#cspeirce — Public Fediverse posts

Icon, Likeness, Likely Story, Likelihood, Probability • 3

Re: Peirce List • Phyllis Chiasson

A more complete excerpt and the translator’s notes are very helpful here.

A probability (εικος) is not the same as a sign (σηµειον).  The former is a generally accepted premiss ;  for that which people know to happen or not to happen, or to be or not to be, usually in a particular way, is a probability :  e.g., that the envious are malevolent or that those who are loved are affectionate.  A sign, however, means a demonstrative premiss which is necessary or generally accepted.1  That which coexists with something else, or before or after whose happening something else has happened, is a sign of that something’s having happened or being.

An enthymeme is a syllogism from probabilities or signs ;  and a sign can be taken in three ways — in just as many ways as there are of taking the middle term in the several figures :  either as in the first figure or as in the second or as in the third.

• E.g., the proof that a woman is pregnant because she has milk is by the first figure ;  for the middle term is ‘having milk’.  A stands for ‘pregnant’, B for ‘having milk’, and C for ‘woman’.
• The proof that the wise are good because Pittacus was good is by the third figure.  A stands for ‘good’, B for ‘the wise’, and C for Pittacus.  Then it is true to predicate both A and B of C ;  only we do not state the latter, because we know it, whereas we formally assume the former.
• The proof that a woman is pregnant because she is sallow is intended to be by the middle figure ;  for since sallowness is a characteristic of woman in pregnancy, and is associated with this particular woman, they suppose that she is proved to be pregnant.  A stands for ‘sallowness’, B for ‘being pregnant’, C for ‘woman’.

If only one premiss is stated, we get only a sign ;  but if the other premiss is assumed as well, we get a syllogism,2 e.g., that Pittacus is high-minded, because those who love honour are high-minded, and Pittacus loves honour ;  or again that the wise are good, because Pittacus is good and also wise.

In this way syllogisms can be effected ;  but whereas a syllogism in the first figure cannot be refuted if it is true, since it is universal, a syllogism in the last figure can be refuted even if the conclusion is true, because the syllogism is neither universal nor relevant to our purpose.3  For if Pittacus is good, it is not necessary for this reason that all other wise men are good.  A syllogism in the middle figure is always and in every way refutable, since we never get a syllogism with the terms in this relation4 ;  for it does not necessarily follow, if a pregnant woman is sallow, and this woman is sallow, that she is pregnant.  Thus truth can be found in all signs, but they differ in the ways which have been described.

We must either classify signs in this way, and regard their middle term as an index (τεκµηριον)5 (for the name ‘index’ is given to that which causes us to know, and the middle term is especially of this nature), or describe the arguments drawn from the extremes6 as ‘signs’, and that which is drawn from the middle as an ‘index’.  For the conclusion which is reached through the first figure is most generally accepted and most true.  (Aristotle, Prior Analytics 2.27, 70a3–70b6).

Translator’s Notes

1. If referable to one phenomenon only, a sign has objective necessity ;  if to more than one, its value is a matter of opinion.
2. Strictly an enthymeme.
3. If the signs of an enthymeme in the first figure are true, the conclusion is inevitable.  Aristotle does not mean that the conclusion is universal, but that the universality of the major premiss implies the validity of the minor and conclusion.  The example (<all> those who have honour, etc.) quoted for the third figure contains no universal premiss or sign, and fails to establish a universal conclusion.
4. i.e. when both premisses are affirmative.
5. Signs may be classified as irrefutable (1st figure) and refutable (2nd and 3rd figures), and the name ‘index’ may be attached to their middle terms, either in all figures or (more probably) only in the first, where the middle is distinctively middle.
6. Alternatively the name ‘sign’ may be restricted to the 2nd and 3rd figures, and may be replaced by ‘index’ in the first.

Reference

• Aristotle, “Prior Analytics”, Hugh Tredennick (trans.), pp. 181–531 in Aristotle, Volume 1, Loeb Classical Library, William Heinemann, London, UK, 1938.

Resource

• Theme One Program • User Guide • Appendix A

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Icon, Likeness, Likely Story, Likelihood, Probability • 3

Re: Peirce List • Phyllis Chiasson

A more complete excerpt and the translator’s notes are very helpful here.

A probability (εικος) is not the same as a sign (σηµειον).  The former is a generally accepted premiss ;  for that which people know to happen or not to happen, or to be or not to be, usually in a particular way, is a probability :  e.g., that the envious are malevolent or that those who are loved are affectionate.  A sign, however, means a demonstrative premiss which is necessary or generally accepted.1  That which coexists with something else, or before or after whose happening something else has happened, is a sign of that something’s having happened or being.

An enthymeme is a syllogism from probabilities or signs ;  and a sign can be taken in three ways — in just as many ways as there are of taking the middle term in the several figures :  either as in the first figure or as in the second or as in the third.

• E.g., the proof that a woman is pregnant because she has milk is by the first figure ;  for the middle term is ‘having milk’.  A stands for ‘pregnant’, B for ‘having milk’, and C for ‘woman’.
• The proof that the wise are good because Pittacus was good is by the third figure.  A stands for ‘good’, B for ‘the wise’, and C for Pittacus.  Then it is true to predicate both A and B of C ;  only we do not state the latter, because we know it, whereas we formally assume the former.
• The proof that a woman is pregnant because she is sallow is intended to be by the middle figure ;  for since sallowness is a characteristic of woman in pregnancy, and is associated with this particular woman, they suppose that she is proved to be pregnant.  A stands for ‘sallowness’, B for ‘being pregnant’, C for ‘woman’.

If only one premiss is stated, we get only a sign ;  but if the other premiss is assumed as well, we get a syllogism,2 e.g., that Pittacus is high-minded, because those who love honour are high-minded, and Pittacus loves honour ;  or again that the wise are good, because Pittacus is good and also wise.

In this way syllogisms can be effected ;  but whereas a syllogism in the first figure cannot be refuted if it is true, since it is universal, a syllogism in the last figure can be refuted even if the conclusion is true, because the syllogism is neither universal nor relevant to our purpose.3  For if Pittacus is good, it is not necessary for this reason that all other wise men are good.  A syllogism in the middle figure is always and in every way refutable, since we never get a syllogism with the terms in this relation4 ;  for it does not necessarily follow, if a pregnant woman is sallow, and this woman is sallow, that she is pregnant.  Thus truth can be found in all signs, but they differ in the ways which have been described.

We must either classify signs in this way, and regard their middle term as an index (τεκµηριον)5 (for the name ‘index’ is given to that which causes us to know, and the middle term is especially of this nature), or describe the arguments drawn from the extremes6 as ‘signs’, and that which is drawn from the middle as an ‘index’.  For the conclusion which is reached through the first figure is most generally accepted and most true.  (Aristotle, Prior Analytics 2.27, 70a3–70b6).

Translator’s Notes

1. If referable to one phenomenon only, a sign has objective necessity ;  if to more than one, its value is a matter of opinion.
2. Strictly an enthymeme.
3. If the signs of an enthymeme in the first figure are true, the conclusion is inevitable.  Aristotle does not mean that the conclusion is universal, but that the universality of the major premiss implies the validity of the minor and conclusion.  The example (<all> those who have honour, etc.) quoted for the third figure contains no universal premiss or sign, and fails to establish a universal conclusion.
4. i.e. when both premisses are affirmative.
5. Signs may be classified as irrefutable (1st figure) and refutable (2nd and 3rd figures), and the name ‘index’ may be attached to their middle terms, either in all figures or (more probably) only in the first, where the middle is distinctively middle.
6. Alternatively the name ‘sign’ may be restricted to the 2nd and 3rd figures, and may be replaced by ‘index’ in the first.

Reference

• Aristotle, “Prior Analytics”, Hugh Tredennick (trans.), pp. 181–531 in Aristotle, Volume 1, Loeb Classical Library, William Heinemann, London, UK, 1938.

Resource

• Theme One Program • User Guide • Appendix A

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Icon, Likeness, Likely Story, Likelihood, Probability • 2

Re: Peirce List • Phyllis Chiasson

I’m still a bit fuzzy on how Aristotle’s account relates to Peirce’s usage, though I’m pretty sure Peirce must have taken Aristotle’s usage into account, but it does seem that Aristotle drew some sort of distinction here, using a term “tekmerion” which gets translated as “index” to make the following remark later on in that chapter.

We must either classify signs in this way, and regard their middle term as an index [τεκµηριον] (for the name ‘index’ is given to that which causes us to know, and the middle term is especially of this nature), or describe the arguments drawn from the extremes as ‘signs’, and that which is drawn from the middle as an ‘index’.  For the conclusion which is reached through the first figure is most generally accepted and most true.  (Aristotle, Prior Analytics, 2.27.70b1–6).

Reference

• Aristotle, “Prior Analytics”, Hugh Tredennick (trans.), pp. 181–531 in Aristotle, Volume 1, Loeb Classical Library, William Heinemann, London, UK, 1938.

Resource

• Theme One Program • User Guide • Appendix A

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Icon, Likeness, Likely Story, Likelihood, Probability • 2

Re: Peirce List • Phyllis Chiasson

I’m still a bit fuzzy on how Aristotle’s account relates to Peirce’s usage, though I’m pretty sure Peirce must have taken Aristotle’s usage into account, but it does seem that Aristotle drew some sort of distinction here, using a term “tekmerion” which gets translated as “index” to make the following remark later on in that chapter.

We must either classify signs in this way, and regard their middle term as an index [τεκµηριον] (for the name ‘index’ is given to that which causes us to know, and the middle term is especially of this nature), or describe the arguments drawn from the extremes as ‘signs’, and that which is drawn from the middle as an ‘index’.  For the conclusion which is reached through the first figure is most generally accepted and most true.  (Aristotle, Prior Analytics, 2.27.70b1–6).

Reference

• Aristotle, “Prior Analytics”, Hugh Tredennick (trans.), pp. 181–531 in Aristotle, Volume 1, Loeb Classical Library, William Heinemann, London, UK, 1938.

Resource

• Theme One Program • User Guide • Appendix A

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Icon, Likeness, Likely Story, Likelihood, Probability • 1

Here’s a likely locus classicus for “icon” in its logical sense —

A probability (εικος) is not the same as a sign (σηµειον).  The former is a generally accepted premiss;  for that which people know to happen or not to happen, or to be or not to be, usually in a particular way, is a probability:  e.g., that the envious are malevolent or that those who are loved are affectionate.  A sign, however, means a demonstrative premiss which is necessary or generally accepted.  That which coexists with something else, or before or after whose happening something else has happened, is a sign of that something’s having happened or being.  (Aristotle, Prior Analytics, 2.27.70a3–10).

Reference

• Aristotle, “Prior Analytics”, Hugh Tredennick (trans.), pp. 181–531 in Aristotle, Volume 1, Loeb Classical Library, William Heinemann, London, UK, 1938.

Resource

• Theme One Program • User Guide • Appendix A

Related Discussion

• Peirce List • Benjamin Udell • Michael Shapiro

Icon, Likeness, Likely Story, Likelihood, Probability • 1

Here’s a likely locus classicus for “icon” in its logical sense —

A probability (εικος) is not the same as a sign (σηµειον).  The former is a generally accepted premiss;  for that which people know to happen or not to happen, or to be or not to be, usually in a particular way, is a probability:  e.g., that the envious are malevolent or that those who are loved are affectionate.  A sign, however, means a demonstrative premiss which is necessary or generally accepted.  That which coexists with something else, or before or after whose happening something else has happened, is a sign of that something’s having happened or being.  (Aristotle, Prior Analytics, 2.27.70a3–10).

Reference

• Aristotle, “Prior Analytics”, Hugh Tredennick (trans.), pp. 181–531 in Aristotle, Volume 1, Loeb Classical Library, William Heinemann, London, UK, 1938.

Resource

• Theme One Program • User Guide • Appendix A

Related Discussion

• Peirce List • Benjamin Udell • Michael Shapiro

Reflection On Recursion • Discussion 1

Re: Reflection On Recursion • 1
Re: Laws of Form • John Mingers

Also the work of Maturana and Varela on autopoiesis and the neurophysiology of cognition which also has recursion at its heart.

Thanks, John.  Yes, we certainly find the whole array of self concepts coming into play here — selfhood, autopoiesis or self creation, self reference and self transformation, just to name a few.  But one thing I need to emphasize from the start is how radically different such concepts appear when viewed under x‑rays of Peirce’s pragmatic semiotics.

I forget where I first heard it, but it’s fairly common observation that the persistence of a recurring problem is a symptom of how unlikely it is to be solved in the paradigm where it keeps occurring.

After a while, it simply becomes time to change the paradigm …

Just by way of a first example, take the very idea of “self‑reference”.  The moment we place it in the medium of triadic sign relations we realize signs do not refer to anything at all except insofar as an interpreter refers them.  And when we think to ask, “What is this that we call an interpreter?”, the pragmatic theory of signs tells us we do not know when we turn out the light but under the x‑ray of the pragmatic maxim the sum of its effects is effectively modeled by an extended triadic sign relation.

Everything I’ll be working at here will be done within a framework like that.

Regards,
Jon

Resources

• Inquiry Driven Systems • Inquiry Into Inquiry
• Reflective Interpretive Frameworks
• The Phenomenology of Reflection
• Higher Order Sign Relations

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Reflection On Recursion • Discussion 1

Re: Reflection On Recursion • 1
Re: Laws of Form • John Mingers

Also the work of Maturana and Varela on autopoiesis and the neurophysiology of cognition which also has recursion at its heart.

Thanks, John.  Yes, we certainly find the whole array of self concepts coming into play here — selfhood, autopoiesis or self creation, self reference and self transformation, just to name a few.  But one thing I need to emphasize from the start is how radically different such concepts appear when viewed under x‑rays of Peirce’s pragmatic semiotics.

I forget where I first heard it, but it’s fairly common observation that the persistence of a recurring problem is a symptom of how unlikely it is to be solved in the paradigm where it keeps occurring.

After a while, it simply becomes time to change the paradigm …

Just by way of a first example, take the very idea of “self‑reference”.  The moment we place it in the medium of triadic sign relations we realize signs do not refer to anything at all except insofar as an interpreter refers them.  And when we think to ask, “What is this that we call an interpreter?”, the pragmatic theory of signs tells us we do not know when we turn out the light but under the x‑ray of the pragmatic maxim the sum of its effects is effectively modeled by an extended triadic sign relation.

Everything I’ll be working at here will be done within a framework like that.

Regards,
Jon

Resources

• Inquiry Driven Systems • Inquiry Into Inquiry
• Reflective Interpretive Frameworks
• The Phenomenology of Reflection
• Higher Order Sign Relations

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Reflection On Recursion • 4

A feature of special note in the recursion diagram is the function traversing the square from one triadic node to the other.  It preserves an image of the object all the while its precedent is being retrieved and processed — thus it injects a measure of parallel process and a modicum of extra memory over and above that afforded by the serial composition of functions.

Resources

• Inquiry Driven Systems • Inquiry Into Inquiry
• Reflective Interpretive Frameworks
• The Phenomenology of Reflection
• Higher Order Sign Relations

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Reflection On Recursion • 4

A feature of special note in the recursion diagram is the function traversing the square from one triadic node to the other.  It preserves an image of the object all the while its precedent is being retrieved and processed — thus it injects a measure of parallel process and a modicum of extra memory over and above that afforded by the serial composition of functions.

Resources

• Inquiry Driven Systems • Inquiry Into Inquiry
• Reflective Interpretive Frameworks
• The Phenomenology of Reflection
• Higher Order Sign Relations

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Reflection On Recursion • 3

One other feature of syntactic recursion deserves to be brought into higher relief.  Evidence of it can be found in the recursion diagram by examining the places where three paths meet.  On the descending side there is the point where three paths diverge.  On the ascending side there is the point where the middlemost of the three divergent paths joins the upshot arrow in medias res.

The arrows of the diagram represent functions, a species of dyadic relations, but nodes of degree three signify aspects of triadic relations somewhere in the mix.

• The three arrows from the initial node represent a function such that
• The three arrows at the penultimate node represent a function such that

For the sake of a first approach, many questions about triadic relations which might arise at this point can be safely left to later discussions, since the current level of generality is comprehensible enough in functional terms.

Resources

• Inquiry Driven Systems • Inquiry Into Inquiry
• Reflective Interpretive Frameworks
• The Phenomenology of Reflection
• Higher Order Sign Relations

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Reflection On Recursion • 3

One other feature of syntactic recursion deserves to be brought into higher relief.  Evidence of it can be found in the recursion diagram by examining the places where three paths meet.  On the descending side there is the point where three paths diverge.  On the ascending side there is the point where the middlemost of the three divergent paths joins the upshot arrow in medias res.

The arrows of the diagram represent functions, a species of dyadic relations, but nodes of degree three signify aspects of triadic relations somewhere in the mix.

• The three arrows from the initial node represent a function such that
• The three arrows at the penultimate node represent a function such that

For the sake of a first approach, many questions about triadic relations which might arise at this point can be safely left to later discussions, since the current level of generality is comprehensible enough in functional terms.

Resources

• Inquiry Driven Systems • Inquiry Into Inquiry
• Reflective Interpretive Frameworks
• The Phenomenology of Reflection
• Higher Order Sign Relations

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Reflection On Recursion • 2

Turning to the form of a simple recursive function the clause we used to define it earns the title of “syntactic recursion” due to the way the function name occurring in the defined phrase re‑occurs in the defining phrase

It needs to be clear there is no circle in the definition — each instance of the type is defined in terms of an instance one step simpler until the base case is reached and fixed by fiat.  Instead of a circle then we have two gyres, the gyre down via the predecessor function and the gyre up via the modifier function

Resources

• Inquiry Driven Systems • Inquiry Into Inquiry
• Reflective Interpretive Frameworks
• The Phenomenology of Reflection
• Higher Order Sign Relations

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon (1) (2) (3)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Reflection On Recursion • 2

Turning to the form of a simple recursive function the clause we used to define it earns the title of “syntactic recursion” due to the way the function name occurring in the defined phrase re‑occurs in the defining phrase

It needs to be clear there is no circle in the definition — each instance of the type is defined in terms of an instance one step simpler until the base case is reached and fixed by fiat.  Instead of a circle then we have two gyres, the gyre down via the predecessor function and the gyre up via the modifier function

Resources

• Inquiry Driven Systems • Inquiry Into Inquiry
• Reflective Interpretive Frameworks
• The Phenomenology of Reflection
• Higher Order Sign Relations

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon (1) (2) (3)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Differential Logic • 18

Tangent and Remainder Maps

If we follow the classical line which singles out linear functions as ideals of simplicity then we may complete the analytic series of the proposition in the following way.

The next venn diagram shows the differential proposition we get by extracting the linear approximation to the difference map at each cell or point of the universe   What results is the logical analogue of what would ordinarily be called the differential of but since the adjective differential is being attached to just about everything in sight the alternative name tangent map is commonly used for whenever it’s necessary to single it out.

To be clear about what’s being indicated here, it’s a visual way of summarizing the following data.

To understand the extended interpretations, that is, the conjunctions of basic and differential features which are being indicated here, it may help to note the following equivalences.

Capping the analysis of the proposition in terms of succeeding orders of linear propositions, the final venn diagram of the series shows the remainder map which happens to be linear in pairs of variables.

Reading the arrows off the map produces the following data.

In short, is a constant field, having the value at each cell.

Resources

• Logic Syllabus
• Minimal Negation Operator
• Survey of Differential Logic
• Survey of Animated Logical Graphs

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon (1) (2)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Differential Logic • 18

Tangent and Remainder Maps

If we follow the classical line which singles out linear functions as ideals of simplicity then we may complete the analytic series of the proposition in the following way.

The next venn diagram shows the differential proposition we get by extracting the linear approximation to the difference map at each cell or point of the universe   What results is the logical analogue of what would ordinarily be called the differential of but since the adjective differential is being attached to just about everything in sight the alternative name tangent map is commonly used for whenever it’s necessary to single it out.

To be clear about what’s being indicated here, it’s a visual way of summarizing the following data.

To understand the extended interpretations, that is, the conjunctions of basic and differential features which are being indicated here, it may help to note the following equivalences.

Capping the analysis of the proposition in terms of succeeding orders of linear propositions, the final venn diagram of the series shows the remainder map which happens to be linear in pairs of variables.

Reading the arrows off the map produces the following data.

In short, is a constant field, having the value at each cell.

Resources

• Logic Syllabus
• Minimal Negation Operator
• Survey of Differential Logic
• Survey of Animated Logical Graphs

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon (1) (2)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Differential Logic • 17

Enlargement and Difference Maps

Continuing with the example the following venn diagram shows the enlargement or shift map in the same style of field picture we drew for the tacit extension

A very important conceptual transition has just occurred here, almost tacitly, as it were.  Generally speaking, having a set of mathematical objects of compatible types, in this case the two differential fields and both of the type is very useful, because it allows us to consider those fields as integral mathematical objects which can be operated on and combined in the ways we usually associate with algebras.

In the present case one notices the tacit extension and the enlargement are in a sense dual to each other.  The tacit extension indicates all the arrows out of the region where is true and the enlargement indicates all the arrows into the region where is true.  The only arc they have in common is the no‑change loop at   If we add the two sets of arcs in mod 2 fashion then the loop of multiplicity 2 zeroes out, leaving the 6 arrows of shown in the following venn diagram.

Resources

• Logic Syllabus
• Minimal Negation Operator
• Survey of Differential Logic
• Survey of Animated Logical Graphs

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon (1) (2)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Differential Logic • 17

Enlargement and Difference Maps

Continuing with the example the following venn diagram shows the enlargement or shift map in the same style of field picture we drew for the tacit extension

A very important conceptual transition has just occurred here, almost tacitly, as it were.  Generally speaking, having a set of mathematical objects of compatible types, in this case the two differential fields and both of the type is very useful, because it allows us to consider those fields as integral mathematical objects which can be operated on and combined in the ways we usually associate with algebras.

In the present case one notices the tacit extension and the enlargement are in a sense dual to each other.  The tacit extension indicates all the arrows out of the region where is true and the enlargement indicates all the arrows into the region where is true.  The only arc they have in common is the no‑change loop at   If we add the two sets of arcs in mod 2 fashion then the loop of multiplicity 2 zeroes out, leaving the 6 arrows of shown in the following venn diagram.

Resources

• Logic Syllabus
• Minimal Negation Operator
• Survey of Differential Logic
• Survey of Animated Logical Graphs

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon (1) (2)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Differential Logic • 15

Differential Fields

The structure of a differential field may be described as follows.  With each point of there is associated an object of the following type:  a proposition about changes in that is, a proposition   In that frame of reference, if is the universe generated by the set of coordinate propositions then is the differential universe generated by the set of differential propositions   The differential propositions and may thus be interpreted as indicating and respectively.

A differential operator of the first order type we are currently considering, takes a proposition and gives back a differential proposition   In the field view of the scene, we see the proposition as a scalar field and we see the differential proposition as a vector field, specifically, a field of propositions about contemplated changes in

The field of changes produced by on is shown in the following venn diagram.

The differential field specifies the changes which need to be made from each point of in order to reach one of the models of the proposition that is, in order to satisfy the proposition

The field of changes produced by on is shown in the following venn diagram.

The differential field specifies the changes which need to be made from each point of in order to feel a change in the felt value of the field

Resources

• Logic Syllabus
• Minimal Negation Operator
• Survey of Differential Logic
• Survey of Animated Logical Graphs

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon (1) (2)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Differential Logic • 15

Differential Fields

The structure of a differential field may be described as follows.  With each point of there is associated an object of the following type:  a proposition about changes in that is, a proposition   In that frame of reference, if is the universe generated by the set of coordinate propositions then is the differential universe generated by the set of differential propositions   The differential propositions and may thus be interpreted as indicating and respectively.

A differential operator of the first order type we are currently considering, takes a proposition and gives back a differential proposition   In the field view of the scene, we see the proposition as a scalar field and we see the differential proposition as a vector field, specifically, a field of propositions about contemplated changes in

The field of changes produced by on is shown in the following venn diagram.

The differential field specifies the changes which need to be made from each point of in order to reach one of the models of the proposition that is, in order to satisfy the proposition

The field of changes produced by on is shown in the following venn diagram.

The differential field specifies the changes which need to be made from each point of in order to feel a change in the felt value of the field

Resources

• Logic Syllabus
• Minimal Negation Operator
• Survey of Differential Logic
• Survey of Animated Logical Graphs

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon (1) (2)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Differential Logic • 14

Field Picture

Let us summarize the outlook on differential logic we’ve reached so far.  We’ve been considering a class of operators on universes of discourse, each of which takes us from considering one universe of discourse to considering a larger universe of discourse   An operator of that general type, namely, acts on each proposition of the source universe to produce a proposition of the target universe

The operators we’ve examined so far are the enlargement or shift operator and the difference operator   The operators and act on propositions in that is, propositions of the form which amount to propositions about the subject matter of and they produce propositions of the form which amount to propositions about specified collections of changes conceivably occurring in

At this point we find ourselves in need of visual representations, suitable arrays of concrete pictures to anchor our more earthy intuitions and help us keep our wits about us as we venture into ever more rarefied airs of abstraction.

One good picture comes to us by way of the field concept.  Given a space a field of a specified type over is formed by associating with each point of an object of type   If that sounds like the same thing as a function from to the space of things of type — it is nothing but — and yet it does seem helpful to vary the mental images and take advantage of the figures of speech most naturally springing to mind under the emblem of the field idea.

In the field picture a proposition becomes a scalar field, that is, a field of values in

For example, consider the logical conjunction shown in the following venn diagram.

Each of the operators takes us from considering propositions here viewed as scalar fields over to considering the corresponding differential fields over analogous to what in real analysis are usually called vector fields over

Resources

• Logic Syllabus
• Minimal Negation Operator
• Survey of Differential Logic
• Survey of Animated Logical Graphs

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon (1) (2)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Differential Logic • 14

Field Picture

Let us summarize the outlook on differential logic we’ve reached so far.  We’ve been considering a class of operators on universes of discourse, each of which takes us from considering one universe of discourse to considering a larger universe of discourse   An operator of that general type, namely, acts on each proposition of the source universe to produce a proposition of the target universe

The operators we’ve examined so far are the enlargement or shift operator and the difference operator   The operators and act on propositions in that is, propositions of the form which amount to propositions about the subject matter of and they produce propositions of the form which amount to propositions about specified collections of changes conceivably occurring in

At this point we find ourselves in need of visual representations, suitable arrays of concrete pictures to anchor our more earthy intuitions and help us keep our wits about us as we venture into ever more rarefied airs of abstraction.

One good picture comes to us by way of the field concept.  Given a space a field of a specified type over is formed by associating with each point of an object of type   If that sounds like the same thing as a function from to the space of things of type — it is nothing but — and yet it does seem helpful to vary the mental images and take advantage of the figures of speech most naturally springing to mind under the emblem of the field idea.

In the field picture a proposition becomes a scalar field, that is, a field of values in

For example, consider the logical conjunction shown in the following venn diagram.

Each of the operators takes us from considering propositions here viewed as scalar fields over to considering the corresponding differential fields over analogous to what in real analysis are usually called vector fields over

Resources

• Logic Syllabus
• Minimal Negation Operator
• Survey of Differential Logic
• Survey of Animated Logical Graphs

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon (1) (2)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Differential Logic • 13

Transforms Expanded over Ordinary and Differential Variables

Two views of how the difference operator acts on the set of sixteen functions are shown below.  Table A5 shows the expansion of over the set of ordinary variables and Table A6 shows the expansion of over the set of differential variables.

Difference Map Expanded over Ordinary Variables

Difference Map Expanded over Differential Variables

Resources

• Logic Syllabus
• Minimal Negation Operator
• Survey of Differential Logic
• Survey of Animated Logical Graphs

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon (1) (2)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

Differential Logic • 13

Transforms Expanded over Ordinary and Differential Variables

Two views of how the difference operator acts on the set of sixteen functions are shown below.  Table A5 shows the expansion of over the set of ordinary variables and Table A6 shows the expansion of over the set of differential variables.

Difference Map Expanded over Ordinary Variables

Difference Map Expanded over Differential Variables

Resources

• Logic Syllabus
• Minimal Negation Operator
• Survey of Differential Logic
• Survey of Animated Logical Graphs

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon (1) (2)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

This thing all things devours:
Birds, beasts, trees, flowers;
Gnaws iron, bites steel;
Grinds hard stones to meal;
Slays king, ruins town,
And beats high mountain down.

— Tolkien • The Hobbit

Talking about time is a waste of time. Time is merely an abstraction from process and what is needed are better languages and better pictures for describing process in all its variety. In the sciences the big breakthrough in describing process came with the differential and integral calculus, that made it possible to shuttle between quantitative measures of state and quantitative measures of change. But every inquiry into a new phenomenon begins with the slimmest grasp of its qualitative features and labors long and hard to reach as far as a tentative logical description. What can avail us in the mean time, still tuning up before the first measure, to reason about change in qualitative terms?

Et sic deinceps … (So it begins …)

#Animata, #CSPeirce, #Change, #Cybernetics, #DifferentialLogic, #GraphTheory, #LawsOfForm, #Logic, #LogicalGraphs, #Mathematics, #Paradox, #Peirce, #Process, #ProcessThinking, #SpencerBrown, #SystemsTheory, #Time, #Tolkien

This thing all things devours:
Birds, beasts, trees, flowers;
Gnaws iron, bites steel;
Grinds hard stones to meal;
Slays king, ruins town,
And beats high mountain down.

— Tolkien • The Hobbit

Talking about time is a waste of time. Time is merely an abstraction from process and what is needed are better languages and better pictures for describing process in all its variety. In the sciences the big breakthrough in describing process came with the differential and integral calculus, that made it possible to shuttle between quantitative measures of state and quantitative measures of change. But every inquiry into a new phenomenon begins with the slimmest grasp of its qualitative features and labors long and hard to reach as far as a tentative logical description. What can avail us in the mean time, still tuning up before the first measure, to reason about change in qualitative terms?

Et sic deinceps … (So it begins …)

#Animata, #CSPeirce, #Change, #Cybernetics, #DifferentialLogic, #GraphTheory, #LawsOfForm, #Logic, #LogicalGraphs, #Mathematics, #Paradox, #Peirce, #Process, #ProcessThinking, #SpencerBrown, #SystemsTheory, #Time, #Tolkien

When reading about Brouwer's concept of Two-ity and how that's related to the problem of real numbers I had the vague idea that I had seen something similar before ... Peircean Thirdness.

That's not fully correct but it touches a few deep common points. And things that I failed to understand the significance of when reading Susan Haack Deviant Logic. Brower, Peirce, there is a lot to explore.

Other people had similar thoughts.
#SusanHaack #Brouwer #CSPeirce #logic

https://faculty.washington.edu/conormw/Papers/Peirce_Brouwer.pdf

When reading about Brouwer's concept of Two-ity and how that's related to the problem of real numbers I had the vague idea that I had seen something similar before ... Peircean Thirdness.

That's not fully correct but it touches a few deep common points. And things that I failed to understand the significance of when reading Susan Haack Deviant Logic. Brower, Peirce, there is a lot to explore.

Other people had similar thoughts.
#SusanHaack #Brouwer #CSPeirce #logic

https://faculty.washington.edu/conormw/Papers/Peirce_Brouwer.pdf

Sign Relations • Signs and Inquiry

There is a close relationship between the pragmatic theory of signs and the pragmatic theory of inquiry.  In fact, the correspondence between the two studies exhibits so many congruences and parallels it is often best to treat them as integral parts of one and the same subject.  In a very real sense, inquiry is the process by which sign relations come to be established and continue to evolve.  In other words, inquiry, “thinking” in its best sense, “is a term denoting the various ways in which things acquire significance” (Dewey, 38).

Tracing the passage of inquiry through the medium of signs calls for an active, intricate form of cooperation between the converging modes of investigation.  Its proper character is best understood by realizing the theory of inquiry is adapted to study the developmental aspects of sign relations, a subject the theory of signs is specialized to treat from comparative and structural points of view.

References

• Dewey, J. (1910), How We Think, D.C. Heath, Boston, MA.  Reprinted (1991), Prometheus Books, Buffalo, NY.  Online.
• Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), pp. 40–52.  Archive.  Journal.  Online (doc) (pdf).

Resources

• Sign Relation • OEIS • MyWikiBiz • Wikiversity
• Survey of Semiotics, Semiosis, Sign Relations
• Survey of Inquiry Driven Systems

cc: Academia.edu • Laws of Form • Research Gate • Syscoi
cc: Cybernetics • Structural Modeling • Systems Science

#CSPeirce #Connotation #Denotation #Inquiry #Logic #LogicOfRelatives #Mathematics #RelationTheory #Semiosis #SemioticEquivalenceRelations #Semiotics #SignRelations #TriadicRelations

Sign Relations • Signs and Inquiry

There is a close relationship between the pragmatic theory of signs and the pragmatic theory of inquiry.  In fact, the correspondence between the two studies exhibits so many congruences and parallels it is often best to treat them as integral parts of one and the same subject.  In a very real sense, inquiry is the process by which sign relations come to be established and continue to evolve.  In other words, inquiry, “thinking” in its best sense, “is a term denoting the various ways in which things acquire significance” (Dewey, 38).

Tracing the passage of inquiry through the medium of signs calls for an active, intricate form of cooperation between the converging modes of investigation.  Its proper character is best understood by realizing the theory of inquiry is adapted to study the developmental aspects of sign relations, a subject the theory of signs is specialized to treat from comparative and structural points of view.

References

• Dewey, J. (1910), How We Think, D.C. Heath, Boston, MA.  Reprinted (1991), Prometheus Books, Buffalo, NY.  Online.
• Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), pp. 40–52.  Archive.  Journal.  Online (doc) (pdf).

Resources

• Sign Relation • OEIS • MyWikiBiz • Wikiversity
• Survey of Semiotics, Semiosis, Sign Relations
• Survey of Inquiry Driven Systems

cc: Academia.edu • Laws of Form • Research Gate • Syscoi
cc: Cybernetics • Structural Modeling • Systems Science

#CSPeirce #Connotation #Denotation #Inquiry #Logic #LogicOfRelatives #Mathematics #RelationTheory #Semiosis #SemioticEquivalenceRelations #Semiotics #SignRelations #TriadicRelations

Sign Relations • Definition

One of Peirce’s clearest and most complete definitions of a sign is one he gives in the context of providing a definition for logic, and so it is informative to view it in that setting.

Logic will here be defined as formal semiotic.  A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time.  Namely, a sign is something, A, which brings something, B, its interpretant sign determined or created by it, into the same sort of correspondence with something, C, its object, as that in which itself stands to C.

It is from this definition, together with a definition of “formal”, that I deduce mathematically the principles of logic.  I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non‑psychological conception of logic has virtually been quite generally held, though not generally recognized.

— C.S. Peirce, New Elements of Mathematics, vol. 4, 20–21

In the general discussion of diverse theories of signs, the question arises whether signhood is an absolute, essential, indelible, or ontological property of a thing, or whether it is a relational, interpretive, and mutable role a thing may be said to have only within a particular context of relationships.

Peirce’s definition of a sign defines it in relation to its objects and its interpretant signs, and thus defines signhood in relative terms, by means of a predicate with three places.  In that definition, signhood is a role in a triadic relation, a role a thing bears or plays in a determinate context of relationships — it is not an absolute or non‑relative property of a thing‑in‑itself, one it possesses independently of all relationships to other things.

Some of the terms Peirce uses in his definition of a sign may need to be elaborated for the contemporary reader.

• Correspondence.  From the way Peirce uses the term throughout his work, it is clear he means what he elsewhere calls a “triple correspondence”, and thus this is just another way of referring to the whole triadic sign relation itself.  In particular, his use of the term should not be taken to imply a dyadic correspondence, like the kinds of “mirror image” correspondence between realities and representations bandied about in contemporary controversies about “correspondence theories of truth”.

• Determination.  Peirce’s concept of determination is broader in several directions than the sense of the word referring to strictly deterministic causal‑temporal processes.  First, and especially in this context, he is invoking a more general concept of determination, what is called a formal or informational determination, as in saying “two points determine a line”, rather than the more special cases of causal and temporal determinisms.  Second, he characteristically allows for what is called determination in measure, that is, an order of determinism admitting a full spectrum of more and less determined relationships.

• Non‑psychological.  Peirce’s “non‑psychological conception of logic” must be distinguished from any variety of anti‑psychologism.  He was quite interested in matters of psychology and had much of import to say about them.  But logic and psychology operate on different planes of study even when they have occasion to view the same data, as logic is a normative science where psychology is a descriptive science, and so they have very different aims, methods, and rationales.

Reference

• Peirce, C.S. (1902), “Parts of Carnegie Application” (L 75), in Carolyn Eisele (ed., 1976), The New Elements of Mathematics by Charles S. Peirce, vol. 4, 13–73.  Online.

Resources

• Sign Relation • OEIS • MyWikiBiz • Wikiversity
• Survey of Semiotics, Semiosis, Sign Relations

cc: Academia.edu • Laws of Form • Research Gate • Syscoi
cc: Cybernetics • Structural Modeling • Systems Science

#CSPeirce #Connotation #Denotation #Inquiry #Logic #LogicOfRelatives #Mathematics #RelationTheory #Semiosis #SemioticEquivalenceRelations #Semiotics #SignRelations #TriadicRelations

Sign Relations • Definition

One of Peirce’s clearest and most complete definitions of a sign is one he gives in the context of providing a definition for logic, and so it is informative to view it in that setting.

Logic will here be defined as formal semiotic.  A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time.  Namely, a sign is something, A, which brings something, B, its interpretant sign determined or created by it, into the same sort of correspondence with something, C, its object, as that in which itself stands to C.

It is from this definition, together with a definition of “formal”, that I deduce mathematically the principles of logic.  I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non‑psychological conception of logic has virtually been quite generally held, though not generally recognized.

— C.S. Peirce, New Elements of Mathematics, vol. 4, 20–21

In the general discussion of diverse theories of signs, the question arises whether signhood is an absolute, essential, indelible, or ontological property of a thing, or whether it is a relational, interpretive, and mutable role a thing may be said to have only within a particular context of relationships.

Peirce’s definition of a sign defines it in relation to its objects and its interpretant signs, and thus defines signhood in relative terms, by means of a predicate with three places.  In that definition, signhood is a role in a triadic relation, a role a thing bears or plays in a determinate context of relationships — it is not an absolute or non‑relative property of a thing‑in‑itself, one it possesses independently of all relationships to other things.

Some of the terms Peirce uses in his definition of a sign may need to be elaborated for the contemporary reader.

• Correspondence.  From the way Peirce uses the term throughout his work, it is clear he means what he elsewhere calls a “triple correspondence”, and thus this is just another way of referring to the whole triadic sign relation itself.  In particular, his use of the term should not be taken to imply a dyadic correspondence, like the kinds of “mirror image” correspondence between realities and representations bandied about in contemporary controversies about “correspondence theories of truth”.

• Determination.  Peirce’s concept of determination is broader in several directions than the sense of the word referring to strictly deterministic causal‑temporal processes.  First, and especially in this context, he is invoking a more general concept of determination, what is called a formal or informational determination, as in saying “two points determine a line”, rather than the more special cases of causal and temporal determinisms.  Second, he characteristically allows for what is called determination in measure, that is, an order of determinism admitting a full spectrum of more and less determined relationships.

• Non‑psychological.  Peirce’s “non‑psychological conception of logic” must be distinguished from any variety of anti‑psychologism.  He was quite interested in matters of psychology and had much of import to say about them.  But logic and psychology operate on different planes of study even when they have occasion to view the same data, as logic is a normative science where psychology is a descriptive science, and so they have very different aims, methods, and rationales.

Reference

• Peirce, C.S. (1902), “Parts of Carnegie Application” (L 75), in Carolyn Eisele (ed., 1976), The New Elements of Mathematics by Charles S. Peirce, vol. 4, 13–73.  Online.

Resources

• Sign Relation • OEIS • MyWikiBiz • Wikiversity
• Survey of Semiotics, Semiosis, Sign Relations

cc: Academia.edu • Laws of Form • Research Gate • Syscoi
cc: Cybernetics • Structural Modeling • Systems Science

#CSPeirce #Connotation #Denotation #Inquiry #Logic #LogicOfRelatives #Mathematics #RelationTheory #Semiosis #SemioticEquivalenceRelations #Semiotics #SignRelations #TriadicRelations

Survey of Semiotics, Semiosis, Sign Relations • 6

C.S. Peirce defines logic as “formal semiotic”, using formal to highlight the place of logic as a normative science, over and above the descriptive study of signs and their role in wider fields of play.  Understanding logic as Peirce understands it thus requires a companion study of semiotics, semiosis, and sign relations.

What follows is a Survey of blog and wiki resources on the theory of signs, variously known as semeiotic or semiotics, and the actions referred to as semiosis which transform signs among themselves in relation to their objects, all as based on C.S. Peirce’s concept of triadic sign relations.

Elements

• Semeiotic
• Sign Relations

Blog Series

• Semeiotic

• Sign Relations • Anthesis • Definition • Signs and Inquiry • Examples • Dyadic Aspects • Denotation • Connotation • Ennotation • Semiotic Equivalence Relations (1) (2)
• Comments • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12)
• Discussions • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12) • (13) • (14)

• Peircean Semiotics and Triadic Sign Relations • (1) • (2) • (3)

• Zeroth Law Of Semiotics
• Comments • (1) • (2) • (3) • (4) • (5) • (6) • (7)
• Discussions • (1) • (2) • (3) • (4)

• All Liar, No Paradox
• Comments • (1)
• Discussions • (1) • (2)

• Sign Relational Manifolds • (1) • (2) • (3) • (4) • (5)
• Discussions • (1) • (2) • (3)

• Semiotics, Semiosis, Sign Relations • (1) • (2) • (3)
• Comments • (1) • (2) • (3) • (4)
• Discussions • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12) • (13) • (14) • (15) • (16) • (17) • (18) • (19)

• Sign Relations, Triadic Relations, Relation Theory • (1) • (2) • (3) • (4)
• Discussions • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12)

• Higher Order Sign Relations • (1)
• Discussions • (1) • (2) • (3) • (4) • (5)

Blog Dialogs

• Signs Of Signs • (1) • (2) • (3) • (4)

• Icon Index Symbol • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12) • (13) • (14) • (15) • (16) • (17) • (18) • (19) • (20)

• Sign Relations, Triadic Relations, Relations • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11)

• Systems of Interpretation • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9)
• Discussions • (1)

Sources

• C.S. Peirce • On the Definition of Logic
• C.S. Peirce • Logic as Semiotic
• C.S. Peirce • Objective Logic

• C.S. Peirce • Algebra of Logic ∫ Philosophy of Notation • (1) • (2)
• C.S. Peirce • Algebra of Logic 1885 • Selections • (1) • (2) • (3) • (4)

Topics

• Interpreter and Interpretant
• Selections • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10)
• Discussions • (1) • (2) • (3) • (4)

• Tone, Token, Type
• Discussions • (1)

Excursions

• Semiositis • (1)
• Signspiel • (1)
• Skiourosemiosis • (1)

References

• Awbrey, S.M., and Awbrey, J.L. (2001), “Conceptual Barriers to Creating Integrative Universities”, Organization : The Interdisciplinary Journal of Organization, Theory, and Society 8(2), Sage Publications, London, UK, 269–284.  Abstract.  Online.
• Awbrey, S.M., and Awbrey, J.L. (September 1999), “Organizations of Learning or Learning Organizations : The Challenge of Creating Integrative Universities for the Next Century”, Second International Conference of the Journal ‘Organization’, Re‑Organizing Knowledge, Trans‑Forming Institutions : Knowing, Knowledge, and the University in the 21st Century, University of Massachusetts, Amherst, MA.  Online.
• Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), 40–52.  Archive.  Journal.  Online (doc) (pdf).
• Awbrey, J.L., and Awbrey, S.M. (1992), “Interpretation as Action : The Risk of Inquiry”, The Eleventh International Human Science Research Conference, Oakland University, Rochester, Michigan.

cc: FB | Semeiotics • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#CSPeirce #IconIndexSymbol #Inquiry #Logic #LogicOfRelatives #Mathematics #RelationTheory #Semiosis #Semiotics #SignRelations #TriadicRelations #Triadicity #Visualization

Differential Propositional Calculus • 10

Special Classes of Propositions (cont.)

Let’s pause at this point and get a better sense of how our special classes of propositions are structured and how they relate to propositions in general.  We can do this by recruiting our visual imaginations and drawing up a sufficient budget of venn diagrams for each family of propositions.  The case for 3 variables is exemplary enough for a start.

Linear Propositions

The linear propositions, may be written as sums:

One thing to keep in mind about these sums is that the values in are added “modulo 2”, that is, in such a way that

In a universe of discourse based on three boolean variables, the linear propositions take the shapes shown in Figure 8.

At the top is the venn diagram for the linear proposition of rank 3, which may be expressed by any one of the following three forms.

Next are the venn diagrams for the three linear propositions of rank 2, which may be expressed by the following three forms, respectively.

Next are the three linear propositions of rank 1, which are none other than the three basic propositions,

At the bottom is the linear proposition of rank 0, the everywhere false proposition or the constant function, which may be expressed by the form or by a simple

Resources

• Logic Syllabus
• Survey of Differential Logic

cc: Academia.edu • Cybernetics • Structural Modeling • Systems Science
cc: Conceptual Graphs • Laws of Form • Mathstodon • Research Gate

#Amphecks #Animata #BooleanAlgebra #BooleanFunctions #CSPeirce #CactusGraphs #CategoryTheory #Change #Cybernetics #DifferentialAnalyticTuringAutomata #DifferentialCalculus #DifferentialLogic #DiscreteDynamics #EquationalInference #FunctionalLogic #GraphTheory #Hologrammautomaton #IndicatorFunctions #InquiryDrivenSystems #Leibniz #Logic #LogicalGraphs #Mathematics #MinimalNegationOperators #PropositionalCalculus #Time #Topology #Visualization

Differential Propositional Calculus • 10

Special Classes of Propositions (cont.)

Let’s pause at this point and get a better sense of how our special classes of propositions are structured and how they relate to propositions in general.  We can do this by recruiting our visual imaginations and drawing up a sufficient budget of venn diagrams for each family of propositions.  The case for 3 variables is exemplary enough for a start.

Linear Propositions

The linear propositions, may be written as sums:

One thing to keep in mind about these sums is that the values in are added “modulo 2”, that is, in such a way that

In a universe of discourse based on three boolean variables, the linear propositions take the shapes shown in Figure 8.

At the top is the venn diagram for the linear proposition of rank 3, which may be expressed by any one of the following three forms.

Next are the venn diagrams for the three linear propositions of rank 2, which may be expressed by the following three forms, respectively.

Next are the three linear propositions of rank 1, which are none other than the three basic propositions,

At the bottom is the linear proposition of rank 0, the everywhere false proposition or the constant function, which may be expressed by the form or by a simple

Resources

• Logic Syllabus
• Survey of Differential Logic

cc: Academia.edu • Cybernetics • Structural Modeling • Systems Science
cc: Conceptual Graphs • Laws of Form • Mathstodon • Research Gate

#Amphecks #Animata #BooleanAlgebra #BooleanFunctions #CSPeirce #CactusGraphs #CategoryTheory #Change #Cybernetics #DifferentialAnalyticTuringAutomata #DifferentialCalculus #DifferentialLogic #DiscreteDynamics #EquationalInference #FunctionalLogic #GraphTheory #Hologrammautomaton #IndicatorFunctions #InquiryDrivenSystems #Leibniz #Logic #LogicalGraphs #Mathematics #MinimalNegationOperators #PropositionalCalculus #Time #Topology #Visualization

Pragmatic Maxim • Introduction
• https://inquiryintoinquiry.com/2023/08/07/pragmatic-maxim-2/

The “pragmatic maxim”, also known as the “maxim of pragmatism” or the “maxim of pragmaticism”, is a maxim of logic formulated by Charles Sanders Peirce. Serving as a practical recommendation or regulative principle in the normative science of logic, its function is to guide the conduct of thought toward the achievement of its purpose, advising the addressee on an optimal way of “attaining clearness of apprehension”.

Charles Sanders Peirce
• https://mywikibiz.com/Charles_Sanders_Peirce

Normative Science
• https://mywikibiz.com/Normative_science

#Peirce #Logic #Inquiry #PragmaticMaxim #RequlativePrinciple
#CSPeirce #CharlesSandersPeirce #NormativeScience #Semiotics

Pragmatic Maxim • Introduction
• https://inquiryintoinquiry.com/2023/08/07/pragmatic-maxim-2/

The “pragmatic maxim”, also known as the “maxim of pragmatism” or the “maxim of pragmaticism”, is a maxim of logic formulated by Charles Sanders Peirce. Serving as a practical recommendation or regulative principle in the normative science of logic, its function is to guide the conduct of thought toward the achievement of its purpose, advising the addressee on an optimal way of “attaining clearness of apprehension”.

Charles Sanders Peirce
• https://mywikibiz.com/Charles_Sanders_Peirce

Normative Science
• https://mywikibiz.com/Normative_science

#Peirce #Logic #Inquiry #PragmaticMaxim #RequlativePrinciple
#CSPeirce #CharlesSandersPeirce #NormativeScience #Semiotics

Pragmatic Maxim • Et Sic Deinceps
• http://inquiryintoinquiry.com/2023/08/07/pragmatic-maxim-2/

The pragmatic maxim is a guideline for the practice of inquiry formulated by Charles Sanders Peirce. Serving as a practical recommendation or regulative principle in the normative science of logic, its function is to guide the conduct of thought toward the achievement of its purpose, advising the addressee on an optimal way of “attaining clearness of apprehension”.

#Peirce #Logic #Inquiry #PragmaticMaxim #RequlativePrinciple
#CSPeirce #CharlesSandersPeirce #NormativeScience #Semiotics

Pragmatic Maxim • Et Sic Deinceps
• http://inquiryintoinquiry.com/2023/08/07/pragmatic-maxim-2/

The pragmatic maxim is a guideline for the practice of inquiry formulated by Charles Sanders Peirce. Serving as a practical recommendation or regulative principle in the normative science of logic, its function is to guide the conduct of thought toward the achievement of its purpose, advising the addressee on an optimal way of “attaining clearness of apprehension”.

#Peirce #Logic #Inquiry #PragmaticMaxim #RequlativePrinciple
#CSPeirce #CharlesSandersPeirce #NormativeScience #Semiotics

Functional Logic • Inquiry and Analogy • 5

Inquiry and Analogy • Aristotle’s “Paradigm” • Reasoning by Analogy

Aristotle examines the subject of analogical inference or “reasoning by example” under the heading of the Greek word παραδειγμα, from which comes the English word paradigm.  In its original sense the word suggests a kind of “side‑show”, or a parallel comparison of cases.

We have an Example (παραδειγμα, or analogy) when the major extreme is shown to be applicable to the middle term by means of a term similar to the third.  It must be known both that the middle applies to the third term and that the first applies to the term similar to the third.

E.g., let A be “bad”, B “to make war on neighbors”, C “Athens against Thebes”, and D “Thebes against Phocis”.  Then if we require to prove that war against Thebes is bad, we must be satisfied that war against neighbors is bad.  Evidence of this can be drawn from similar examples, e.g., that war by Thebes against Phocis is bad.  Then since war against neighbors is bad, and war against Thebes is against neighbors, it is evident that war against Thebes is bad.

Aristotle, “Prior Analytics” 2.24, Hugh Tredennick (trans.)

Figure 6 shows the logical relationships involved in Aristotle’s example of analogy.

Resources

• Logic Syllabus
• Boolean Function
• Boolean-Valued Function
• Logical Conjunction
• Minimal Negation Operator
• Functional Logic • Part 1 • Part 2 • Part 3
• Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#Abduction #Analogy #Argument #Aristotle #CSPeirce #Constraint #Deduction #Determination #DiagrammaticReasoning #Diagrams #DifferentialLogic #FunctionalLogic #Hypothesis #Indication #Induction #Inference #Information #Inquiry #Logic #LogicOfScience #Mathematics #PragmaticSemioticInformation #ProbableReasoning #PropositionalCalculus #Propositions #Reasoning #Retroduction #Semiotics #SignRelations #Syllogism #TriadicRelations #Visualization

“It is easy to speak with precision upon a general theme. Only, one must commonly surrender all ambition to be certain. It is equally easy to be certain. One has only to be sufficiently vague.”

— C.S. Peirce

#cspeirce #pragmatism

“It is easy to speak with precision upon a general theme. Only, one must commonly surrender all ambition to be certain. It is equally easy to be certain. One has only to be sufficiently vague.”

— C.S. Peirce

#cspeirce #pragmatism

@Inquiry @Harishjosev

Yes, another version of #CSPeirce’s triad is #WRAshby’s distinction between #Machine (object, observed phenomenon), #System (a finite set of variables and their relationships selected by the #Observer), and the #Model created by the observer to describe or/and #Control the machine.

However, I was speaking about the different phenomena (machines) and their controllability.
Mere #Physical (static or dynamic) #Structures driven by known (discoverable, invariant) #NaturalLaws as opposed to emergent #DynamicalSystems that are under the influence of arbitrary, local #Rules as explained by #HHPattee.

@Inquiry @Harishjosev

Yes, another version of #CSPeirce’s triad is #WRAshby’s distinction between #Machine (object, observed phenomenon), #System (a finite set of variables and their relationships selected by the #Observer), and the #Model created by the observer to describe or/and #Control the machine.

However, I was speaking about the different phenomena (machines) and their controllability.
Mere #Physical (static or dynamic) #Structures driven by known (discoverable, invariant) #NaturalLaws as opposed to emergent #DynamicalSystems that are under the influence of arbitrary, local #Rules as explained by #HHPattee.

In this new post for the Erraticus Blog, Jon Alan Schmidt was gracious enough to take some time challenging @ironick17 and me on our views of truth and the great chasm that exists between #CSPeirce and #williamjames. #pragmatism #philosophy

https://erraticus.co/2022/11/28/truth-as-pragmatisms-only-hope/

@algersanpin @philosophy "abduction", also know as "retroduction" .... or as the way our knowledge progresses from a prior "what we believe we know" through a scientific process and logic to a posterior "what we believe we know".

I'm curious if the author's take on #pragmatism is in the spirit of #CSPeirce #pragmaticism

@algersanpin @philosophy "abduction", also know as "retroduction" .... or as the way our knowledge progresses from a prior "what we believe we know" through a scientific process and logic to a posterior "what we believe we know".

I'm curious if the author's take on #pragmatism is in the spirit of #CSPeirce #pragmaticism

@emilyherring

Maybe you should try writing in the way of #CSPeirce ?

De Tienne, André. "The Peirce papers: How to pick up manuscripts that fell to the floor." Text 10 (1997): 259-282.

#CategoryTheory meets #CSPeirce 's notion of Firstness, Secondness and Thirdness.

#CategoryTheory meets #CSPeirce 's notion of Firstness, Secondness and Thirdness.

@emilyherring I would have answered C.S. Peirce #cspeirce

Propositions As Types Analogy • 1

Re: R.J. Lipton • Mathematical Tricks

One of my favorite mathematical tricks — it almost seems too tricky to be true — is the Propositions As Types Analogy.

And I see hints the 2‑part analogy can be extended to a 3‑part analogy, as follows.

See my working notes on the Propositions As Types Analogy for more information.

#Animata #CSPeirce #CombinatorCalculus #CombinatoryLogic #CurryHowardIsomorphism #GraphTheory #LambdaCalculus #Logic #LogicalGraphs #Mathematics #ProofTheory #PropositionsAsTypesAnalogy #TypeTheory