126 



REASONING. 



general theorem which stands at the 

 head of the demonstration, is not the 

 proposition actually demonstrated. 

 One instance only is demonstrated : 

 but the process by which this is done 

 is a process which, when we consider 

 its nature, we perceive might be 

 exactly copied in an indefinite number 

 of other instances ; in every instance 

 which conforms to certain conditions. 

 The contrivance of general language 

 furnishing us with terms which con- 

 note these conditions, we are able to 

 assert this indefinite multitude of 

 truths in a single expression, and this 

 expression is the general theorem. 

 By dropping the use of diagrams, and 

 substituting, in the demonstrations, 

 general phrases for the letters of the 

 alphabet, we might prove the general 

 theorem directly, that is, we might 

 demonstrate all the cases at once ; 

 and to do this we must, of course, 

 employ as our premises the axioms 

 and definitions in their general form. 

 But this oidy means, that if we can 

 prove an individual conclusion by 

 assuming an individual fact, then in 

 whatever case we are warranted in 

 making an exactly similar assumption, 

 we may draw an exactly similar con- 

 clusion. The definition is a sort of 

 notice to ourselves and others what 

 assumptions we think ourselves en- 

 titled to make. And so in all cases, 

 the general propositions, whether 

 called definitions, axioms, or laws of 

 nature, which we lay down at the 

 beginning of our reasonings, are 

 merely abridged statements, in a 

 kind of shorthand, of the particular 

 facts, which, as occasion arises, we 

 either think we may pnxjeed on as 

 proved, or intend to assume. In any 

 one demonstration it is enough if we 

 assume for a particular case, suitably 

 selected, what by the statement of the 

 definition or principle we announce 

 that we intend to assume in all cases 

 which may arise. The definition of 

 the circle, therefore, is to one of 

 Euclid's demonstrations exactly what, 

 according to Stewart, the axioms are ; 

 that is, the demonstration does not 



depend on it, but yet if we deny it 

 the demonstration fails. The proof 

 does not rest on the general assump- 

 tion, but on a similar assumption con- 

 fined to the particular case : that case, 

 however, being chosen as a specimen 

 or paradigm of the whole class of cases 

 included in the theorem, there can be 

 nogroundformaking the assumption in 

 that case which does not exist in every 

 other ; and to deny the assumption 

 as a general truth is to deny the right 

 of making it in the particular instance. 

 There are, undoubtedly, the most 

 ample reasons for stating both the 

 principles and the theorems in their 

 general form, and these will be ex- 

 plained presently, so far as explana- 

 tion is requisite. But, that unprac- 

 tised learners, even in making use of 

 one theorem to demonstrate another, 

 reason rather from particular to parti- 

 cular than from the general proposi- 

 tion, is manifest from the difficulty 

 they find in applying a theorem to a 

 case in which the configuration of the 

 diagram is extremely unlike that of 

 the diagram by which the original 

 theorem was demonstrated. A diflB- 

 culty which, except in cases of unusual 

 mental power, long practice can alone 

 remove, and removeschiefly byrender- 

 ing us familiar with all the configura- 

 tions consistent with the general con- 

 ditions of the theorem. 



§ 4. From the considerations now 

 adduced the following conclusions 

 seem to be established. All inference 

 is from particulars to particulars : 

 General propositions are merely re- 

 gistersof such inferences already made, 

 and short formulae for making more : 

 The major premise of a syllogism, con- 

 sequently, is a formula of this descrip- 

 tion ; and the conclusion is not an 

 inference drawn /?-om the formula, but 

 an inference drawn according to the 

 formula ; the real logical antecedent 

 or premise being the particular facts 

 from which thegeneral proposition was 

 collected by induction. Those facts, 

 : and the individual instances which 

 I supplied them, may have been for- 



