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<p>It is plain that both the regular non-invertible addition and the invertible addition satisfy the absolute conditions. But the notation has other recommendations. The conception of ''taking together'' involved in these processes is strongly analogous to that of summation, the sum of 2 and 5, for example, being the number of a collection which consists of a collection of two and a collection of five. Any logical equation or inequality in which no operation but addition is involved may be converted into a numerical equation or inequality by substituting the numbers of the several terms for the terms themselves — provided all the terms summed are mutually exclusive.</p>
<p>It is plain that both the regular non-invertible addition and the invertible addition satisfy the absolute conditions. But the notation has other recommendations. The conception of ''taking together'' involved in these processes is strongly analogous to that of summation, the sum of 2 and 5, for example, being the number of a collection which consists of a collection of two and a collection of five. Any logical equation or inequality in which no operation but addition is involved may be converted into a numerical equation or inequality by substituting the numbers of the several terms for the terms themselves — provided all the terms summed are mutually exclusive.</p>
<p>Addition being taken in this sense, ''nothing'' is to be denoted by 'zero', for then:</p>
<p>Addition being taken in this sense, ''nothing'' is to be denoted by ''zero'', for then</p>
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| align="center" | <math>x ~+\!\!,~ 0 ~=~ x</math>
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<p>whatever is denoted by ''x''; and this is the definition of ''zero''. This interpretation is given by Boole, and is very neat, on account of the resemblance between the ordinary conception of ''zero'' and that of nothing, and because we shall thus have</p>
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<p>whatever is denoted by <math>~x~</math>; and this is the definition of ''zero''. This interpretation is given by Boole, and is very neat, on account of the resemblance between the ordinary conception of ''zero'' and that of nothing, and because we shall thus have</p>
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| align="center" | <math>[0] ~=~ 0.</math>
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<p>(Peirce, CP 3.67).</p>
<p>(Peirce, CP 3.67).</p>
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