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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 1960's, 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 ''Ex'' 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 ''En'' 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 ''En'' says that truth is an active condition, while ''Ex'' 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 1960's, 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.

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.

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 ''Ex'', 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>\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:

{| align="center" style="text-align:center; width:90%"

{| align="center" style="text-align:center; width:90%"

|}

|}

* En, for which blank = false and cross = true, calls this "equivalence".

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

* Ex, for which blank = true and cross = false, calls this "distinction".

* <math>\operatorname{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:

|}

|}

For the sake of simplicity in discussing this example, I will revert to the ''existential interpretation'' (''Ex'') of logical graphs and their corresponding parse strings.

For the sake of simplicity in discussing this example, I will revert to the ''existential interpretation'' (<math>\operatorname{Ex}</math>) of logical graphs and their corresponding parse strings.

Under ''Ex'' the expression "(p (q))(p (r))" interprets as the vernacular expression "p &rArr; q &and; p &rArr; r", so this is the reading that we'll want to keep in mind for the present.

Under <math>\operatorname{Ex}</math> the expression "(p (q))(p (r))" interprets as the vernacular expression "p &rArr; q &and; p &rArr; r", so this is the reading that we'll want to keep in mind for the present.

Where brevity is required, and it occasionally is, we may invoke the propositional expression "(p (q))(p (r))" under the name "f" by making use of the following definition:  "f = (p (q))(p (r))".

Where brevity is required, and it occasionally is, we may invoke the propositional expression "(p (q))(p (r))" under the name "f" by making use of the following definition:  "f = (p (q))(p (r))".

In other words, "p is equivalent to p and q and r".

In other words, "p is equivalent to p and q and r".

Let's pause to refresh ourselves with a few morsels of lemmas bread.  One lemma that I can see just far enough ahead to see our imminent need of is the principle that I canonize as the ''Emptiness Rule''.  It says that a bare lobe expression like "(&hellip; , &hellip;)", 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 "()" that ExG interprets as denoting the logical value ''false''.  To depict the rule in graphical form, we have this continuing sequence of equations:

Let's pause to refresh ourselves with a few morsels of lemmas bread.  One lemma that I can see just far enough ahead to see our imminent need of is the principle that I canonize as the ''Emptiness Rule''.  It says that a bare lobe expression like "(&hellip; , &hellip;)", 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 "()" that <math>\operatorname{Ex}</math> interprets as denoting the logical value ''false''.  To depict the rule in graphical form, we have this continuing sequence of equations:

{| align="center" style="text-align:center; width:90%"

{| align="center" style="text-align:center; width:90%"

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

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

Under ExG we have the following interpretations:

Under <math>\operatorname{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 ''Ex'' 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>\operatorname{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.