Studies in Logic

This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1883 edition. Excerpt: ...like the above are (ab...gh...l) (h...lm...q)l-(ab...rj) (m...q) (ab M) (llm?)--(ab km...q)-In the first, whero ono premise is particular, inferenco can take placo independently of any number of middlo terms, provided each term is positive in both premises, or negative in both. In the second formula, when both premises aro universal, inference can take placo independently of only ono middle term, and this must bo of different quality in the two premises. By an obvious substitution these two formulas aro reduced to tho formulas previously given invelving only two terms in each promise. Thus.. That is, the premises of the first mean "that which is common (x) to a, b, ...g, has something in common with the common part (y) of h, ... I'," and " the common part (y) of A, ... Z has nothing in common with m, ... g." Whenco the inferenco is (xy-1), or (ai...g) (m... j)_1. The premises of the second mean "whatever may bo common (z) to a, b, ... k, has nothing in common with I;" and " whatever may be common (z) to m, ... q, has nothing in common with non-J." Whenco the infereuco is (xz)-1, or (flii...km... q)-(abc) means (ab) (ac) (be), .: (abc)-1" (ab)-1 + (ac)-l + (bc)-l. Thus any ono of theso propositions is redueible to a function of Do Morgan's eight. 5. Note on De Morgan's Twenty Propositions It is proposed in this section to consider a simplo methed of deriving and writing Do Morgan's Twenty Propositions. Lot A = all of A, a = part of A, A = all of noiwl, and a = part of noiwl, where pari of is understood to mean less than the whole of. Let a second term B be modified in the same way. Then, by affirming and denying identity between each modification of the first term and each modification of..."