Editors' Note: The following explication was done according to the guidelines provided by [Delaware 2019]. In brief, notes, remarks, and explanations are added to an excerpt from a primary source, such as "Algebra of Logic," to foster readers' understanding of the technical content and intellectual context embedded in a writer's text by answering readers' questions before they have thought to ask them. Thus, on this page, all comments in square brackets were added to Ladd's text by the author.
From “On the Algebra of Logic” by Christine Ladd [1883, pp. 37–39]: [Note: Throughout, Ladd used “proposition” to refer to a relationship between any number of terms, what has been defined as a “statement” above. Also, throughout, parentheses have been added for clarity, particularly to make clear groupings on either side of the copula.]
On Elimination. – In (24ʹ)
[derived in the section above and accompanied by an example quoted here without proof. The symbol \(\therefore\) is read “therefore.”
“… if the premises are both universal,
\(\begin{array}{ll} (24ʹ) \,\,\,\,& a \overline{\vee} b \\ & c \overline{\vee} d \\ & \therefore ac \overline{\vee} (b + d) \end{array} \)
If no bankers [\(a\)] have souls [\(b\)] and no poets [\(c\)] have bodies [\(d\)], then no banker-poets [\(ac\)] have either souls or bodies [\(b + d\)]."]
there is no elimination [all terms in both of the premises, namely \(a\), \(b\), \(c\), \(d\), appear in the conclusion], and in (24°)
[derived earlier in the thesis and quoted here with example:
“…if the premises are one universal and one particular, \[\begin{array}{ll} (24^{\circ}) \,\,\,\,& a \overline{\vee} b \\ & ac \vee (b + d) \\ & \therefore c \vee d \end{array} \] If no Africans [\(a\)] are brave [\(b\)] and some African chiefs [\(ac\)] are either brave or deceitful [\(b + d\)], then some chiefs [\(c\)] are deceitful [\(d\)].”]
there is elimination of the whole of the first premise [no Africans are brave, \(a \overline{\vee} b\)] and part of the second [some African chiefs are brave, \(ac\vee b\) ]. The most common object [goal] in reasoning is to eliminate a single term at a time—namely, one which occurs in both premises [as seen in the classic "All Greeks are mortal" example of a valid syllogism, with the elimination of the middle term “men”]. Each of these inferences gives rise to a form of argument, as a special case, by which that object [elimination of a single term at a time] is accomplished,—the premises being on the one hand both universal [as in (24ʹ)], and on the other hand one universal and the other particular [as in (24°)]. The inconsistency \(I\)
[derived by Ladd-Franklin earlier in the thesis (p. 34), stated with large parentheses added: \[I. \,\,\, (a \overline{\vee} b)(c\overline{\vee}d)\overline{\vee}\left (ac \vee (b+d)\right )\] Meaning, the conjunction of statements “no \(a\) is \(b\)” and “no \(c\) is \(d\)” is inconsistent with the conclusion that “some \(ac\) is \(b\) or \(d\).” Or, as Ladd-Franklin explains, “it is not possible that a combination of several qualities should be found in any classes from each of which some one of those qualities is absent. If, for example, culture [\(a\)] is never found in business men [\(b\)] nor respectability [\(c\)] among artists [\(d\)], then it is impossible that cultured respectability [\(ac\)] should be found among either business men or artists [\(b + d\)]” [Ladd 1883, p. 35]]
becomes when \(d\) is equal to \(\overline{b}\) [\(d\) is not-\(b\)] and hence \(b + d\) [what is either \(b\) or \(d\), means, with the substitution, what is \(b\) or not-\(b\)] equal to [\(b +\overline{b} =\)] \(\infty\). \[\left ( (a \overline{\vee} b)(c\overline{\vee}\overline{b})(ac\vee \infty)\right )\overline{\vee}\]
[large parentheses added. This formula indicates that the combination of statements, “\(a\) is not \(b\)” (\(a \overline{\vee} b)\) and “\(c\) is not not-\(b\)” (\(c\overline{\vee}\overline{b})\) and “the conjunction of \(a\) and \(c\) exists” (\(ac \vee\infty\)) is inconsistent (\(\overline{\vee} \) ), meaning that given the first two statements as premises, the third statement does not follow as a valid conclusion.]
or
\[II. \left ( (a \overline{\vee} b)(\overline{b}\overline{\vee}c)(c \vee a )\right ) \overline{\vee}\]
[large parentheses added. The argument consisting of the first premise statement "\(a\) is not \(b\)” and the second premise statement “not-\(b\) is not \(c\)” (\(\overline{b}\overline{\vee}c\)) equivalent to (\(c\overline{\vee}\overline{b}\)) above, and the conclusion statement “some \(c\) is \(a\)” (\(c \vee a \)) is inconsistent (\(\overline{\vee}\)) because, since by the first premise no \(a\) is \( b\), then not-\(b\) would include \(a\), and by the second premise not-\(b\) is inconsistent with \(c\), so the conclusion statement (\(c \vee a \)) does not follow (\(\overline{\vee}\)), meaning \(a\) cannot coexist with \(c\), previously written (\(ac\overline{\vee}\infty\)).]
Given any two of these propositions, the third proposition, with which it is inconsistent, is free from the term common to the two given propositions; \(a\), \(b\), and \(c\), are, of course, expressions [statements] of any degree of complexity. The propositions \(m a\overline{\vee}(x+y)\) [what is both \(m\) and \(a\) is inconsistent with \(x\) or \(y\)], \(\overline{x}\,\overline{y}\overline{\vee}(c+n)\) [what is both not-\(x\) and not-\(y\) is inconsistent with \(c\) or \(n\)], for instance, are inconsistent with \(ma\vee(c+n)\) [what is both \(m\) and \(a\) is consistent with \(c\) or \(n\); this cannot be a valid conclusion because \(ma\) is not consistent with \(x\) or \(y\), meaning what is not-\(x\) and not-\(y\), including \(ma\), must be inconsistent with \(c\) or \(n\).]; any number of terms may be eliminated at once by combining them in such a way that they shall make up a complete universe [\(\infty\), the universe of discourse].
When any two of the [three mutually] inconsistent propositions in II. are taken as the premises, the negative of the remaining one is the [valid] conclusion.
There are, therefore, two distinct forms of inference with elimination of a middle term, special cases of (24ʹ) [when the premises are both universal] and (24°) [when one premise is universal and the other particular]. If we write \(x\) for the middle term, we have
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(25ʹ) \[ \mbox{[i]}\,\, a\overline{\vee}x\] [\(a\) is inconsistent with \(x\)] \[ \mbox{[ii]}\,\, b \overline{\vee} \overline{x}\] [\(b\) is inconsistent with not-\(x\), so \(b\) is consistent with \(x\)] \[ \mbox{[iii]}\,\, \therefore ab \overline{\vee}\] [therefore, there is no \(a\) and \(b\), or \(a\) is inconsistent with \(b\)] The premises are [can be rewritten as] \[ \mbox{[i*]}\,\, a(b+\overline{b})x\overline{\vee}\] [\(a=a(b+\overline{b})=a\infty\)] \[ \mbox{[ii*]}\,\, (a+\overline{a})b\overline{x}\overline{\vee}\] [\(b=(a+\overline{a})b=\infty b\)] and together they affirm that \[ \mbox{[iv]}\,\, (ab(x+\overline{x})+a\overline{b}x+\overline{a}b\overline{x})\overline{\vee}\] or [equivalently] \[ \mbox{[iv*]}\,\, (ab+a\overline{b}x+\overline{a}b\overline{x})\overline{\vee}\] [Applying the distributive laws, known to Ladd, to [i*] and [ii*], we obtain [i*] \(a(b+\overline{b})x\overline{\vee}=(abx+a\overline{b})x\overline{\vee} = (abx+a\overline{b}x)\overline{\vee}\), and [ii*] \((a+\overline{a})b\overline{x}\overline{\vee}= (ab\overline{x}+\overline{a}b\overline{x})\overline{\vee}\). Then, adding, we obtain (\(abx + ab\overline{x} +a\overline{b}x+\overline{a}b\overline{x})\overline{\vee} \), which simplifies to [iv]. Since \(x+\overline{x} = \infty\) , it can be removed, so we finally reach [iv*].] Dropping the information concerning \(x\) [in [iv*]], there remains \[ ab\overline{\vee}. \,\,\,\, \mbox{[the conclusion, [iii].}] \] The information given by the conclusion is thus exactly one half of the information given by the premises (Jevons) [Ladd is referencing a condition for elimination given by a developer of an existing algebra of logic, William Stanley Jevons (1835–1882). The condition that has been satisfied is the removal of one of the two terms (one half of the information), namely \(x\), from each premise, leaving only \(a\) and \(b\) in the conclusion]. |
(25°) \[ \mbox{[i]}\,\, a\overline{\vee}x\] [\(a\) is inconsistent with \(x\)] \[ \mbox{[ii]}\,\, b \vee x\] [\(b\) is consistent with \(x\)] \[ \mbox{[iii]}\,\,\therefore b \overline{a}\vee\] [therefore, \(b\) and not-\(a\) are consistent] The second premise is [can be rewritten as] \[ \mbox{[ii*]}\,\, bx(ax+\overline{ax})\vee\] [because \(ax+\overline{ax} =\infty\)] Which becomes, since there is no \(ax\), [as known from [i]] \[ \mbox{[ii*]}\,\, bx(ax +\overline{ax})\vee = bx (\overline{ax})\vee = bx(\overline{a}+\overline{x})\vee \] [By De Morgan’s law which states that the negation of (\(a\) and \(x\)) is (not-\(a\) or not-\(x\))] or [equivalently] \[bx\overline{a}\vee.\] [In [ii*] \(bx(\overline{a}+\overline{x})\vee = (bx\overline{a}+bx\overline{x})\vee\). Since \(x\overline{x}=0\), that term can be removed, leaving only \(bx\overline{a}\vee\).] Dropping the information concerning \(x\) there remains \[b\overline{a}\vee \,\,\,\, \mbox{[the conclusion, [iii].}] \] This conclusion is equivalent to \[b\overline{a}\vee(x+\overline{x})\] [because \(x+\overline{x}=\infty\)] but the [rewritten and simplified] premises permit the conclusion \[b\overline{a} \vee x;\] [if \(b\overline{a}\) is consistent with \(x\) or not-\(x\), then \(b\overline{a}\) is consistent with \(x\) is a valid conclusion] hence the amount of information retained is exactly one half of the (particular) information given by the premises.
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Elimination is therefore merely a particular case of dropping irrelevant information.