X = of wood,
Y = table,
there is no reason why, in our symbols, XY should not be just as correct an expression for “table of wood ” as YX. In this case indeed we might substitute for “of wood ” the corresponding adjective “wooden,” but we should often fail to find any adjective answering exactly to a phrase. There is no single word by which we could express the notion “of specific gravity 10·5:” but logically we may consider these words as forming an adjective; and denoting this by S and metal by P, we may say that SP means “metal of specific gravity 10·5.” It is one of many advantages in these blank letter-symbols that they enable us completely to neglect all grammatical peculiarities and to fix our attention solely on the purely logical relations involved. Investigation will probably show that the rules of grammar are mainly founded upon traditional usage and have little logical signification. This indeed is sufficiently proved by the wide grammatical differences which exist between languages, though the logical foundation must be the same.
Symbolic Expression of the Law of Contradiction.
The synthesis of terms is subject to the all-important Law of Thought, described in a previous section (p. [5]) and called the Law of Contradiction, It is self-evident that no quality can be both present and absent at the same time and place. This fundamental condition of all thought and of all existence is expressed symbolically by a rule that a term and its negative shall never be allowed to come into combination. Such combined terms as Aa, Bb, Cc, &c., are self-contradictory and devoid of all intelligible meaning. If they could represent anything, it would be what cannot exist, and cannot even be imagined in the mind. They can therefore only enter into our consideration to suffer immediate exclusion. The criterion of false reasoning, as we shall find, is that it involves self-contradiction, the affirming and denying of the same statement. We might represent the object of all reasoning as the separation of the consistent and possible from the inconsistent and impossible; and we cannot make any statement except a truism without implying that certain combinations of terms are contradictory and excluded from thought. To assert that “all A’s are B’s” is equivalent to the assertion that “A’s which are not B’s cannot exist.”
It will be convenient to have the means of indicating the exclusion of the self-contradictory, and we may use the familiar sign for nothing, the cipher 0. Thus the second law of thought may be symbolised in the forms
Aa = 0 ABb = 0 ABCa = 0
We may variously describe the meaning of 0 in logic as the non-existent, the impossible, the self-inconsistent, the inconceivable. Close analogy exists between this meaning and its mathematical signification.
Certain Special Conditions of Logical Symbols.
In order that we may argue and infer truly we must treat our logical symbols according to the fundamental laws of Identity and Difference. But in thus using our symbols we shall frequently meet with combinations of which the meaning will not at first sight be apparent. If in one case we learn that an object is “yellow and round,” and in another case that it is “round and yellow,” there arises the question whether these two descriptions are identical in meaning or not. Again, if we proved that an object was “round round,” the meaning of such an expression would be open to doubt. Accordingly we must take notice, before proceeding further, of certain special laws which govern the combination of logical terms.
In the first place the combination of a logical term with itself is without effect, just as the repetition of a statement does not alter the meaning of the statement; “a round round object” is simply “a round object.” What is yellow yellow is merely yellow; metallic metals cannot differ from metals, nor circular circles from circles. In our symbolic language we may similarly hold that AA is identical with A, or