Supplement: I think, in my below text there are a lot of mistakes: I mixed up tuples with products, I am not sure when to use round and when winged brackets, and the set of unordered triples does not consist of three, but of six sets of ordered ones, and I dont know what else. I must read your wikipedia papers about relations first, I think. It is very interesting, eg. it seems to me, that other than with dyadic products, with triadic ones there are a lot of different kinds of symmetry, rotational and linear. Happy Christmas and happy new Year!
Jon, list,
about ordered and unordered pairs: In the mathematical books I had read in, there was only the way of writing ordered pairs. And symmetry was only explained by the example of a subset of a product of two same sets (A x A). I had thought then, if you have two different sets, A and B, symmetry makes no sense, lest you look at a subset of {(A x B) U (B x A)}. You wrote, the short way of writing this, is {A x B}. That would be an aggregate of unordered pairs. Now, if you have three sets, A, B, C, then how do you write the unordered cartesian product? Should be {A, B, C} written in a triangle, with one "x" in the centre, or "{A x B x C}"? (short form) ? Or {(A x B x C) U (B x C x A) U (C x A x B)} (long form)?
I still find relation reduction interesting. I must read your paper again, about the projective reduction, and try to find out, whether it is possible to projectively reduce a triadic relation of R, O, I to three dyadic ones, but not R-O, O-I, I-R, but R-R, R-O, R-I. Because I think, this is the Peircean way. Have you tried that? Im not sure, when I will have time resp. overcome my laziness to, well, just take a piece of paper and start. To just start is always the most difficult part, like with the tax declaration, but other than with that, there is no penalty if I dont.
Best regards,
Helmut
Inquiry Blog
http://inquiryintoinquiry.com/2015/12/08/relations-their-relatives-16/
http://inquiryintoinquiry.com/2015/12/10/relations-their-relatives-17/
http://inquiryintoinquiry.com/2015/12/12/relations-their-relatives-18/
Peirce List
JA:http://permalink.gmane.org/gmane.science.philosophy.peirce/17890
GF:http://permalink.gmane.org/gmane.science.philosophy.peirce/17894
JBD:http://permalink.gmane.org/gmane.science.philosophy.peirce/17902
JA:http://permalink.gmane.org/gmane.science.philosophy.peirce/17907
HR:http://permalink.gmane.org/gmane.science.philosophy.peirce/17911
GF:http://permalink.gmane.org/gmane.science.philosophy.peirce/17916
JA:http://permalink.gmane.org/gmane.science.philosophy.peirce/17955
JA:http://permalink.gmane.org/gmane.science.philosophy.peirce/17956
HR:http://permalink.gmane.org/gmane.science.philosophy.peirce/17958
JA:http://permalink.gmane.org/gmane.science.philosophy.peirce/17991
JA:http://permalink.gmane.org/gmane.science.philosophy.peirce/18002
HR:http://permalink.gmane.org/gmane.science.philosophy.peirce/18003
HR:http://permalink.gmane.org/gmane.science.philosophy.peirce/18007
Helmut, List,
I would not want the dyadic case to detain us too long,
as often happens when we frame a simple example for the
purpose of illustration and then fail to rise beyond it.
I raised the example of biblical brothers simply as a way of
illustrating the distinction between a relation proper, like
that symbolized by the formula “x is y's brother” and any of
its elementary relations, like the ordered pair (Cain, Abel).
There are, however, a few more points that could be
illustrated within the scope of this simple example.
Recall that we had a universe of discourse X consisting of
biblical figures and a 2-place relation B forming a subset
of the cartesian product X x X such that (x, y) is in B if
and only if x is a brother of y.
The “biblical brother relation” B would contain a large number of
elementary dyadic relations, or ordered pairs (x, y), for example:
(Abel, Cain), (Isaac, Ishmael), (Esau, Jacob), (Benjamin, Joseph), …
(Cain, Abel), (Ishmael, Isaac), (Jacob, Esau), (Joseph, Benjamin), …
Because B is a symmetric relation, each unordered pair {x, y}
makes its appearance as two ordered pairs, (x, y) and (y, x).
The extension of the elder brother relation E would have the pairs:
(Cain, Abel), (Ishmael, Isaac), (Esau, Jacob), (Joseph, Benjamin), …
Peirce regarded a set of tuples as an “aggregate” or “logical sum”
and would have written the above subset of B in the following way:
B = Abel:Cain +, Isaac:Ishmael +, Esau:Jacob +, Benjamin:Joseph +, …
+, Cain:Abel +, Ishmael:Isaac +, Jacob:Esau +, Joseph:Benjamin +, …
So what does all this -- the distinction between relations in general
and elementary relations plus the analysis of relations in general
as sets or sums of elementary relations -- imply for the case of
triadic relations in general and sign relations in particular?
It means that non-trivial examples of triadic relations are aggregates,
logical sums, or sets of many elementary triadic relations or triples.
As a result, the classification of single triples and their components
gets us only so far in the classification of triadic relations proper,
and except in very special cases not very far at all.
Regards,
Jon
On 12/12/2015 4:32 AM, Helmut Raulien wrote:
> Supplement: I suspect, that my below consideration is non-Peircean, as far as I
> know, because I ony know examples by Peirce, that are about relatives, that is
> terms, i.e. language. Language, of course, can only be inter-subjective. An
> intra-subjective consideration as below may be weird or incalculable, but I
> guess, it can be interesting: In the mentioned (Alice loves Bob) case, it shows
> a difference between language and reality: Language suggests, that "loves" in
> "Alice loves Bob" denotes a relation between Alice and Bob. But a closer look
> shows, that in fact it is about a relation inside Alices mind (Bob might be a
> movie star, whom Alice only has seen on a TV screen).
> Jon, List,
> Thank you. I am happy, that I now am more or less clear about the difference eg.
> between relation and relative term, and general and elementary. I find it
> complicated to apply the mathematical relation concept to realworld situations.
> There seem to be relations (and relative terms) of the mind, and others of the
> material-energetic world. Eg. if there is a wall made of bricks, one can tell
> the relations each brick has towards another brick, and so define the topology
> of the wall with relations from relative terms like "is above of", "is north
> of", and so on. But if Alice loves Bob, then this is a relation in Alices mind
> (a subset of a product of the set of all aspects in Alices mind with itself).
> And "Alice and Bob love each other" perhaps is a relation between the relations
> in Alices mind, and those in Bobs mind. But which are these aspects of the mind?
> Not very easy, all this, I mean, at least at this intra-subjective level. Maybe
> it leads astray to some sort of obsolete reductionism, I dont know.
> Best,
> Helmut
--
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http://inquiryintoinquiry.com/2015/12/08/relations-their-relatives-16/
http://inquiryintoinquiry.com/2015/12/10/relations-their-relatives-17/
http://inquiryintoinquiry.com/2015/12/12/relations-their-relatives-18/
Peirce List
JA:http://permalink.gmane.org/gmane.science.philosophy.peirce/17890
GF:http://permalink.gmane.org/gmane.science.philosophy.peirce/17894
JBD:http://permalink.gmane.org/gmane.science.philosophy.peirce/17902
JA:http://permalink.gmane.org/gmane.science.philosophy.peirce/17907
HR:http://permalink.gmane.org/gmane.science.philosophy.peirce/17911
GF:http://permalink.gmane.org/gmane.science.philosophy.peirce/17916
JA:http://permalink.gmane.org/gmane.science.philosophy.peirce/17955
JA:http://permalink.gmane.org/gmane.science.philosophy.peirce/17956
HR:http://permalink.gmane.org/gmane.science.philosophy.peirce/17958
JA:http://permalink.gmane.org/gmane.science.philosophy.peirce/17991
JA:http://permalink.gmane.org/gmane.science.philosophy.peirce/18002
HR:http://permalink.gmane.org/gmane.science.philosophy.peirce/18003
HR:http://permalink.gmane.org/gmane.science.philosophy.peirce/18007
Helmut, List,
I would not want the dyadic case to detain us too long,
as often happens when we frame a simple example for the
purpose of illustration and then fail to rise beyond it.
I raised the example of biblical brothers simply as a way of
illustrating the distinction between a relation proper, like
that symbolized by the formula “x is y's brother” and any of
its elementary relations, like the ordered pair (Cain, Abel).
There are, however, a few more points that could be
illustrated within the scope of this simple example.
Recall that we had a universe of discourse X consisting of
biblical figures and a 2-place relation B forming a subset
of the cartesian product X x X such that (x, y) is in B if
and only if x is a brother of y.
The “biblical brother relation” B would contain a large number of
elementary dyadic relations, or ordered pairs (x, y), for example:
(Abel, Cain), (Isaac, Ishmael), (Esau, Jacob), (Benjamin, Joseph), …
(Cain, Abel), (Ishmael, Isaac), (Jacob, Esau), (Joseph, Benjamin), …
Because B is a symmetric relation, each unordered pair {x, y}
makes its appearance as two ordered pairs, (x, y) and (y, x).
The extension of the elder brother relation E would have the pairs:
(Cain, Abel), (Ishmael, Isaac), (Esau, Jacob), (Joseph, Benjamin), …
Peirce regarded a set of tuples as an “aggregate” or “logical sum”
and would have written the above subset of B in the following way:
B = Abel:Cain +, Isaac:Ishmael +, Esau:Jacob +, Benjamin:Joseph +, …
+, Cain:Abel +, Ishmael:Isaac +, Jacob:Esau +, Joseph:Benjamin +, …
So what does all this -- the distinction between relations in general
and elementary relations plus the analysis of relations in general
as sets or sums of elementary relations -- imply for the case of
triadic relations in general and sign relations in particular?
It means that non-trivial examples of triadic relations are aggregates,
logical sums, or sets of many elementary triadic relations or triples.
As a result, the classification of single triples and their components
gets us only so far in the classification of triadic relations proper,
and except in very special cases not very far at all.
Regards,
Jon
On 12/12/2015 4:32 AM, Helmut Raulien wrote:
> Supplement: I suspect, that my below consideration is non-Peircean, as far as I
> know, because I ony know examples by Peirce, that are about relatives, that is
> terms, i.e. language. Language, of course, can only be inter-subjective. An
> intra-subjective consideration as below may be weird or incalculable, but I
> guess, it can be interesting: In the mentioned (Alice loves Bob) case, it shows
> a difference between language and reality: Language suggests, that "loves" in
> "Alice loves Bob" denotes a relation between Alice and Bob. But a closer look
> shows, that in fact it is about a relation inside Alices mind (Bob might be a
> movie star, whom Alice only has seen on a TV screen).
> Jon, List,
> Thank you. I am happy, that I now am more or less clear about the difference eg.
> between relation and relative term, and general and elementary. I find it
> complicated to apply the mathematical relation concept to realworld situations.
> There seem to be relations (and relative terms) of the mind, and others of the
> material-energetic world. Eg. if there is a wall made of bricks, one can tell
> the relations each brick has towards another brick, and so define the topology
> of the wall with relations from relative terms like "is above of", "is north
> of", and so on. But if Alice loves Bob, then this is a relation in Alices mind
> (a subset of a product of the set of all aspects in Alices mind with itself).
> And "Alice and Bob love each other" perhaps is a relation between the relations
> in Alices mind, and those in Bobs mind. But which are these aspects of the mind?
> Not very easy, all this, I mean, at least at this intra-subjective level. Maybe
> it leads astray to some sort of obsolete reductionism, I dont know.
> Best,
> Helmut
--
academia: http://independent.academia.edu/JonAwbrey
my word press blog: http://inquiryintoinquiry.com/
inquiry list: http://stderr.org/pipermail/inquiry/
isw: http://intersci.ss.uci.edu/wiki/index.php/JLA
oeiswiki: http://www.oeis.org/wiki/User:Jon_Awbrey
facebook page: https://www.facebook.com/JonnyCache
-----------------------------
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