second figure; all the others are in the mood Celarent of the first figure.

It should be noted that in the Aristotelian Sorites the conclusion of a Prosyllogism becomes the minor premiss, while in the Goclenian it becomes the major premiss, in the next Episyllogism, throughout the whole train of reasoning. We shall conclude with an Aristotelian Sorites in the 3rd figure :VI. All A is B,

All B is C,

All C is D,

All A is E,

.. Some D is E.

Analysis of VI.

(1) All A is B,

All B is C,
.. All A is C,

(2) All A is C,
All C is D,

... All A is D,

(3) All A is D,

All A is E,

.. Some D is E.

Here the 3rd Syllogism is in Darapti in the 3rd figure, and the others in Barbara.

§ 5. Questions and exercises.

1. Analyse and test the following trains of reasoning:

(1)

66

Bucephalus is a horse; a horse is a quadruped; a quadruped is an animal; an animal is a substance: therefore Bucephalus is a substance."

(2) "If Harpagon be avaricious, he is intent on gain; if intent on gain, he is discontented; if discontented, he is unhappy; now Harpagon is avaricious: he is, therefore, unhappy."

(3) "Whatever promotes happiness is good; whatever perfects the soul promotes happiness: therefore whatever perfects the soul is good; misfortune which happens to the good, serves either to disci

pline or to improve the soul: hence misfortune which befalls the good is good."

(4) "Sentient beings seek happiness; all finite beings are sentient; all men are finite beings; Caius is a man: therefore he seeks happiness."

(5) "That which thinks is active; that which is active has strength; that which has strength is a substance; the soul thinks: therefore it is a substance."

(6) A is equal to B; B is equal to C; C is equal to D; D is equal to E: therefore A is equal to E.

(7) A is greater than B; B is greater than C; C is greater than D; D is greater than E: therefore A is greater than E.

(8) A is the cause of B; B is the cause of C; C is the cause of D; D is the cause of E: therefore A is the cause of E.

(9) A lies above B; B lies above C; C lies above D: therefore A lies above D.

(10) A co-exists with B; B co-exists with C; C co-exists with D: therefore A co-exists with D.

(11) A is a mark of B; B is a mark of C; C is a mark of D: therefore A is a mark of D.

(12) If a gas is heated, its temperature rises; if its temperature rises, its elastic force increases; if its elastic force increases, the pressure on the walls of the containing vessel increases: therefore if a gas is heated, the pressure on the walls of the containing vessel increases.

2. Analyse the demonstration of the 20th Proposition in Todhunter's Euclid, p. 23, into the constituent syllogisms.

3. Prove both synthetically and analytically the 18th Proposition of Euclid, Book I, Todhunter, p. 22.

4. Analyse into fully-expressed syllogisms both the construction and the demonstration of the 32nd Proposition of Euclid, Book I.

5. Distinguish between the Analytical Method in Deductive Logic and Analysis as employed in Geometry.

CHAPTER VII.

OF FALLACIES.

§ 1. I.-A General Outline.

A Fallacy, in the proper sense of the term, is a transgression of a rule of inference. A fallacious reasoning is, in fact, an apparent reasoning involving the breach of some rule or other of the various kinds of inference. Thus there are as many different kinds of Fallacies as of Reasoning or Inference.

The breach of the laws of Inductive Reasoning gives rise to the Fallacies of Induction with which we have nothing to do here. The breach of the rules of Deductive Inference gives rise, first, to the Fallacies of Immediate Inference, when the rule transgressed is a rule of Immediate Inference, and, secondly, to the Fallacies of Mediate Inference, when the rule violated is one of Syllogism or of any other kind of Mediate Deductive Reasoning. Thus we have the following classes of Fallacies in the sense we have defined above :—

Permutation.

Contraposition.

Fallacies of Mediate
Inference.
1

I. Syllogistic:

Undistributed Middle.

Illicit Process.

Four Terms.

&c. &c.

Opposition.

Subalternation.

Modal Consequence. II. Non-Syllogistic, e. g.,

Change of Relation.

Mathematical.

In a wider sense a Fallacy is a transgression of any logical rule whatever. In this sense we have in Deductive Logic the Fallacies or Faults of Division and Definition; and in Inductive Logic those of Classification, Hypothesis, &c. The violation of the rules to which every logical division and definition ought to conform gives rise to the faults of division and definition, such as cross division, incomplete division, definition by accidental qualities, &c. To this class belong also the fallacies arising from ambiguity in language, such as those of Ambiguous Middle, of Division, Composition, &c. These are transgressions of the logical rule that our thoughts should be expressed and reasonings conducted in clear and unambiguous language.

In the widest sense, the word fallacy may be taken to mean an error of any kind, whether of Intuition, Perception, Observation, Division, Definition, Inference, &c. In this sense it includes, besides those mentioned above, the fallacies of Irrelevancy or Irrelevant Conclusion, technically called Ignoratio Elenchi, of Petitio Principii (begging the question), of False Premiss, and also those which Mill calls Fallacies of Simple Inspection, or of Erroneous First Principles and Axioms.

It is not necessary that we should describe and explain in detail each of the fallacies mentioned above, for most of them have been already made evident in explaining and illustrating the rules. In the following pages we shall notice and illustrate the more frequent and important kinds only.