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That is to say, if something ''x'' is said to be ''z'', then we may guess fairly surely that ''x'' is really an orange, in other words, that ''x'' has all of the additional features that would be summed up quite succinctly in the much more constrained term ''y'' = ''an orange''.

That is to say, if something <math>x\!</math> is said to be <math>z,\!</math> then we may guess fairly surely that <math>x\!</math> is really an orange, in other words, that <math>x\!</math> has all of the additional features that would be summed up quite succinctly in the much more constrained term <math>y,\!</math> where <math>y\!</math> means "an orange".

Figure 1 depicts the situation that is being contemplated here.

Figure&nbsp;1 depicts the situation that is being contemplated here.

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Figure 1.  Conjunctive Term z, Taken as Predicate

Figure 1.  Conjunctive Term z, Taken as Predicate

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What Peirce is saying about ''z'' not being a genuinely useful symbol can be explained in terms of the gap between the logical conjunction ''z'', in lattice terms, the ''greatest lower bound'' (''glb'') of the conjoined terms, ''z'' = ''glb''{''t''_''j''&nbsp;:&nbsp;''j''&nbsp;=&nbsp;1&nbsp;to&nbsp;6}, and what we might regard as the "natural conjunction" or the "natural glb" of these terms, namely, ''y''&nbsp;=&nbsp;''an&nbsp;orange''.  That is to say, there is an extra measure of constraint that goes into forming the natural kinds lattice from the free lattice that logic and set theory would otherwise impose.  The local manifestations of this global information are meted out over the structure of the natural lattice by just such abductive gaps as the one between ''z'' and ''y''.

What Peirce is saying about <math>z\!</math> not being a genuinely useful symbol can be explained in terms of the gap between the logical conjunction <math>z,\!</math> in lattice terms, the ''greatest lower bound'' (''glb'') of the conjoined terms, <math>z = \operatorname{glb}( \{ t_j : j = 1 ~\text{to}~ 6 \}),</math> and what we might regard as the "natural conjunction" or the "natural glb" of these terms, namely, <math>y = \text{an orange}.\!</math> That is to say, there is an extra measure of constraint that goes into forming the natural kinds lattice from the free lattice that logic and set theory would otherwise impose.  The local manifestations of this global information are meted out over the structure of the natural lattice by just such abductive gaps as the one between <math>z\!</math> and <math>y.\!</math>

===Discussion===

===Discussion===