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Suppose that <math>z\!</math> is the logical conjunction of these six terms:

Suppose that <math>z\!</math> is the logical conjunction of the above six terms:

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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>

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===

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Suppose that <math>u\!</math> is the logical disjunction of these four terms:

Suppose that <math>u\!</math> is the logical disjunction of the above four terms:

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In a similar but dual fashion to the preceding consideration, there is a gap between the the logical disjunction <math>u,\!</math> in lattice terminology, the ''least upper bound'' (''lub'') of the disjoined terms, <math>u = \operatorname{lub} ( \{ s_j : j = 1 ~\text{to}~ 4 \}),</math> and what we might regard as the "natural disjunction" or the "natural lub", namely, <math>v = \text{cloven-hoofed}.\!</math>

In a similar but dual fashion to the preceding consideration, there is a gap between the the logical disjunction <math>u,\!</math> in lattice terminology, the ''least upper bound'' (''lub'') of the disjoined terms, <math>u = \operatorname{lub} ( \{ s_j : j = 1 ~\text{to}~ 4 \}),</math> and what we might regard as the ''natural disjunction'' or the ''natural lub'', namely, <math>v := \text{cloven-hoofed}.\!</math>

Once again, the sheer implausibility of imagining that the disjunctive term <math>u\!</math> would ever be embedded exactly as such in a lattice of natural kinds, leads to the evident ''naturalness'' of the induction to <math>v \Rightarrow w,</math> namely, the rule that cloven-hoofed animals are herbivorous.

Once again, the sheer implausibility of imagining that the disjunctive term <math>u\!</math> would ever be embedded exactly as such in a lattice of natural kinds, leads to the evident ''naturalness'' of the induction to <math>v \Rightarrow w,</math> namely, the rule that cloven-hoofed animals are herbivorous.

<p>For this purpose, I must call your attention to the differences there are in the manner in which different representations stand for their objects.</p>

<p>For this purpose, I must call your attention to the differences there are in the manner in which different representations stand for their objects.</p>

<p>In the first place there are likenesses or copies — such as ''statues'', ''pictures'', ''emblems'', ''hieroglyphics'', and the like.  Such representations stand for their objects only so far as they have an actual resemblance to them — that is agree with them in some characters.  The peculiarity of such representations is that they do not determine their objects — they stand for anything more or less;  for they stand for whatever they resemble and they resemble everything more or less.</p>

<p>In the first place there are likenesses or copies &mdash; such as ''statues'', ''pictures'', ''emblems'', ''hieroglyphics'', and the like.  Such representations stand for their objects only so far as they have an actual resemblance to them &mdash; that is agree with them in some characters.  The peculiarity of such representations is that they do not determine their objects &mdash; they stand for anything more or less;  for they stand for whatever they resemble and they resemble everything more or less.</p>

<p>The second kind of representations are such as are set up by a convention of men or a decree of God.  Such are ''tallies'', ''proper names'', &c.  The peculiarity of these ''conventional signs'' is that they represent no character of their objects.  Likenesses denote nothing in particular;  ''conventional signs'' connote nothing in particular.</p>

<p>The second kind of representations are such as are set up by a convention of men or a decree of God.  Such are ''tallies'', ''proper names'', &c.  The peculiarity of these ''conventional signs'' is that they represent no character of their objects.  Likenesses denote nothing in particular;  ''conventional signs'' connote nothing in particular.</p>

<p>The third and last kind of representations are ''symbols'' or general representations.  They connote attributes and so connote them as to determine what they denote.  To this class belong all ''words'' and all ''conceptions''.  Most combinations of words are also symbols.  A proposition, an argument, even a whole book may be, and should be, a single symbol. (Peirce 1866, "Lowell Lecture 7", CE 1, 467–468).</p>

<p>The third and last kind of representations are ''symbols'' or general representations.  They connote attributes and so connote them as to determine what they denote.  To this class belong all ''words'' and all ''conceptions''.  Most combinations of words are also symbols.  A proposition, an argument, even a whole book may be, and should be, a single symbol.</p>

 

<p>(Peirce 1866, Lowell Lecture 7, CE 1, 467&ndash;468).</p>

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In order to draw out these themes, and to see how Peirce was led and often inspired to develop their main motives, let us bring together our previous Figures, abstracting away from all of those distractingly ephemeral details about defunct stockyards full of imaginary beasts, and see if we can see what is really going to go on here.

In order to draw out these themes, and to see how Peirce was led and often inspired to develop their main motives, let us bring together our previous Figures, abstracting away from all of those distractingly ephemeral details about defunct stockyards full of imaginary beasts, and see if we can see what is really going to go on here.

Figure 3 shows an abductive step of inquiry, as it is taken on the cue of an iconic sign.

Figure&nbsp;3 shows an abductive step of inquiry, as it is taken on the cue of an iconic sign.

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Figure 3.  Conjunctive Predicate z, Abduction of Case (x (y))

Figure 3.  Conjunctive Predicate z, Abduction of Case (x (y))

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Figure 4 depicts an inductive step of inquiry, as it is taken on the cue of an indicial sign.

Figure&nbsp;4 depicts an inductive step of inquiry, as it is taken on the cue of an indicial sign.

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Figure 4.  Disjunctive Subject u, Induction of Rule (v (w))

Figure 4.  Disjunctive Subject u, Induction of Rule (v (w))

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I have up to this point followed Peirce's suggestions somewhat unthinkingly, but I can tell you now that previous unfortunate experience has led me concurrently to remain suspicious of all attempts to conflate the types of signs and the roles of terms in arguments quite so facilely, so I will keep that as a topic for future inquiry.

I have up to this point followed Peirce's suggestions somewhat unthinkingly, but I can tell you now that previous unfortunate experience has led me concurrently to remain suspicious of all attempts to conflate the types of signs and the roles of terms in arguments quite so facilely, so I will keep that as a topic for future inquiry.