80 PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, 



Section I. 



ON THE APPROXIMATION OP LOGICAL AND ALGEBRAICAL MODES OF THOUGHT. 



Throughout this paper I use the word logic in the purely technical sense. The progress 

 of algebra as distinguished from arithmetic, is marked by the gradual approach to the following 

 theorem, that every pair of opposite relations is undistinguishable from every other pair, in the 

 instruments of operation which are required. So early did this principle gain some practical 

 acceptance, that no attempt was ever persisted in (even if made, which is more than I know) to 

 signify different oppositions by different pairs of symbols : + and - were found instrumentally 

 adequate to all the wants of the mechanism of the science. I do not say that this was a benefit : 

 I only state it as a fact. An algebraist, who may be required, should the proper problem 

 occur, to interpret - (- ( + (-«))) as the removal from an expression of all traces of a loss in- 

 curred in an ascent made at a time prior to a certain epoch — may have gained the power of 

 making such interpretation very slowly, in consequence of never having sufficiently distinguished 

 differences, as a preliminary to, or a concurrent with, abstraction by observation of resem- 

 blances. There may be many for whom it would have been better that the above symbol had 

 been - (*(+ (J«)))> or the like, until identity of rules had suggested identity of symbols : and I 

 am sure that I was of the number. 



The forms of thought which have not immediate relation to magnitude have been otherwise 

 treated : consideration of differences has predominated, that of resemblances has been almost 

 ignored. In the single case in which algebra was forestalled, the old maxim that two 

 negatives make an affirmative, so loose was the treatment that the penalty of algebra was 

 incurred. The two negatives, which are only instrumentally the equivalent of an affirmative, 

 are two signs as different in origin and character as the two negative signs in — (—a). In 

 " man is not (not-animal)," the first negative disconnects, the second describes the predicate 

 disconnected. 



In many cases, the difference of symbols, so much wanted by the beginner in algebra, 

 far from encouraging abstraction of resemblances, stimulated differences of interpretation, as in 

 • not unwilling,' which means less than willing : the double negative has, in common language, 

 deteriorated into an affirmative of a lower degree. 



The suggestions of symbolic notation have led me to more recognition than is usually made 

 of harmonies which exist among various pairs of opponent notions common in logical thought. 

 I select the following ; — affirmative and negative — universal and particular — the subjective 



distinction of possible and impossible — the objective distinction of existent and non-existent 



necessary and not necessary — sufficient and insufficient — conjunctive and disjunctive — con- 

 vertible and inconvertible — conclusive and inconclusive — singular and plural — definite and 

 indefinite : omitting true and false, the most general of all, as most obviously capable of 

 forming one element of the distinctive definition of any pair. I believe that any of these 

 oppositions might be interchanged and used for each other : but not always without what would 

 be called forcing. This, however, is not a conclusive objection : a forced analogy may only 

 deserve that name because we have not been accustomed to the comparisons which it suggests, or 



