ; as power of laughing, which is the property of man as a rational being.

SPECIES AND GENUS. 29

18. An Accident is something that may be, i may not be joined to the essential difference, as tall or short, living in London, horn in Paris.

1. This tree is a Pine, with very thick boughs.

2. This is a small magnet, having the power of attraction, and it turns upon a pivot in a direction north and south.

3. This vessel is a ship, having three masts, full rigged, and very long.

4. A republic is a government in which the people have sway and choose their own rulers.

5. This seminary of learning is a college.

6. A lame animal, a blind stag.

7. General terms are names of classes.

8. A bird with blue feathers and broad wings.

9. Mercury is the planet nearest the Sun.

10. A circle is a figure whose circumference is in every part equally distant from the center.

11. A whale is the largest of fish, and is often seen in our waters.

12. A plant is an organized being, destitute 1 of sensation.

SECTION IY.

DIVISION.

24. Division is the distinct enumeration of the several things signified by a term ; as,

New England is divided into Maine, New Hampshire, Vermont, Massachnsetts, Rhode Island, and Connecticut.

Bark signifies the rind of a tree, a small ship, and the cry of a dog.

25. Rule of Division. The several parts must not he contained in one another, and all together must he exactly equal to the thing divided.

1. The year is divided into Summer, Autumn, Winter, and Spring.

2. Metals are divided into gold, copper, and iron.

3. The Human race is divided into Americans, Africans, Asiatics, Chinese, and Europeans.

* Examples, it will be seen, are given for criticism.

DEFINITION. 31

4. King signifies a ruler and also a man's name.

5. Mercury is the name of a heathen deity, a planet, a plant, and quicksilver.

6. The globe consists of land and water.

7. Form signifies shape and ceremony.

8. Trump is a trumpet and a winning card.

9. Orchard is an inclosure for apple-trees and fruit-trees.

10. A solid body has length and breadth.

11. A tree consists of trunk, branches, and leaves.

SECTION" V.

DEFINITION.

26. A Definition is an expression explaining a term, so as to distinguish or separate it from everything else.

27. A Nominal Definition distinguishes the meaning of a term by an equivalent term, which is better known ; as,

When all the equivalent terms are given it is one kind of division.

28. A Physical Definition lays down the real parts of the essence ; as,

Injustice is the intentional violation of another's rights. A plant has leaves, stalks, roots.

This last is another kind of division.

29. A Logical Definition assigns the Genus and the Essential Difference of the thing defined; as,

Man is a rational animal.

30. Accidental Definition, or Description, assigns the accidents or properties of the thing defined ; as,

Man is an animal that uses fire to dress his food. Columbus was a native of Genoa.

31. It will be observed that more than one of these kinds of definition will coincide in matters strictly scientific.

32. Rule of Definition. A Definition must be adequate, that is, not too extensive nor too narrow, for the tiling defined j plainer, and contained in a suitable number of proper, not figurative terms.

-33. The chief concern of Logic with defini-

DEFINITIONS. 33

! tions is, that words be not used in different \ senses in argument.

EXAMPLES.

1. Pleiades — the seven stars.

2. A king is the ruler of a people.

3. A square is a figure having four sides and four right angles.

4. Surprise is a state of mind produced by some unexpected occurrence.

5. A circle is a figure whose circumference is in every point equally distant from the center.

6. Species is a term for a class.

7. "Wine is the juice of the grape.

8. "Whiteness is the color arising from the prevalence of brightness.

9. A plant is an organized being destitute of sensation.

10. Logic is the science of inference.

11. Man is a risible animal.

12. Mercury is the planet nearest the Sun.

13. A Church is a congregation of faithful

men, in which the word of God is preached,

14. Punishment is pain inflicted for a crime, in order to correct the offender.

15. Courage is boldness and endurance in time of peril.

16. Sin is voluntary transgression of a known law.

17. Sin is any transgression of God's law.

Part II. Of Propositions

The second part of Logic treats of Propositions.

Section I

1. A Proposition is a judgment expressed in words, or a sentence, whereof one part is affirmed or denied of the other ; as,

Man is an animal.

Moses and Thomas are not Statesmen.

All animals are mortal.

2. The Subject of a proposition is that part, of which something is affirmed or denied ; as, in the last example all animals is the subject.

3. The Predicate of a proposition is that which is affirmed or denied of the subject; as, in the same example, mortal is the predicate.

4. The Copula is the verb by which the two terms are connected. It is the present tense of the verb to be, with or without the particle not

5. Sometimes one part of a proposition is contained , in another ; as, the wind blows, I think; which imply, I am thinking, the wind is

6. The subject of a proposition usually stands first, and the predicate last; but this order is sometimes inverted, as,

In the "West are extensive Prairies.

EXAMPLES.

1. Matter is divisible.

2. Man is not infallible.

3. Christopher Columbus was the discoverer

of America.

4. A wise man rules his own spirit.

5. George Washington was a brave but pru-

dent general.

6. The world exhibits marks of a great con-

vulsion.

7. I see — he feels — you walk — they run.

9. Blessed are the pure in heart.

DISTINCTION OF PEOPOSITIONS. 37

Section II. Distinction Of Peopositions

7. Propositions are distinguished into affirmative and negative, which is a distinction with respect to Quality.

8. An Affirmative Proposition is one in which the predicate is declared to agree with the subject ; as,

Man is a fallible creature.

9. A Negative Proposition is one in which the predicate is declared to disagree with the subject; as,

The world is not eternal ; no miser is happy.

10. Propositions are also distingnished into Universal and Particular, which is a distinction in respect to Quantity.

11. A Universal Proposition is one in which the predicate is asserted of the whole of the subject.

The signs of universality are all, every, no, neither, and the like, which are expressed or understood ; also, proper names, as John, London; and common names with a singular or a definite sign ; as, That hoy, five hooks.

No discontented man is happy. •

Those stars revolve about the sun.

Exceptions. In negative propositions the idiom of the English language makes all and every particular or singular terms ; as, all the sailors were not drowned, does not mean that all the sailors escaped drowning, but that some of them only escaped drowning ; or, it denies drowning of the totality as a collective unit.

To. state a universality here you must say : None of the sailors were drowned, or simply, The sailors were not drowned.

12. A Particular Proposition is one in which the predicate is asserted of an indefinite part of the subject.

The signs of particularity are some, many, few, several, and the like ; as,

Some culprits were not punished.

13. A Universal Affirmative Proposition is one in which the predicate is said to agree

DISTINCTION OF PROPOSITIONS. 39

with the whole of the subject ; as, in the above example — All men are mortal. This is not only a universal proposition, but a universal affirmative proposition.

All tyrants deserve death.

14. A Universal Negative Proposition is one in which the predicate is said to disagree with the whole of the subject; as, no discontented man is happy; which is not only a universal proposition, but a universal negative proposition.

No sins are excusable.

15. A Particular Affirmative Proposition is one in which the predicate is asserted to agree with only some part of the subject; as, some islands are fertile; which is not only a particular proposition, but a particular affirmative proposition.

Several men were drowned.

16. A Particular Negat^e Proposition is one in which the predicate is asserted to disagree with only some part of the subject ; as, some culprits were not punished; which is not only a particular proposition, but a particular negative proposition.

17. A, E, I, O are symbols employed to represent these propositions : thus,

A stands for universal affirmative.

E stands for universal negative.

I stands for particular affirmative.

O stands for particular negative.

EXAMPLES.

1. These sailors were not drowned.

3. Some animals are sagacious.

4. Men are unaccountable beings.

5. Every effect must have an adequate

cause.

6. Few men become suddenly rich.

7. Many criminals are not brought to pun-

ishment.

8. Planets are bodies moving in orbits about

the sun.

9. Islands are surrounded by water.

10. The Christian religion is attested by

miracles.

11. Every sinner will be punished.

DISTINCTIONS OF PROPOSITIONS. 41

12. That man was the inventor of lightning

rods.

13. Not all his foes could alarm him.

14. No flower, is always in bloom.

15. All laws are not useful.

16. Every soldier was not killed.

Section III. Further Distinctions Of Propositions

18. Propositions are distinguished into Categorical and Hypothetical.

19. The Categorical asserts simply that the predicate agrees or disagrees with the subject; as,

Truth is invaluable.

The eye is a natural telescope.

20. The Hypothetical asserts with a condition, or with an alternative ; as,

If It storms, the ship will not sail. It is summer or winter.

21. Hypothetical Propositions are divided into Conditional and Disjunctive.

22. A Conditional Proposition is one whose parts are limited by the particle if, or some word expressing a condition ; as,

If there be no fire, there will be no heat, Cesar deserved death, if he was a tyrant.

23. A Conditional Proposition contains two, and only two, Categorical Propositions, whereof one follows from the other.

If the Bible is true, it ought to be studied.

The first is called the antecedent, and that which results from it is called the consequent.

24. A Disjunctive Proposition asserts that a subject agrees with one of two or more predicates, or a predicate with one of two or more subjects; as,

It is either day or night.

Prosperity or adversity will be your lot.

25. A Disjunctive may easily be converted into a Conditional ; thus,

It is either day or night.

If it is not day, it is night.

TEEMS OF PROPOSITIONS. 43

EXAMPLES.

1. Man is free, or he is not responsible.

2. This proposition is either true or false.

3. The earth must move, if the sun be fixed.

4. If the harvest is large, corn will be cheap.

5. If there be no providence, prayer avails not.

6. If the boat goes, the letter will probably