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Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)
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==Selection 1==
==Selection 1==
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<p>The letters of the alphabet will denote logical signs.</p>
<p>The letters of the alphabet will denote logical signs.</p>
One might also call attention to the following two statements:
One might also call attention to the following two statements:
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<p>Now logical terms are of three grand classes.</p>
<p>Now logical terms are of three grand classes.</p>
==Selection 2==
==Selection 2==
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<p>'''Numbers Corresponding to Letters'''</p>
<p>'''Numbers Corresponding to Letters'''</p>
==Selection 3==
==Selection 3==
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<p>'''The Signs of Inclusion, Equality, Etc.'''</p>
<p>'''The Signs of Inclusion, Equality, Etc.'''</p>
==Selection 4==
==Selection 4==
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<p>'''The Signs for Addition'''</p>
<p>'''The Signs for Addition'''</p>
==Selection 5==
==Selection 5==
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<p>'''The Signs for Multiplication'''</p>
<p>'''The Signs for Multiplication'''</p>
==Selection 6==
==Selection 6==
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<p>'''The Signs for Multiplication''' (cont.)</p>
<p>'''The Signs for Multiplication''' (cont.)</p>
==Selection 7==
==Selection 7==
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<p>'''The Signs for Multiplication''' (cont.)</p>
<p>'''The Signs for Multiplication''' (cont.)</p>
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==Selection 8==
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<p>'''The Signs for Multiplication''' (cont.)</p>
<p>'''The Signs for Multiplication''' (cont.)</p>
The foregoing has hopefully filled in enough background that we can begin to make sense of the more mysterious parts of CP 3.73.
The foregoing has hopefully filled in enough background that we can begin to make sense of the more mysterious parts of CP 3.73.
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<p>Thus far, we have considered the multiplication of relative terms only. Since our conception of multiplication is the application of a relation, we can only multiply absolute terms by considering them as relatives.</p>
<p>Thus far, we have considered the multiplication of relative terms only. Since our conception of multiplication is the application of a relation, we can only multiply absolute terms by considering them as relatives.</p>
==Selection 9==
==Selection 9==
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<p>'''The Signs for Multiplication''' (cont.)</p>
<p>'''The Signs for Multiplication''' (cont.)</p>
Let us backtrack a few years, and consider how Boole explained his twin conceptions of "selective operations" and "selective symbols".
Let us backtrack a few years, and consider how Boole explained his twin conceptions of "selective operations" and "selective symbols".
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<p>Let us then suppose that the universe of our discourse is the actual universe, so that words are to be used in the full extent of their meaning, and let us consider the two mental operations implied by the words "white" and "men". The word "men" implies the operation of selecting in thought from its subject, the universe, all men; and the resulting conception, ''men'', becomes the subject of the next operation. The operation implied by the word "white" is that of selecting from its subject, "men", all of that class which are white. The final resulting conception is that of "white men".</p>
<p>Let us then suppose that the universe of our discourse is the actual universe, so that words are to be used in the full extent of their meaning, and let us consider the two mental operations implied by the words "white" and "men". The word "men" implies the operation of selecting in thought from its subject, the universe, all men; and the resulting conception, ''men'', becomes the subject of the next operation. The operation implied by the word "white" is that of selecting from its subject, "men", all of that class which are white. The final resulting conception is that of "white men".</p>
In setting up his discussion of selective operations and their corresponding selective symbols, Boole writes this:
In setting up his discussion of selective operations and their corresponding selective symbols, Boole writes this:
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The operation which we really perform is one of ''selection according to a prescribed principle or idea''. To what faculties of the mind such an operation would be referred, according to the received classification of its powers, it is not important to inquire, but I suppose that it would be considered as dependent upon the two faculties of Conception or Imagination, and Attention. To the one of these faculties might be referred the formation of the general conception; to the other the fixing of the mental regard upon those individuals within the prescribed universe of discourse which answer to the conception. If, however, as seems not improbable, the power of Attention is nothing more than the power of continuing the exercise of any other faculty of the mind, we might properly regard the whole of the mental process above described as referrible to the mental faculty of Imagination or Conception, the first step of the process being the conception of the Universe itself, and each succeeding step limiting in a definite manner the conception thus formed. Adopting this view, I shall describe each such step, or any definite combination of such steps, as a ''definite act of conception''. (Boole, ''Laws of Thought'', 43).
The operation which we really perform is one of ''selection according to a prescribed principle or idea''. To what faculties of the mind such an operation would be referred, according to the received classification of its powers, it is not important to inquire, but I suppose that it would be considered as dependent upon the two faculties of Conception or Imagination, and Attention. To the one of these faculties might be referred the formation of the general conception; to the other the fixing of the mental regard upon those individuals within the prescribed universe of discourse which answer to the conception. If, however, as seems not improbable, the power of Attention is nothing more than the power of continuing the exercise of any other faculty of the mind, we might properly regard the whole of the mental process above described as referrible to the mental faculty of Imagination or Conception, the first step of the process being the conception of the Universe itself, and each succeeding step limiting in a definite manner the conception thus formed. Adopting this view, I shall describe each such step, or any definite combination of such steps, as a ''definite act of conception''. (Boole, ''Laws of Thought'', 43).
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<p>'''The Signs for Multiplication''' (cont.)<p>
<p>'''The Signs for Multiplication''' (cont.)<p>
We have sufficiently covered the application of the comma functor, or the diagonal extension, to absolute terms, so let us return to where we were in working our way through CP 3.73, and see whether we can validate Peirce's statements about the "commifications" of 2-adic relative terms that yield their 3-adic diagonal extensions.
We have sufficiently covered the application of the comma functor, or the diagonal extension, to absolute terms, so let us return to where we were in working our way through CP 3.73, and see whether we can validate Peirce's statements about the "commifications" of 2-adic relative terms that yield their 3-adic diagonal extensions.
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<p>But not only may any absolute term be thus regarded as a relative term, but any relative term may in the same way be regarded as a relative with one correlate more. It is convenient to take this additional correlate as the first one.</p>
<p>But not only may any absolute term be thus regarded as a relative term, but any relative term may in the same way be regarded as a relative with one correlate more. It is convenient to take this additional correlate as the first one.</p>
I return to where we were in unpacking the contents of CP 3.73. Peirce remarks that the comma operator can be iterated at will:
I return to where we were in unpacking the contents of CP 3.73. Peirce remarks that the comma operator can be iterated at will:
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<p>In point of fact, since a comma may be added in this way to any relative term, it may be added to one of these very relatives formed by a comma, and thus by the addition of two commas an absolute term becomes a relative of two correlates.</p>
<p>In point of fact, since a comma may be added in this way to any relative term, it may be added to one of these very relatives formed by a comma, and thus by the addition of two commas an absolute term becomes a relative of two correlates.</p>
==Selection 11==
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<p>'''The Signs for Multiplication''' (concl.)</p>
<p>'''The Signs for Multiplication''' (concl.)</p>
'''NOF 1'''
'''NOF 1'''
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<p>I propose to assign to all logical terms, numbers; to an absolute term, the number of individuals it denotes; to a relative term, the average number of things so related to one individual.</p>
<p>I propose to assign to all logical terms, numbers; to an absolute term, the number of individuals it denotes; to a relative term, the average number of things so related to one individual.</p>
'''NOF 2'''
'''NOF 2'''
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But not only do the significations of '=' and '<' here adopted fulfill all absolute requirements, but they have the supererogatory virtue of being very nearly the same as the common significations. Equality is, in fact, nothing but the identity of two numbers; numbers that are equal are those which are predicable of the same collections, just as terms that are identical are those which are predicable of the same classes. So, to write 5 < 7 is to say that 5 is part of 7, just as to write ''f'' < ''m'' is to say that Frenchmen are part of men. Indeed, if ''f'' < ''m'', then the number of Frenchmen is less than the number of men, and if ''v'' = ''p'', then the number of Vice-Presidents is equal to the number of Presidents of the Senate; so that the numbers may always be substituted for the terms themselves, in case no signs of operation occur in the equations or inequalities. (Peirce, CP 3.66).
But not only do the significations of '=' and '<' here adopted fulfill all absolute requirements, but they have the supererogatory virtue of being very nearly the same as the common significations. Equality is, in fact, nothing but the identity of two numbers; numbers that are equal are those which are predicable of the same collections, just as terms that are identical are those which are predicable of the same classes. So, to write 5 < 7 is to say that 5 is part of 7, just as to write ''f'' < ''m'' is to say that Frenchmen are part of men. Indeed, if ''f'' < ''m'', then the number of Frenchmen is less than the number of men, and if ''v'' = ''p'', then the number of Vice-Presidents is equal to the number of Presidents of the Senate; so that the numbers may always be substituted for the terms themselves, in case no signs of operation occur in the equations or inequalities. (Peirce, CP 3.66).
'''NOF 3'''
'''NOF 3'''
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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>
'''NOF 4'''
'''NOF 4'''
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<p>The conception of multiplication we have adopted is that of the application of one relation to another. …</p>
<p>The conception of multiplication we have adopted is that of the application of one relation to another. …</p>
Let's continue to work our way through the rest of the first set of definitions, making up appropriate examples as we go.
Let's continue to work our way through the rest of the first set of definitions, making up appropriate examples as we go.
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Let ''P'' ⊆ ''X'' × ''Y'' be an arbitrary 2-adic relation. The following properties of ''P'' can be defined:
Let ''P'' ⊆ ''X'' × ''Y'' be an arbitrary 2-adic relation. The following properties of ''P'' can be defined:
</pre>
</pre>
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<p>If ''P'' ⊆ ''X'' × ''Y'' is tubular at ''X'', then ''P'' is known as a "partial function" or a "pre-function" from ''X'' to ''Y'', frequently signalized by renaming ''P'' with an alternative lower case name, say "''p''", and writing ''p'' : ''X'' ~> ''Y''.</p>
<p>If ''P'' ⊆ ''X'' × ''Y'' is tubular at ''X'', then ''P'' is known as a "partial function" or a "pre-function" from ''X'' to ''Y'', frequently signalized by renaming ''P'' with an alternative lower case name, say "''p''", and writing ''p'' : ''X'' ~> ''Y''.</p>
Now let's re-examine the ''numerical incidence properties'' of relations, concentrating on the definitions of the assorted regularity conditions.
Now let's re-examine the ''numerical incidence properties'' of relations, concentrating on the definitions of the assorted regularity conditions.
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<p>For instance, L is said to be "''c''-regular at ''j''" if and only if the cardinality of the local flag ''L''<sub>''x''.''j''</sub> is ''c'' for all ''x'' in ''X'<sub>''j''</sub>, coded in symbols, if and only if |''L''<sub>''x''.''j''</sub>| = ''c'' for all ''x'' in ''X<sub>''j''</sub>.</p>
<p>For instance, L is said to be "''c''-regular at ''j''" if and only if the cardinality of the local flag ''L''<sub>''x''.''j''</sub> is ''c'' for all ''x'' in ''X'<sub>''j''</sub>, coded in symbols, if and only if |''L''<sub>''x''.''j''</sub>| = ''c'' for all ''x'' in ''X<sub>''j''</sub>.</p>
Among the vast variety of conceivable regularities affecting 2-adic relations, we pay special attention to the ''c''-regularity conditions where ''c'' is equal to 1.
Among the vast variety of conceivable regularities affecting 2-adic relations, we pay special attention to the ''c''-regularity conditions where ''c'' is equal to 1.
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<p>Let ''P'' ⊆ ''X'' × ''Y'' be an arbitrary 2-adic relation. The following properties of P can be defined:</p>
<p>Let ''P'' ⊆ ''X'' × ''Y'' be an arbitrary 2-adic relation. The following properties of P can be defined:</p>
Also, we introduced a few bits of additional terminology and special-purpose notations for working with tubular relations:
Also, we introduced a few bits of additional terminology and special-purpose notations for working with tubular relations:
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Thus, we arrive by way of this winding stair at the very special stamps of 2-adic relations ''P'' ⊆ ''X'' × ''Y'' that are "total prefunctions" at ''X'' (or ''Y''), "total and tubular" at ''X'' (or ''Y''), or "1-regular" at ''X'' (or ''Y''), more often celebrated as "functions" at ''X'' (or ''Y'').
Thus, we arrive by way of this winding stair at the very special stamps of 2-adic relations ''P'' ⊆ ''X'' × ''Y'' that are "total prefunctions" at ''X'' (or ''Y''), "total and tubular" at ''X'' (or ''Y''), or "1-regular" at ''X'' (or ''Y''), more often celebrated as "functions" at ''X'' (or ''Y'').
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<p>If ''P'' is a pre-function ''P'' : ''X'' ~> ''Y'' that happens to be total at ''X'', then ''P'' is known as a "function" from ''X'' to ''Y'', typically indicated as ''P'' : ''X'' → ''Y''.</p>
<p>If ''P'' is a pre-function ''P'' : ''X'' ~> ''Y'' that happens to be total at ''X'', then ''P'' is known as a "function" from ''X'' to ''Y'', typically indicated as ''P'' : ''X'' → ''Y''.</p>
In the case of a 2-adic relation ''F'' ⊆ ''X'' × ''Y'' that has the qualifications of a function ''f'' : ''X'' → ''Y'', there are a number of further differentia that arise:
In the case of a 2-adic relation ''F'' ⊆ ''X'' × ''Y'' that has the qualifications of a function ''f'' : ''X'' → ''Y'', there are a number of further differentia that arise:
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First, a correction. Ignore for now the gloss that I gave in regard to Figure 19:
First, a correction. Ignore for now the gloss that I gave in regard to Figure 19:
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Here, I have used arrowheads to indicate the relational domains at which each of the relations ''J'', ''K'', ''L'' happens to be functional.
Here, I have used arrowheads to indicate the relational domains at which each of the relations ''J'', ''K'', ''L'' happens to be functional.
* [http://stderr.org/pipermail/inquiry/2004-November/001814.html LOR.COM 11.2].
* [http://stderr.org/pipermail/inquiry/2004-November/001814.html LOR.COM 11.2].
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<p>I propose to assign to all logical terms, numbers; to an absolute term, the number of individuals it denotes; to a relative term, the average number of things so related to one individual.<p>
<p>I propose to assign to all logical terms, numbers; to an absolute term, the number of individuals it denotes; to a relative term, the average number of things so related to one individual.<p>
With that little bit of encouragement and exhortation, let us return to the nitty gritty details of the text.
With that little bit of encouragement and exhortation, let us return to the nitty gritty details of the text.
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But not only do the significations of "=" and "<" here adopted fulfill all absolute requirements, but they have the supererogatory virtue of being very nearly the same as the common significations. Equality is, in fact, nothing but the identity of two numbers; numbers that are equal are those which are predicable of the same collections, just as terms that are identical are those which are predicable of the same classes. So, to write 5 < 7 is to say that 5 is part of 7, just as to write ''f'' < ''m'' is to say that Frenchmen are part of men. Indeed, if ''f'' < ''m'', then the number of Frenchmen is less than the number of men, and if ''v'' = ''p'', then the number of Vice-Presidents is equal to the number of Presidents of the Senate; so that the numbers may always be substituted for the terms themselves, in case no signs of operation occur in the equations or inequalities. (Peirce, CP 3.66).
But not only do the significations of "=" and "<" here adopted fulfill all absolute requirements, but they have the supererogatory virtue of being very nearly the same as the common significations. Equality is, in fact, nothing but the identity of two numbers; numbers that are equal are those which are predicable of the same collections, just as terms that are identical are those which are predicable of the same classes. So, to write 5 < 7 is to say that 5 is part of 7, just as to write ''f'' < ''m'' is to say that Frenchmen are part of men. Indeed, if ''f'' < ''m'', then the number of Frenchmen is less than the number of men, and if ''v'' = ''p'', then the number of Vice-Presidents is equal to the number of Presidents of the Senate; so that the numbers may always be substituted for the terms themselves, in case no signs of operation occur in the equations or inequalities. (Peirce, CP 3.66).
Peirce next takes up the action of the "number of" map on the two types of, loosely speaking, "additive" operations that we normally consider in logic.
Peirce next takes up the action of the "number of" map on the two types of, loosely speaking, "additive" operations that we normally consider in logic.
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It is plain that both the regular non-invertible addition and the invertible addition satisfy the absolute conditions. (CP 3.67).
It is plain that both the regular non-invertible addition and the invertible addition satisfy the absolute conditions. (CP 3.67).
The "invertible addition" is signified in algebra by "+", corresponding to what we'd call the exclusive disjunction of logical terms or the symmetric difference of their sets, ignoring many details and nuances that are often important, of course.
The "invertible addition" is signified in algebra by "+", corresponding to what we'd call the exclusive disjunction of logical terms or the symmetric difference of their sets, ignoring many details and nuances that are often important, of course.
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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. (CP 3.67).
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. (CP 3.67).
This is why Peirce trims his discussion of this point with the following hedge:
This is why Peirce trims his discussion of this point with the following hedge:
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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. (CP 3.67).
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. (CP 3.67).
Finally, a morphism with respect to addition, even a contingently qualified one, must do the right stuff on behalf of the additive identity:
Finally, a morphism with respect to addition, even a contingently qualified one, must do the right stuff on behalf of the additive identity:
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<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>
We arrive at the last, for the time being, of Peirce's statements about the "number of" map.
We arrive at the last, for the time being, of Peirce's statements about the "number of" map.
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<p>The conception of multiplication we have adopted is that of the application of one relation to another. …</p>
<p>The conception of multiplication we have adopted is that of the application of one relation to another. …</p>
Proviso for [''xy''] = [''x''][''y''] —
Proviso for [''xy''] = [''x''][''y''] —
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there are just as many ''x''’s per ''y'' as there are ''per'' things[,] things of the universe …
there are just as many ''x''’s per ''y'' as there are ''per'' things[,] things of the universe …
I have placed angle brackets around a comma that CP shows but CE omits, not that it helps much either way. So let us resort to the example:
I have placed angle brackets around a comma that CP shows but CE omits, not that it helps much either way. So let us resort to the example:
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<p>For instance, if our universe is perfect men, and there are as many teeth to a Frenchman (perfect understood) as there are to any one of the universe, then:</p>
<p>For instance, if our universe is perfect men, and there are as many teeth to a Frenchman (perfect understood) as there are to any one of the universe, then:</p>
Now let's see if we can use this picture to make sense of the following statement:
Now let's see if we can use this picture to make sense of the following statement:
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<p>For instance, if our universe is perfect men, and there are as many teeth to a Frenchman (perfect understood) as there are to any one of the universe, then:</p>
<p>For instance, if our universe is perfect men, and there are as many teeth to a Frenchman (perfect understood) as there are to any one of the universe, then:</p>
One more example and one more general observation, and then we will be all caught up with our homework on Peirce's "number of" function.
One more example and one more general observation, and then we will be all caught up with our homework on Peirce's "number of" function.
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<p>So if men are just as apt to be black as things in general:</p>
<p>So if men are just as apt to be black as things in general:</p>
Let's look at that last example from a different angle.
Let's look at that last example from a different angle.
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<p>So if men are just as apt to be black as things in general:</p>
<p>So if men are just as apt to be black as things in general:</p>
Let me try to sum up as succinctly as possible the lesson that we ought to take away from Peirce's last "number of" example, since I know that the account that I have given of it so far may appear to have wandered rather widely.
Let me try to sum up as succinctly as possible the lesson that we ought to take away from Peirce's last "number of" example, since I know that the account that I have given of it so far may appear to have wandered rather widely.
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<p>So if men are just as apt to be black as things in general:</p>
<p>So if men are just as apt to be black as things in general:</p>
And so we come to the end of the "number of" examples that we found on our agenda at this point in the text:
And so we come to the end of the "number of" examples that we found on our agenda at this point in the text:
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<p>It is to be observed that:</p>
<p>It is to be observed that:</p>
==Selection 12==
==Selection 12==
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<p>'''The Sign of Involution'''</p>
<p>'''The Sign of Involution'''</p>
Let us make a few preliminary observations about the "logical sign of involution", as Peirce uses it here:
Let us make a few preliminary observations about the "logical sign of involution", as Peirce uses it here:
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<p>'''The Sign of Involution'''</p>
<p>'''The Sign of Involution'''</p>