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==Selection 2==
==Selection 2==
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<p>'''Numbers Corresponding to Letters'''</p>
<p>'''Numbers Corresponding to Letters'''</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. Thus in a universe of perfect men (''men''), the number of "tooth of" would be 32. The number of a relative with two correlates would be the average number of things so related to a pair of individuals; and so on for relatives of higher numbers of correlates. I propose to denote the number of a logical term by enclosing the term in square brackets, thus <nowiki>[</nowiki>''t''<nowiki>]</nowiki>. (Peirce, CP 3.65).</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. Thus in a universe of perfect men (''men''), the number of "tooth of" would be 32. The number of a relative with two correlates would be the average number of things so related to a pair of individuals; and so on for relatives of higher numbers of correlates. I propose to denote the number of a logical term by enclosing the term in square brackets, thus <nowiki>[</nowiki>''t''<nowiki>]</nowiki>. (Peirce, CP 3.65).</p>
|}
Peirce's remarks at CP 3.65 are so replete with remarkable ideas, some of them so taken for granted in mathematical discourse that they usually escape explicit mention, and others so suggestive of things to come in a future remote from his time of writing, and yet so smoothly introduced in passing that it's all too easy to overlook their consequential significance, that I can do no better here than to highlight these ideas in other words, whose main advantage is to be a little more jarring to the mind's sensibilities.
Peirce's remarks at CP 3.65 are so replete with remarkable ideas, some of them so taken for granted in mathematical discourse that they usually escape explicit mention, and others so suggestive of things to come in a future remote from his time of writing, and yet so smoothly introduced in passing that it's all too easy to overlook their consequential significance, that I can do no better here than to highlight these ideas in other words, whose main advantage is to be a little more jarring to the mind's sensibilities.
==Selection 3==
==Selection 3==
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<p>'''The Signs of Inclusion, Equality, Etc.'''</p>
<p>'''The Signs of Inclusion, Equality, Etc.'''</p>
<p>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).</p>
<p>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).</p>
|}
The quantifier mapping from terms to their numbers that Peirce signifies by means of the square bracket notation has one of its principal uses in providing a basis for the computation of frequencies, probabilities, and all of the other statistical measures that can be constructed from these, and thus in affording what may be called a "principle of correspondence" between probability theory and its limiting case in the forms of logic.
The quantifier mapping from terms to their numbers that Peirce signifies by means of the square bracket notation has one of its principal uses in providing a basis for the computation of frequencies, probabilities, and all of the other statistical measures that can be constructed from these, and thus in affording what may be called a "principle of correspondence" between probability theory and its limiting case in the forms of logic.
==Selection 4==
==Selection 4==
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<p>'''The Signs for Addition'''</p>
<p>'''The Signs for Addition'''</p>
(Peirce, CP 3.67).
(Peirce, CP 3.67).
|}
A wealth of issues arise here that I hope to take up in depth at a later point, but for the moment I shall be able to mention only the barest sample of them in passing.
A wealth of issues arise here that I hope to take up in depth at a later point, but for the moment I shall be able to mention only the barest sample of them in passing.
==Selection 5==
==Selection 5==
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<p>'''The Signs for Multiplication'''</p>
<p>'''The Signs for Multiplication'''</p>
<p>whatever relative term 'x' may be. For what is a lover of something identical with anything, is the same as a lover of that thing. (Peirce, CP 3.68).</p>
<p>whatever relative term 'x' may be. For what is a lover of something identical with anything, is the same as a lover of that thing. (Peirce, CP 3.68).</p>
|}
Peirce in 1870 is five years down the road from the Peirce of 1865–1866 who lectured extensively on the role of sign relations in the logic of scientific inquiry, articulating their involvement in the three types of inference, and inventing the concept of "information" to explain what it is that signs convey in the process. By this time, then, the semiotic or sign relational approach to logic is so implicit in his way of working that he does not always take the trouble to point out its distinctive features at each and every turn. So let's take a moment to draw out a few of these characters.
Peirce in 1870 is five years down the road from the Peirce of 1865–1866 who lectured extensively on the role of sign relations in the logic of scientific inquiry, articulating their involvement in the three types of inference, and inventing the concept of "information" to explain what it is that signs convey in the process. By this time, then, the semiotic or sign relational approach to logic is so implicit in his way of working that he does not always take the trouble to point out its distinctive features at each and every turn. So let's take a moment to draw out a few of these characters.
==Selection 6==
==Selection 6==
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<p>'''The Signs for Multiplication''' (cont.)</p>
<p>'''The Signs for Multiplication''' (cont.)</p>
<p>(Peirce, CP 3.69–70).</p>
<p>(Peirce, CP 3.69–70).</p>
|}
Peirce's way of representing sets as sums may seem archaic, but it is quite often used, and is actually the tool of choice in many branches of algebra, combinatorics, computing, and statistics to this very day.
Peirce's way of representing sets as sums may seem archaic, but it is quite often used, and is actually the tool of choice in many branches of algebra, combinatorics, computing, and statistics to this very day.
==Selection 7==
==Selection 7==
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<p>'''The Signs for Multiplication''' (cont.)</p>
<p>'''The Signs for Multiplication''' (cont.)</p>
<p>I would suggest that such a notation may be found useful in treating other cases of non-associative multiplication. By comparing this with what was said above [in CP 3.55] concerning functional multiplication, it appears that multiplication by a conjugative term is functional, and that the letter denoting such a term is a symbol of operation. I am therefore using two alphabets, the Greek and Kennerly, where only one was necessary. But it is convenient to use both. (Peirce, CP 3.71–72).</p>
<p>I would suggest that such a notation may be found useful in treating other cases of non-associative multiplication. By comparing this with what was said above [in CP 3.55] concerning functional multiplication, it appears that multiplication by a conjugative term is functional, and that the letter denoting such a term is a symbol of operation. I am therefore using two alphabets, the Greek and Kennerly, where only one was necessary. But it is convenient to use both. (Peirce, CP 3.71–72).</p>
|}
NB. On account of the fact that various listservers balk at Peirce's "marks of reference" I will make the following substitutions in Peirce's text:
NB. On account of the fact that various listservers balk at Peirce's "marks of reference" I will make the following substitutions in Peirce's text:
==Selection 8==
==Selection 8==
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<p>'''The Signs for Multiplication''' (cont.)</p>
<p>'''The Signs for Multiplication''' (cont.)</p>
<p>(Peirce, CP 3.73).</p>
<p>(Peirce, CP 3.73).</p>
|}
===Commentary Note 8.1===
===Commentary Note 8.1===
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>
<p>(Peirce, CP 3.73).</p>
<p>(Peirce, CP 3.73).</p>
|}
In any system where elements are organized according to types, there tend to be any number of ways in which elements of one type are naturally associated with elements of another type. If the association is anything like a logical equivalence, but with the first type being "lower" and the second type being "higher" in some sense, then one frequently speaks of a "semantic ascent" from the lower to the higher type.
In any system where elements are organized according to types, there tend to be any number of ways in which elements of one type are naturally associated with elements of another type. If the association is anything like a logical equivalence, but with the first type being "lower" and the second type being "higher" in some sense, then one frequently speaks of a "semantic ascent" from the lower to the higher type.
==Selection 9==
==Selection 9==
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<p>'''The Signs for Multiplication''' (cont.)</p>
<p>'''The Signs for Multiplication''' (cont.)</p>
<p>(Peirce, CP 3.74).</p>
<p>(Peirce, CP 3.74).</p>
|}
===Commentary Note 9.1===
===Commentary Note 9.1===
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>
<p>It is equally clear that the mental operation above described is of such a nature that its effect is not altered by repetition. Suppose that by a definite act of conception the attention has been fixed upon men, and that by another exercise of the same faculty we limit it to those of the race who are white. Then any further repetition of the latter mental act, by which the attention is limited to white objects, does not in any way modify the conception arrived at, viz., that of white men. This is also an example of a general law of the mind, and it has its formal expression in the law ((2) Chap. II) of the literal symbols [namely, ''x''<sup>2</sup> = ''x'']. (Boole, ''Laws of Thought'', 44–45).</p>
<p>It is equally clear that the mental operation above described is of such a nature that its effect is not altered by repetition. Suppose that by a definite act of conception the attention has been fixed upon men, and that by another exercise of the same faculty we limit it to those of the race who are white. Then any further repetition of the latter mental act, by which the attention is limited to white objects, does not in any way modify the conception arrived at, viz., that of white men. This is also an example of a general law of the mind, and it has its formal expression in the law ((2) Chap. II) of the literal symbols [namely, ''x''<sup>2</sup> = ''x'']. (Boole, ''Laws of Thought'', 44–45).</p>
|}
===Commentary Note 9.2===
===Commentary Note 9.2===
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).
|}
===Commentary Note 9.3===
===Commentary Note 9.3===
==Selection 10==
==Selection 10==
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<p>'''The Signs for Multiplication''' (cont.)<p>
<p>'''The Signs for Multiplication''' (cont.)<p>
<p>(Peirce, CP 3.75).</p>
<p>(Peirce, CP 3.75).</p>
|}
===Commentary Note 10.1===
===Commentary Note 10.1===
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>
<p>The comma here after 'l' should not be considered as altering at all the meaning of 'l', but as only a subjacent sign, serving to alter the arrangement of the correlates. (Peirce, CP 3.73).</p>
<p>The comma here after 'l' should not be considered as altering at all the meaning of 'l', but as only a subjacent sign, serving to alter the arrangement of the correlates. (Peirce, CP 3.73).</p>
|}
Just to plant our feet on a more solid stage, let's apply this idea to the Othello example.
Just to plant our feet on a more solid stage, let's apply this idea to the Othello example.
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>
<p>Thus we see that, after one comma is added, the addition of another does not change the meaning at all, so that whatever has one comma after it must be regarded as having an infinite number. (Peirce, CP 3.73).</p>
<p>Thus we see that, after one comma is added, the addition of another does not change the meaning at all, so that whatever has one comma after it must be regarded as having an infinite number. (Peirce, CP 3.73).</p>
|}
Again, let us check whether this makes sense on the stage of our small but dramatic model.
Again, let us check whether this makes sense on the stage of our small but dramatic model.
==Selection 11==
==Selection 11==
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<p>'''The Signs for Multiplication''' (concl.)</p>
<p>'''The Signs for Multiplication''' (concl.)</p>
<p>Our logical multiplication, then, satisfies the essential conditions of multiplication, has a unity, has a conception similar to that of admitted multiplications, and contains numerical multiplication as a case under it. (Peirce, CP 3.76).</p>
<p>Our logical multiplication, then, satisfies the essential conditions of multiplication, has a unity, has a conception similar to that of admitted multiplications, and contains numerical multiplication as a case under it. (Peirce, CP 3.76).</p>
|}
===Commentary Note 11.1===
===Commentary Note 11.1===
'''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>
<p>I propose to denote the number of a logical term by enclosing the term in square brackets, thus [''t'']. (Peirce, CP 3.65).</p>
<p>I propose to denote the number of a logical term by enclosing the term in square brackets, thus [''t'']. (Peirce, CP 3.65).</p>
|}
'''NOF 2'''
'''NOF 2'''
<blockquote>
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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'''
<blockquote>
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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>(Peirce, CP 3.67).</p>
<p>(Peirce, CP 3.67).</p>
|}
'''NOF 4'''
'''NOF 4'''
<blockquote>
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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>
<p>Our logical multiplication, then, satisfies the essential conditions of multiplication, has a unity, has a conception similar to that of admitted multiplications, and contains numerical multiplication as a case under it. (Peirce, CP 3.76).</p>
<p>Our logical multiplication, then, satisfies the essential conditions of multiplication, has a unity, has a conception similar to that of admitted multiplications, and contains numerical multiplication as a case under it. (Peirce, CP 3.76).</p>
|}
Before I can discuss Peirce's "number of" function in greater detail I will need to deal with an expositional difficulty that I have been very carefully dancing around all this time, but that will no longer abide its assigned place under the rug.
Before I can discuss Peirce's "number of" function in greater detail I will need to deal with an expositional difficulty that I have been very carefully dancing around all this time, but that will no longer abide its assigned place under the rug.
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.
<blockquote>
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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:
| ''P'' is (≤1)-regular at ''Y''.
| ''P'' is (≤1)-regular at ''Y''.
|}
|}
|}
''E''<sub>1</sub> exemplifies the quality of "totality at ''X''".
''E''<sub>1</sub> exemplifies the quality of "totality at ''X''".
</pre>
</pre>
<blockquote>
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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>
<p>''P'' is a "pre-function" ''P'' : ''X'' <~ ''Y'' iff ''P'' is tubular at ''Y''.</p>
<p>''P'' is a "pre-function" ''P'' : ''X'' <~ ''Y'' iff ''P'' is tubular at ''Y''.</p>
|}
So, ''E''<sub>3</sub> is a pre-function ''e''<sub>3</sub> : ''X'' ~> ''Y'', and ''E''<sub>4</sub> is a pre-function ''e''<sub>4</sub> : ''X'' <~ ''Y''.
So, ''E''<sub>3</sub> is a pre-function ''e''<sub>3</sub> : ''X'' ~> ''Y'', and ''E''<sub>4</sub> is a pre-function ''e''<sub>4</sub> : ''X'' <~ ''Y''.
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.
<blockquote>
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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>
| |''L''<sub>''x''.''j''</sub>| ≥ ''c'' for all ''x'' in ''X''<sub>''j''</sub>.
| |''L''<sub>''x''.''j''</sub>| ≥ ''c'' for all ''x'' in ''X''<sub>''j''</sub>.
|}
|}
|}
Clearly, if any relation is (≤''c'')-regular on one of its domains ''X''<sub>''j''</sub> and also (≥''c'')-regular on the same domain, then it must be (=''c'')-regular on the affected domain ''X''<sub>''j''</sub>, in effect, ''c''-regular at ''j''.
Clearly, if any relation is (≤''c'')-regular on one of its domains ''X''<sub>''j''</sub> and also (≥''c'')-regular on the same domain, then it must be (=''c'')-regular on the affected domain ''X''<sub>''j''</sub>, in effect, ''c''-regular at ''j''.
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.
<blockquote>
{| align="center" cellpadding="8" width="90%" <!--QUOTE-->
|
<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>
| ''P'' is (≤1)-regular at ''Y''.
| ''P'' is (≤1)-regular at ''Y''.
|}
|}
|}
We have already looked at 2-adic relations that separately exemplify each of these regularities.
We have already looked at 2-adic relations that separately exemplify each of these regularities.
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:
<blockquote>
{| align="center" cellpadding="8" width="90%" <!--QUOTE-->
|
{| cellpadding="4"
{| cellpadding="4"
| ''P'' is a "pre-function" ''P'' : ''X'' ~> ''Y''
| ''P'' is a "pre-function" ''P'' : ''X'' ~> ''Y''
| ''P'' is tubular at ''Y''.
| ''P'' is tubular at ''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'').
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'').
<blockquote>
{| align="center" cellpadding="8" width="90%" <!--QUOTE-->
|
<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>
| ''P'' is 1-regular at ''Y''.
| ''P'' is 1-regular at ''Y''.
|}
|}
|}
For example, let ''X'' = ''Y'' = {0, …, 9} and let ''F'' ⊆ ''X'' × ''Y'' be the 2-adic relation that is depicted in the bigraph below:
For example, let ''X'' = ''Y'' = {0, …, 9} and let ''F'' ⊆ ''X'' × ''Y'' be the 2-adic relation that is depicted in the bigraph below:
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:
<blockquote>
{| align="center" cellpadding="8" width="90%" <!--QUOTE-->
|
{| cellpadding="4"
{| cellpadding="4"
| ''f'' is "surjective"
| ''f'' is "surjective"
| ''f'' is 1-regular at ''Y''.
| ''f'' is 1-regular at ''Y''.
|}
|}
|}
For example, or more precisely, contra example, the function ''f'' : ''X'' → ''Y'' that is depicted below is neither total at ''Y'' nor tubular at ''Y'', and so it cannot enjoy any of the properties of being sur-, or in-, or bi-jective.
For example, or more precisely, contra example, the function ''f'' : ''X'' → ''Y'' that is depicted below is neither total at ''Y'' nor tubular at ''Y'', and so it cannot enjoy any of the properties of being sur-, or in-, or bi-jective.
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:
<blockquote>
{| align="center" cellpadding="8" width="90%" <!--QUOTE-->
|
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.
|}
It is more like the feathers of the arrows that serve to mark the relational domains at which the relations ''J'', ''K'', ''L'' are functional, but it would take yet another construction to make this precise, as the feathers are not uniquely appointed but many splintered.
It is more like the feathers of the arrows that serve to mark the relational domains at which the relations ''J'', ''K'', ''L'' are functional, but it would take yet another construction to make this precise, as the feathers are not uniquely appointed but many splintered.
* [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].
<blockquote>
{| align="center" cellpadding="8" width="90%" <!--QUOTE-->
|
<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>
<p>I propose to denote the number of a logical term by enclosing the term in square brackets, thus [''t'']. (Peirce, CP 3.65).</p>
<p>I propose to denote the number of a logical term by enclosing the term in square brackets, thus [''t'']. (Peirce, CP 3.65).</p>
|}
We may formalize the role of the "number of" function by assigning it a local habitation and a name ''v'' : ''S'' → '''R''', where ''S'' is a suitable set of signs, called the ''syntactic domain'', that is ample enough to hold all of the terms that we might wish to number in a given discussion, and where '''R''' is the real number domain.
We may formalize the role of the "number of" function by assigning it a local habitation and a name ''v'' : ''S'' → '''R''', where ''S'' is a suitable set of signs, called the ''syntactic domain'', that is ample enough to hold all of the terms that we might wish to number in a given discussion, and where '''R''' is the real number domain.
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.
<blockquote>
{| align="center" cellpadding="8" width="90%" <!--QUOTE-->
|
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 is here remarking on the principle that the measure ''v'' on terms "preserves" or "respects" the prevailing implication, inclusion, or subsumption relations that impose an ordering on those terms.
Peirce is here remarking on the principle that the measure ''v'' on terms "preserves" or "respects" the prevailing implication, inclusion, or subsumption relations that impose an ordering on those terms.
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.
<blockquote>
{| align="center" cellpadding="8" width="90%" <!--QUOTE-->
|
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 "regular non-invertible addition" is signified by "+,", corresponding to what we'd call the inclusive disjunction of logical terms or the union of their extensions as sets.
The "regular non-invertible addition" is signified by "+,", corresponding to what we'd call the inclusive disjunction of logical terms or the union of their extensions as sets.
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.
<blockquote>
{| align="center" cellpadding="8" width="90%" <!--QUOTE-->
|
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).
|}
A full interpretation of this remark will require us to pick up the precise technical sense in which Peirce is using the word "collection", and that will take us back to his logical reconstruction of certain aspects of number theory, all of which I am putting off to another time, but it is still possible to get a rough sense of what he's saying relative to the present frame of discussion.
A full interpretation of this remark will require us to pick up the precise technical sense in which Peirce is using the word "collection", and that will take us back to his logical reconstruction of certain aspects of number theory, all of which I am putting off to another time, but it is still possible to get a rough sense of what he's saying relative to the present frame of discussion.
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:
<blockquote>
{| align="center" cellpadding="8" width="90%" <!--QUOTE-->
|
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:
<blockquote>
{| align="center" cellpadding="8" width="90%" <!--QUOTE-->
|
<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>
<p>(Peirce, CP 3.67).</p>
<p>(Peirce, CP 3.67).</p>
|}
With respect to the nullity 0 in ''S'' and the number 0 in '''R''', we have:
With respect to the nullity 0 in ''S'' and the number 0 in '''R''', we have:
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.
<blockquote>
{| align="center" cellpadding="8" width="90%" <!--QUOTE-->
|
<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>
<p>(Peirce, CP 3.76).</p>
<p>(Peirce, CP 3.76).</p>
|}
Peirce is here observing what we might dub a "contingent morphism" or a "skeptraphotic arrow", if you will. Provided that a certain condition, to be named and, what is more hopeful, to be clarified in short order, happens to be satisfied, we would find it holding that the "number of" map ''v'' : ''S'' → '''R''' such that ''vs'' = [''s''] serves to preserve the multiplication of relative terms, that is as much to say, the composition of relations, in the form: [''xy''] = [''x''][''y''].
Peirce is here observing what we might dub a "contingent morphism" or a "skeptraphotic arrow", if you will. Provided that a certain condition, to be named and, what is more hopeful, to be clarified in short order, happens to be satisfied, we would find it holding that the "number of" map ''v'' : ''S'' → '''R''' such that ''vs'' = [''s''] serves to preserve the multiplication of relative terms, that is as much to say, the composition of relations, in the form: [''xy''] = [''x''][''y''].
Proviso for [''xy''] = [''x''][''y''] —
Proviso for [''xy''] = [''x''][''y''] —
<blockquote>
{| align="center" cellpadding="8" width="90%" <!--QUOTE-->
|
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:
<blockquote>
{| align="center" cellpadding="8" width="90%" <!--QUOTE-->
|
<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>
<p>holds arithmetically. (CP 3.76).</p>
<p>holds arithmetically. (CP 3.76).</p>
|}
Now that is something that we can sink our teeth into, and trace the bigraph representation of the situation. In order to do this, it will help to recall our first examination of the "tooth of" relation, and to adjust the picture that we sketched of it on that occasion.
Now that is something that we can sink our teeth into, and trace the bigraph representation of the situation. In order to do this, it will help to recall our first examination of the "tooth of" relation, and to adjust the picture that we sketched of it on that occasion.
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:
<blockquote>
{| align="center" cellpadding="8" width="90%" <!--QUOTE-->
|
<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>
<p>holds arithmetically. (CP 3.76).</p>
<p>holds arithmetically. (CP 3.76).</p>
|}
In the lingua franca of statistics, Peirce is saying this: That if the population of Frenchmen is a "fair sample" of the general population with regard to dentition, then the morphic equation [''tf''] = [''t''][''f''], whose transpose gives [''t''] = [''tf'']/[''f''], is every bite as true as the defining equation in this circumstance, namely, [''t''] = [''tm'']/[''m''].
In the lingua franca of statistics, Peirce is saying this: That if the population of Frenchmen is a "fair sample" of the general population with regard to dentition, then the morphic equation [''tf''] = [''t''][''f''], whose transpose gives [''t''] = [''tf'']/[''f''], is every bite as true as the defining equation in this circumstance, namely, [''t''] = [''tm'']/[''m''].
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.
<blockquote>
{| align="center" cellpadding="8" width="90%" <!--QUOTE-->
|
<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>
<p>(Peirce, CP 3.76).</p>
<p>(Peirce, CP 3.76).</p>
|}
The protasis, "men are just as apt to be black as things in general", is elliptic in structure, and presents us with a potential ambiguity. If we had no further clue to its meaning, it might be read as either of the following:
The protasis, "men are just as apt to be black as things in general", is elliptic in structure, and presents us with a potential ambiguity. If we had no further clue to its meaning, it might be read as either of the following:
Let's look at that last example from a different angle.
Let's look at that last example from a different angle.
<blockquote>
{| align="center" cellpadding="8" width="90%" <!--QUOTE-->
|
<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>
<p>(Peirce, CP 3.76).
<p>(Peirce, CP 3.76).
|}
In different lights the formula [''m'',''b''] = [''m'',][''b''] presents itself as an "aimed arrow", "fair sample", or "independence" condition.
In different lights the formula [''m'',''b''] = [''m'',][''b''] presents itself as an "aimed arrow", "fair sample", or "independence" condition.
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.
<blockquote>
{| align="center" cellpadding="8" width="90%" <!--QUOTE-->
|
<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>
<p>C.S. Peirce, CP 3.76</p>
<p>C.S. Peirce, CP 3.76</p>
|}
In different lights the formula [''m'',''b''] = [''m'',][''b''] presents itself as an "aimed arrow", "fair sample", or "independence" condition. I had taken the tack of illustrating this polymorphous theme in bas relief, that is, via detour through a universe of discourse where it fails. Here's a brief reminder of the Othello example:
In different lights the formula [''m'',''b''] = [''m'',][''b''] presents itself as an "aimed arrow", "fair sample", or "independence" condition. I had taken the tack of illustrating this polymorphous theme in bas relief, that is, via detour through a universe of discourse where it fails. Here's a brief reminder of the Othello example:
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:
<blockquote>
{| align="center" cellpadding="8" width="90%" <!--QUOTE-->
|
<p>It is to be observed that:</p>
<p>It is to be observed that:</p>
<p>C.S. Peirce, CP 3.76</p>
<p>C.S. Peirce, CP 3.76</p>
|}
There appears to be a problem with the printing of the text at this point. Let us first recall the conventions that I am using in this transcription: `1` for the "antique 1" that Peirce defines as !1!<sub>∞</sub> = "something", and !1! for the "bold 1" that signifies the ordinary 2-identity relation.
There appears to be a problem with the printing of the text at this point. Let us first recall the conventions that I am using in this transcription: `1` for the "antique 1" that Peirce defines as !1!<sub>∞</sub> = "something", and !1! for the "bold 1" that signifies the ordinary 2-identity relation.
==Selection 12==
==Selection 12==
<blockquote>
{| align="center" cellpadding="8" width="90%" <!--QUOTE-->
|
<p>'''The Sign of Involution'''</p>
<p>'''The Sign of Involution'''</p>
<p>(C.S. Peirce, CP 3.77).</p>
<p>(C.S. Peirce, CP 3.77).</p>
|}
===Commentary Note 12===
===Commentary Note 12===
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:
<blockquote>
{| align="center" cellpadding="8" width="90%" <!--QUOTE-->
|
<p>'''The Sign of Involution'''</p>
<p>'''The Sign of Involution'''</p>
<p>(C.S. Peirce, CP 3.77).</p>
<p>(C.S. Peirce, CP 3.77).</p>
|}
In arithmetic, the "involution" ''x''<sup>''y''</sup>, or the "exponentiation" of ''x'' to the power of ''y'', is the iterated multiplication of the factor ''x'', repeated as many times as there are ones making up the exponent ''y''.
In arithmetic, the "involution" ''x''<sup>''y''</sup>, or the "exponentiation" of ''x'' to the power of ''y'', is the iterated multiplication of the factor ''x'', repeated as many times as there are ones making up the exponent ''y''.
