Changes

To consider how a system of logical graphs, taken together as a semiotic domain, might bear an iconic relationship to a system of logical objects that make up our object domain, we will next need to consider what our logical objects are.

To consider how a system of logical graphs, taken together as a semiotic domain, might bear an iconic relationship to a system of logical objects that make up our object domain, we will next need to consider what our logical objects are.

A popular answer, if by popular one means that both Peirce and Frege agreed on it, is to say that our ultimate logical objects are without loss of generality most conveniently referred to as Truth and Falsity.  If nothing else, it serves the end of beginning simply to go along with this thought for a while, and so we can start with an object domain that consists of just two ''objects'' or ''values'', to wit, <math>O = \mathbb{B} = \{ \operatorname{false}, \operatorname{true} \}.</math>

A popular answer, if by popular one means that both Peirce and Frege agreed on it, is to say that our ultimate logical objects are without loss of generality most conveniently referred to as Truth and Falsity.  If nothing else, it serves the end of beginning simply to go along with this thought for a while, and so we can start with an object domain that consists of just two ''objects'' or ''values'', to wit, <math>O = \mathbb{B} = \{ \mathrm{false}, \mathrm{true} \}.</math>

Given those two categories of structured individuals, namely, <math>O = \mathbb{B} = \{ \operatorname{false}, \operatorname{true} \}</math> and <math>S = \{ \text{logical graphs} \},\!</math> the next task is to consider the brands of morphisms from <math>S\!</math> to <math>O\!</math> that we might reasonably have in mind when we speak of the ''arrows of interpretation''.

Given those two categories of structured individuals, namely, <math>O = \mathbb{B} = \{ \mathrm{false}, \mathrm{true} \}</math> and <math>S = \{ \text{logical graphs} \},\!</math> the next task is to consider the brands of morphisms from <math>S\!</math> to <math>O\!</math> that we might reasonably have in mind when we speak of the ''arrows of interpretation''.

With the aim of embedding our consideration of logical graphs, as seems most fitting, within Peirce's theory of triadic sign relations, we have declared the first layers of our object, sign, and interpretant domains.  As we often do in formal studies, we've taken the sign and interpretant domains to be the same set, <math>S = I,\!</math> calling it the ''semiotic domain'', or, as I see that I've done in some other notes, the ''syntactic domain''.

With the aim of embedding our consideration of logical graphs, as seems most fitting, within Peirce's theory of triadic sign relations, we have declared the first layers of our object, sign, and interpretant domains.  As we often do in formal studies, we've taken the sign and interpretant domains to be the same set, <math>S = I,\!</math> calling it the ''semiotic domain'', or, as I see that I've done in some other notes, the ''syntactic domain''.

As agents of systems, whether that system is our own physiology or our own society, we move through what we commonly imagine to be a continuous manifold of states, but with distinctions being drawn in that space that are every bit as compelling to us, and often quite literally, as the difference between life and death.  So the relation of discretion to continuity is not one of those issues that we can take lightly, or simply dissolve by choosing a side and ignoring the other, as we may imagine in abstraction.  I'll try to get back to this point later, one in a long list of cautionary notes that experience tells me has to be attached to every tale of our pilgrimage, but for now we must get under way.

As agents of systems, whether that system is our own physiology or our own society, we move through what we commonly imagine to be a continuous manifold of states, but with distinctions being drawn in that space that are every bit as compelling to us, and often quite literally, as the difference between life and death.  So the relation of discretion to continuity is not one of those issues that we can take lightly, or simply dissolve by choosing a side and ignoring the other, as we may imagine in abstraction.  I'll try to get back to this point later, one in a long list of cautionary notes that experience tells me has to be attached to every tale of our pilgrimage, but for now we must get under way.

Returning to <math>\operatorname{En}</math> and <math>\operatorname{Ex},</math> the two most popular interpretations of logical graphs, ones that happen to be dual to each other in a certain sense, let's see how they fly as ''hermeneutic arrows'' from the syntactic domain <math>S\!</math> to the object domain <math>O,\!</math> at any rate, as their trajectories can be spied in the radar of what George Spencer Brown called the ''primary arithmetic''.

Returning to <math>\mathrm{En}</math> and <math>\mathrm{Ex},</math> the two most popular interpretations of logical graphs, ones that happen to be dual to each other in a certain sense, let's see how they fly as ''hermeneutic arrows'' from the syntactic domain <math>S\!</math> to the object domain <math>O,\!</math> at any rate, as their trajectories can be spied in the radar of what George Spencer Brown called the ''primary arithmetic''.

Taking <math>\operatorname{En}\!</math> and <math>\operatorname{Ex}\!</math> as arrows of the form <math>\operatorname{En}, \operatorname{Ex} : S \to O,</math> at the level of arithmetic taking <math>S = \{ \text{rooted trees} \}\!</math> and <math>O = \{ \operatorname{falsity}, \operatorname{truth} \},\!</math> it is possible to factor each arrow across the domain <math>S_0\!</math> that consists of a single rooted node plus a single rooted edge, in other words, the domain of formal constants <math>S_0 = \{ \ominus, \vert \} = \{</math>[[Image:Rooted Node.jpg|16px]],&nbsp;[[Image:Rooted Edge.jpg|12px]]<math>\}.\!</math>  This allows each arrow to be broken into a purely syntactic part <math>\operatorname{En}_\text{syn}, \operatorname{Ex}_\text{syn} : S \to S_0</math> and a purely semantic part <math>\operatorname{En}_\text{sem}, \operatorname{Ex}_\text{sem} : S_0 \to O.</math>

Taking <math>\mathrm{En}\!</math> and <math>\mathrm{Ex}\!</math> as arrows of the form <math>\mathrm{En}, \mathrm{Ex} : S \to O,</math> at the level of arithmetic taking <math>S = \{ \text{rooted trees} \}\!</math> and <math>O = \{ \mathrm{falsity}, \mathrm{truth} \},\!</math> it is possible to factor each arrow across the domain <math>S_0\!</math> that consists of a single rooted node plus a single rooted edge, in other words, the domain of formal constants <math>S_0 = \{ \ominus, \vert \} = \{</math>[[Image:Rooted Node.jpg|16px]],&nbsp;[[Image:Rooted Edge.jpg|12px]]<math>\}.\!</math>  This allows each arrow to be broken into a purely syntactic part <math>\mathrm{En}_\text{syn}, \mathrm{Ex}_\text{syn} : S \to S_0</math> and a purely semantic part <math>\mathrm{En}_\text{sem}, \mathrm{Ex}_\text{sem} : S_0 \to O.</math>

As things work out, the syntactic factors are formally the same, leaving our dualing interpretations to differ in their semantic components alone.  Specifically, we have the following mappings:

As things work out, the syntactic factors are formally the same, leaving our dualing interpretations to differ in their semantic components alone.  Specifically, we have the following mappings:

{| cellpadding="6"

{| cellpadding="6"

| width="5%" | &nbsp;

| width="5%" | &nbsp;

| width="5%" | <math>\operatorname{En}_\text{sem} :</math>

| width="5%" | <math>\mathrm{En}_\text{sem} :</math>

| width="5%" align="center" | [[Image:Rooted Node.jpg|16px]]

| width="5%" align="center" | [[Image:Rooted Node.jpg|16px]]

| width="5%" | <math>\mapsto</math>

| width="5%" | <math>\mapsto</math>

| <math>\operatorname{false},</math>

| <math>\mathrm{false},</math>

|-

|-

| &nbsp;

| &nbsp;

| align="center" | [[Image:Rooted Edge.jpg|12px]]

| align="center" | [[Image:Rooted Edge.jpg|12px]]

| <math>\mapsto</math>

| <math>\mapsto</math>

| <math>\operatorname{true}.</math>

| <math>\mathrm{true}.</math>

|-

|-

| &nbsp;

| &nbsp;

| <math>\operatorname{Ex}_\text{sem} :</math>

| <math>\mathrm{Ex}_\text{sem} :</math>

| align="center" | [[Image:Rooted Node.jpg|16px]]

| align="center" | [[Image:Rooted Node.jpg|16px]]

| <math>\mapsto</math>

| <math>\mapsto</math>

| <math>\operatorname{true},</math>

| <math>\mathrm{true},</math>

|-

|-

| &nbsp;

| &nbsp;

| align="center" | [[Image:Rooted Edge.jpg|12px]]

| align="center" | [[Image:Rooted Edge.jpg|12px]]

| <math>\mapsto</math>

| <math>\mapsto</math>

| <math>\operatorname{false}.</math>

| <math>\mathrm{false}.</math>

|}

|}

On the other side of the ledger, because the syntactic factors, <math>\operatorname{En}_\text{syn}</math> and <math>\operatorname{Ex}_\text{syn},</math> are indiscernible from each other, there is a syntactic contribution to the overall interpretation process that can be most readily investigated on purely formal grounds.  That will be the task to face when next we meet on these lists.

On the other side of the ledger, because the syntactic factors, <math>\mathrm{En}_\text{syn}</math> and <math>\mathrm{Ex}_\text{syn},</math> are indiscernible from each other, there is a syntactic contribution to the overall interpretation process that can be most readily investigated on purely formal grounds.  That will be the task to face when next we meet on these lists.

Cast into the form of a 3-adic sign relation, the situation before us can now be given the following shape:

Cast into the form of a 3-adic sign relation, the situation before us can now be given the following shape:

|}

|}

The interpretation maps <math>\operatorname{En}, \operatorname{Ex} : Y \to X</math> are factored into (1) a common syntactic part and (2) a couple of distinct semantic parts:

The interpretation maps <math>\mathrm{En}, \mathrm{Ex} : Y \to X</math> are factored into (1) a common syntactic part and (2) a couple of distinct semantic parts:

{| align="center" cellpadding="10" width="90%"

{| align="center" cellpadding="10" width="90%"

<math>\begin{array}{ll}

<math>\begin{array}{ll}

1. &

1. &

\operatorname{En}_\text{syn} = \operatorname{Ex}_\text{syn} = \operatorname{E}_\text{syn} : Y \to Y_0

\mathrm{En}_\text{syn} = \mathrm{Ex}_\text{syn} = \mathrm{E}_\text{syn} : Y \to Y_0

\\[10pt]

\\[10pt]

2. &

2. &

\operatorname{En}_\text{sem}, \operatorname{Ex}_\text{sem} : Y_0 \to X

\mathrm{En}_\text{sem}, \mathrm{Ex}_\text{sem} : Y_0 \to X

\end{array}</math>

\end{array}</math>

|}

|}

The functional images of the syntactic reduction map <math>\operatorname{E}_\text{syn} : Y \to Y_0</math> are the two simplest signs or the most reduced pair of expressions, regarded as the rooted trees [[Image:Rooted Node.jpg|16px]] and [[Image:Rooted Edge.jpg|12px]], and these may be treated as the canonical representatives of their respective equivalence classes.

The functional images of the syntactic reduction map <math>\mathrm{E}_\text{syn} : Y \to Y_0</math> are the two simplest signs or the most reduced pair of expressions, regarded as the rooted trees [[Image:Rooted Node.jpg|16px]] and [[Image:Rooted Edge.jpg|12px]], and these may be treated as the canonical representatives of their respective equivalence classes.

The more Peirce-sistent among you, on contemplating that last picture, will naturally ask, "What happened to the irreducible 3-adicity of sign relations in this portrayal of logical graphs?"

The more Peirce-sistent among you, on contemplating that last picture, will naturally ask, "What happened to the irreducible 3-adicity of sign relations in this portrayal of logical graphs?"

|}

|}

The answer is that the last bastion of 3-adic irreducibility presides precisely in the duality of the dual interpretations <math>\operatorname{En}_\text{sem}</math> and <math>\operatorname{Ex}_\text{sem}.</math>  To see this, consider the consequences of there being, contrary to all that we've assumed up to this point, some ultimately compelling reason to assert that the clean slate, the empty medium, the vacuum potential, whatever one wants to call it, is inherently more meaningful of either Falsity or Truth.  This would issue in a conviction forthwith that the 3-adic sign relation involved in this case decomposes as a composition of a couple of functions, that is to say, reduces to a 2-adic relation.

The answer is that the last bastion of 3-adic irreducibility presides precisely in the duality of the dual interpretations <math>\mathrm{En}_\text{sem}</math> and <math>\mathrm{Ex}_\text{sem}.</math>  To see this, consider the consequences of there being, contrary to all that we've assumed up to this point, some ultimately compelling reason to assert that the clean slate, the empty medium, the vacuum potential, whatever one wants to call it, is inherently more meaningful of either Falsity or Truth.  This would issue in a conviction forthwith that the 3-adic sign relation involved in this case decomposes as a composition of a couple of functions, that is to say, reduces to a 2-adic relation.

The duality of interpretation for logical graphs tells us that the empty medium, the tabula rasa, what Peirce called the ''Sheet of Assertion'' (SA) is a genuine symbol, not to be found among the degenerate species of signs that make up icons and indices, nor, as the SA has no parts, can it number icons or indices among its parts.  What goes for the medium must go for all of the signs that it mediates.  Thus we have the kinds of signs that Peirce in one place called "pure symbols", naming a selection of signs for basic logical operators specifically among them.

The duality of interpretation for logical graphs tells us that the empty medium, the tabula rasa, what Peirce called the ''Sheet of Assertion'' (SA) is a genuine symbol, not to be found among the degenerate species of signs that make up icons and indices, nor, as the SA has no parts, can it number icons or indices among its parts.  What goes for the medium must go for all of the signs that it mediates.  Thus we have the kinds of signs that Peirce in one place called "pure symbols", naming a selection of signs for basic logical operators specifically among them.

Thus, if you find youself in an argument with another interpreter who swears to the influence of some quality common to the object and the sign and that really does affect his or her conduct in regard to the two of them, then that argument is almost certainly bound to be utterly futile.  I am sure we've all been there.

Thus, if you find youself in an argument with another interpreter who swears to the influence of some quality common to the object and the sign and that really does affect his or her conduct in regard to the two of them, then that argument is almost certainly bound to be utterly futile.  I am sure we've all been there.

When I first became acquainted with the Entish and Extish hermenautics of logical graphs, back in the late great 1960s, I was struck in the spirit of those times by what I imagined to be their Zen and Zenoic sensibilities, the ''tao is silent'' wit of the Zen mind being the empty mind, that seems to go along with the <math>\operatorname{Ex}\!</math> interpretation, and the way from ''the way that's marked is not the true way'' to ''the mark that's marked is not the remarkable mark'' and to ''the sign that's signed is not the significant sign'' of the <math>\operatorname{En}\!</math> interpretation, reminding us that the sign is not the object, no matter how apt the image.  And later, when my discovery of the cactus graph extension of logical graphs led to the leimons of neural pools, where <math>\operatorname{En}\!</math> says that truth is an active condition, while <math>\operatorname{Ex}\!</math> says that sooth is a quiescent mind, all these themes got reinforced more still.

When I first became acquainted with the Entish and Extish hermenautics of logical graphs, back in the late great 1960s, I was struck in the spirit of those times by what I imagined to be their Zen and Zenoic sensibilities, the ''tao is silent'' wit of the Zen mind being the empty mind, that seems to go along with the <math>\mathrm{Ex}\!</math> interpretation, and the way from ''the way that's marked is not the true way'' to ''the mark that's marked is not the remarkable mark'' and to ''the sign that's signed is not the significant sign'' of the <math>\mathrm{En}\!</math> interpretation, reminding us that the sign is not the object, no matter how apt the image.  And later, when my discovery of the cactus graph extension of logical graphs led to the leimons of neural pools, where <math>\mathrm{En}\!</math> says that truth is an active condition, while <math>\mathrm{Ex}\!</math> says that sooth is a quiescent mind, all these themes got reinforced more still.

We hold these truths to be self-iconic, but they come in complementary couples, in consort to the flip-side of the tao.

We hold these truths to be self-iconic, but they come in complementary couples, in consort to the flip-side of the tao.

|}

|}

The object domain <math>O\!</math> is the boolean domain <math>\mathbb{B} = \{ \operatorname{falsity}, \operatorname{truth} \},</math> the semiotic domain <math>S\!</math> is any of the spaces isomorphic to the set of rooted trees, matched-up parentheses, or unlabeled alpha graphs, and we treat a couple of ''denotation maps'' <math>\operatorname{den}_\text{en}, \operatorname{den}_\text{ex} : S \to O.</math>

The object domain <math>O\!</math> is the boolean domain <math>\mathbb{B} = \{ \mathrm{falsity}, \mathrm{truth} \},</math> the semiotic domain <math>S\!</math> is any of the spaces isomorphic to the set of rooted trees, matched-up parentheses, or unlabeled alpha graphs, and we treat a couple of ''denotation maps'' <math>\mathrm{den}_\text{en}, \mathrm{den}_\text{ex} : S \to O.</math>

Either one of the denotation maps induces the same partition of <math>S\!</math> into RECs, a partition whose structure is suggested by the following two sets of strings:

Either one of the denotation maps induces the same partition of <math>S\!</math> into RECs, a partition whose structure is suggested by the following two sets of strings:

|}

|}

Next time we'll apply this general scheme to the <math>\operatorname{En}\!</math> and <math>\operatorname{Ex}\!</math> interpretations of logical graphs, and see how it helps us to sort out the varieties of iconic mapping that are involved in that setting.

Next time we'll apply this general scheme to the <math>\mathrm{En}\!</math> and <math>\mathrm{Ex}\!</math> interpretations of logical graphs, and see how it helps us to sort out the varieties of iconic mapping that are involved in that setting.

Corresponding to the Entitative and Existential interpretations of the primary arithmetic, there are two distinct mappings from the sign domain <math>S,\!</math> containing the topological equivalents of bare and rooted trees, onto the object domain <math>O,\!</math> containing the two objects whose conventional, ordinary, or meta-language names are ''falsity'' and ''truth'', respectively.

Corresponding to the Entitative and Existential interpretations of the primary arithmetic, there are two distinct mappings from the sign domain <math>S,\!</math> containing the topological equivalents of bare and rooted trees, onto the object domain <math>O,\!</math> containing the two objects whose conventional, ordinary, or meta-language names are ''falsity'' and ''truth'', respectively.

The idea of mappings that preserve 3-adic relations should ring a few bells here.

The idea of mappings that preserve 3-adic relations should ring a few bells here.

Once again into the breach between the interpretations <math>\operatorname{En}, \operatorname{Ex} : S \to O,</math> drawing but a single Figure in the sand and relying on the reader to recall:

Once again into the breach between the interpretations <math>\mathrm{En}, \mathrm{Ex} : S \to O,</math> drawing but a single Figure in the sand and relying on the reader to recall:

{| align="center" cellpadding="10" width="90%"

{| align="center" cellpadding="10" width="90%"

|

|

<p><math>\operatorname{En}\!</math> maps every tree on the left side of <math>S\!</math> to the left side of <math>O.\!</math></p>

<p><math>\mathrm{En}\!</math> maps every tree on the left side of <math>S\!</math> to the left side of <math>O.\!</math></p>

<p><math>\operatorname{En}\!</math> maps every tree on the right side of <math>S\!</math> to the right side of <math>O.\!</math></p>

<p><math>\mathrm{En}\!</math> maps every tree on the right side of <math>S\!</math> to the right side of <math>O.\!</math></p>

|-

|-

|

|

<p><math>\operatorname{Ex}\!</math> maps every tree on the left side of <math>S\!</math> to the right side of <math>O.\!</math></p>

<p><math>\mathrm{Ex}\!</math> maps every tree on the left side of <math>S\!</math> to the right side of <math>O.\!</math></p>

<p><math>\operatorname{Ex}\!</math> maps every tree on the right side of <math>S\!</math> to the left side of <math>O.\!</math></p>

<p><math>\mathrm{Ex}\!</math> maps every tree on the right side of <math>S\!</math> to the left side of <math>O.\!</math></p>

|}

|}

Those who wish to say that these logical signs are iconic of their logical objects must not only find some reason that logic itself singles out one interpretation over the other, but, even if they succeed in that, they must further make us believe that every sign for Truth is iconic of Truth, while every sign for Falsity is iconic of Falsity.

Those who wish to say that these logical signs are iconic of their logical objects must not only find some reason that logic itself singles out one interpretation over the other, but, even if they succeed in that, they must further make us believe that every sign for Truth is iconic of Truth, while every sign for Falsity is iconic of Falsity.

One of the questions that arises at this point, where we have a very small object domain <math>O = \{ \operatorname{falsity}, \operatorname{truth} \}</math> and a very large sign domain <math>S \cong \{ \text{rooted trees} \},</math> is the following:

One of the questions that arises at this point, where we have a very small object domain <math>O = \{ \mathrm{falsity}, \mathrm{truth} \}</math> and a very large sign domain <math>S \cong \{ \text{rooted trees} \},</math> is the following:

:* Why do we have so many ways of saying the same thing?

:* Why do we have so many ways of saying the same thing?

What sorts of sign relation are implicated in this sign process?  For simplicity, let's answer for the existential interpretation.

What sorts of sign relation are implicated in this sign process?  For simplicity, let's answer for the existential interpretation.

In <math>\operatorname{Ex},</math> all four of the listed signs are expressions of Falsity, and, viewed within the special type of semiotic procedure that is being considered here, each sign interprets its predecessor in the sequence.  Thus we might begin by drawing up this Table:

In <math>\mathrm{Ex},</math> all four of the listed signs are expressions of Falsity, and, viewed within the special type of semiotic procedure that is being considered here, each sign interprets its predecessor in the sequence.  Thus we might begin by drawing up this Table:

{| align="center" cellpadding="10"

{| align="center" cellpadding="10"

|}

|}

* <math>\operatorname{En},</math> for which blank = false and cross = true, calls this "equivalence".

* <math>\mathrm{En},</math> for which blank = false and cross = true, calls this "equivalence".

* <math>\operatorname{Ex},</math> for which blank = true and cross = false, calls this "distinction".

* <math>\mathrm{Ex},</math> for which blank = true and cross = false, calls this "distinction".

The step of controlled reflection that we just took can be iterated just as far as we wish to take it, as suggested by the following set:

The step of controlled reflection that we just took can be iterated just as far as we wish to take it, as suggested by the following set:

|}

|}

Remembering that a blank node is the graphical equivalent of a logical value <math>\operatorname{true},</math> the resulting DNF may be read as follows:

Remembering that a blank node is the graphical equivalent of a logical value <math>\mathrm{true},</math> the resulting DNF may be read as follows:

{| align="center" cellpadding="10" style="text-align:center; width:60%"

{| align="center" cellpadding="10" style="text-align:center; width:60%"

|}

|}

This can be read as <math>{}^{\backprime\backprime} \operatorname{not}~ p ~\operatorname{without}~ q ~\operatorname{and~not}~ q ~\operatorname{without}~ p {}^{\prime\prime},</math> in symbols, <math>(p \Rightarrow q) \land (q \Rightarrow p).</math>

This can be read as <math>{}^{\backprime\backprime} \mathrm{not}~ p ~\mathrm{without}~ q ~\mathrm{and~not}~ q ~\mathrm{without}~ p {}^{\prime\prime},</math> in symbols, <math>(p \Rightarrow q) \land (q \Rightarrow p).</math>

Graphing the topological dual form, one obtains the following rooted tree:

Graphing the topological dual form, one obtains the following rooted tree:

|}

|}

For the sake of simplicity in discussing this example, let's stick with the existential interpretation (<math>\operatorname{Ex}\!</math>) of logical graphs and their corresponding parse strings.  Under <math>\operatorname{Ex}\!</math> the formal expression <math>{\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}}\!</math> translates into the vernacular expression <math>{}^{\backprime\backprime} p ~\operatorname{implies}~ q ~\operatorname{and}~ p ~\operatorname{implies}~ r {}^{\prime\prime},</math> in symbols, <math>(p \Rightarrow q) \land (p \Rightarrow r),</math> so this is the reading that we'll want to keep in mind for the present.  Where brevity is required, we may refer to the propositional expression <math>{\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}}\!</math> under the name <math>f\!</math> by making use of the following definition:

For the sake of simplicity in discussing this example, let's stick with the existential interpretation (<math>\mathrm{Ex}\!</math>) of logical graphs and their corresponding parse strings.  Under <math>\mathrm{Ex}\!</math> the formal expression <math>{\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}}\!</math> translates into the vernacular expression <math>{}^{\backprime\backprime} p ~\mathrm{implies}~ q ~\mathrm{and}~ p ~\mathrm{implies}~ r {}^{\prime\prime},</math> in symbols, <math>(p \Rightarrow q) \land (p \Rightarrow r),</math> so this is the reading that we'll want to keep in mind for the present.  Where brevity is required, we may refer to the propositional expression <math>{\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}}\!</math> under the name <math>f\!</math> by making use of the following definition:

{| align="center" cellpadding="8"

{| align="center" cellpadding="8"

Since the expression <math>{\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}}\!</math> involves just three variables, it may be worth the trouble to draw a venn diagram of the situation.  There are in fact two different ways to execute the picture.

Since the expression <math>{\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}}\!</math> involves just three variables, it may be worth the trouble to draw a venn diagram of the situation.  There are in fact two different ways to execute the picture.

Figure&nbsp;27 indicates the points of the universe of discourse <math>X\!</math> for which the proposition <math>f : X \to \mathbb{B}</math> has the value 1, here interpreted as the logical value <math>\operatorname{true}.</math>  In this ''paint by numbers'' style of picture, one simply paints over the cells of a generic template for the universe <math>X,\!</math> going according to some previously adopted convention, for instance:  Let the cells that get the value 0 under <math>f\!</math> remain untinted and let the cells that get the value 1 under <math>f\!</math> be painted or shaded.  In doing this, it may be good to remind ourselves that the value of the picture as a whole is not in the ''paints'', in other words, the <math>0, 1\!</math> in <math>\mathbb{B},</math> but in the pattern of regions that they indicate.

Figure&nbsp;27 indicates the points of the universe of discourse <math>X\!</math> for which the proposition <math>f : X \to \mathbb{B}</math> has the value 1, here interpreted as the logical value <math>\mathrm{true}.</math>  In this ''paint by numbers'' style of picture, one simply paints over the cells of a generic template for the universe <math>X,\!</math> going according to some previously adopted convention, for instance:  Let the cells that get the value 0 under <math>f\!</math> remain untinted and let the cells that get the value 1 under <math>f\!</math> be painted or shaded.  In doing this, it may be good to remind ourselves that the value of the picture as a whole is not in the ''paints'', in other words, the <math>0, 1\!</math> in <math>\mathbb{B},</math> but in the pattern of regions that they indicate.

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There are a number of standard ways in mathematics and statistics for talking about the subset <math>W\!</math> of the functional domain <math>X\!</math> that gets painted with the value <math>z \in \mathbb{B}</math> by the indicator function <math>f : X \to \mathbb{B}.</math>  The region <math>W \subseteq X</math> is called by a variety of names in different settings, for example, the ''antecedent'', the ''fiber'', the ''inverse image'', the ''level set'', or the ''pre-image'' in <math>X\!</math> of <math>z\!</math> under <math>f.\!</math>  It is notated and defined as <math>W = f^{-1}(z).\!</math>  Here, <math>f^{-1}\!</math> is called the ''converse relation'' or the ''inverse relation'' &mdash; it is not in general an inverse function &mdash; corresponding to the function <math>f.\!</math>  Whenever possible in simple examples, we use lower case letters for functions <math>f : X \to \mathbb{B},</math> and it is sometimes useful to employ capital letters for subsets of <math>X,\!</math> if possible, in such a way that <math>F\!</math> is the fiber of 1 under <math>f,\!</math> in other words, <math>F = f^{-1}(1).\!</math>

There are a number of standard ways in mathematics and statistics for talking about the subset <math>W\!</math> of the functional domain <math>X\!</math> that gets painted with the value <math>z \in \mathbb{B}</math> by the indicator function <math>f : X \to \mathbb{B}.</math>  The region <math>W \subseteq X</math> is called by a variety of names in different settings, for example, the ''antecedent'', the ''fiber'', the ''inverse image'', the ''level set'', or the ''pre-image'' in <math>X\!</math> of <math>z\!</math> under <math>f.\!</math>  It is notated and defined as <math>W = f^{-1}(z).\!</math>  Here, <math>f^{-1}\!</math> is called the ''converse relation'' or the ''inverse relation'' &mdash; it is not in general an inverse function &mdash; corresponding to the function <math>f.\!</math>  Whenever possible in simple examples, we use lower case letters for functions <math>f : X \to \mathbb{B},</math> and it is sometimes useful to employ capital letters for subsets of <math>X,\!</math> if possible, in such a way that <math>F\!</math> is the fiber of 1 under <math>f,\!</math> in other words, <math>F = f^{-1}(1).\!</math>

The easiest way to see the sense of the venn diagram is to notice that the expression <math>\texttt{(} p \texttt{(} q \texttt{))},</math> read as <math>p \Rightarrow q,</math> can also be read as <math>{}^{\backprime\backprime} \operatorname{not}~ p ~\operatorname{without}~ q {}^{\prime\prime}.</math>  Its assertion effectively excludes any tincture of truth from the region of <math>P\!</math> that lies outside the region <math>Q.\!</math>  In a similar manner, the expression <math>\texttt{(} p \texttt{(} r \texttt{))},</math> read as <math>p \Rightarrow r,</math> can also be read as <math>{}^{\backprime\backprime} \operatorname{not}~ p ~\operatorname{without}~ r {}^{\prime\prime}.</math>  Asserting it effectively excludes any tincture of truth from the region of <math>P\!</math> that lies outside the region <math>R.\!</math>

The easiest way to see the sense of the venn diagram is to notice that the expression <math>\texttt{(} p \texttt{(} q \texttt{))},</math> read as <math>p \Rightarrow q,</math> can also be read as <math>{}^{\backprime\backprime} \mathrm{not}~ p ~\mathrm{without}~ q {}^{\prime\prime}.</math>  Its assertion effectively excludes any tincture of truth from the region of <math>P\!</math> that lies outside the region <math>Q.\!</math>  In a similar manner, the expression <math>\texttt{(} p \texttt{(} r \texttt{))},</math> read as <math>p \Rightarrow r,</math> can also be read as <math>{}^{\backprime\backprime} \mathrm{not}~ p ~\mathrm{without}~ r {}^{\prime\prime}.</math>  Asserting it effectively excludes any tincture of truth from the region of <math>P\!</math> that lies outside the region <math>R.\!</math>

Figure&nbsp;28 shows the other standard way of drawing a venn diagram for such a proposition.  In this ''punctured soap film'' style of picture &mdash; others may elect to give it the more dignified title of a ''logical quotient topology'' &mdash; one begins with Figure&nbsp;27 and then proceeds to collapse the fiber of 0 under <math>X\!</math> down to the point of vanishing utterly from the realm of active contemplation, arriving at the following picture:

Figure&nbsp;28 shows the other standard way of drawing a venn diagram for such a proposition.  In this ''punctured soap film'' style of picture &mdash; others may elect to give it the more dignified title of a ''logical quotient topology'' &mdash; one begins with Figure&nbsp;27 and then proceeds to collapse the fiber of 0 under <math>X\!</math> down to the point of vanishing utterly from the realm of active contemplation, arriving at the following picture:

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Remembering that a blank node is the graphical equivalent of a logical value <math>\operatorname{true},</math> the resulting DNF may be read as follows:

Remembering that a blank node is the graphical equivalent of a logical value <math>\mathrm{true},</math> the resulting DNF may be read as follows:

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This can be read to say <math>{}^{\backprime\backprime} \operatorname{either}~ p q r ~\operatorname{or}~ \operatorname{not}~ p {}^{\prime\prime},</math> which gives us yet another equivalent for the expression <math>{\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}}\!</math> and the expression <math>\texttt{(} p \texttt{(} q r \texttt{))}.</math>  Still another way of writing the same thing would be as follows:

This can be read to say <math>{}^{\backprime\backprime} \mathrm{either}~ p q r ~\mathrm{or}~ \mathrm{not}~ p {}^{\prime\prime},</math> which gives us yet another equivalent for the expression <math>{\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}}\!</math> and the expression <math>\texttt{(} p \texttt{(} q r \texttt{))}.</math>  Still another way of writing the same thing would be as follows:

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In other words, <math>{}^{\backprime\backprime} p ~\operatorname{is~equivalent~to}~ p ~\operatorname{and}~ q ~\operatorname{and}~ r {}^{\prime\prime}.</math>

In other words, <math>{}^{\backprime\backprime} p ~\mathrm{is~equivalent~to}~ p ~\mathrm{and}~ q ~\mathrm{and}~ r {}^{\prime\prime}.</math>

One lemma that suggests itself at this point is a principle that may be canonized as the ''Emptiness Rule''.  It says that a bare lobe expression like <math>\texttt{( \_, \_, \ldots )},</math> with any number of places for arguments but nothing but blanks as filler, is logically tantamount to the proto-typical expression of its type, namely, the constant expression <math>\texttt{(~)}</math> that <math>\operatorname{Ex}\!</math> interprets as denoting the logical value <math>\operatorname{false}.</math>  To depict the rule in graphical form, we have the continuing sequence of equations:

One lemma that suggests itself at this point is a principle that may be canonized as the ''Emptiness Rule''.  It says that a bare lobe expression like <math>\texttt{( \_, \_, \ldots )},</math> with any number of places for arguments but nothing but blanks as filler, is logically tantamount to the proto-typical expression of its type, namely, the constant expression <math>\texttt{(~)}</math> that <math>\mathrm{Ex}\!</math> interprets as denoting the logical value <math>\mathrm{false}.</math>  To depict the rule in graphical form, we have the continuing sequence of equations:

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: ''e''₅ = "(( (p (q))(p (r)) , (p (q r)) ))"

: ''e''₅ = "(( (p (q))(p (r)) , (p (q r)) ))"

Under <math>\operatorname{Ex}\!</math> we have the following interpretations:

Under <math>\mathrm{Ex}\!</math> we have the following interpretations:

: ''e''₀ expresses the logical constant "false"

: ''e''₀ expresses the logical constant "false"

Proof 2 lit on by burning the candle at both ends, changing ''e''₂ into a normal form that reduced to ''e''₄, changing ''e''₃ into a normal form that reduced to e_4, in this way tethering ''e''₂ and ''e''₃ to a common point.  We got that (p (q))(p (r)) is equal to (p q r, (p)), then we got that (p (q r)) is equal to (p q r, (p)), so we got that (p (q))(p (r)) is equal to (p (q r)).

Proof 2 lit on by burning the candle at both ends, changing ''e''₂ into a normal form that reduced to ''e''₄, changing ''e''₃ into a normal form that reduced to e_4, in this way tethering ''e''₂ and ''e''₃ to a common point.  We got that (p (q))(p (r)) is equal to (p q r, (p)), then we got that (p (q r)) is equal to (p q r, (p)), so we got that (p (q))(p (r)) is equal to (p (q r)).

Proof 3 took the path of reflection, expressing the meta-equation between ''e''₂ and ''e''₃ via the object equation ''e''₅, then taking ''e''₅ as ''s''₁ and exchanging it by dint of value preserving steps for ''e''₁ as ''s''_''n''.  Thus we went from "(( (p (q))(p (r)) , (p (q r)) ))" to the blank expression that <math>\operatorname{Ex}\!</math> recognizes as true.

Proof 3 took the path of reflection, expressing the meta-equation between ''e''₂ and ''e''₃ via the object equation ''e''₅, then taking ''e''₅ as ''s''₁ and exchanging it by dint of value preserving steps for ''e''₁ as ''s''_''n''.  Thus we went from "(( (p (q))(p (r)) , (p (q r)) ))" to the blank expression that <math>\mathrm{Ex}\!</math> recognizes as true.

I need to say something about the concept of ''reflection'' that I've been using according to my informal intuitions about it at numerous points in this discussion.  This is, of course, distinct from the use of the word "reflection" to license an application of the double negation theorem.

I need to say something about the concept of ''reflection'' that I've been using according to my informal intuitions about it at numerous points in this discussion.  This is, of course, distinct from the use of the word "reflection" to license an application of the double negation theorem.