then the definition is of the form A = BC. If we take the series of eight combinations of three letters in the Logical Alphabet (p. 94) and strike out those which are inconsistent with the definition, we have the following result:-- ABC *a*B*c* *ab*C *abc.* For the description of the class C we have C = ABC ꖌ *ab*C, that is, “a rectilinear figure is either a triangle and three-sided, or not a triangle and not three-sided.” For the class *b* we have *b* = *ab*C ꖌ *abc*. To the second side of this we may apply the process of simplification by abstraction described in the last section; for by the Law of Duality *ab* = *ab*C ꖌ *abc*; and as we have two propositions identical in the second side of each we may substitute, getting *b* = *ab*, or what is not three-sided is not a triangle (whether it be rectilinear or not). *Second Example.* Let us treat by this method the following argument:-- “Blende is not an elementary substance; elementary substances are those which are undecomposable; blende, therefore, is decomposable.” Taking our letters thus-- A = blende, B = elementary substance, C = undecomposable, the premises are of the forms A = A*b*, (1) B = C. (2) No immediate substitution can be made; but if we take the contrapositive of (2) (see p. 86), namely *b* = *c*, (3) we can substitute in (1) obtaining the conclusion A = A*c*. But the same result may be obtained by taking the eight combinations of A, B, C, of the Logical Alphabet; it will be found that only three combinations, namely, A*bc* *a*BC *abc*, are consistent with the premises, whence it results that A = A*bc*, or by the process of Ellipsis before described (p. 57) A = A*c*. *Third Example.* As a somewhat more complex example I take the argument thus stated, one which could not be thrown into the syllogistic form:-- “All metals except gold and silver are opaque; therefore what is not opaque is either gold or silver or is not-metal.” There is more implied in this statement than is distinctly asserted, the full meaning being as follows: All metals not gold or silver are opaque, (1) Gold is not opaque but is a metal, (2) Silver is not opaque but is a metal, (3) Gold is not silver. (4) Taking our letters thus-- A = metal C = silver B = gold D = opaque, we may state the premises in the forms A*bc* = A*bc*D (1) B = AB*d* (2) C = AC*d* (3) B = B*c*. (4) To obtain a complete solution of the question we take the sixteen combinations of A, B, C, D, and striking out those which are inconsistent with the premises, there remain only AB*cd* A*b*C*d* A*bc*D *abc*D *abcd*. The expression for not-opaque things consists of the three combinations containing *d*, thus