Our inductions will often be embarrassed by exceptions, real or apparent. We might affirm that all gems are incombustible were not diamonds undoubtedly combustible. Nothing seems more evident than that all the metals are opaque until we examine them in fine films, when gold and silver are found to be transparent. All plants absorb carbonic acid except certain fungi; all the bodies of the planetary system have a progressive motion from west to east, except the satellites of Uranus and Neptune. Even some of the profoundest laws of matter are not quite universal; all solids expand by heat except india-rubber, and possibly a few other substances; all liquids which have been tested expand by heat except water below 4° C. and fused bismuth; all gases have a coefficient of expansion increasing with the temperature, except hydrogen. In a later chapter I shall consider how such anomalous cases may be regarded and classified; here we have only to express them in a consistent manner by our notation. Let us take the case of the transparency of metals, and assign the terms thus:-- A = metal D = iron B = gold E, F, &c. = copper, lead, &c. C = silver X = opaque. Our premises will be A = B ꖌ C ꖌ D ꖌ E, &c. B = B*x* C = C*x* D = DX E = EX, and so on for the rest of the metals. Now evidently A*bc* = (D ꖌ E ꖌ F ꖌ ...)*bc*, and by substitution as before we shall obtain A*bc* = A*bc*X, or in words, “All metals not gold nor silver are opaque;” at the same time we have A(B ꖌ C) = AB ꖌ AC = AB*x* ꖌ AC*x* = A(B ꖌ C)*x*, or “Metals which are either gold or silver are not opaque.”