The Law of Forms: Finding Equivalent Forms

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0.43 The Law of FormsThis law answers the following problem: An equality being given, to find for any term (simple or complex) a determination equivalent to this equality. In other words, the question is to find all the _forms_ equivalent to this equality, any term at all being given as its first member.We know that any equality can be reduced to a form in which the second member is 0 or 1; _i.e._, to one of the two equivalent forms\[N=0,\qquad N^{\prime}=1.\]The function \(N\) is what Poretsky calls the _logical zero_ of the given equality; \(N^{\prime}\) is its logical _whole_.50They are called “logical” to distinguish them from the identical zero and whole, i.e., to indicate that these two terms are not equal to 0 and 1 respectively except by virtue of the data of the problem.Let \(U\) be any term; then the determination of \(U\):\[U=N^{\prime}U+NU^{\prime}\]is equivalent to the proposed equality; for we know it is equivalent to the equality\[(NU+NU^{\prime}=0)=(N=0).\]Let us recall the signification of the determination\[U=N^{\prime}U+NU^{\prime}.\]It denotes that the term \(U\) is contained in \(N^{\prime}\) and contains \(N\). This is easily understood, since, by hypothesis, \(N\) is equal to \(0\) and \(N^{\prime}\) to \(1\). Therefore we can formulate the _law of forms_ in the following way:_To obtain all the forms equivalent to a given equality, it is sufficient to express that any term contains the logical zero of this equality and is contained in its logical whole._The number of forms of a given equality is unlimited; for any term gives rise to a form, and to a form different from the others, since it has a different first member. But if we are limited to the universe of discourse determined by \(n\) simple terms, the number of forms becomes finite and determinate. For, in this limited universe, there are \(2^{n}\) constituents. Now, all the terms in this universe that can be conceived and defined are sums of some of these constituents. Their number is, therefore, equal to the number of combinations that can be made with \(2^{n}\) constituents, namely \(2^{2^{n}}\) (including \(0\), the combination of \(0\) constituent, and \(1\), the combination of all the constituents). This will also be the number of different forms of any equality in the universe in question. 0.44 The Law of ConsequencesWe shall now pass to the law of consequences. Generalizing the conception of Boole, who made deduction consist in the elimination of middle terms, Poretsky makes it consist in the elimination of known terms (_connaissances_). This conception is explained and justified as follows.All problems in which the data are expressed by logical equalities or inclusions can be reduced to a single logical equality by means of the formula51We employ capitals to denote complex terms (logical functions) in contrast to simple terms denoted by small letters \((a,b,c,\ldots)\)\[(A=0)(B=0)(C=0)\ldots=(A+B+C\ldots=0).\]In this logical equality, which sums up all the data of the problem, we develop the first member with respect to all the simple terms which appear in it (and not with respect to the unknown quantities). Let \(n\) be the number of simple terms; then the number of the constituents of the development of \(1\) is \(2^{n}\). Let \(m\) (\(\leq 2^{n}\)) be the number of those constituents appearing in the first member of the equality. All possible consequences of this equality (in the universe of the \(n\) terms in question) may be obtained by forming all the additive combinations of these \(m\) constituents, and equating them to \(0\); and this is done in virtue of the formula\[(A+B=0)

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