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<font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

<font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

A '''logical graph''' is a [[graph theory|graph-theoretic]] structure in one of the systems of graphical [[syntax]] that [http://mywikibiz.com/Charles_Sanders_Peirce Charles Sanders Peirce] developed for [[logic]].

A '''logical graph''' is a [[graph theory|graph-theoretic]] structure in one of the systems of graphical [[syntax]] that [http://mywikibiz.com/Charles_Sanders_Peirce Charles Sanders Peirce] developed for [[logic]].

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| width="60%" | &nbsp;

| width="40%" | ''Wollust ward dem Wurm gegeben …''

| width="40%" | ''Wollust ward dem Wurm gegeben &hellip;''

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| align="right" | &mdash; Friedrich Schiller, ''An die Freude''

| align="right" | &mdash; Friedrich Schiller, ''An die Freude''

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==In lieu of a beginning==

==In lieu of a beginning==

Consider the formal equations indicated in Figures 1 and 2.

Consider the formal equations indicated in Figures&nbsp;1 and 2.

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| width="33%" | <math>\texttt{(~(~)~)}</math>

| width="33%" | <math>\texttt{(~(~)~)}</math>

| width="34%" | <math>=\!</math>

| width="34%" | <math>=\!</math>

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| width="33%" | &nbsp;

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Simple intuition, or a simple inductive proof, assures us that any rooted tree can be reduced by way of the arithmetic initials either to a root node [[Image:Rooted Node.jpg|16px]] or else to a rooted edge [[Image:Rooted Edge.jpg|12px]] .

Simple intuition, or a simple inductive proof, assures us that any rooted tree can be reduced by way of the arithmetic initials either to a root node [[Image:Rooted Node.jpg|16px]] or else to a rooted edge [[Image:Rooted Edge.jpg|12px]]&nbsp;.

For example, consider the reduction that proceeds as follows:

For example, consider the reduction that proceeds as follows:

There is one other reason for lingering a while longer in these primitive forests, and this is that an acquaintance with "bare trees", those as yet unadorned with literal or numerical labels, will provide a firm basis for understanding what's really at issue in such problems as the "ontological status of variables".

There is one other reason for lingering a while longer in these primitive forests, and this is that an acquaintance with "bare trees", those as yet unadorned with literal or numerical labels, will provide a firm basis for understanding what's really at issue in such problems as the "ontological status of variables".

It is probably best to illustrate this theme in the setting of a concrete case, which we can do by reviewing the previous example of reductive evaluation shown in Figure 16.

It is probably best to illustrate this theme in the setting of a concrete case, which we can do by reviewing the previous example of reductive evaluation shown in Figure&nbsp;16.

The observation of several ''semioses'', or sign-transformations, of roughly this shape will most likely lead an observer with any observational facility whatever to notice that it doesn't really matter what sorts of branches happen to sprout from the side of the root aside from the lone edge that also grows there — the end result will always be the same.

The observation of several ''semioses'', or sign-transformations, of roughly this shape will most likely lead an observer with any observational facility whatever to notice that it doesn't really matter what sorts of branches happen to sprout from the side of the root aside from the lone edge that also grows there &mdash; the end result will always be the same.

Our observer might think to summarize the results of many such observations by introducing a label or variable to signify any shape of branch whatever, writing something like the following:

Our observer might think to summarize the results of many such observations by introducing a label or variable to signify any shape of branch whatever, writing something like the following:

====C₁. Double negation====

====C₁. Double negation====

The first theorem goes under the names of ''Consequence 1'' <math>(C_1)\!</math>, the ''double negation theorem'' (DNT), or ''Reflection''.

The first theorem goes under the names of ''Consequence&nbsp;1'' <math>(C_1)\!</math>, the ''double negation theorem'' (DNT), or ''Reflection''.

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====C₂. Generation theorem====

====C₂. Generation theorem====

One theorem of frequent use goes under the nickname of the ''weed and seed theorem'' (WAST).  The proof is just an exercise in mathematical induction, once a suitable basis is laid down, and it will be left as an exercise for the reader.  What the WAST says is that a label can be freely distributed or freely erased anywhere in a subtree whose root is labeled with that label.  The second in our list of frequently used theorems is in fact the base case of this weed and seed theorem.  In LOF, it goes by the names of ''Consequence 2'' <math>(C_2)\!</math> or ''Generation''.

One theorem of frequent use goes under the nickname of the ''weed and seed theorem'' (WAST).  The proof is just an exercise in mathematical induction, once a suitable basis is laid down, and it will be left as an exercise for the reader.  What the WAST says is that a label can be freely distributed or freely erased anywhere in a subtree whose root is labeled with that label.  The second in our list of frequently used theorems is in fact the base case of this weed and seed theorem.  In LOF, it goes by the names of ''Consequence&nbsp;2'' <math>(C_2)\!</math> or ''Generation''.

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====C₃. Dominant form theorem====

====C₃. Dominant form theorem====

The third of the frequently used theorems of service to this survey is one that Spencer-Brown annotates as ''Consequence 3'' <math>(C_3)\!</math> or ''Integration''.  A better mnemonic might be ''dominance and recession theorem'' (DART), but perhaps the brevity of ''dominant form theorem'' (DFT) is sufficient reminder of its double-edged role in proofs.

The third of the frequently used theorems of service to this survey is one that Spencer-Brown annotates as ''Consequence&nbsp;3'' <math>(C_3)\!</math> or ''Integration''.  A better mnemonic might be ''dominance and recession theorem'' (DART), but perhaps the brevity of ''dominant form theorem'' (DFT) is sufficient reminder of its double-edged role in proofs.

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The existential graph representation of Peirce's law is shown in Figure 33.

The existential graph representation of Peirce's law is shown in Figure&nbsp;33.

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A graphical proof of Peirce's law is shown in Figure 34.

A graphical proof of Peirce's law is shown in Figure&nbsp;34.

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a b + a c + b c

a b + a c + b c

\end{matrix}</math>

\end{matrix}</math>

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==Bibliography==

==Bibliography==

* [[Gottfried Leibniz|Leibniz, G.W.]] (1679–1686 ?), "Addenda to the Specimen of the Universal Calculus", pp. 40–46 in G.H.R. Parkinson (ed. and trans., 1966), ''Leibniz : Logical Papers'', Oxford University Press, London, UK.

* [[Gottfried Leibniz|Leibniz, G.W.]] (1679&ndash;1686 ?), "Addenda to the Specimen of the Universal Calculus", pp. 40&ndash;46 in G.H.R. Parkinson (ed. and trans., 1966), ''Leibniz : Logical Papers'', Oxford University Press, London, UK.

* [[Charles Peirce (Bibliography)|Peirce, C.S., Bibliography]].

* [[Charles Peirce (Bibliography)|Peirce, C.S., Bibliography]].

* [[Charles Peirce|Peirce, C.S.]] (1931–1935, 1958), ''Collected Papers of Charles Sanders Peirce'', vols. 1–6, [[Charles Hartshorne]] and [[Paul Weiss]] (eds.), vols. 7–8, [[Arthur W. Burks]] (ed.), Harvard University Press, Cambridge, MA.  Cited as (CP volume.paragraph).

* [[Charles Peirce|Peirce, C.S.]] (1931&ndash;1935, 1958), ''Collected Papers of Charles Sanders Peirce'', vols. 1&ndash;6, [[Charles Hartshorne]] and [[Paul Weiss]] (eds.), vols. 7&ndash;8, [[Arthur W. Burks]] (ed.), Harvard University Press, Cambridge, MA.  Cited as (CP&nbsp;volume.paragraph).

* Peirce, C.S. (1981–), ''Writings of Charles S. Peirce : A Chronological Edition'', [[Peirce Edition Project]] (eds.), Indiana University Press, Bloomington and Indianopolis, IN.  Cited as (CE volume, page).

* Peirce, C.S. (1981&ndash;), ''Writings of Charles S. Peirce : A Chronological Edition'', [[Peirce Edition Project]] (eds.), Indiana University Press, Bloomington and Indianopolis, IN.  Cited as (CE&nbsp;volume, page).

* Peirce, C.S. (1885), "On the Algebra of Logic : A Contribution to the Philosophy of Notation", ''American Journal of Mathematics'' 7 (1885), 180–202.  Reprinted as CP 3.359–403 and CE 5, 162–190.

* Peirce, C.S. (1885), "On the Algebra of Logic : A Contribution to the Philosophy of Notation", ''American Journal of Mathematics'' 7 (1885), 180&ndash;202.  Reprinted as CP&nbsp;3.359&ndash;403 and CE&nbsp;5, 162&ndash;190.

* Peirce, C.S. (''c.'' 1886), "Qualitative Logic", MS 736.  Published as pp. 101–115 in Carolyn Eisele (ed., 1976), ''The New Elements of Mathematics by Charles S. Peirce, Volume 4, Mathematical Philosophy'', Mouton, The Hague.

* Peirce, C.S. (''c.'' 1886), "Qualitative Logic", MS 736.  Published as pp. 101&ndash;115 in Carolyn Eisele (ed., 1976), ''The New Elements of Mathematics by Charles S. Peirce, Volume 4, Mathematical Philosophy'', Mouton, The Hague.

* Peirce, C.S. (1886 a), "Qualitative Logic", MS 582.  Published as pp. 323–371 in ''Writings of Charles S. Peirce : A Chronological Edition, Volume 5, 1884–1886'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1993.

* Peirce, C.S. (1886 a), "Qualitative Logic", MS 582.  Published as pp. 323&ndash;371 in ''Writings of Charles S. Peirce : A Chronological Edition, Volume 5, 1884&ndash;1886'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1993.

* Peirce, C.S. (1886 b), "The Logic of Relatives : Qualitative and Quantitative", MS 584.  Published as pp. 372–378 in ''Writings of Charles S. Peirce : A Chronological Edition, Volume 5, 1884–1886'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1993.

* Peirce, C.S. (1886 b), "The Logic of Relatives : Qualitative and Quantitative", MS 584.  Published as pp. 372&ndash;378 in ''Writings of Charles S. Peirce : A Chronological Edition, Volume 5, 1884&ndash;1886'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1993.

* [[George Spencer Brown|Spencer Brown, George]] (1969), ''[[Laws of Form]]'', George Allen and Unwin, London, UK.

* [[George Spencer Brown|Spencer Brown, George]] (1969), ''[[Laws of Form]]'', George Allen and Unwin, London, UK.

* [http://dr-dau.net/index.shtml Dau, Frithjof]

* [http://dr-dau.net/index.shtml Dau, Frithjof]

** [http://dr-dau.net/eg_readings.shtml Peirce's Existential Graphs : Readings and Links]

** [http://dr-dau.net/eg_readings.shtml Peirce's Existential Graphs : Readings and Links]

** [http://dr-dau.net/pc.shtml Existential Graphs as Moving Pictures of Thought] — Computer Animated Proof of Leibniz's Praeclarum Theorema

** [http://dr-dau.net/pc.shtml Existential Graphs as Moving Pictures of Thought] &mdash; Computer Animated Proof of Leibniz's Praeclarum Theorema

* [http://www.math.uic.edu/~kauffman/ Kauffman, Louis H.]

* [http://www.math.uic.edu/~kauffman/ Kauffman, Louis H.]