Our first premise informs us that iron is a metal, and if we substitute this description in (γ) and (δ) we shall have self-contradictory combinations. Our second premise likewise informs us that metal is element, and applying this description to (β) we again have self-contradiction, so that there remains only (α) as a description of iron--our inference is Iron = iron, metal, element. To represent this process of reasoning in general symbols, let A = iron B = metal C = element, The premises of the problem take the forms A = AB (1) B = BC. (2) By the Law of Duality we have A = AB ꖌ A*b* (3) A = AC ꖌ A*c*. (4) Now, if we insert for A in the second side of (3) its description in (4), we obtain what I shall call the *development of A with respect to B and C*, namely A = ABC ꖌ AB*c* ꖌ A*b*C ꖌ A*bc*. (5) Wherever the letters A or B appear in the second side of (5) substitute their equivalents given in (1) and (2), and the results stated at full length are A = ABC ꖌ ABC*c* ꖌ AB*b*C ꖌ AB*b*C*c*. The last three alternatives break the Law of Contradiction, so that A = ABC ꖌ 0 ꖌ 0 ꖌ 0 = ABC. This conclusion is, indeed, no more than we could obtain by the direct process of substitution, that is by substituting for B in (1), its description in (2) as in p. 55; it is the characteristic of the Indirect process that it gives all possible logical conclusions, both those which we have previously obtained, and an immense number of others or which the ancient logic took little or no account. From the same premises, for instance, we can obtain a description of the class *not-element* or *c*. By the Law of Duality we can develop *c* into four alternatives, thus *c* = AB*c* ꖌ A*bc* ꖌ *a*B*c* ꖌ *abc*. If we substitute for A and B as before, we get *c* = ABC*c* ꖌ AB*bc* ꖌ *a*BC*c* ꖌ *abc*, and, striking out the terms which break the Law of Contradiction, there remains *c* = *abc*, or what is not element is also not iron and not metal. This Indirect Method of Inference thus furnishes a complete solution of the following problem--*Given any number of logical premises or conditions, required the description of any class of objects, or of any term, as governed by those conditions.* The steps of the process of inference may thus be concisely stated-- 1. By the Law of Duality develop the utmost number of alternatives which may exist in the description of the required class or term as regards the terms involved in the premises. 2. For each term in these alternatives substitute its description as given in the premises. 3. Strike out every alternative which is then found to break the Law of Contradiction. 4. The remaining terms may be equated to the term in question as the desired description. *Mr. Venn’s Problem.*