345 



LOGIC. 



LOGIC. 



34! 



are declared to be of so different a nature from all other processes and 

 relations as to make the difference of form and matter. To us the pure 

 form of the proposition, divested of all matter, is the assertion or 

 denial of the following : x stands in relation L to y. The proposition 

 " X is X " is brought under the general form by two introductions of 

 matter : L is identity, and X and Y are the same symbols. 



What is the test of distinction between a consequence which is true 

 by its form, and one which is only true by its matter ? When the 

 assertion is true by its form, it can only be refused admission by 

 impossibility in its matter ; and when it can only be refused admission 

 by impossibility in its matter, it is true by its form : and by impossi- 

 bility we mean incompatibility with the conditions of the universe 

 understood. If the universe be the whole sphere of possible thought, 

 the impossible is that which contradicts itself, and nothing else ; a 

 five-footed quadruped, for example. But if it be the terrestrial sphere 

 of actual existence, the impossible is only the non-existent on earth ; a 

 rational quadruped, for example. If the universe be limited to the 

 idea of animal, a stone is an impossibility. For instance, " A is A," and 

 " A is not not-A," are true by form, in our view ; how can they be 

 denied 1 or rather, when can we doubt of admitting them ? for assertion 

 must be opposed to non-assertion, not to denial. Only when A contra- 

 dicts itself. A horse which is not a horse, and a cow which is not a 

 cow, are they the same things, or different ? and must they be either 

 one or the other ? We leave the logicians to settle it ; for ourselves we 

 cannot tell : we do not see how to affirm, and we do not see how to 

 deny. But we do see that " A is A " can only be questioned, whatever 

 the result may be, when the matter of A is impossible; and we 

 recognise it as therefore true in form. 



Again, let x be an existing animal ; it follows that the tail of x is 

 tlie tail of an animal. Is this consequence formal or material ? Formal, 

 because this it true whatever a tail may be, so long as there is a tail ; 

 and it cannot be refused assertion except when x has no tail. A guinea- 

 pig, for instance, puts this proposition out of the pale of assertion, and 

 equally out of that of denial ; the tail of a non-tailed animal is beyond 

 us. There are those who will assert that the tail of a non-tailed 

 animal is no tail ; but we are satisfied that the tail of a non-tailed ani- 

 mal is just as much a tail as no-tall ; and therein lies the contradiction 

 and the impossibility. If we were clear that the tail of a nun-toiled 

 animal ia no tail, we should know how to think of it ; for we can think 

 of every subject which comes under a predicate. This very guinea- 

 pig l-i here produced by us because it has been brought forward by an 

 acute thinker to prove that the consequence above given is material, 

 and nut formal. It has been made to prove that ' x is an animal, there- 

 fore the tail of x is the tail of an animal ' is a special inference gamed 

 from our material knowledge of the thing thought about, and not a 

 general inference necessitated by the universal laws of thinking. We 

 answer that though the lack of tail in a guinea-pig is not a law of logic, 

 but a material incident of the object, yet we know the comcquence to 

 be necessitated by the laws of thinking, because we must go to im- 

 possible matter, we must make the tail of x a non-existence, before we 

 can refuse to assert it. 



The establishment of ordinary syllogism requires two principles. 

 First, the convertibility of the terms of a proposition : " A is B " gives 

 " B is A." Secondly, the transit/tenets of the connecting relation : " A 

 is B " and " B is c " gives " A is c." The logicians affirm that the three 

 principles of identity, difference, and excluded middle, are sufficient for 

 the evolution of all the laws of syllogism. We are not aware that any 

 one has ever made this good by actually deducing the two principles 

 last named from the higher three. This must be done, if at all, with- 

 out syllogism, for syllogism must not be taken for granted as a step 

 towards its own establishment ; anil if, as we believe, it cannot be done, 

 then the three principles do not possess that dominant and all-sufficient 

 character which has been claimed for them. 



The syllogism is the determination of the relation which exists 

 between two objects of thought by means of the relation in which 

 each of them stands to some third object, which is the middle term. 

 In our view of the subject the pure form of the syllogism, when its 

 premises are alwolutely asserted, is as follows : x is in the relation L 

 to Y ; Y is in the relation M to z ; therefore x is in the relation " L of 

 M," compounded of L and M, to z. In ordinary logic, which admits 

 only the relation of identity, the actual composition of the relation is 

 made by our consciousness of its Intuitive character. Thus x is Y and 

 v is '/., tell us that x is that which is z. But an identical of an 

 identical is an identical ; whence x is z. Again, x is Y and Y is not z ; 

 here x is that which is uot-z : or x is not z. The full syllogism is a 

 collection of such singular cases : thus every x is Y, and no Y is z, 

 repeat* " x ia a Y, that Y is a not-z," as often as there are xs in 

 'ice, and gives " every x is not-'/.," or no x is z. 



The requisites of the copular relation, in the system of ordinary 

 syllogism, are traniititenfet and ctjutcrliliility. Any relation which 

 possesses these qualities may take the place of " is " in the common 

 yllogisin, without impeachment of its validity. Thus the word " fel- 

 low " may have transitive senses, and is always convertible : choose a 

 transitive signification, under which we may say that when x is a 

 fellow of Y, and Y of z, then x is a fellow of z. Take a common syllo- 

 gism, such as 110 Y ia any Z, some Yd are xs; whence some xs are not 

 any ZD. For " is " read " is a fellow of," and for " i not " read " is not 

 v of ; " the inference remains valid. Against admitting this 



extension it is argued that the inference is one of matter, not of form ; 

 but though every instance is material, or has its own matter, we main- 

 tain that the above inference depends for its validity only on those 

 qualities which must be seen in identity before the common syllogism 

 can be established. 



The logicians are well aware that there are many copula: which give 

 inference by comparison with a middle term, and which are not what 

 they call syllogisms ; for instance, " A equals B, B equals c, therefore 

 A equals c." They reduce such arguments to syllogism by stating 

 the requisite combination of relation in a major premise, and affirming 

 the case to come under the combination in a minor premise. As in 

 " An equal of an equal of c is an equal of o ; A is an equal of an equal 

 of c ; whence A is an equal of c." But A and are really compared in 

 the mind with a middle term B, and thence with one another. When 

 A=B, B=C, are made to give A = c, the process of the human mind is 

 guided by steps in which = is as truly a copula, or connecting relation, 

 as "is" in " A is B, B is c, whence A is c." The transitive character of 

 the copula is as much the dictator of the result in the first case as in 

 the second. 



Dismissing, for further consideration in RELATION", the extension to 

 wider relations than those contained under " is," we shall proceed to 

 point out the questions which have arisen as to the common forms of 

 enunciation, and the systems of enunciation and- of syllogism which 

 have been proposed. After a few words on the common system, we 

 shall notice the different proposals of Mr. Boole, Sir Wm. Hamilton, 

 and Mr. De Morgan. 



The common proposition is derived from the notion of assertion or 

 denial, applied to all or to some, giving the universal affirmative and 

 negative, every x is Y, no x is Y ; and the particular affirmative am! 

 negative, some xs are YS, some xs are not YS. [SYLLOGISM.] Here 

 some, as in all logical writings, means no more than not-none, one at 

 least, many it may be, even all, possibly. When we say that some x's 

 are not YS, we only mean to deny that the class x is entirely included 

 in the class Y ; the exclusion may be partial or may be total. The 

 logical opposition of quantity, then, is not that of whole and not-whole 

 or part ; but of quantity asserted to be the whole, and quantity not 

 asserted to be the whole. 



The old writers on logic hardly speak of universal and particular 

 terms ; they apply these adjectives to the propositions. They say that 

 a prafoatioH is made universal by prefixing the word omua to the 

 xi'iiji.rt. Later writers quantify both subject and predicate. They find 

 out that " all xs are YS " means tliat " all xs are some YS ; " that " no 

 x is Y " means that " every x is no one of all the YS ; " that " some xs 

 are Ys " is " some xs are some YS ; " and that " some, xs are not YS " 

 is " some xs are not any of all the YS." And hence the well-known 

 rule that while the quantity of the subject is indicated, the quantity 

 of the predicate is determined by the quality of the proposition ; 

 particular in affirmatives, universal in negatives. But conversion of 

 term and quantity is not allowed : " all xs are some YS " is never seen 

 as "some. YS are all xs." The usual forms of language are made to 

 dictate restrictions to thought. We shall presently return to this 

 point. 



Mr. Boole's generalisation of the forms of logic is by far the boldest 

 and most original of those of which we have to treat. It cannot be 

 separated from Mathematics, since it not only demands algebra, but 

 such taste for thought about the notation of algebra as is rarely 

 acquired without much aud deep practice. When the ideas thrown 

 out by Mr. Boole shall have borne their full fruit, algebra, though 

 only founded on ideas of number in the first instance, will appear like 

 a sectional model of the whole form of thought. Its forms, considered 

 apart from their matter, will be seen to contain all the forms of 

 thought in general. The anti-mathematical logician says that it makes 

 thought a branch of algebra, instead of algebra a branch of thought. 

 It makes nothing ; it Jiads : and it finds the laws of thought symbolised 

 in the forms of algebra. [ALGEBRA, p. 199, 9.] We caunot attempt, 

 in this article, to do more than show, by one single case, how common 

 algebraical operation is transformation from one logical equivalent to 

 another. We shall choose the conversion of "to be both not-A and 

 not-B is impossible " into " every thing is either A or B or both." Let 

 A and B represent two objects of thought. Let 1 represent the uni- 

 verse, all that exists ; let represent the impossible, something that 

 does not exist. Let = represent identity. Let A + B represent the 

 class containing both A and B, with all the common part, if any, 

 counted twice ; let A B signify what is left of the class A, when B, 

 which it contains, is withdrawn. Let A B represent the common part 

 of the notions A and B. Then 1 A and 1 B represent all that is not 

 A and all that is not B ; and the non-existence of everything which is 

 both not A and not B is symbolised by 



(1 - A) (1 - B) = 



The common rules of algebra transform this into 

 A + B AB = 1. 



Now, A + B-AB is simply the aggregate of A and B, without repe- 

 tition, the part reckoned twice in A + B, if any, being withdrawn once. 

 Accordingly the second equation means that A and B between them 

 contain the whole universe ; or that everything is either A or B, or 



