ART. I. -- DEFINITION AND ELEMENTS OF REASONING.
43. Reasoning is the third operation of the intellect, by which, from the relation existing between two judgments, it infers a third as the result of the other two. -- There are two kinds of judgments. Some are self-evident, and on that account are called analytical or immediate. Others are not self-evident, and are called deductive or mediate; the relation between the predicate and the subject cannot be perceived without comparing them with a third idea. The act by which we seek to determine the relation of two terms by comparing them with a third is reasoning. The verbal expression of a reasoning is called an argument, and is, therefore, defined as a discourse in which one proposition is inferred from another.
44. The elements of reasoning are three ideas and three judgments, and the relation existing between these ideas and judgments. -- Every reasoning must contain three ideas, since its end is to establish the relation between subject and predicate by means of a third idea. Again, it must contain three judgments: two to show the relation of the subject and predicate with a middle term, a third to point out the relation of the predicate with the subject. The three ideas and the three judgments constitute the matter of reasoning, their connection constitutes its form.
45. The truth of a reasoning may be considered in respect both to matter and form. -- That a reasoning may be materially true, it suffices that the premises and the conclusion be separately true; but that it be formally true, the connection between conclusion and premises must likewise be true; hence it is clear that a reasoning may be materially true and formally false, and vice versa. Thus the reasoning, "Every man is mortal; every man is an animal: therefore every animal is mortal," is materially true, but formally false; while the reasoning, "All substances are spiritual; color is a substance: therefore color is spiritual," is materially false, but formally true.
46. All reasoning is based on one of these two axioms: 1. Two things which agree with a third wholly or in part, agree with each other wholly or in part; 2. Two things, one of which agrees wholly or in part with a third, with which the other does not agree, do not agree with each other. -- The first axiom is the principle of affirmative reasoning; the second is the basis of negative reasoning.
ART. II. -- DIVISION OF REASONING.
47. Reasoning considered in respect to its form, is deductive or inductive; in respect to its matter, it is categorical or hypothetical. -- In every reasoning a predicate is affirmed or denied of a subject, because, after comparing each of them with a middle term, we know whether or not the middle term contains the other two. Now, as one thing may be in another as a part in the whole, or as the whole in the sum of its parts, reasoning is of two kinds, according as we proceed from the whole to its parts, or from the parts to the whole; that is, according as we proceed from genera to species and from species to individuals, or from individuals to species and from species to genera. The first is deductive reasoning, the second is inductive. Reasoning is also deductive if it proceeds from effects to their cause, as from signs to the thing signified, and it is thus we attain to a knowledge of God.
Reasoning considered in respect to the judgments entering into it, is categorical or hypothetical according as the judgments are categorical or hypothetical. But whether reasoning be inductive or deductive, categorical or hypothetical, the truth of the conclusion is always mediate and deduced. Hence the regular form of all reasoning is deduction, or the syllogism.
ART. III. -- CATEGORICAL SYLLOGISMS AND THEIR RULES.
48. The syllogism is that form of argument in which the two extremes of a proposition are compared affirmatively or negatively with a third term in order to conclude their agreement or disagreement. -- It is easily seen from this definition that the syllogism must contain three terms and three propositions. The subject of the deduced proposition is called the minor term or minor extreme; the predicate is called the major term or major extreme, because the predicate, when not identical with the subject, has always a greater extension than the subject. The term with which the extremes are compared is called the middle term. The two propositions in which the two extremes are compared with the middle term are called premises or antecedent; that which contains the major term is called the major premise; that which contains the minor term is called the minor premise. The proposition which is deduced from the other two, and in which the minor term is compared with the major, is called the conclusion or consequent.
49. Syllogisms are subject to the following eight rules:
I. A syllogism should contain only three terms.
II. No term should have a greater extension in the conclusion than it has in the premises.
III. The middle term should be taken universally at least once in the premises.
IV. The conclusion should not contain the middle term.
V. Nothing can be concluded from two negative premises.
VI. Two affirmative premises cannot give a negative conclusion.
VII. The conclusion always follows the weaker part.
VIII. From two particular premises nothing can be concluded.
I. The first rule is derived from the very essence of the syllogism, which consists in establishing the relation between two terms by means of a third. This rule is commonly violated by using one of the terms in two different senses; as, "Every spirit is endowed with intelligence; but alcohol is a spirit; therefore alcohol is endowed with intelligence."
II. The conclusion should not be more extended than the premises; otherwise, we should have a consequent not contained in the antecedent, an effect which exceeds its cause; as, "Eagles are animals; but eagles fly in the air; therefore all animals fly in the air."
III. The middle term must be taken at least once universally; otherwise, being twice particular, it would be equivalent to two different terms, and we should have a syllogism containing four terms; as, "Some animals are endowed with reason; but a horse is an animal; therefore a horse is endowed with reason."
IV. The middle term should not be found in the conclusion; because, being used as a term of comparison, for the purpose of finding agreement or disagreement between the other two terms, its proper place is in the premises, where this relation is established. Its appearance in the conclusion either repeats what has already been expressed; as, "All crimes are detestable; treason is a crime; therefore crime is detestable:" or sometimes introduces a fourth term into the syllogism; as, "English Catholics were persecuted by Queen Elizabeth; Shakespeare was an English Catholic; therefore Shakespeare was an English Catholic persecuted by Queen Elizabeth."
V. Two negatives give no conclusion; for in that case we simply see that the term chosen for the middle cannot serve to establish any relation between the extremes; hence the antecedent is null, and no consequent can be drawn from it; as, "Shepherds are not learned; but Peter is not a shepherd." It cannot he concluded that Peter is or is not learned.
VI. A negative cannot be inferred from two affirmatives, for two things identical with a third cannot but be identical with each other.
VII. The conclusion always follows the weaker or worse part; that is, if one of the premises is negative the conclusion must be negative; if particular, the conclusion must be particular. In the first place, it is evident that, if one of two things is identical with a third, and the other is not, the two things cannot be identical with each other; thus, in the syllogism, "No spiritual substance is mortal; the human soul is a spiritual substance," we must conclude," The human soul is not mortal." In the second place, if one of two premises is particular, the conclusion cannot be universal, otherwise it will have a term more extended here than in the premises; as in the syllogism, "Some men are rational animals; some men are poets; therefore all rational animals are poets."
VIII. Two particulars afford no conclusion; because if both are affirmative, the middle term is necessarily twice particular ; as, "Some students are industrious; some industrious persons are successful; therefore some students are successful." If one of the two is negative, the conclusion must contain a universal term, which is particular in the premises; as, "Some heroes are young men; some young men are not pious; therefore all heroes are not pious, or, No heroes are pious."
All these rules may be reduced to the following Rule of Modern Logicians: The conclusion must be contained in one of the premises, and the other premise must show that it is contained therein.
ART. IV. -- MODES AND FIGURES OF THE SYLLOGISM.
50. The mode of a syllogism is its form according to the quantity and quality of the three propositions which enter into it. -- Propositions considered in respect both to their quantity and quality, are of four kinds: 1. Universal affirmative; 2. Universal negative; 3. Particular affirmative; 4. Particular negative. Logicians have designated these four kinds of propositions by the letters A, E, I, O, respectively. It is evident that these four propositions, combined in threes, give sixty-four possible combinations; but applying to these the rules of the syllogism, there will be found only ten valid modes. These are: AAA, AAI, AEE, AII, AOO, EAE, EAO, EIO, IAI, OAO.
51. The figure of a syllogism is its form according to the position of the middle term in the premises. -- The middle term may be: 1. Subject of major and predicate of minor; 2. Predicate of both; 3. Subject of both; 4. Predicate of major and subject of minor. There are, then, four figures; but many logicians make no account of the fourth, or turn it into the first. Each figure is susceptible of the ten modes, if no regard is had to the rules of syllogism, because the propositions may preserve their quality and quantity without changing the place of the middle term.
52. There are only nineteen conclusive modes; they are designated by the following lines:
Barbara, Celarent, Darii, Ferioque prioris;{1}
Cesare, Camestres, Festino, Baroco secundae;
Tertia Darapti, Disamis Felapton, Datisi,
Bocardo,{2} ferison habet. Quarta insuper addit
Bramantip, Camenes, Dimaris, Fesapno, Fresison.
Applying the rules of the syllogism to these modes, we see that the first figure, in which the middle term is subject of the major and predicate of the minor, excludes: 1. Modes whose minor is negative 2. Modes whose major is particular; 3. AAI, EAO as useless. The second figure, in which the middle term is used twice as predicate, excludes: 1. Modes whose two premises are affirmative 2. Those in which the major is particular; 3. EAO as useless. The third figure, in which the middle term occurs twice as subject, excludes: 1. Modes in which the minor is negative; 2. Modes in which the conclusion is universal. The fourth figure, in which the middle term is predicate of the major and subject of the minor, excludes: 1. Modes having an affirmative major with a particular minor; 2. Modes having an affirmative minor with a universal conclusion; 3. OAO as contrary to the second rule. There remain only the following nineteen valid modes:
All these modes may be converted into the four modes of the first figure, which on that account are called perfect. They are summed up in the four lines already given, which, by a happy disposition of vowels and consonants, designate at once a particular mode, the perfect mode into which it may be reduced, and the divers operations by which the reduction is effected. The three vowels of each word indicate the mode; the initial consonant shows to what mode of the first figure this mode may be reduced, to that, namely, which begins with the same consonant; the consonants, S, P, C, M, denote the operation to be performed in order to effect the reduction. S indicates that the proposition designated by the vowel before it must be converted simply; P, that it must be converted per accidens: C, that the syllogism must be reduced per impossibile; F, by infinitation or obversion; M signifies that the order of the premises must be reversed; P in Bramantip, that from the premises a universal conclusion may be drawn.{3}
Thus the syllogism, "No material being is simple; some simple beings are human souls; therefore some human souls are not material;" is designated by Fresison of the fourth figure; for the mode is seen from the three vowels EIO, and the figure is known by the position of the middle term. This mode may be reduced to that mode of the first figure that begins with F, viz., Ferio. The letter S following E and I in Fresison indicates that the premises represented by these two letters are to be converted simply. Hence the syllogism becomes: "No simple being is material; some human souls are simple; therefore some human souls are not material."
Other modes are similiarly reduced except Baroco and Bocardo, which must be reduced per impossibile.
ART V. -- HYPOTHETICAL SYLLOGISMS AND THEIR RULES.
53. A hypothetical syllogism is that in which the major premise is hypothetical. -- If the major premise is a disjunctive proposition, the syllogism is called disjunctive. If the major premise is a conjunctive proposition, the syllogism is conjunctive; if it is conditional, the syllogism is conditional. A syllogism of whatever kind, besides the rules peculiar to it, is subject to the eight rules of the categorical syllogism.
54. A disjunctive syllogism is subject to the two following rules 1. One of the incompatible predicates being, affirmed in the minor, all the others must be denied in the conclusion; 2. All the predicates but one being denied in the minor, that one must be affirmed disjunctively in the conclusion. -- It is evident that, for the legitimacy of the conclusion of a disjunctive syllogism, the disjunctive premise must make a complete enumeration of all the predicates that can agree with the subject. Hence this syllogism is false: "The rich must either squander their money or hoard it; but they should not hoard it; therefore they should squander it." The disjunction is not complete; it has omitted a third member, which is "to expend money prudently."
55. A conjunctive or copulative syllogism, from the affirmation of one of the members, infers the negation of all the others; but not vice versa. -- It is clear that the conclusiveness of this syllogism requires that the members enumerated in the conjunctive proposition be opposed to one another in such a way that they cannot agree with the same subject at the same time; as, "No one can serve God and Mammon; but many serve Mammon; therefore many do not serve God." From this example it is clear that if the minor were negative, as, "But the spendthrift does not serve Mammon," we could not infer the affirmative: "Therefore he serves God,"{4} unless the opposition be contradictory.
56. A conditional syllogism concludes in two ways: 1. From the affirmation of the antecedent it infers the affirmation of the consequent; 2. From the negation of the consequent it infers the negation of the antecedent; but not vice versa. -- In fact, the antecedent contains the reason of the consequent; therefore the affirmation of the first implies that of the second, as the negation of the second implies that of the first; as, "If Christ arose from the dead, He is God; but He did arise from the dead; therefore He is God." But since an effect may depend on several causes, the reverse of the rules laid down would not give a logical conclusion; as, "If Peter is studious, he merits a reward; but he is not studious; therefore he does not merit a reward." It is clear that a reward may be merited for some other reason than that of being studious. If the antecedent is always the sole reason of the consequent, then we may conclude from the affirmation or negation of the consequent; as, "If he is a man, he is endowed with reason; but he is endowed with reason; therefore he is a man."
ART. VI. -- ABRIDGED AND COMPOUND SYLLOGISMS, OR ENTHTYMEME, PROSYLLOGI5M, EPICHIREMA, SORITES, AND DILEMMA.
57. An enthymeme is an abridged syllogism, one premise of which is understood; as, "God is Just; therefore God will reward the good."
58. A prosyllogism is a syllogism composed of two syllogisms, the conclusion of the first becoming the major of the second; as, "Every act of virtue will be rewarded by God; but humility is a virtue; therefore every act of humility will be rewarded by God; but the bearing of injuries is an act of humility; therefore the bearing of injuries will be rewarded by God."
59. An epichirema is a syllogism in which at least one of the premises is accompanied with proof; as, "God should be adored; but Jesus Christ is God, as His life and miracles attest; therefore Jesus Christ should be adored."{5}
60. A sorites is a form of reasoning composed of several propositions so connected that the predicate of the first becomes the subject of the second, and so on, until the predicate of the last is joined to the subject of the first. -- This form of reasoning may be separated into as many syllogisms as there are propositions less two, It rests on the principle that whatever is said of the predicate may be said of the subject; as, "Sin offends God; whatever offends God separates us from Him; whatever separates us from God deprives us of the sovereign good; whatever deprives us of the sovereign good is the greatest of evils; therefore sin is the greatest of evils." To be conclusive: 1. There should be no negative premise with the affirmative premises; otherwise in the resolution of the sorites there would be a negative premise with an affirmative conclusion, or the conclusion would have a greater extension. The middle term may be negative, and hence one of the premises may be apparently negative; 2. The premise immediately preceding the conclusion can be negative, and then the conclusion will be negative; 3. All the premises except the first must be universal, otherwise one of the middle terms would be taken twice particularly. If the first premise is particular, the conclusion will be particular.
61. A dilemma is a compound syllogism in which each member of a disjunctive major premise is taken in a minor consisting of several conditional propositions, and serves to conclude against the adversary. -- In this form of reasoning care must be taken: 1. That the disjunction of the major be complete; 2. That no member of the minor can be retorted in an opposite sense. Ex. "A general said to a soldier who had allowed the enemy to pass: 'Either you were at your post or you were not; if you were, you deserve death for neglecting to give notice of the enemy; if you were not, you deserve death for breach of discipline.' " A dilemma may also have for major a proposition with a disjunctive consequent, the minor denying each member of the consequent, the conclusion, therefore, denying the antecedent.
62. To these arguments may be added the Example, a species of reasoning in which one proposition is drawn from another to which it has a relation of resemblance, of opposition, or of superiority. -- This argument may be reduced to a syllogism whose major is confirmed by a particular fact bearing on the conclusion which we wish to infer. Ex. 1. "Our Lord pardoned St. Peter on account of his repentance; therefore He will pardon you, if, having imitated St. Peter in his fault, you likewise imitate him in his repentance." -- 2. "Louis XIV. and Napoleon I. caused great evils on account of their love of war; it is therefore desirable that a people have a sovereign who loves peace." -- 3. "Behold the fowls of the air, for they sow not, neither do they reap, nor gather into barns; yet your heavenly Father feedeth them. Are not you of much more value than they ?" (St. Matt. vi. 26.) In the first example we conclude a pari; in the second, a contrario or ab opposito; in the third, a fortiori.
When the example is drawn from the words and actions of an adversary and is used against him, it is called argumentum ad hominem.
ART. VII. -- INDUCTION.
63. Induction is that process in which the mind, after affirming or denying an attribute of each part of a whole, pronounces the same judgment of the whole. -- As has been said already, the reasoning process is twofold: it proceeds either from the whole to the parts which compose it, or from the parts to the whole which they constitute. In the first case we have deduction, in the second induction. "The Gospel has penetrated into Europe, Asia, Africa, America, and Oceanica; but these five Grand Divisions make up the whole known world; therefore the Gospel has penetrated into all the known world," is an example of induction; whence it becomes manifest that the middle term in an inductive syllogism is simply the enumeration of the parts. These parts united are in reality identical with the whole, though logically distinct from it; they can, consequently perform the function of a middle term.
64. The legitimacy of an inductive syllogism rests on the principle that, the sum of the parts being identical with the whole, whatever is affirmed or denied of all the parts may be confirmed or denied of the whole. -- Hence that the inductive syllogism may be rigorously conclusive, it is essential that the enumeration of the parts composing the whole be complete. But this enumeration may be actually or virtually complete: actually, when what has been predicated of the whole has been verified in each of its parts; virtually, when the predicate has been verified only in a certain number of the parts, and we regard it as applicable to the others on the principle that natural causes always produce the same effects, since they operate necessarily, or on the principle that the laws of nature are constant, universal, and uniform. In virtue of this principle, the mind regards that which is constant in a certain number of beings as essential to their nature. Hence, knowing that whatever proceeds from the nature of a being is always verified in that being and in all others having the same physical nature, we conclude that a quality Which has been verified in some beings must be found, under the same circumstances, in all beings the same physical nature. having When induction is really incomplete and inadequate, it does not authorize a universal and absolute conclusion.{6} It gives only a greater or less degree of probability, in direct ratio to the number of parts in which the predicate has been verified.
ART. VIII. -- PROBABLE OR DIALECTIC SYLLOGISM.
65. A probable syllogism is that in which at least one of the premises is only probable, and which, therefore, gives only a probable conclusion. -- Apart from the sciences and in the affairs of life, we cannot ordinarily arrive at complete certitude; we must be satisfied with probability. The argument which is thus concerned with contingent matter and with things known only in part is called a probable argument, and its expression, a probable syllogism.
66. Whilst we argue in probable matter, we must endeavor to attain the highest possible degree of probability. -- Hence: 1. We must be assured of the possibility of the thing; 2. We must, as far as we can, establish the certainty of all the circumstances; 3. We must ascertain that there are more and better reasons on one side than on the other. Used in this way, the probable syllogism often prepares the way to complete certitude and to science properly so called; at all events. it gives solidity to the mind, prevents it from advancing anything rashly, and from judging before the fact be well ascertained.
ART. IX. -- SOPHISTICAL SYLLOGISM.
67. A sophism is a syllogism which leads into error, and yet has the appearance of truth. -- The better to enable us to arrive at truth by means of reasoning, logic not only lays down the rules to which a syllogism must conform to be conclusive, but, moreover, exposes the artifices by which our minds are liable to be led into error, and thus enables us the better to defend ourselves against them. These artifices are called sophisms when they suppose in him who makes use of them the desire to deceive; they are called paralogisms when they are employed through inadvertence or through ignorance of the rules of reasoning; in either case they may be called fallacies. Taken together, they constitute the art of sophistry, which was particularly taught and practised by Greek orators, in order that, by enabling them to support at pleasure all causes and parties, it might be to them a means of acquiring wealth and influence.
68. Fallacies are divided into those in diction and those extra-diction, according as they lead into error by an abuse of words or by other captious arguments. -- Fallacies in diction are six in number: (a) Equivocation, (b) Amphibology, (c) Fallacy of composition, (d) Fallacy of division, (e) Fallacy of accent, (f) Fallacy of figure of diction. -- Fallacies extra-diction are seven in number: (a) Fallacy of accident, (b) Passing from the absolute to the relative and vice versa, (c) Pretended cause, (d) Evading the Question, or Irrelevant conclusion, (e) Fallacy of consequent, (f) Begging the question and vicious circle, (g) Fallacy of many questions.
69. The principal fallacies in diction are six:
(a) Equivocation, which consists in using the same term with different meanings; as, "The dog barks; but the dog is a constellation; therefore a constellation barks."
(b) Amphibology, which consists in making use of a phrase in a twofold sense; as, "I say, Pyrrhus, you the Romans will conquer." -- "And thus the son his aged sire addressed."
(c) The fallacy of composition, which arises when things which are separately true are taken as collectively true; as, "The Gospel says the blind see; but that the blind should see is a contradiction; therefore the Gospel contains contradictions."
(d) The fallacy of division, which is the reverse of the preceding; as, "According to the Scriptures, the impious shall not enter the kingdom of heaven; therefore it is useless for the impious to repent." -- "Five is one number; two and three are five; therefore two and three are one number."
(e) The fallacy of accent, which changes the meaning of a word by changing the accent; as, "He conjured me not to betray my country; therefore he practised the black art."{7}
(f) The fallacy of figure of diction, which consists in assuming to be true literally that which is true only figuratively; as, "That is a babbling brook; but only children babble; therefore that brook is a child."
The principal fallacies extra-diction are seven:
(a) The fallacy of accident, which attributes or denies to an individual a real accident that is rightly affirmed or denied only of the species; as, "Spaniards are haughty; therefore, St. Theresa was haughty." With this fallacy may be classed that which is called imperfect enumeration.
(b) The fallacy of passing from the absolute to the relative, and vice versa, which occurs when we argue from what is true absolutely to what is true only in some respect, and vice versa; as, "We must obey our parents; but my parents command me not to adore God; therefore," etc.: or, "John is a good penman; therefore he is good."
(c) The fallacy of pretended cause, which occurs when we assign as the cause of an effect what is not really such; as, "Inebriety is bad; but wine inebriates; therefore wine is bad."
(d) Evading the question, or irrelevant conclusion, which occurs when we prove something which is not in question; as would be the case if "a minister of state, being pressed to modify certain laws, should demonstrate the necessity of law."
(e) The fallacy of consequent, which occurs when in a conditional syllogism the consequent is not inferred from the antecedent, but the antecedent from the consequent; as, "If that is a man, it is an animal; but it is not a man; therefore it is not an animal."
(f) Begging the question, which occurs when we assume, in fact or in principle, the thing in question, or that which requires to be proved; as would be the case if we should undertake to prove that the earth revolves about the sun thus: "The sun is at rest; therefore the earth revolves about it." When this fallacy proves two disputed propositions, each by the other, it is called a vicious circle; as if "after relying on the veracity of a witness to prove a fact, I should rely on the truth of the fact to prove the veracity of the witness."
(g) The fallacy of many questions, or of interrogation, occurs when several questions requiring different answers are asked, and the answer given to one is assumed as applicable to the others; as, "Are virtue and vice good or evil?" Whether we answer yes or no, we fall into error.
We may also classify among fallacies all reasonings in which any one of the rules of the syllogism is violated.
ART. X. -- UTILITY OF THE SYLLOGISM.
70. The use of the syllogism gives clearness, strength, and flexibility to the mind. -- By the use of the syllogism the mind discerns more readily the value of a reasoning and detects more easily the vices of a fallacy. As gymnastics strengthens the body and makes it supple, so the use of the syllogistic art gives solidity, flexibility, and precision to the mind. For if the errors that are rife to-day be stripped of their wordy covering and reduced to this severe form of reasoning, they will appear as the rankest sophisms. It is evident, however, that, though the use of the syllogism pre-. sents these great advantages, its abuse may easily generate stiffness and subtlety, and impede the prog ress of intelligence instead of aiding it.
{1} Or Fakofo.
{2} Or Dokamok.
{3} These rules are contained in the couplet:
S vult simpliciter verti P vero per accid:
M vult transponi; C per impossibile duci.
{4} The minor of a conjunctive syllogism always affirms one of the two incompatibles expressed in the major.
{5} In the days of Aristotle an enthymeme was a "syllogism drawn from probabilities and signs of the conclusion;" and an epichirema, a dialectical syllogism in which the conclusion is reached after a careful examination of objections and difficulties. See Logic, Stonyhurst Series, pp. 356, 359.
{6} "In spite of this, these methods of incomplete induction cannot be passed over in the present day. They are too important a factor in the present condition of human society to admit of our neglecting them. . . . Besides, we must understand and appreciate them in order to protest against their abuse. . . . Mill and his followers drag down all the a priori laws to the level of the a posteriori, or rather deny the existence of a priori laws at all. This is the fatal result of the neglect of scholastic methods, which began at the Reformation, and has been carried further day by day." Logic Stonyhurst Series, p. 387.
{7} The fallacy of accent also includes the mistaking of one word for another having the same pronunciation but a different spelling; as if I should say that "there were small islands in the church, because it has many aisles."