Page:An Investigation of the Laws of Thought (1854, Boole, investigationofl00boolrich).djvu/47

tioned by the intimate laws of thought, but for reasons of convenience not exercised in the ordinary use of language.

Thirdly, The law expressed by (1) may be characterized by saying that the literal symbols $\scriptstyle {x}$, $\scriptstyle {y}$, $\scriptstyle {z}$, are commutative, like the symbols of Algebra. In saying this, it is not affirmed that the process of multiplication in Algebra, of which the fundamental law is expressed by the equation $\scriptstyle {xy=yx}$, possesses in itself any analogy with that process of logical combination which $\scriptstyle {xy}$ has been made to represent above; but only that if the arithmetical and the logical process are expressed in the same manner, their symbolical expressions will be subject to the same formal law. The evidence of that subjection is in the two cases quite distinct.

9. As the combination of two literal symbols in the form $\scriptstyle {xy}$ expresses the whole of that class of objects to which the names or qualities represented by $\scriptstyle {x}$ and $\scriptstyle {y}$ are together applicable, it follows that if the two symbols have exactly the same signification, their combination expresses no more than either of the symbols taken alone would do. In such case we should therefore have

$\scriptstyle {xy=x}$.

As $\scriptstyle {y}$ is, however, supposed to have the same meaning as $\scriptstyle {x}$, we may replace it in the above equation by $\scriptstyle {x}$, and we thus get

$\scriptstyle {xx=x}$.

Now in common Algebra the combination $\scriptstyle {xx}$ is more briefly represented by $\scriptstyle {x^{2}}$. Let us adopt the same principle of notation here; for the mode of expressing a particular succession of mental operations is a thing in itself quite as arbitrary as the mode of expressing a single idea or operation (II. 3). In accordance with this notation, then, the above equation assumes the form (2)$\scriptstyle {x^{2}=x}$, and is, in fact, the expression of a second general law of those symbols by which names, qualities, or descriptions, are symbolically represented.