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(+ \overline{\underline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}})
 
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{{DISPLAYTITLE:Propositional Equation Reasoning Systems}}
+
==Variations on a theme of transitivity==
* '''Note.''' The MathJax parser is not rendering this page properly.<br>Until it can be fixed please see the [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems InterSciWiki version].
 
  
'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''
+
The next Example is extremely important, and for reasons that reach well beyond the level of propositional calculus as it is ordinarily conceived.  But it's slightly tricky to get all of the details right, so it will be worth taking the trouble to look at it from several different angles and as it appears in diverse frames, genres, or styles of representation.
  
This article develops elementary facts about a family of formal calculi described as '''propositional equation reasoning systems''' ('''PERS''').  This work follows on the ''alpha graphs'' that Charles Sanders Peirce devised as a graphical syntax for [[propositional calculus]] and also on the ''calculus of indications'' that George Spencer Brown presented in his ''Laws of Form''.
+
In discussing this Example, it is useful to observe that the implication relation indicated by the propositional form <math>x \Rightarrow y\!</math> is equivalent to an order relation <math>x \le y\!</math> on the boolean values <math>0, 1 \in \mathbb{B},\!</math> where <math>0\!</math> is taken to be less than <math>1.\!</math>
  
==Formal development==
+
{| align="center" cellpadding="8" width="90%"
 
+
| width="1%" | <big>&bull;</big>
The first order of business is to give the exact forms of the axioms that we use, devolving from Peirce's &ldquo;[[Logical Graphs]]&rdquo; via Spencer-Brown's ''Laws of Form'' (LOF)In formal proofs, we use a variation of the annotation scheme from LOF to mark each step of the proof according to which axiom, or ''initial'', is being invoked to justify the corresponding step of syntactic transformation, whether it applies to graphs or to strings.
+
| colspan="3" | '''Example 2Transitivity'''
 
+
|-
===Axioms===
+
| &nbsp;
 
+
| width="1%" | &nbsp;
The axioms are just four in number, divided into the ''arithmetic initials'', <math>I_1\!</math> and <math>I_2,\!</math> and the ''algebraic initials'', <math>J_1\!</math> and <math>J_2.\!</math>
+
| colspan="2" | ''Information Reducing Inference''
 
 
{| align="center" cellpadding="10"
 
| [[Image:PERS_Figure_01.jpg|500px]] || (1)
 
 
|-
 
|-
| [[Image:PERS_Figure_02.jpg|500px]] || (2)
+
| &nbsp;
 +
| &nbsp;
 +
| width="1%" | &nbsp;
 +
|
 +
<math>\begin{array}{l}
 +
~ p \le q
 +
\\
 +
~ q \le r
 +
\\
 +
\overline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}
 +
\\
 +
~ p \le r
 +
\end{array}</math>
 
|-
 
|-
| [[Image:PERS_Figure_03.jpg|500px]] || (3)
+
| &nbsp;
 +
| &nbsp;
 +
| colspan="2" | ''Information Preserving Inference''
 
|-
 
|-
| [[Image:PERS_Figure_04.jpg|500px]] || (4)
+
| &nbsp;
|}
+
| &nbsp;
 
+
| &nbsp;
One way of assigning logical meaning to the initial equations is known as the ''entitative interpretation'' <math>(\mathrm{En}).\!</math>  Under <math>\mathrm{En},\!</math> the axioms read as follows:
 
 
 
{| align="center" cellpadding="10"
 
 
|
 
|
<math>\begin{matrix}
+
<math>\begin{array}{l}
I_1
+
~ p \le q
& : &
 
\mathrm{true} ~\mathrm{or}~ \mathrm{true}
 
& = &
 
\mathrm{true}
 
 
\\
 
\\
I_2
+
~ q \le r
& : &
 
\mathrm{not}~ \mathrm{true}
 
& = &
 
\mathrm{false}
 
 
\\
 
\\
J_1
+
\overline{\underline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}
& : &
 
a ~\mathrm{or}~ \mathrm{not}~ a
 
& = &
 
\mathrm{true}
 
 
\\
 
\\
J_2
+
~ p \le q \le r
& : &
+
\end{array}</math>
(a ~\mathrm{or}~ b) ~\mathrm{and}~ (a ~\mathrm{or}~ c)
 
& = &
 
a ~\mathrm{or}~ (b ~\mathrm{and}~ c)
 
\end{matrix}</math>
 
 
|}
 
|}
  
Another way of assigning logical meaning to the initial equations is known as the ''existential interpretation'' <math>(\mathrm{Ex}).\!</math>  Under <math>\mathrm{Ex},\!</math> the axioms read as follows:
+
In stating the information-preserving analogue of transitivity, I have taken advantage of a common idiom in the use of order relation symbols, one that represents their logical conjunction by way of a concatenated syntax.  Thus, <math>p \le q \le r\!</math> means <math>p \le q ~\mathrm{and}~ q \le r.\!</math>  The claim that this 3-adic order relation holds among the three propositions <math>p, q, r\!</math> is a stronger claim &mdash; conveys more information &mdash; than the claim that the 2-adic relation <math>p \le r\!</math> holds between the two propositions <math>p\!</math> and <math>r.\!</math>
  
{| align="center" cellpadding="10"
+
To study the differences between these two versions of transitivity within what is locally a familiar context, let's view the propositional forms involved as if they were elementary cellular automaton rules, resulting in the following Table.
 +
 
 +
<br>
 +
 
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 +
|+ <math>\text{Table 51.}~~\text{Composite and Compiled Order Relations}\!</math>
 +
|- style="background:#f0f0ff"
 +
|
 +
<p><math>\mathcal{L}_1\!</math></p>
 +
<p><math>\text{Decimal}\!</math></p>
 +
|
 +
<p><math>\mathcal{L}_2\!</math></p>
 +
<p><math>\text{Binary}\!</math></p>
 +
|
 +
<p><math>\mathcal{L}_3\!</math></p>
 +
<p><math>\text{Vector}\!</math></p>
 +
|
 +
<p><math>\mathcal{L}_4\!</math></p>
 +
<p><math>\text{Cactus}\!</math></p>
 +
|
 +
<p><math>\mathcal{L}_5\!</math></p>
 +
<p><math>\text{Order}\!</math></p>
 +
|- style="background:#f0f0ff"
 +
| &nbsp;
 +
| align="right" | <math>p\colon\!</math>
 +
| <math>1~1~1~1~0~0~0~0\!</math>
 +
| &nbsp;
 +
| &nbsp;
 +
|- style="background:#f0f0ff"
 +
| &nbsp;
 +
| align="right" | <math>q\colon\!</math>
 +
| <math>1~1~0~0~1~1~0~0\!</math>
 +
| &nbsp;
 +
| &nbsp;
 +
|- style="background:#f0f0ff"
 +
| &nbsp;
 +
| align="right" | <math>r\colon\!</math>
 +
| <math>1~0~1~0~1~0~1~0\!</math>
 +
| &nbsp;
 +
| &nbsp;
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_{207}
 +
\\[4pt]
 +
f_{187}
 +
\\[4pt]
 +
f_{175}
 +
\\[4pt]
 +
f_{139}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{11001111}
 +
\\[4pt]
 +
f_{10111011}
 +
\\[4pt]
 +
f_{10101111}
 +
\\[4pt]
 +
f_{10001011}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
1~1~0~0~1~1~1~1
 +
\\[4pt]
 +
1~0~1~1~1~0~1~1
 +
\\[4pt]
 +
1~0~1~0~1~1~1~1
 +
\\[4pt]
 +
1~0~0~0~1~0~1~1
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{(} p \texttt{ (} q \texttt{))}
 +
\\[4pt]
 +
\texttt{(} q \texttt{ (} r \texttt{))}
 +
\\[4pt]
 +
\texttt{(} p \texttt{ (} r \texttt{))}
 +
\\[4pt]
 +
\texttt{(} p \texttt{ (} q \texttt{)) (} q \texttt{ (} r \texttt{))}
 +
\end{matrix}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
I_1
+
p \le q
& : &
+
\\[4pt]
\mathrm{false} ~\mathrm{and}~ \mathrm{false}
+
q \le r
& = &
+
\\[4pt]
\mathrm{false}
+
p \le r
\\
+
\\[4pt]
I_2
+
p \le q \le r
& : &
 
\mathrm{not}~ \mathrm{false}
 
& = &
 
\mathrm{true}
 
\\
 
J_1
 
& : &
 
a ~\mathrm{and}~ \mathrm{not}~ a
 
& = &
 
\mathrm{false}
 
\\
 
J_2
 
& : &
 
(a ~\mathrm{and}~ b) ~\mathrm{or}~ (a ~\mathrm{and}~ c)
 
& = &
 
a ~\mathrm{and}~ (b ~\mathrm{or}~ c)
 
 
\end{matrix}</math>
 
\end{matrix}</math>
 
|}
 
|}
  
All of the axioms in this set have the form of equations.  This means that all of the inference licensed by them are reversible.  The proof annotation scheme employed below makes use of a double bar &#9552;&#9552;&#9552;&#9552;&#9552; to mark this fact, but it will often be left to the reader to decide which of the two possible ways of applying the axiom is the one that is called for in a particular case.
+
<br>
  
Peirce introduced these formal equations at a level of abstraction that is one step higher than their customary interpretations as propositional calculi, which two readings he called the ''Entitative'' and the ''Existential'' interpretations, here referred to as <math>\mathrm{En}\!</math> and <math>\mathrm{Ex},\!</math> respectively.  The early CSP, as in his essay on &ldquo;Qualitative Logic&rdquo;, and also GSB, emphasized the <math>\mathrm{En}\!</math> interpretation, while the later CSP developed mostly the <math>\mathrm{Ex}\!</math> interpretation.
+
Taking up another angle of incidence by way of extra perspective, let us now reflect on the venn diagrams of our four propositions.
  
===Frequently used theorems===
+
{| align="center" cellpadding="6" style="text-align:center"
 +
| [[Image:Venn Diagram (P (Q)).jpg|500px]] || (52)
 +
|-
 +
| <math>f_{207}(p, q, r) ~=~ \texttt{(} p \texttt{ (} q \texttt{))}\!</math>
 +
|-
 +
| &nbsp;
 +
|-
 +
| [[Image:Venn Diagram (Q (R)).jpg|500px]] || (53)
 +
|-
 +
| <math>f_{187}(p, q, r) ~=~ \texttt{(} q \texttt{ (} r \texttt{))}\!</math>
 +
|-
 +
| &nbsp;
 +
|-
 +
| [[Image:Venn Diagram (P (R)).jpg|500px]] || (54)
 +
|-
 +
| <math>f_{175}(p, q, r) ~=~ \texttt{(} p \texttt{ (} r \texttt{))}\!</math>
 +
|-
 +
| &nbsp;
 +
|-
 +
| [[Image:Venn Diagram (P (Q)) (Q (R)).jpg|500px]] || (55)
 +
|-
 +
| <math>f_{139}(p, q, r) ~=~ \texttt{(} p \texttt{ (} q \texttt{)) (} q \texttt{ (} r \texttt{))}\!</math>
 +
|}
  
====C<sub>1</sub>. Double negation====
+
Among other things, these images make it visually obvious that the constraint on the three boolean variables <math>p, q, r\!</math> that is indicated by asserting either of the forms <math>\texttt{(} p \texttt{(} q \texttt{))(} q \texttt{(} r \texttt{))}\!</math> or <math>p \le q \le r\!</math> implies a constraint on the two boolean variables <math>p, r\!</math> that is indicated by either of the forms <math>\texttt{(} p \texttt{(} r \texttt{))}\!</math> or <math>p \le r,\!</math> but that it imposes additional constraints on these variables that are not captured by the illative conclusion.
  
The first theorem goes under the names of ''Consequence&nbsp;1'' <math>(C_1),\!</math> the ''double negation theorem'' (DNT), or ''Reflection''.
+
One way to view a proposition <math>f : \mathbb{B}^k \to \mathbb{B}\!</math> is to consider its ''fiber of truth'', <math>f^{-1}(1) \subseteq \mathbb{B}^k,\!</math> and to regard it as a <math>k\!</math>-adic relation <math>L \subseteq \mathbb{B}^k.\!</math>
  
{| align="center" cellpadding="10"
+
By way of general definition, the ''fiber'' of a function <math>f : X \to Y\!</math> at a given value <math>y\!</math> of its co-domain <math>Y\!</math> is the ''antecedent'' (also known as the ''inverse image'' or ''pre-image'') of <math>y\!</math> under <math>f.\!</math>  This is a subset, possibly empty, of the domain <math>X,\!</math> notated as <math>f^{-1}(y) \subseteq X.\!</math>
| [[Image:Double Negation 1.0 Splash Page.png|500px]] || (5)
+
 
|}
+
In particular, if <math>f\!</math> is a proposition <math>f : X \to \mathbb{B},\!</math> then the fiber of truth <math>f^{-1}(1)\!</math> is the subset of <math>X\!</math> that is ''indicated'' by the proposition <math>f.\!</math>  Whenever we ''assert'' a proposition <math>f : X \to \mathbb{B},\!</math> we are saying that what it indicates is all that happens to be the case in the relevant universe of discourse <math>X.\!</math>  Because the fiber of truth is used so often in logical contexts, it is convenient to define the more compact notation <math>[| f |] = f^{-1}(1).\!</math>
 +
 
 +
Using this panoply of notions and notations, we may treat the fiber of truth of each proposition <math>f : \mathbb{B}^3 \to \mathbb{B}\!</math> as if it were a relational data table of the shape <math>\{ (p, q, r) \} \subseteq \mathbb{B}^3,\!</math> where the triples <math>(p, q, r)\!</math> are bit-tuples indicated by the proposition <math>f.\!</math>
 +
 
 +
Thus we obtain the following four relational data tables for the propositions that we are looking at in Example&nbsp;2.
  
The proof that follows is adapted from the one that was given by George Spencer Brown in his book ''Laws of Form'' (LOF) and credited to two of his students, John Dawes and D.A. Utting.
+
<br>
  
{| align="center" cellpadding="8"
+
{| align="center" cellpadding="8" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:40%"
|
+
|+ style="height:30px" | <math>\text{Table 56.} ~~ [| f_{207} |] ~=~ [| p \le q |]\!</math>
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center"
+
|- style="height:40px; background:#f0f0ff"
 +
| style="border-bottom:1px solid black" | <math>p\!</math>
 +
| style="border-bottom:1px solid black" | <math>q\!</math>
 +
| style="border-bottom:1px solid black" | <math>r\!</math>
 
|-
 
|-
| [[Image:Double Negation 1.0 Marquee Title.png|500px]]
+
| <math>0\!</math> || <math>0\!</math> || <math>0\!</math>
 
|-
 
|-
| [[Image:Double Negation 1.0 Storyboard 1.png|500px]]
+
| <math>0\!</math> || <math>0\!</math> || <math>1\!</math>
 
|-
 
|-
| [[Image:Equational Inference I2 Elicit (( )).png|500px]]
+
| <math>0\!</math> || <math>1\!</math> || <math>0\!</math>
 
|-
 
|-
| [[Image:Double Negation 1.0 Storyboard 2.png|500px]]
+
| <math>0\!</math> || <math>1\!</math> || <math>1\!</math>
 
|-
 
|-
| [[Image:Equational Inference J1 Insert (a).png|500px]]
+
| <math>1\!</math> || <math>1\!</math> || <math>0\!</math>
 
|-
 
|-
| [[Image:Double Negation 1.0 Storyboard 3.png|500px]]
+
| <math>1\!</math> || <math>1\!</math> || <math>1\!</math>
 +
|}
 +
 
 +
<br>
 +
 
 +
{| align="center" cellpadding="8" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:40%"
 +
|+ style="height:30px" | <math>\text{Table 57.} ~~ [| f_{187} |] ~=~ [| q \le r |]\!</math>
 +
|- style="height:40px; background:#f0f0ff"
 +
| style="border-bottom:1px solid black" | <math>p\!</math>
 +
| style="border-bottom:1px solid black" | <math>q\!</math>
 +
| style="border-bottom:1px solid black" | <math>r\!</math>
 
|-
 
|-
| [[Image:Equational Inference J2 Distribute ((a)).png|500px]]
+
| <math>0\!</math> || <math>0\!</math> || <math>0\!</math>
 
|-
 
|-
| [[Image:Double Negation 1.0 Storyboard 4.png|500px]]
+
| <math>0\!</math> || <math>0\!</math> || <math>1\!</math>
 
|-
 
|-
| [[Image:Equational Inference J1 Delete (a).png|500px]]
+
| <math>0\!</math> || <math>1\!</math> || <math>1\!</math>
 
|-
 
|-
| [[Image:Double Negation 1.0 Storyboard 5.png|500px]]
+
| <math>1\!</math> || <math>0\!</math> || <math>0\!</math>
 
|-
 
|-
| [[Image:Equational Inference J1 Insert a.png|500px]]
+
| <math>1\!</math> || <math>0\!</math> || <math>1\!</math>
 
|-
 
|-
| [[Image:Double Negation 1.0 Storyboard 6.png|500px]]
+
| <math>1\!</math> || <math>1\!</math> || <math>1\!</math>
 +
|}
 +
 
 +
<br>
 +
 
 +
{| align="center" cellpadding="8" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:40%"
 +
|+ style="height:30px" | <math>\text{Table 58.} ~~ [| f_{175} |] ~=~ [| p \le r |]\!</math>
 +
|- style="height:40px; background:#f0f0ff"
 +
| style="border-bottom:1px solid black" | <math>p\!</math>
 +
| style="border-bottom:1px solid black" | <math>q\!</math>
 +
| style="border-bottom:1px solid black" | <math>r\!</math>
 
|-
 
|-
| [[Image:Equational Inference J2 Collect a.png|500px]]
+
| <math>0\!</math> || <math>0\!</math> || <math>0\!</math>
 
|-
 
|-
| [[Image:Double Negation 1.0 Storyboard 7.png|500px]]
+
| <math>0\!</math> || <math>0\!</math> || <math>1\!</math>
 
|-
 
|-
| [[Image:Equational Inference J1 Delete ((a)).png|500px]]
+
| <math>0\!</math> || <math>1\!</math> || <math>0\!</math>
 
|-
 
|-
| [[Image:Double Negation 1.0 Storyboard 8.png|500px]]
+
| <math>0\!</math> || <math>1\!</math> || <math>1\!</math>
 
|-
 
|-
| [[Image:Equational Inference I2 Cancel (( )).png|500px]]
+
| <math>1\!</math> || <math>0\!</math> || <math>1\!</math>
 
|-
 
|-
| [[Image:Double Negation 1.0 Storyboard 9.png|500px]]
+
| <math>1\!</math> || <math>1\!</math> || <math>1\!</math>
|-
 
| [[Image:Equational Inference Marquee QED.png|500px]]
 
|}
 
| (6)
 
 
|}
 
|}
  
The steps of this proof are replayed in the following animation.
+
<br>
  
{| align="center" cellpadding="8"
+
{| align="center" cellpadding="8" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:40%"
|
+
|+ style="height:30px" | <math>\text{Table 59.} ~~ [| f_{139} |] ~=~ [| p \le q \le r |]\!</math>
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center"
+
|- style="height:40px; background:#f0f0ff"
 +
| style="border-bottom:1px solid black" | <math>p\!</math>
 +
| style="border-bottom:1px solid black" | <math>q\!</math>
 +
| style="border-bottom:1px solid black" | <math>r\!</math>
 +
|-
 +
| <math>0\!</math> || <math>0\!</math> || <math>0\!</math>
 +
|-
 +
| <math>0\!</math> || <math>0\!</math> || <math>1\!</math>
 +
|-
 +
| <math>0\!</math> || <math>1\!</math> || <math>1\!</math>
 
|-
 
|-
| [[Image:Double Negation 2.0 Animation.gif]]
+
| <math>1\!</math> || <math>1\!</math> || <math>1\!</math>
|}
 
| (7)
 
 
|}
 
|}
  
====C<sub>2</sub>.  Generation theorem====
+
<br>
 +
 
 +
In the medium of these unassuming examples, we begin to see the activities of logical inference and methodical inquiry as ''information clarifying operations''.
 +
 
 +
First, we drew a distinction between information preserving and information reducing processes and we noted the related distinction between equational and implicational inferencesI will use the acronyms EROI and IROI, respectively, for the equational and implicational analogues of the various rules of inference.
  
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''.
+
For example, we considered the brands of ''information fusion'' that are involved in a couple of standard rules of inference, taken in both their equational and their illative variants.
  
{| align="center" cellpadding="10"
+
In particular, let us assume that we begin from a state of uncertainty about the universe of discourse <math>X = \mathbb{B}^3\!</math> that is standardly represented by a uniform distribution <math>u : X \to \mathbb{B}\!</math> such that <math>u(x) = 1\!</math> for all <math>x\!</math> in <math>X,\!</math> in short, by the constant proposition <math>1 : X \to \mathbb{B}.\!</math>  This amounts to the ''maximum entropy sign state'' (MESS). As a measure of uncertainty, let us use either the multiplicative measure given by the cardinality of <math>X,\!</math> commonly notated as <math>|X|,\!</math> or else the additive measure given by <math>{\log_2 |X|}.\!</math>  In this frame we have <math>{|X| = 8}\!</math> and <math>{\log_2 |X| = 3},\!</math> to wit, 3 bits of doubt.
| [[Image:Generation Theorem 1.0 Splash Page.png|500px]] || (8)
 
|}
 
  
Here is a proof of the Generation Theorem.
+
Let us now consider the various rules of inference for transitivity in the light of their performance as information-developing actions.
  
{| align="center" cellpadding="8"
+
{| align="center" cellpadding="4" width="90%"
|
+
| <big>&bull;</big>
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center"
+
| colspan="3" | '''Transitive Law''' (Implicational Inference)
 
|-
 
|-
| [[Image:Generation Theorem 1.0 Marquee Title.png|500px]]
+
| width="1%" | &nbsp;
 +
| width="1%" | &nbsp;
 +
| colspan="2" |
 +
<math>\begin{array}{l}
 +
~ p \le q
 +
\\
 +
~ q \le r
 +
\\
 +
\overline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}
 +
\\
 +
~ p \le r
 +
\end{array}</math>
 
|-
 
|-
| [[Image:Generation Theorem 1.0 Storyboard 1.png|500px]]
+
| valign="top" | <big>&bull;</big>
 +
| colspan="3" | By itself, the information <math>p \le q\!</math> would reduce our uncertainty from <math>\log 8\!</math> bits to <math>\log 6\!</math> bits.
 
|-
 
|-
| [[Image:Equational Inference C1 Reflect a(b).png|500px]]
+
| valign="top" | <big>&bull;</big>
 +
| colspan="3" | By itself, the information <math>q \le r\!</math> would reduce our uncertainty from <math>\log 8\!</math> bits to <math>\log 6\!</math> bits.
 
|-
 
|-
| [[Image:Generation Theorem 1.0 Storyboard 2.png|500px]]
+
| valign="top" | <big>&bull;</big>
|-
+
| colspan="3" | By itself, the information <math>p \le r\!</math> would reduce our uncertainty from <math>\log 8\!</math> bits to <math>\log 6\!</math> bits.
| [[Image:Equational Inference I2 Elicit (( )).png|500px]]
 
|-
 
| [[Image:Generation Theorem 1.0 Storyboard 3.png|500px]]
 
|-
 
| [[Image:Equational Inference J1 Insert a.png|500px]]
 
|-
 
| [[Image:Generation Theorem 1.0 Storyboard 4.png|500px]]
 
|-
 
| [[Image:Equational Inference J2 Collect a.png|500px]]
 
|-
 
| [[Image:Generation Theorem 1.0 Storyboard 5.png|500px]]
 
|-
 
| [[Image:Equational Inference C1 Reflect a, b.png|500px]]
 
|-
 
| [[Image:Generation Theorem 1.0 Storyboard 6.png|500px]]
 
|-
 
| [[Image:Equational Inference Marquee QED.png|500px]]
 
|}
 
| (9)
 
 
|}
 
|}
  
The steps of this proof are replayed in the following animation.
+
In this situation the application of the implicational rule of inference for transitivity to the information <math>p \le q\!</math> and the information <math>q \le r\!</math> to get the information <math>p \le r\!</math> does not increase the measure of information beyond what any one of the three propositions has independently of the other two.  In a sense, then, the implicational rule operates only to move the information around without changing its measure in the slightest bit.
  
{| align="center" cellpadding="8"
+
{| align="center" cellpadding="4" width="90%"
|
+
| <big>&bull;</big>
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center"
+
| colspan="3" | '''Transitive Law''' (Equational Inference)
 +
|-
 +
| width="1%" | &nbsp;
 +
| width="1%" | &nbsp;
 +
| colspan="2" |
 +
<math>\begin{array}{l}
 +
~ p \le q
 +
\\
 +
~ q \le r
 +
\\
 +
\overline{\underline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}
 +
\\
 +
~ p \le q \le r
 +
\end{array}</math>
 +
|-
 +
| valign="top" | <big>&bull;</big>
 +
| colspan="3" | The contents and the measures of information that are associated with the propositions <math>p \le q\!</math> and <math>q \le r\!</math> are the same as before.
 
|-
 
|-
| [[Image:Generation Theorem 2.0 Animation.gif]]
+
| valign="top" | <big>&bull;</big>
|}
+
| colspan="3" | On its own, the information <math>p \le q \le r\!</math> would reduce our uncertainty from log(8) = 3 bits to log(4) = 2 bits, a reduction of 1 bit.
| (10)
 
 
|}
 
|}
  
====C<sub>3</sub>Dominant form theorem====
+
These are just some of the initial observations that can be made about the dimensions of information and uncertainty in the conduct of logical inference, and there are many issues to be taken up as we get to the thick of itIn particular, we are taking propositions far too literally at the outset, reading their spots at face value, as it were, without yet considering their species character as fallible signs.
  
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.
+
For ease of reference in the rest of this discussion, let us refer to the propositional form <math>f : \mathbb{B}^3 \to \mathbb{B}\!</math> such that <math>f(p, q, r) = f_{139}(p, q, r) = \texttt{(} p \texttt{(} q \texttt{))(} q \texttt{(} r \texttt{))}\!</math> as the ''syllogism map'', written as <math>\mathrm{syll} : \mathbb{B}^3 \to \mathbb{B},\!</math> and let us refer to its fiber of truth <math>[| \mathrm{syll} |] = \mathrm{syll}^{-1}(1)\!</math> as the ''syllogism relation'', written as <math>\mathrm{Syll} \subseteq \mathbb{B}^3.\!</math>  Table&nbsp;60 shows <math>\mathrm{Syll}\!</math> as a relational dataset.
  
{| align="center" cellpadding="10"
+
<br>
| [[Image:Dominant Form 1.0 Splash Page.png|500px]] || (11)
 
|}
 
 
 
Here is a proof of the Dominant Form Theorem.
 
  
{| align="center" cellpadding="8"
+
{| align="center" cellpadding="8" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:40%"
|
+
|+ style="height:30px" | <math>\text{Table 60.} ~~ \text{Syllogism Relation}\!</math>
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center"
+
|- style="height:40px; background:#f0f0ff"
 +
| style="border-bottom:1px solid black" | <math>p\!</math>
 +
| style="border-bottom:1px solid black" | <math>q\!</math>
 +
| style="border-bottom:1px solid black" | <math>r\!</math>
 
|-
 
|-
| [[Image:Dominant Form 1.0 Marquee Title.png|500px]]
+
| <math>0\!</math> || <math>0\!</math> || <math>0\!</math>
 
|-
 
|-
| [[Image:Dominant Form 1.0 Storyboard 1.png|500px]]
+
| <math>0\!</math> || <math>0\!</math> || <math>1\!</math>
 
|-
 
|-
| [[Image:Equational Inference C2 Regenerate a.png|500px]]
+
| <math>0\!</math> || <math>1\!</math> || <math>1\!</math>
 
|-
 
|-
| [[Image:Dominant Form 1.0 Storyboard 2.png|500px]]
+
| <math>1\!</math> || <math>1\!</math> || <math>1\!</math>
|-
 
| [[Image:Equational Inference J1 Delete a.png|500px]]
 
|-
 
| [[Image:Dominant Form 1.0 Storyboard 3.png|500px]]
 
|-
 
| [[Image:Equational Inference Marquee QED.png|500px]]
 
|}
 
| (12)
 
 
|}
 
|}
  
The following animation provides an instant re*play.
+
<br>
 +
 
 +
One of the first questions that we might ask about a 3-adic relation, in this case <math>\mathrm{Syll},\!</math> is whether it is ''determined by'' its 2-adic projections.  I will illustrate what this means in the present case.
 +
 
 +
Table&nbsp;61 repeats the relation <math>\mathrm{Syll}\!</math> in the first column, listing its 3-tuples in bit-string form, followed by the 2-adic or ''planar'' projections of <math>\mathrm{Syll}\!</math> in the next three columns.  For instance, <math>\mathrm{Syll}_{pq}\!</math> is the 2-adic projection of <math>\mathrm{Syll}\!</math> on the <math>pq\!</math> plane that is arrived at by deleting the <math>r\!</math> column and counting each 2-tuple that results just one time.  Likewise, <math>\mathrm{Syll}_{pr}\!</math> is obtained by deleting the <math>q\!</math> column and <math>\mathrm{Syll}_{qr}\!</math> is derived by deleting the <math>p\!</math> column, ignoring whatever duplicate pairs may result.  The final row of the right three columns gives the propositions of the form <math>f : \mathbb{B}^2 \to \mathbb{B}\!</math> that indicate the 2-adic relations that result from these projections.
 +
 
 +
<br>
  
{| align="center" cellpadding="8"
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
 +
|+ style="height:30px" | <math>\text{Table 61.} ~~ \text{Dyadic Projections of the Syllogism Relation}\!</math>
 +
|- style="height:40px; background:#f0f0ff"
 +
| <math>\mathrm{Syll}\!</math>
 +
| <math>\mathrm{Syll}_{pq}\!</math>
 +
| <math>\mathrm{Syll}_{pr}\!</math>
 +
| <math>\mathrm{Syll}_{qr}\!</math>
 +
|-
 
|
 
|
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center"
+
<math>\begin{matrix}
|-
+
0~0~0 \\ 0~0~1 \\ 0~1~1 \\ 1~1~1
| [[Image:Dominant Form 2.0 Animation.gif]]
+
\end{matrix}</math>
|}
+
|
| (13)
+
<math>\begin{matrix}
 +
0~0 \\ 0~0 \\ 0~1 \\ 1~1
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0~0 \\ 0~1 \\ 0~1 \\ 1~1
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0~0 \\ 0~1 \\ 1~1 \\ 1~1
 +
\end{matrix}</math>
 +
|- style="height:40px; background:#f0f0ff"
 +
| <math>p \le q \le r\!</math>
 +
| <math>\texttt{(} p \texttt{ (} q \texttt{))}\!</math>
 +
| <math>\texttt{(} p \texttt{ (} r \texttt{))}\!</math>
 +
| <math>\texttt{(} q \texttt{ (} r \texttt{))}\!</math>
 
|}
 
|}
  
===Exemplary proofs===
+
<br>
 +
 
 +
Let us make the simple observation that taking a projection, in our framework, deleting a column from a relational table, is like taking a derivative in differential calculus.  What it means is that our attempt to return to the integral from whence the derivative was derived will in general encounter an indefinite variation on account of the circumstance that real information may have been destroyed by the derivation.
  
Based on the axioms given at the outest, and aided by the theorems recorded so far, it is possible to prove a multitude of much more complex theoremsA couple of all-time favorites are given next.
+
One will find that some relations can be reconstructed from various types of derivatives and projections, others cannotThe reconstuctible relations are said to be ''reducible'' to the types of reductive data in question, while the others are said to be ''irreducible'' with respect to those means.
  
====Peirce's law====
+
The analogies between derivation, differentiation, implication, projection, and others sorts of information reducing operation will undergo extensive development in the remainder and sequel of the present discussion.
  
: ''Main article'' : [[Peirce's law]]
+
We were in the middle of discussing the relationships between information preserving rules of inference and information destroying rules of inference &mdash; folks of a 3-basket philosophical bent will no doubt be asking, "And what of information creating rules of inference?", but there I must wait for some signs of enlightenment, desiring not to tread on the rules of that succession.
  
Peirce's law is commonly written in the following form:
+
The contrast between the information destroying and the information preserving versions of the transitive rule of inference led us to examine the relationships among several boolean functions, namely, those that qualify locally as the elementary cellular automata rules <math>f_{139}, f_{175}, f_{187}, f_{207}.\!</math>
  
{| align="center" cellpadding="10"
+
The function <math>f_{139} : \mathbb{B}^3 \to \mathbb{B}\!</math> and its fiber <math>[| f_{139} |] \subseteq \mathbb{B}^3\!</math> appeared to be key to many structures in this setting, and so I singled them out under the new names of <math>\mathrm{syll} : \mathbb{B}^3 \to \mathbb{B}\!</math> and <math>\mathrm{Syll} \subseteq \mathbb{B}^3,\!</math> respectively.
| <math>((p \Rightarrow q) \Rightarrow p) \Rightarrow p\!</math>
 
|}
 
  
The existential graph representation of Peirce's law is shown below.
+
Managing the conceptual complexity of our considerations at this juncture put us in need of some conceptual tools that I broke off to develop in my notes on "Reductions Among Relations".  The main items that we need right away from that thread are the definitions of relational projections and their inverses, the tacit extensions.
  
{| align="center" cellpadding="10"
+
But the more I survey the problem setting the more it looks like we need better ways to bring our visual intuitions to play on the scene, and so let us next lay out some visual schemata that are designed to facilitate that.
| [[Image:Peirce's Law 1.0 Splash Page.png|500px]] || (14)
 
|}
 
  
A graphical proof of Peirce's law is shown next.
+
Figure&nbsp;62 shows the familiar picture of a boolean 3-cube, where the points of <math>\mathbb{B}^3\!</math> are coordinated as bit strings of length three.  Looking at the functions <math>f : \mathbb{B}^3 \to \mathbb{B}\!</math> and the relations <math>L \subseteq \mathbb{B}^3\!</math> on this pattern, one views the construction of either type of object as a matter of coloring the nodes of the 3-cube with choices from a pair of colors that stipulate which points are in the relation <math>{L = [| f |]}\!</math> and which points are out of it.  Bowing to common convention, we may use the color <math>1\!</math> for points that are ''in'' a given relation and the color <math>0\!</math> for points that are ''out'' of the same relation.  However, it will be more convenient here to indicate the former case by writing the coordinates in the place of the node and to indicate the latter case by plotting the point as an unlabeled node "o".
  
{| align="center" cellpadding="8"
+
{| align="center" border="0" cellpadding="10"
 
|
 
|
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center"
+
<pre>
|-
+
o-------------------------------------------------o
| [[Image:Peirce's Law 1.0 Marquee Title.png|500px]]
+
|                                                 |
|-
+
|                       111                      |
| [[Image:Peirce's Law 1.0 Storyboard 1.png|500px]]
+
|                       /|\                      |
|-
+
|                     / | \                      |
| [[Image:Equational Inference Band Collect p.png|500px]]
+
|                     /  |  \                    |
|-
+
|                   /  |  \                    |
| [[Image:Peirce's Law 1.0 Storyboard 2.png|500px]]
+
|                   /    |    \                  |
|-
+
|                 /    |    \                  |
| [[Image:Equational Inference Band Quit ((q)).png|500px]]
+
|                 /      |      \                |
|-
+
|               110    101    011              |
| [[Image:Peirce's Law 1.0 Storyboard 3.png|500px]]
+
|               |\    / \    /|                |
|-
+
|               | \  /  \  / |                |
| [[Image:Equational Inference Band Cancel (( )).png|500px]]
+
|               |  \ /    \ /  |                |
|-
+
|               |  \      /  |                |
| [[Image:Peirce's Law 1.0 Storyboard 4.png|500px]]
+
|               |  / \    / \  |                |
|-
+
|               | /  \  /  \ |                |
| [[Image:Equational Inference Band Delete p.png|500px]]
+
|               |/    \ /    \|                |
|-
+
|               100    010    001              |
| [[Image:Peirce's Law 1.0 Storyboard 5.png|500px]]
+
|                 \      |      /                |
|-
+
|                 \    |    /                  |
| [[Image:Equational Inference Band Cancel (( )).png|500px]]
+
|                   \    |    /                  |
|-
+
|                   \  |  /                    |
| [[Image:Peirce's Law 1.0 Storyboard 6.png|500px]]
+
|                     \  |  /                    |
|-
+
|                     \ | /                      |
| [[Image:Equational Inference Marquee QED.png|500px]]
+
|                       \|/                      |
|}
+
|                       000                      |
| (15)
+
|                                                 |
 +
o-------------------------------------------------o
 +
Figure 62.  Boolean 3-Cube B^3
 +
</pre>
 +
| (62)
 
|}
 
|}
  
The following animation replays the steps of the proof.
+
Table&nbsp;63 shows the 3-adic relation <math>\mathrm{Syll} \subseteq \mathbb{B}^3\!</math> again, and Figure&nbsp;64 shows it plotted on a 3-cube template.
  
{| align="center" cellpadding="8"
+
{| align="center" border="0" cellpadding="10"
 
|
 
|
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center"
+
<pre>
|-
+
Table 63.  Syll c B^3
| [[Image:Peirce's Law 2.0 Animation.gif]]
+
o-----------------------o
 +
|  p      q      r  |
 +
o-----------------------o
 +
|   0      0       0   |
 +
|  0      0      1  |
 +
|  0      1      1  |
 +
|  1      1      1  |
 +
o-----------------------o
 +
</pre>
 +
| (63)
 
|}
 
|}
| (16)
+
 
 +
{| align="center" border="0" cellpadding="10"
 +
|
 +
<pre>
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                      111                      |
 +
|                      /|\                      |
 +
|                      / | \                      |
 +
|                    /  |  \                    |
 +
|                    /  |  \                    |
 +
|                  /    |    \                  |
 +
|                  /    |    \                  |
 +
|                /      |      \                |
 +
|                o      o      011              |
 +
|                |\    / \    /|                |
 +
|                | \  /  \  / |                |
 +
|                |  \ /    \ /  |                |
 +
|                |  \      /  |                |
 +
|                |  / \    / \  |                |
 +
|                | /  \  /  \ |                |
 +
|                |/    \ /    \|                |
 +
|                o      o      001              |
 +
|                \      |      /                |
 +
|                  \    |    /                  |
 +
|                  \    |    /                  |
 +
|                    \  |  /                    |
 +
|                    \  |  /                    |
 +
|                      \ | /                      |
 +
|                      \|/                      |
 +
|                      000                      |
 +
|                                                |
 +
o-------------------------------------------------o
 +
Figure 64.  Triadic Relation Syll c B^3
 +
</pre>
 +
| (64)
 
|}
 
|}
  
====Praeclarum theorema====
+
We return once more to the plane projections of <math>\mathrm{Syll} \subseteq \mathbb{B}^3.\!</math>
  
An illustrious example of a propositional theorem is the ''praeclarum theorema'', the ''admirable'', ''shining'', or ''splendid'' theorem of Leibniz.
+
{| align="center" border="0" cellpadding="10"
 +
|
 +
<pre>
 +
Table 65. Syll c B^3
 +
o-----------------------o
 +
|  p      q      r  |
 +
o-----------------------o
 +
|  0      0      0  |
 +
|  0      0      1  |
 +
|  0      1      1  |
 +
|  1      1      1  |
 +
o-----------------------o
 +
</pre>
 +
| (65)
 +
|}
  
{| align="center" cellpadding="10" width="90%"
+
{| align="center" border="0" cellpadding="10"
 
|
 
|
<p>If ''a'' is ''b'' and ''d'' is ''c'', then ''ad'' will be ''bc''.</p>
+
<pre>
 +
Table 66. Dyadic Projections of Syll
 +
o-----------o o-----------o o-----------o
 +
|  Syll_12  | |  Syll_13  | |  Syll_23  |
 +
o-----------o o-----------o o-----------o
 +
|  p  q  | |  p  r  | |  q  r  |
 +
o-----------o o-----------o o-----------o
 +
|  0  0  | |  0  0  | |  0  0  |
 +
|  0  1  | |  0  1  | |  0  1  |
 +
|  1  1  | |  1  1  | |  1  1  |
 +
o-----------o o-----------o o-----------o
 +
|  (p (q))  | |  (p (r))  | |  (q (r))  |
 +
o-----------o o-----------o o-----------o
 +
</pre>
 +
| (66)
 +
|}
  
<p>This is a fine theorem, which is proved in this way:</p>
+
In showing the 2-adic projections of a 3-adic relation <math>L \subseteq \mathbb{B}^3,\!</math> I will translate the coordinates of the points in each relation to the plane of the projection, there dotting out with a dot "." the bit of the bit string that is out of place on that plane.
  
<p>''a'' is ''b'', therefore ''ad'' is ''bd'' (by what precedes),</p>
+
Figure&nbsp;67 shows <math>\mathrm{Syll}\!</math> and its three 2-adic projections:
  
<p>''d'' is ''c'', therefore ''bd'' is ''bc'' (again by what precedes),</p>
+
{| align="center" border="0" cellpadding="10"
 +
|
 +
<pre>
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                      111                      |
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|                /      |      \                |
 +
|                o      o      011              |
 +
|                |\    / \    /|                |
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|                | \  /  \  / |                |
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|                |  / \    / \  |                |
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|                | /  \  /  \ |                |
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|                |/    \ /    \|                |
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|                o      o      001              |
 +
|                \      |      /                |
 +
|    11.          \    |    /          .11    |
 +
|      |\          \    |    /          /|      |
 +
|      | \          \  |  /          / |      |
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|      |  \          \  |  /          /  |      |
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|      |  \          \ | /          /  |      |
 +
|      |    \          \|/          /    |      |
 +
|      |    \          000          /    |      |
 +
|      |      \                    /      |      |
 +
|      o      01.                  o      .01    |
 +
|      \      |                  |      /      |
 +
|        \    |                  |    /        |
 +
|        \    |                  |    /        |
 +
|          \  |        1.1        |  /          |
 +
|          \  |        / \        |  /          |
 +
|            \ |      /  \      | /            |
 +
|            \|      /    \      |/            |
 +
|            00.    /      \    .00            |
 +
|                  /        \                  |
 +
|                  /          \                  |
 +
|                /            \                |
 +
|                o              0.1              |
 +
|                \            /                |
 +
|                  \          /                  |
 +
|                  \        /                  |
 +
|                    \      /                    |
 +
|                    \    /                    |
 +
|                      \  /                      |
 +
|                      \ /                      |
 +
|                      0.0                      |
 +
|                                                |
 +
o-------------------------------------------------o
 +
Figure 67.  Syll c B^3 and its Dyadic Projections
 +
</pre>
 +
| (67)
 +
|}
  
<p>''ad'' is ''bd'', and ''bd'' is ''bc'', therefore ''ad'' is ''bc''.  Q.E.D.</p>
+
We now compute the tacit extensions of the 2-adic projections of <math>\mathrm{Syll},\!</math> alias <math>f_{139},\!</math> and this makes manifest its relationship to the other functions and fibers, namely, <math>f_{175}, f_{187}, f_{207}.\!</math>
  
<p>(Leibniz, ''Logical Papers'', p. 41).</p>
+
{| align="center" border="0" cellpadding="10"
 +
|
 +
<pre>
 +
Table 68.  Syll c B^3
 +
o-----------------------o
 +
p       q      r  |
 +
o-----------------------o
 +
|  0      0      0  |
 +
|  0      0      1  |
 +
|  0      1      1  |
 +
|  1      1      1  |
 +
o-----------------------o
 +
</pre>
 +
| (68)
 
|}
 
|}
  
Under the existential interpretation, the praeclarum theorema is represented by means of the following logical graph.
+
{| align="center" border="0" cellpadding="10"
 +
|
 +
<pre>
 +
Table 69.  Dyadic Projections of Syll
 +
o-----------o o-----------o o-----------o
 +
|  Syll_12  | |  Syll_13  | |  Syll_23  |
 +
o-----------o o-----------o o-----------o
 +
|  p  q  | |  p  r  | |  q  r  |
 +
o-----------o o-----------o o-----------o
 +
|  0  0  | |  0  0  | |  0  0  |
 +
|  0  1  | |  0  1  | |  0  1  |
 +
|  1  1  | |  1  1  | |  1  1  |
 +
o-----------o o-----------o o-----------o
 +
|  (p (q))  | |  (p (r))  | |  (q (r))  |
 +
o-----------o o-----------o o-----------o
 +
</pre>
 +
| (69)
 +
|}
  
{| align="center" cellpadding="10"
+
{| align="center" border="0" cellpadding="10"
| [[Image:Praeclarum Theorema 1.0 Splash Page.png|500px]] || (17)
+
|
 +
<pre>
 +
Table 70.  Tacit Extensions of Projections of Syll
 +
o---------------o o---------------o o---------------o
 +
|  te(Syll_12)  | |  te(Syll_13)  | |  te(Syll_23)  |
 +
o---------------o o---------------o o---------------o
 +
|  p  q  r  | |  p  q  r  | |  p  q  r  |
 +
o---------------o o---------------o o---------------o
 +
|  0  0  0  | |  0  0  0  | |  0  0  0  |
 +
|  0  0  1  | |  0  1  0  | |  1  0  0  |
 +
|  0  1  0  | |  0  0  1  | |  0  0  1  |
 +
|  0  1  1  | |  0  1  1  | |  1  0  1  |
 +
|  1  1  0  | |  1  0  1  | |  0  1  1  |
 +
|  1  1  1  | |  1  1  1  | |  1  1  1  |
 +
o---------------o o---------------o o---------------o
 +
| [| (p (q)) |] | | [| (p (r)) |] | | [| (q (r)) |] |
 +
o---------------o o---------------o o---------------o
 +
| [| f_207  |] | | [|  f_175  |] | | [|  f_187  |] |
 +
o---------------o o---------------o o---------------o
 +
</pre>
 +
| (70)
 
|}
 
|}
  
And here's a neat proof of that nice theorem.
+
{| align="center" border="0" cellpadding="10"
 +
|
 +
<pre>
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                      111                      |
 +
|                      /|\                      |
 +
|                      / | \                      |
 +
|                    /  |  \                    |
 +
|                    /  |  \                    |
 +
|                  /    |    \                  |
 +
|                  /    |    \                  |
 +
|                /      |      \                |
 +
|              110      o      011              |
 +
|                |\    / \    /|                |
 +
|                | \  /  \  / |                |
 +
|                |  \ /    \ /  |                |
 +
|                |  \      /  |                |
 +
|                |  / \    / \  |                |
 +
|                | /  \  /  \ |                |
 +
|                |/    \ /    \|                |
 +
|                o      010    001              |
 +
|                \      |      /                |
 +
|    11.         \    |    /                  |
 +
|      |\          \    |    /                  |
 +
|      | \          \  |  /                    |
 +
|      |  \          \  |  /                    |
 +
|      |  \          \ | /                      |
 +
|      |    \          \|/                      |
 +
|      |    \          000                      |
 +
|      |      \                                  |
 +
|      o      01.                                |
 +
|      \      |                                  |
 +
|        \    |                                  |
 +
|        \    |                                  |
 +
|          \  |                                  |
 +
|          \  |                                  |
 +
|            \ |                                  |
 +
|            \|                                  |
 +
|            00.                                |
 +
|                                                |
 +
o-------------------------------------------------o
 +
Figure 71.  Tacit Extension te_12_3 (Syll_12)
 +
</pre>
 +
| (71)
 +
|}
  
{| align="center" cellpadding="8"
+
{| align="center" border="0" cellpadding="10"
 
|
 
|
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center"
+
<pre>
|-
+
o-------------------------------------------------o
| [[Image:Praeclarum Theorema 1.0 Marquee Title.png|500px]]
+
|                                                |
|-
+
|                      111                      |
| [[Image:Praeclarum Theorema 1.0 Storyboard 1.png|500px]]
+
|                      /|\                      |
|-
+
|                      / | \                      |
| [[Image:Equational Inference Rule Reflect ad(bc).png|500px]]
+
|                     /  |  \                    |
|-
+
|                    /  |  \                    |
| [[Image:Praeclarum Theorema 1.0 Storyboard 2.png|500px]]
+
|                  /    |    \                  |
|-
+
|                  /    |    \                  |
| [[Image:Equational Inference Rule Weed a, d.png|500px]]
+
|                /      |      \                |
|-
+
|                o      101    011              |
| [[Image:Praeclarum Theorema 1.0 Storyboard 3.png|500px]]
+
|                |\    / \    /|                |
|-
+
|                | \  /  \  / |                |
| [[Image:Equational Inference Rule Reflect b, c.png|500px]]
+
|                |  \ /    \ /  |                |
|-
+
|                |  \      /  |                |
| [[Image:Praeclarum Theorema 1.0 Storyboard 4.png|500px]]
+
|                |  / \    / \  |                |
|-
+
|                | /  \  /  \ |                |
| [[Image:Equational Inference Rule Weed bc.png|500px]]
+
|                |/    \ /    \|                |
|-
+
|                o      010    001              |
| [[Image:Praeclarum Theorema 1.0 Storyboard 5.png|500px]]
+
|                 \      |      /                |
|-
+
|                 \    |    /                  |
| [[Image:Equational Inference Rule Quit abcd.png|500px]]
+
|                   \    |    /                  |
|-
+
|                   \  |  /                    |
| [[Image:Praeclarum Theorema 1.0 Storyboard 6.png|500px]]
+
|                     \  |  /                    |
|-
+
|                     \ | /                      |
| [[Image:Equational Inference Rule Cancel (( )).png|500px]]
+
|                       \|/                      |
|-
+
|                       000                      |
| [[Image:Praeclarum Theorema 1.0 Storyboard 7.png|500px]]
+
|                                                 |
|-
+
|                                                 |
| [[Image:Equational Inference Marquee QED.png|500px]]
+
|                       1.1                      |
 +
|                       / \                      |
 +
|                     /  \                      |
 +
|                     /    \                    |
 +
|                   /      \                    |
 +
|                   /        \                  |
 +
|                 /          \                  |
 +
|                 /            \                |
 +
|               o              0.1              |
 +
|                 \            /                |
 +
|                 \          /                  |
 +
|                   \        /                  |
 +
|                   \      /                    |
 +
|                     \    /                    |
 +
|                     \  /                      |
 +
|                       \ /                      |
 +
|                       0.0                       |
 +
|                                                 |
 +
o-------------------------------------------------o
 +
Figure 72. Tacit Extension te_13_2 (Syll_13)
 +
</pre>
 +
| (72)
 
|}
 
|}
| (18)
+
 
 +
{| align="center" border="0" cellpadding="10"
 +
|
 +
<pre>
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                      111                      |
 +
|                      /|\                      |
 +
|                      / | \                      |
 +
|                    /  |  \                    |
 +
|                    /  |  \                    |
 +
|                  /    |    \                  |
 +
|                  /    |    \                  |
 +
|                /      |      \                |
 +
|                o      101    011              |
 +
|                |\    / \    /|                |
 +
|                | \  /  \  / |                |
 +
|                |  \ /    \ /  |                |
 +
|                |  \      /  |                |
 +
|                |  / \    / \  |                |
 +
|                | /  \  /  \ |                |
 +
|                |/    \ /    \|                |
 +
|              100      o      001              |
 +
|                \      |      /                |
 +
|                  \    |    /          .11    |
 +
|                  \    |    /          /|      |
 +
|                    \  |  /          / |      |
 +
|                    \  |  /          /  |      |
 +
|                      \ | /          /  |      |
 +
|                      \|/          /    |      |
 +
|                      000          /    |      |
 +
|                                  /      |      |
 +
|                                  o      .01    |
 +
|                                  |      /      |
 +
|                                  |    /        |
 +
|                                  |    /        |
 +
|                                  |  /          |
 +
|                                  |  /          |
 +
|                                  | /            |
 +
|                                  |/            |
 +
|                                .00            |
 +
|                                                |
 +
o-------------------------------------------------o
 +
Figure 73.  Tacit Extension te_23_1 (Syll_23)
 +
</pre>
 +
| (73)
 
|}
 
|}
  
The steps of the proof are replayed in the following animation.
+
The reader may wish to contemplate Figure&nbsp;74 and use it to verify the following two facts:
  
{| align="center" cellpadding="8"
+
{| align="center" cellpadding="8" width="90%"
 
|
 
|
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center"
+
<math>\begin{array}{lcc}
|-
+
\mathrm{Syll}
| [[Image:Praeclarum Theorema 2.0 Animation.gif]]
+
& = &
 +
\mathrm{te}(\mathrm{Syll}_{12})
 +
\cap
 +
\mathrm{te}(\mathrm{Syll}_{23})
 +
\\[6pt]
 +
\mathrm{Syll}_{13}
 +
& = &
 +
\mathrm{Syll}_{12}
 +
\circ
 +
\mathrm{Syll}_{23}
 +
\end{array}</math>
 
|}
 
|}
| (19)
+
 
 +
{| align="center" border="0" cellpadding="10"
 +
|
 +
<pre>
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                        *                        |
 +
|                      /|\                      |
 +
|                      / | \                      |
 +
|                    /  |  \                    |
 +
|                    /  |  \                    |
 +
|                  /    |    \                  |
 +
|                  /    |    \                  |
 +
|                /      |      \                |
 +
|                o      o      *                |
 +
|              /|\    / \    /|\              |
 +
|              / | \  /  \  / | \              |
 +
|            /  |  \ /    \ /  |  \            |
 +
|            /  |  \      /  |  \            |
 +
|          /    |  / \    / \  |    \          |
 +
|          /    | /  \  /  \ |    \          |
 +
|        /      |/    \ /    \|      \        |
 +
|        /      o      o      *      \        |
 +
|      /        \    /|      / \      \      |
 +
|      *          \  / |    /  \      *      |
 +
|      |\          \ /  |    /    \    /|      |
 +
|      | \          /  |  /      \  / |      |
 +
|      |  \        / \  |  /        \ /  |      |
 +
|      |  \      /  \ | /          /  |      |
 +
|      |    \    /    \|/          / \  |      |
 +
|      |    \  /      *          /  \ |      |
 +
|      |      \ /      / \        /    \|      |
 +
|      o      *      /  \      o      *      |
 +
|      \      |      /    \      |      /      |
 +
|        \    |    /      \    |    /        |
 +
|        \    |    /        \    |    /        |
 +
|          \  |  /    *    \  |  /          |
 +
|          \  |  /    / \    \  |  /          |
 +
|            \ | /    /  \    \ | /            |
 +
|            \|/    /    \    \|/            |
 +
|              *    /      \    *              |
 +
|                  /        \                  |
 +
|                  /          \                  |
 +
|                /            \                |
 +
|                o              *                |
 +
|                \            /                |
 +
|                  \          /                  |
 +
|                  \        /                  |
 +
|                    \      /                    |
 +
|                    \    /                    |
 +
|                      \  /                      |
 +
|                      \ /                      |
 +
|                        *                        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
Figure 74.  Syll = te(Syll_12) |^| te(Syll_23)
 +
</pre>
 +
| (74)
 
|}
 
|}
  
====Two-thirds majority function====
+
I don't know about you, but I am still puzzled by all of thus stuff, that is to say, by the entanglements of composition and projection and their relationship to the information processing properties of logical inference rules.  What I lack is a single picture that could show me all of the pieces and make the pattern of their informational relationships clear.
  
Consider the following equation in boolean algebra, posted as a [http://mathoverflow.net/questions/9292/newbie-boolean-algebra-question problem for proof] at [http://mathoverflow.net/ MathOverFlow].
+
In accord with my experimental way, I will stick with the case of transitive inference until I have pinned it down thoroughly, but of course the real interest is much more general than that.
  
{| align="center" cellpadding="20"
+
At first sight, the relationships seem easy enough to write out.  Figure&nbsp;75 shows how the various logical expressions are related to each other:  The expressions <math>{}^{\backprime\backprime} \texttt{(} p \texttt{ (} q \texttt{))} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \texttt{(} q \texttt{ (} r \texttt{))} {}^{\prime\prime}\!</math> are conjoined in a purely syntactic fashion &mdash; much in the way that one might compile a theory from axioms without knowing what either the theory or the axioms were about &mdash; and the best way to sum up the state of information implicit in taking them together is just the expression <math>{}^{\backprime\backprime} \texttt{(} p \texttt{ (} q \texttt{)) (} q \texttt{ (} r \texttt{))}{}^{\prime\prime}\!</math> that would the canonical result of an equational or reversible rule of inference.  From that equational inference, one might arrive at the implicational inference <math>{}^{\backprime\backprime} \texttt{(} p \texttt{ (} r \texttt{))} {}^{\prime\prime}\!</math> by the most conventional implication.
 +
 
 +
{| align="center" border="0" cellpadding="10"
 
|
 
|
<math>\begin{matrix}
+
<pre>
a b \bar{c} + a \bar{b} c + \bar{a} b c + a b c
+
o-------------------o        o-------------------o
\\[6pt]
+
|                  |        |                  |
\iff
+
|        q        |        |        r        |
\\[6pt]
+
|        o        |        |        o        |
a b + a c + b c
+
|        |        |        |        |        |
\end{matrix}</math>
+
|      p o        |        |      q o        |
| (20)
+
|        |        |        |        |        |
 +
|        @        |        |        @        |
 +
|                  |        |                  |
 +
o-------------------o        o-------------------o
 +
|      (p (q))      |        |      (q (r))      |
 +
o-------------------o        o-------------------o
 +
|      f_207      |        |      f_187      |
 +
o---------o---------o        o---------o---------o
 +
          \                           /         
 +
            \       Conjunction      /           
 +
            \                       /           
 +
              v                    v             
 +
              o-------------------o             
 +
              |                  |             
 +
              |                  |             
 +
              |                  |             
 +
              |      q  r      |             
 +
              |      o  o      |             
 +
              |      |  |      |             
 +
              |    p o  o q    |             
 +
              |        \ /       |             
 +
              |        @        |             
 +
              |                  |             
 +
              o-------------------o             
 +
              |  (p (q)) (q (r))  |             
 +
              o-------------------o             
 +
              |      f_139      |             
 +
              o---------o---------o             
 +
                        |                       
 +
                    Implication                   
 +
                        |                       
 +
                        v                       
 +
              o---------o---------o             
 +
              |                  |             
 +
              |        r        |             
 +
              |        o        |             
 +
              |        |        |             
 +
              |      p o        |             
 +
              |        |        |             
 +
              |        @        |             
 +
              |                  |             
 +
              o-------------------o             
 +
              |     (p (r))      |             
 +
              o-------------------o             
 +
              |      f_175      |             
 +
              o-------------------o             
 +
                                                 
 +
Figure 75.  Expressive Aspects of Transitive Inference
 +
</pre>
 
|}
 
|}
  
The required equation can be proven in the medium of logical graphs as shown in the following Figure.
+
Most of the customary names for this type of process have turned out to have misleading connotations, and so I will experiment with calling it the ''expressive'' aspect of the various rules for transitive inference, simply to emphasize the fact that rules can be given for it that operate solely on signs and expressions, without necessarily needing to look at the objects that are denoted by these signs and expressions.
 +
 
 +
In the way of many experiments, the word ''expressive'' does not seem to work for what I wanted to say here, since we too often use it to suggest something that expresses an object or a purpose, and I wanted it to imply what is purely a matter of expression, shorn of consideration for anything objective.  Aside from coining a word like ''ennotative'', some other options would be ''connotative'', ''hermeneutic'', ''semiotic'', ''syntactic'' &mdash; each of which works in some range of interpretation but fails in others.  Let's try ''formulaic''.
 +
 
 +
Despite how simple the formulaic aspects of transitive inference might appear on the surface, there are problems that wait for us just beneath the syntactic surface, as we quickly discover if we turn to considering the kinds of objects, abstract and concrete, that these formulas are meant to denote, and all the more so if we try to do this in a context of computational implementations, where the "interpreters" to be addressed take nothing on faith.  Thus we engage the ''denotative semantics'' or the ''model theory'' of these extremely simple programs that we call ''propositions''.
  
{| align="center" cellpadding="8"
+
Figure&nbsp;76 is an attempt to outline the model-theoretic relationships that are involved in our study of transitive inference.  A couple of alternative notations are introduced in this Table:
|
+
 
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center"
+
{| align="center" cellpadding="8" width="90%"
|-
+
| The forms <math>X:Y:Z\!</math> and <math>x:y:z\!</math> are used as alternative notations for the cartesian product <math>X \times Y \times Z\!</math> and the tuple <math>(x, y, z),\!</math> respectively.
| [[Image:Majority Function Example Proof 1 Title.jpg|500px]]
 
|-
 
| [[Image:Majority Function Example 2.0 Proof 1 Frame 1.jpg|500px]]
 
|-
 
| [[Image:Equational Inference Reflect ab, ac, bc ISW.jpg|500px]]
 
|-
 
| [[Image:Majority Function Example 2.0 Proof 1 Frame 2.jpg|500px]]
 
|-
 
| [[Image:Equational Inference Distribute (abc).jpg|500px]]
 
 
|-
 
|-
| [[Image:Majority Function Example 2.0 Proof 1 Frame 3.jpg|500px]]
+
| In situations where we have products like <math>X:Y:Z\!</math> with <math>X = Y = Z = \mathbb{B},\!</math> and relations like <math>{L \subseteq X:Y},\!</math> &nbsp; <math>{M \subseteq X:Z},\!</math> &nbsp; <math>{N \subseteq Y:Z},\!</math> the forms <math>{L \subseteq \mathbb{B}:\mathbb{B}:-},\!</math> &nbsp; <math>{M \subseteq \mathbb{B}:-:\mathbb{B}},\!</math> &nbsp; <math>{N \subseteq -:\mathbb{B}:\mathbb{B}}\!</math> are used to remind us that we are considering particular ways of situating <math>{L, M, N}\!</math> within the product space <math>X:Y:Z.\!</math>
|-
 
| [[Image:Equational Inference Collect ab, ac, bc ISW.jpg|500px]]
 
|-
 
| [[Image:Majority Function Example 2.0 Proof 1 Frame 4.jpg|500px]]
 
|-
 
| [[Image:Equational Inference Quit (a), (b), (c).jpg|500px]]
 
|-
 
| [[Image:Majority Function Example 2.0 Proof 1 Frame 5.jpg|500px]]
 
|-
 
| [[Image:Equational Inference Cancel (( )).jpg|500px]]
 
|-
 
| [[Image:Majority Function Example 2.0 Proof 1 Frame 6.jpg|500px]]
 
|-
 
| [[Image:Equational Inference Weed ab, ac, bc.jpg|500px]]
 
|-
 
| [[Image:Majority Function Example 2.0 Proof 1 Frame 7.jpg|500px]]
 
|-
 
| [[Image:Equational Inference Delete a, b, c ISW.jpg|500px]]
 
|-
 
| [[Image:Majority Function Example 2.0 Proof 1 Frame 8.jpg|500px]]
 
|-
 
| [[Image:Equational Inference Cancel (( )).jpg|500px]]
 
|-
 
| [[Image:Majority Function Example 2.0 Proof 1 Frame 9.jpg|500px]]
 
|-
 
| [[Image:Equational Inference QED.jpg|500px]]
 
 
|}
 
|}
| (21)
+
 
 +
{| align="center" border="0" cellpadding="10"
 +
|
 +
<pre>
 +
o-------------------o        o-------------------o
 +
|                  |        |                  |
 +
|      0:0:0      |        |      0:0:0      |
 +
|      0:0:1      |        |      0:0:1      |
 +
|      0:1:0      |        |      0:1:1      |
 +
|      0:1:1      |        |      1:0:0      |
 +
|      1:1:0      |        |      1:0:1      |
 +
|      1:1:1      |        |      1:1:1      |
 +
|                  |        |                  |
 +
o-------------------o        o-------------------o
 +
|te(Syll_12) c B:B:B|        |te(Syll_23) c B:B:B|
 +
o-------------------o        o-------------------o
 +
|    [| f_207 |]    |        |    [| f_187 |]    |
 +
o----o---------o----o        o----o---------o----o
 +
    ^          \                /          ^   
 +
    |          \ Intersection  /          |   
 +
    |            \            /            |   
 +
    |            v          v            |   
 +
    |        o-------------------o        |   
 +
    |        |                  |        |   
 +
    |        |      0:0:0      |        |   
 +
    |        |      0:0:1      |        |   
 +
    |        |      0:1:1      |        |   
 +
    |        |      1:1:1      |        |   
 +
    |        |                  |        |   
 +
    |        o-------------------o        |   
 +
    |        |    Syll c B:B:B  |        |   
 +
    |        o-------------------o        |   
 +
    |        |    [| f_139 |]    |        |   
 +
    |        o---------o---------o        |   
 +
    |                  |                  |   
 +
    |              Projection              |   
 +
    |                  |                  |   
 +
    |                  v                  |   
 +
    |        o---------o---------o        |   
 +
    |        |                  |        |   
 +
    |        |        0:0        |        |   
 +
    |        |        0:1        |        |   
 +
    |        |        1:1        |        |   
 +
    |        |                  |        |   
 +
    |        o-------------------o        |   
 +
    |        |  Syll_13 c B:~:B  |        |   
 +
    |        o-------------------o        |   
 +
    |        |  [| (p (r)) |]  |        |   
 +
    |        o----o---------o----o        |   
 +
    |            ^          ^            |   
 +
    |            /            \            |   
 +
    |          /  Composition  \          |   
 +
    |          /                \          |   
 +
o----o---------o----o        o----o---------o----o
 +
|                  |        |                  |
 +
|        0:0        |        |        0:0        |
 +
|        0:1        |        |        0:1        |
 +
|        1:1        |        |        1:1        |
 +
|                  |        |                  |
 +
o-------------------o        o-------------------o
 +
|  Syll_12 c B:B:~  |        |  Syll_23 c ~:B:B  |
 +
o-------------------o        o-------------------o
 +
|  [| (p (q)) |]  |        |  [| (q (r)) |]  |
 +
o---------o---------o        o---------o---------o
 +
                                                 
 +
Figure 76.  Denotative Aspects of Transitive Inference
 +
</pre>
 
|}
 
|}
  
Here's an animated recap of the graphical transformations that occur in the above proof:
+
A piece of syntax like <math>{}^{\backprime\backprime} \texttt{(} p \texttt{(} q \texttt{))} {}^{\prime\prime}\!</math> or <math>{}^{\backprime\backprime} p \Rightarrow q {}^{\prime\prime}\!</math> is an abstract description, and abstraction is a process that loses information about the objects described.  So when we go to reverse the abstraction, as we do when we look for models of that description, there is a degree of indefiniteness that comes into play.
  
{| align="center" cellpadding="8"
+
For example, the proposition <math>\texttt{(} p \texttt{(} q \texttt{))}\!</math> is typically assigned the functional type <math>\mathbb{B}^2 \to \mathbb{B},\!</math> but that is only its canonical or its minimal abstract type.  No sooner do we use it in a context that invokes additional variables, as we do when we next consider the proposition <math>\texttt{(} q \texttt{(} r \texttt{))},\!</math> than its type is tacitly adjusted to fit the new context, for instance, acquiring the extended type <math>{\mathbb{B}^3 \to \mathbb{B}}.\!</math>  This is one of those things that most people eventually learn to do without blinking an eye, that is to say, unreflectively, and this is precisely what makes the same facility so much trouble to implement properly in computational form.
 +
 
 +
Both the fibering operation, that takes us from the function <math>\texttt{(} p \texttt{(} q \texttt{))}\!</math> to the relation <math>[| \texttt{(} p \texttt{(} q \texttt{))} |],\!</math> and the tacit extension operation, that takes us from the relation <math>[| \texttt{(} p \texttt{(} q \texttt{))} |] \subseteq \mathbb{B}:\mathbb{B}\!</math> to the relation <math>[| q_{207} |] \subseteq \mathbb{B}:\mathbb{B}:\mathbb{B},\!</math> have this same character of abstraction-undoing or modelling operations that require us to re-interpret the same pieces of syntax under different types.  This accounts for a large part of the apparent ambiguities.
 +
 
 +
Up till now I've concentrated mostly on the abstract types of domains and propositions, things like <math>\mathbb{B}^k\!</math> and <math>\mathbb{B}^k \to \mathbb{B},\!</math> respectively.  This is a little like trying to do physics all in dimensionless quantities without keeping track of the qualitative physical units.  So much abstraction has its obvious limits, not to mention its hidden dangers.
 +
 
 +
To remedy this situation I will start to introduce the concrete types of domains and propositions, once again as they pertain to our current collection of examples.
 +
 
 +
We have been using the lower case letters <math>p, q, r\!</math> for the basic propositions of abstract type <math>\mathbb{B}^3 \to \mathbb{B}\!</math> and the upper case letters <math>P, Q, R\!</math> for the basic regions of the universe of discourse where <math>p, q, r,\!</math> respectively, hold true.
 +
 
 +
The set of signs <math>\mathcal{X} = \{ {}^{\backprime\backprime} p {}^{\prime\prime}, {}^{\backprime\backprime} q {}^{\prime\prime}, {}^{\backprime\backprime} r {}^{\prime\prime} \}\!</math> is the ''alphabet'' for the universe of discourse that is notated as <math>X^\bullet = [\mathcal{X}] = [p, q, r],\!</math> already getting sloppy about quotation marks to single out the signs.
 +
 
 +
The universe <math>{X^\bullet}\!</math> is composed of two different spaces of objects.  The first is the space of positions <math>X = \langle p, q, r \rangle = \{ (p, q, r) \}.\!</math>  The second is the space of propositions <math>X^\uparrow = (X \to \mathbb{B}).\!</math>
 +
 
 +
Let us make the following definitions:
 +
 
 +
{| align="center" cellpadding="8" width="90%"
 
|
 
|
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center"
+
<math>\begin{matrix}
|-
+
P^\ddagger & = & X_p & = & \{ \texttt{(} p \texttt{)}, p \},
| [[Image:Two-Thirds Majority Function 500 x 250 Animation.gif]]
+
\\[4pt]
|}
+
Q^\ddagger & = & X_q & = & \{ \texttt{(} q \texttt{)}, q \},
| (22)
+
\\[4pt]
 +
R^\ddagger & = & X_r & = & \{ \texttt{(} r \texttt{)}, r \}.
 +
\end{matrix}</math>
 
|}
 
|}
  
==Formal extension : Cactus calculus==
+
These are three sets of two abstract signs each, altogether staking out the qualitative dimensions of the universe of discourse <math>X^\bullet.\!</math>
 +
 
 +
Given this framework, the concrete type of the space <math>X\!</math> is <math>P^\ddagger \times Q^\ddagger \times R^\ddagger ~\cong~ \mathbb{B}^3\!</math> and the concrete type of each proposition in <math>X^\uparrow = (X \to \mathbb{B})\!</math> is <math>P^\ddagger \times Q^\ddagger \times R^\ddagger \to \mathbb{B}.~\!</math>  Given the length of the type markers, we will often omit the cartesian product symbols and write just <math>P^\ddagger Q^\ddagger R^\ddagger.\!</math>
 +
 
 +
An abstract reference to a point of <math>X\!</math> is a triple in <math>\mathbb{B}^3.\!</math>  A concrete reference to a point of <math>X\!</math> is a conjunction of signs from the dimensions <math>P^\ddagger, Q^\ddagger, R^\ddagger,\!</math> picking exactly one sign from each dimension.
  
Let us now extend the CSP&ndash;GSB calculus in the following way:
+
To illustrate the use of concrete coordinates for points and concrete types for spaces and propositions, Figure&nbsp;77 translates the contents of Figure&nbsp;76 into the new language.
  
The first extension is the ''reflective extension of logical graphs'', or what may be described as the ''cactus language'', after its principal graph-theoretic data structure. It is generated by generalizing the negation operator <math>\texttt{(_)}\!</math> in a particular manner,
+
{| align="center" border="0" cellpadding="10"
 +
|
 +
<pre>
 +
o-------------------o        o-------------------o
 +
|                  |        |                  |
 +
|    (p)(q)(r)    |        |    (p)(q)(r)    |
 +
|    (p)(q) r      |        |    (p)(q) r      |
 +
|    (p) q (r)    |        |    (p) q  r      |
 +
|    (p) q  r      |        |      p (q)(r)    |
 +
|      p  q (r)    |        |      p (q) r      |
 +
|      p  q  r      |        |      p  q  r      |
 +
|                  |        |                  |
 +
o-------------------o        o-------------------o
 +
|TE(Syll_12) c B:B:B|        |TE(Syll_23) c B:B:B|
 +
o-------------------o        o-------------------o
 +
|    [| f_207 |]    |        |    [| f_187 |]    |
 +
o----o---------o----o        o----o---------o----o
 +
    ^          \                /          ^   
 +
    |          \ Intersection /          |   
 +
    |            \             /            |   
 +
    |            v          v            |   
 +
    |        o-------------------o        |   
 +
    |        |                  |        |   
 +
    |        |    (p)(q)(r)    |        |   
 +
    |        |    (p)(q) r      |        |   
 +
    |        |    (p) q  r      |        |   
 +
    |        |      p  q  r      |        |   
 +
    |        |                  |        |   
 +
    |        o-------------------o        |   
 +
    |        |  Syll c P‡ Q‡ R‡  |        |   
 +
    |        o-------------------o        |   
 +
    |        |    [| f_139 |]    |        |   
 +
    |        o---------o---------o        |   
 +
    |                  |                  |   
 +
    |              Projection              |   
 +
    |                  |                  |   
 +
    |                  v                  |   
 +
    |        o---------o---------o        |   
 +
    |        |                  |        |   
 +
    |        |      (p) (r)      |        |   
 +
    |        |      (p)  r      |        |   
 +
    |        |      p  r      |        |   
 +
    |        |                  |        |   
 +
    |        o-------------------o        |   
 +
    |        |  Syll_13 c P‡ R‡  |        |   
 +
    |        o-------------------o        |   
 +
    |        |  [| (p (r)) |]  |        |   
 +
    |        o----o---------o----o        |   
 +
    |            ^          ^            |   
 +
    |            /            \            |   
 +
    |          /  Composition  \          |   
 +
    |          /                \         |   
 +
o----o---------o----o        o----o---------o----o
 +
|                  |        |                  |
 +
|      (p) (q)      |        |      (q) (r)      |
 +
|      (p)  q      |        |      (q)  r      |
 +
|      p  q      |        |      q  r      |
 +
|                  |        |                  |
 +
o-------------------o        o-------------------o
 +
|  Syll_12 c P‡ Q‡  |        |  Syll_23 c Q‡ R‡  |
 +
o-------------------o        o-------------------o
 +
|  [| (p (q)) |]  |        |  [| (q (r)) |]  |
 +
o---------o---------o        o---------o---------o
 +
                                                   
 +
Figure 77.  Denotative Aspects of Transitive Inference
 +
</pre>
 +
|}

Latest revision as of 19:40, 8 November 2016

Variations on a theme of transitivity

The next Example is extremely important, and for reasons that reach well beyond the level of propositional calculus as it is ordinarily conceived. But it's slightly tricky to get all of the details right, so it will be worth taking the trouble to look at it from several different angles and as it appears in diverse frames, genres, or styles of representation.

In discussing this Example, it is useful to observe that the implication relation indicated by the propositional form \(x \Rightarrow y\!\) is equivalent to an order relation \(x \le y\!\) on the boolean values \(0, 1 \in \mathbb{B},\!\) where \(0\!\) is taken to be less than \(1.\!\)

Example 2. Transitivity
    Information Reducing Inference
     

\(\begin{array}{l} ~ p \le q \\ ~ q \le r \\ \overline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~} \\ ~ p \le r \end{array}\)

    Information Preserving Inference
     

\(\begin{array}{l} ~ p \le q \\ ~ q \le r \\ \overline{\underline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}} \\ ~ p \le q \le r \end{array}\)

In stating the information-preserving analogue of transitivity, I have taken advantage of a common idiom in the use of order relation symbols, one that represents their logical conjunction by way of a concatenated syntax. Thus, \(p \le q \le r\!\) means \(p \le q ~\mathrm{and}~ q \le r.\!\) The claim that this 3-adic order relation holds among the three propositions \(p, q, r\!\) is a stronger claim — conveys more information — than the claim that the 2-adic relation \(p \le r\!\) holds between the two propositions \(p\!\) and \(r.\!\)

To study the differences between these two versions of transitivity within what is locally a familiar context, let's view the propositional forms involved as if they were elementary cellular automaton rules, resulting in the following Table.


\(\text{Table 51.}~~\text{Composite and Compiled Order Relations}\!\)

\(\mathcal{L}_1\!\)

\(\text{Decimal}\!\)

\(\mathcal{L}_2\!\)

\(\text{Binary}\!\)

\(\mathcal{L}_3\!\)

\(\text{Vector}\!\)

\(\mathcal{L}_4\!\)

\(\text{Cactus}\!\)

\(\mathcal{L}_5\!\)

\(\text{Order}\!\)

  \(p\colon\!\) \(1~1~1~1~0~0~0~0\!\)    
  \(q\colon\!\) \(1~1~0~0~1~1~0~0\!\)    
  \(r\colon\!\) \(1~0~1~0~1~0~1~0\!\)    

\(\begin{matrix} f_{207} \\[4pt] f_{187} \\[4pt] f_{175} \\[4pt] f_{139} \end{matrix}\)

\(\begin{matrix} f_{11001111} \\[4pt] f_{10111011} \\[4pt] f_{10101111} \\[4pt] f_{10001011} \end{matrix}\)

\(\begin{matrix} 1~1~0~0~1~1~1~1 \\[4pt] 1~0~1~1~1~0~1~1 \\[4pt] 1~0~1~0~1~1~1~1 \\[4pt] 1~0~0~0~1~0~1~1 \end{matrix}\)

\(\begin{matrix} \texttt{(} p \texttt{ (} q \texttt{))} \\[4pt] \texttt{(} q \texttt{ (} r \texttt{))} \\[4pt] \texttt{(} p \texttt{ (} r \texttt{))} \\[4pt] \texttt{(} p \texttt{ (} q \texttt{)) (} q \texttt{ (} r \texttt{))} \end{matrix}\)

\(\begin{matrix} p \le q \\[4pt] q \le r \\[4pt] p \le r \\[4pt] p \le q \le r \end{matrix}\)


Taking up another angle of incidence by way of extra perspective, let us now reflect on the venn diagrams of our four propositions.

Venn Diagram (P (Q)).jpg (52)
\(f_{207}(p, q, r) ~=~ \texttt{(} p \texttt{ (} q \texttt{))}\!\)
 
Venn Diagram (Q (R)).jpg (53)
\(f_{187}(p, q, r) ~=~ \texttt{(} q \texttt{ (} r \texttt{))}\!\)
 
Venn Diagram (P (R)).jpg (54)
\(f_{175}(p, q, r) ~=~ \texttt{(} p \texttt{ (} r \texttt{))}\!\)
 
Venn Diagram (P (Q)) (Q (R)).jpg (55)
\(f_{139}(p, q, r) ~=~ \texttt{(} p \texttt{ (} q \texttt{)) (} q \texttt{ (} r \texttt{))}\!\)

Among other things, these images make it visually obvious that the constraint on the three boolean variables \(p, q, r\!\) that is indicated by asserting either of the forms \(\texttt{(} p \texttt{(} q \texttt{))(} q \texttt{(} r \texttt{))}\!\) or \(p \le q \le r\!\) implies a constraint on the two boolean variables \(p, r\!\) that is indicated by either of the forms \(\texttt{(} p \texttt{(} r \texttt{))}\!\) or \(p \le r,\!\) but that it imposes additional constraints on these variables that are not captured by the illative conclusion.

One way to view a proposition \(f : \mathbb{B}^k \to \mathbb{B}\!\) is to consider its fiber of truth, \(f^{-1}(1) \subseteq \mathbb{B}^k,\!\) and to regard it as a \(k\!\)-adic relation \(L \subseteq \mathbb{B}^k.\!\)

By way of general definition, the fiber of a function \(f : X \to Y\!\) at a given value \(y\!\) of its co-domain \(Y\!\) is the antecedent (also known as the inverse image or pre-image) of \(y\!\) under \(f.\!\) This is a subset, possibly empty, of the domain \(X,\!\) notated as \(f^{-1}(y) \subseteq X.\!\)

In particular, if \(f\!\) is a proposition \(f : X \to \mathbb{B},\!\) then the fiber of truth \(f^{-1}(1)\!\) is the subset of \(X\!\) that is indicated by the proposition \(f.\!\) Whenever we assert a proposition \(f : X \to \mathbb{B},\!\) we are saying that what it indicates is all that happens to be the case in the relevant universe of discourse \(X.\!\) Because the fiber of truth is used so often in logical contexts, it is convenient to define the more compact notation \([| f |] = f^{-1}(1).\!\)

Using this panoply of notions and notations, we may treat the fiber of truth of each proposition \(f : \mathbb{B}^3 \to \mathbb{B}\!\) as if it were a relational data table of the shape \(\{ (p, q, r) \} \subseteq \mathbb{B}^3,\!\) where the triples \((p, q, r)\!\) are bit-tuples indicated by the proposition \(f.\!\)

Thus we obtain the following four relational data tables for the propositions that we are looking at in Example 2.


\(\text{Table 56.} ~~ [| f_{207} |] ~=~ [| p \le q |]\!\)
\(p\!\) \(q\!\) \(r\!\)
\(0\!\) \(0\!\) \(0\!\)
\(0\!\) \(0\!\) \(1\!\)
\(0\!\) \(1\!\) \(0\!\)
\(0\!\) \(1\!\) \(1\!\)
\(1\!\) \(1\!\) \(0\!\)
\(1\!\) \(1\!\) \(1\!\)


\(\text{Table 57.} ~~ [| f_{187} |] ~=~ [| q \le r |]\!\)
\(p\!\) \(q\!\) \(r\!\)
\(0\!\) \(0\!\) \(0\!\)
\(0\!\) \(0\!\) \(1\!\)
\(0\!\) \(1\!\) \(1\!\)
\(1\!\) \(0\!\) \(0\!\)
\(1\!\) \(0\!\) \(1\!\)
\(1\!\) \(1\!\) \(1\!\)


\(\text{Table 58.} ~~ [| f_{175} |] ~=~ [| p \le r |]\!\)
\(p\!\) \(q\!\) \(r\!\)
\(0\!\) \(0\!\) \(0\!\)
\(0\!\) \(0\!\) \(1\!\)
\(0\!\) \(1\!\) \(0\!\)
\(0\!\) \(1\!\) \(1\!\)
\(1\!\) \(0\!\) \(1\!\)
\(1\!\) \(1\!\) \(1\!\)


\(\text{Table 59.} ~~ [| f_{139} |] ~=~ [| p \le q \le r |]\!\)
\(p\!\) \(q\!\) \(r\!\)
\(0\!\) \(0\!\) \(0\!\)
\(0\!\) \(0\!\) \(1\!\)
\(0\!\) \(1\!\) \(1\!\)
\(1\!\) \(1\!\) \(1\!\)


In the medium of these unassuming examples, we begin to see the activities of logical inference and methodical inquiry as information clarifying operations.

First, we drew a distinction between information preserving and information reducing processes and we noted the related distinction between equational and implicational inferences. I will use the acronyms EROI and IROI, respectively, for the equational and implicational analogues of the various rules of inference.

For example, we considered the brands of information fusion that are involved in a couple of standard rules of inference, taken in both their equational and their illative variants.

In particular, let us assume that we begin from a state of uncertainty about the universe of discourse \(X = \mathbb{B}^3\!\) that is standardly represented by a uniform distribution \(u : X \to \mathbb{B}\!\) such that \(u(x) = 1\!\) for all \(x\!\) in \(X,\!\) in short, by the constant proposition \(1 : X \to \mathbb{B}.\!\) This amounts to the maximum entropy sign state (MESS). As a measure of uncertainty, let us use either the multiplicative measure given by the cardinality of \(X,\!\) commonly notated as \(|X|,\!\) or else the additive measure given by \({\log_2 |X|}.\!\) In this frame we have \({|X| = 8}\!\) and \({\log_2 |X| = 3},\!\) to wit, 3 bits of doubt.

Let us now consider the various rules of inference for transitivity in the light of their performance as information-developing actions.

Transitive Law (Implicational Inference)
   

\(\begin{array}{l} ~ p \le q \\ ~ q \le r \\ \overline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~} \\ ~ p \le r \end{array}\)

By itself, the information \(p \le q\!\) would reduce our uncertainty from \(\log 8\!\) bits to \(\log 6\!\) bits.
By itself, the information \(q \le r\!\) would reduce our uncertainty from \(\log 8\!\) bits to \(\log 6\!\) bits.
By itself, the information \(p \le r\!\) would reduce our uncertainty from \(\log 8\!\) bits to \(\log 6\!\) bits.

In this situation the application of the implicational rule of inference for transitivity to the information \(p \le q\!\) and the information \(q \le r\!\) to get the information \(p \le r\!\) does not increase the measure of information beyond what any one of the three propositions has independently of the other two. In a sense, then, the implicational rule operates only to move the information around without changing its measure in the slightest bit.

Transitive Law (Equational Inference)
   

\(\begin{array}{l} ~ p \le q \\ ~ q \le r \\ \overline{\underline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}} \\ ~ p \le q \le r \end{array}\)

The contents and the measures of information that are associated with the propositions \(p \le q\!\) and \(q \le r\!\) are the same as before.
On its own, the information \(p \le q \le r\!\) would reduce our uncertainty from log(8) = 3 bits to log(4) = 2 bits, a reduction of 1 bit.

These are just some of the initial observations that can be made about the dimensions of information and uncertainty in the conduct of logical inference, and there are many issues to be taken up as we get to the thick of it. In particular, we are taking propositions far too literally at the outset, reading their spots at face value, as it were, without yet considering their species character as fallible signs.

For ease of reference in the rest of this discussion, let us refer to the propositional form \(f : \mathbb{B}^3 \to \mathbb{B}\!\) such that \(f(p, q, r) = f_{139}(p, q, r) = \texttt{(} p \texttt{(} q \texttt{))(} q \texttt{(} r \texttt{))}\!\) as the syllogism map, written as \(\mathrm{syll} : \mathbb{B}^3 \to \mathbb{B},\!\) and let us refer to its fiber of truth \([| \mathrm{syll} |] = \mathrm{syll}^{-1}(1)\!\) as the syllogism relation, written as \(\mathrm{Syll} \subseteq \mathbb{B}^3.\!\) Table 60 shows \(\mathrm{Syll}\!\) as a relational dataset.


\(\text{Table 60.} ~~ \text{Syllogism Relation}\!\)
\(p\!\) \(q\!\) \(r\!\)
\(0\!\) \(0\!\) \(0\!\)
\(0\!\) \(0\!\) \(1\!\)
\(0\!\) \(1\!\) \(1\!\)
\(1\!\) \(1\!\) \(1\!\)


One of the first questions that we might ask about a 3-adic relation, in this case \(\mathrm{Syll},\!\) is whether it is determined by its 2-adic projections. I will illustrate what this means in the present case.

Table 61 repeats the relation \(\mathrm{Syll}\!\) in the first column, listing its 3-tuples in bit-string form, followed by the 2-adic or planar projections of \(\mathrm{Syll}\!\) in the next three columns. For instance, \(\mathrm{Syll}_{pq}\!\) is the 2-adic projection of \(\mathrm{Syll}\!\) on the \(pq\!\) plane that is arrived at by deleting the \(r\!\) column and counting each 2-tuple that results just one time. Likewise, \(\mathrm{Syll}_{pr}\!\) is obtained by deleting the \(q\!\) column and \(\mathrm{Syll}_{qr}\!\) is derived by deleting the \(p\!\) column, ignoring whatever duplicate pairs may result. The final row of the right three columns gives the propositions of the form \(f : \mathbb{B}^2 \to \mathbb{B}\!\) that indicate the 2-adic relations that result from these projections.


\(\text{Table 61.} ~~ \text{Dyadic Projections of the Syllogism Relation}\!\)
\(\mathrm{Syll}\!\) \(\mathrm{Syll}_{pq}\!\) \(\mathrm{Syll}_{pr}\!\) \(\mathrm{Syll}_{qr}\!\)

\(\begin{matrix} 0~0~0 \\ 0~0~1 \\ 0~1~1 \\ 1~1~1 \end{matrix}\)

\(\begin{matrix} 0~0 \\ 0~0 \\ 0~1 \\ 1~1 \end{matrix}\)

\(\begin{matrix} 0~0 \\ 0~1 \\ 0~1 \\ 1~1 \end{matrix}\)

\(\begin{matrix} 0~0 \\ 0~1 \\ 1~1 \\ 1~1 \end{matrix}\)

\(p \le q \le r\!\) \(\texttt{(} p \texttt{ (} q \texttt{))}\!\) \(\texttt{(} p \texttt{ (} r \texttt{))}\!\) \(\texttt{(} q \texttt{ (} r \texttt{))}\!\)


Let us make the simple observation that taking a projection, in our framework, deleting a column from a relational table, is like taking a derivative in differential calculus. What it means is that our attempt to return to the integral from whence the derivative was derived will in general encounter an indefinite variation on account of the circumstance that real information may have been destroyed by the derivation.

One will find that some relations can be reconstructed from various types of derivatives and projections, others cannot. The reconstuctible relations are said to be reducible to the types of reductive data in question, while the others are said to be irreducible with respect to those means.

The analogies between derivation, differentiation, implication, projection, and others sorts of information reducing operation will undergo extensive development in the remainder and sequel of the present discussion.

We were in the middle of discussing the relationships between information preserving rules of inference and information destroying rules of inference — folks of a 3-basket philosophical bent will no doubt be asking, "And what of information creating rules of inference?", but there I must wait for some signs of enlightenment, desiring not to tread on the rules of that succession.

The contrast between the information destroying and the information preserving versions of the transitive rule of inference led us to examine the relationships among several boolean functions, namely, those that qualify locally as the elementary cellular automata rules \(f_{139}, f_{175}, f_{187}, f_{207}.\!\)

The function \(f_{139} : \mathbb{B}^3 \to \mathbb{B}\!\) and its fiber \([| f_{139} |] \subseteq \mathbb{B}^3\!\) appeared to be key to many structures in this setting, and so I singled them out under the new names of \(\mathrm{syll} : \mathbb{B}^3 \to \mathbb{B}\!\) and \(\mathrm{Syll} \subseteq \mathbb{B}^3,\!\) respectively.

Managing the conceptual complexity of our considerations at this juncture put us in need of some conceptual tools that I broke off to develop in my notes on "Reductions Among Relations". The main items that we need right away from that thread are the definitions of relational projections and their inverses, the tacit extensions.

But the more I survey the problem setting the more it looks like we need better ways to bring our visual intuitions to play on the scene, and so let us next lay out some visual schemata that are designed to facilitate that.

Figure 62 shows the familiar picture of a boolean 3-cube, where the points of \(\mathbb{B}^3\!\) are coordinated as bit strings of length three. Looking at the functions \(f : \mathbb{B}^3 \to \mathbb{B}\!\) and the relations \(L \subseteq \mathbb{B}^3\!\) on this pattern, one views the construction of either type of object as a matter of coloring the nodes of the 3-cube with choices from a pair of colors that stipulate which points are in the relation \({L = [| f |]}\!\) and which points are out of it. Bowing to common convention, we may use the color \(1\!\) for points that are in a given relation and the color \(0\!\) for points that are out of the same relation. However, it will be more convenient here to indicate the former case by writing the coordinates in the place of the node and to indicate the latter case by plotting the point as an unlabeled node "o".

o-------------------------------------------------o
|                                                 |
|                       111                       |
|                       /|\                       |
|                      / | \                      |
|                     /  |  \                     |
|                    /   |   \                    |
|                   /    |    \                   |
|                  /     |     \                  |
|                 /      |      \                 |
|               110     101     011               |
|                |\     / \     /|                |
|                | \   /   \   / |                |
|                |  \ /     \ /  |                |
|                |   \       /   |                |
|                |  / \     / \  |                |
|                | /   \   /   \ |                |
|                |/     \ /     \|                |
|               100     010     001               |
|                 \      |      /                 |
|                  \     |     /                  |
|                   \    |    /                   |
|                    \   |   /                    |
|                     \  |  /                     |
|                      \ | /                      |
|                       \|/                       |
|                       000                       |
|                                                 |
o-------------------------------------------------o
Figure 62.  Boolean 3-Cube B^3
(62)

Table 63 shows the 3-adic relation \(\mathrm{Syll} \subseteq \mathbb{B}^3\!\) again, and Figure 64 shows it plotted on a 3-cube template.

Table 63.  Syll c B^3
o-----------------------o
|   p       q       r   |
o-----------------------o
|   0       0       0   |
|   0       0       1   |
|   0       1       1   |
|   1       1       1   |
o-----------------------o
(63)
o-------------------------------------------------o
|                                                 |
|                       111                       |
|                       /|\                       |
|                      / | \                      |
|                     /  |  \                     |
|                    /   |   \                    |
|                   /    |    \                   |
|                  /     |     \                  |
|                 /      |      \                 |
|                o       o      011               |
|                |\     / \     /|                |
|                | \   /   \   / |                |
|                |  \ /     \ /  |                |
|                |   \       /   |                |
|                |  / \     / \  |                |
|                | /   \   /   \ |                |
|                |/     \ /     \|                |
|                o       o      001               |
|                 \      |      /                 |
|                  \     |     /                  |
|                   \    |    /                   |
|                    \   |   /                    |
|                     \  |  /                     |
|                      \ | /                      |
|                       \|/                       |
|                       000                       |
|                                                 |
o-------------------------------------------------o
Figure 64.  Triadic Relation Syll c B^3
(64)

We return once more to the plane projections of \(\mathrm{Syll} \subseteq \mathbb{B}^3.\!\)

Table 65.  Syll c B^3
o-----------------------o
|   p       q       r   |
o-----------------------o
|   0       0       0   |
|   0       0       1   |
|   0       1       1   |
|   1       1       1   |
o-----------------------o
(65)
Table 66.  Dyadic Projections of Syll
o-----------o o-----------o o-----------o
|  Syll_12  | |  Syll_13  | |  Syll_23  |
o-----------o o-----------o o-----------o
|   p   q   | |   p   r   | |   q   r   |
o-----------o o-----------o o-----------o
|   0   0   | |   0   0   | |   0   0   |
|   0   1   | |   0   1   | |   0   1   |
|   1   1   | |   1   1   | |   1   1   |
o-----------o o-----------o o-----------o
|  (p (q))  | |  (p (r))  | |  (q (r))  |
o-----------o o-----------o o-----------o
(66)

In showing the 2-adic projections of a 3-adic relation \(L \subseteq \mathbb{B}^3,\!\) I will translate the coordinates of the points in each relation to the plane of the projection, there dotting out with a dot "." the bit of the bit string that is out of place on that plane.

Figure 67 shows \(\mathrm{Syll}\!\) and its three 2-adic projections:

o-------------------------------------------------o
|                                                 |
|                       111                       |
|                       /|\                       |
|                      / | \                      |
|                     /  |  \                     |
|                    /   |   \                    |
|                   /    |    \                   |
|                  /     |     \                  |
|                 /      |      \                 |
|                o       o      011               |
|                |\     / \     /|                |
|                | \   /   \   / |                |
|                |  \ /     \ /  |                |
|                |   \       /   |                |
|                |  / \     / \  |                |
|                | /   \   /   \ |                |
|                |/     \ /     \|                |
|                o       o      001               |
|                 \      |      /                 |
|     11.          \     |     /          .11     |
|      |\           \    |    /           /|      |
|      | \           \   |   /           / |      |
|      |  \           \  |  /           /  |      |
|      |   \           \ | /           /   |      |
|      |    \           \|/           /    |      |
|      |     \          000          /     |      |
|      |      \                     /      |      |
|      o      01.                  o      .01     |
|       \      |                   |      /       |
|        \     |                   |     /        |
|         \    |                   |    /         |
|          \   |        1.1        |   /          |
|           \  |        / \        |  /           |
|            \ |       /   \       | /            |
|             \|      /     \      |/             |
|             00.    /       \    .00             |
|                   /         \                   |
|                  /           \                  |
|                 /             \                 |
|                o              0.1               |
|                 \             /                 |
|                  \           /                  |
|                   \         /                   |
|                    \       /                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                       0.0                       |
|                                                 |
o-------------------------------------------------o
Figure 67.  Syll c B^3 and its Dyadic Projections
(67)

We now compute the tacit extensions of the 2-adic projections of \(\mathrm{Syll},\!\) alias \(f_{139},\!\) and this makes manifest its relationship to the other functions and fibers, namely, \(f_{175}, f_{187}, f_{207}.\!\)

Table 68.  Syll c B^3
o-----------------------o
|   p       q       r   |
o-----------------------o
|   0       0       0   |
|   0       0       1   |
|   0       1       1   |
|   1       1       1   |
o-----------------------o
(68)
Table 69.  Dyadic Projections of Syll
o-----------o o-----------o o-----------o
|  Syll_12  | |  Syll_13  | |  Syll_23  |
o-----------o o-----------o o-----------o
|   p   q   | |   p   r   | |   q   r   |
o-----------o o-----------o o-----------o
|   0   0   | |   0   0   | |   0   0   |
|   0   1   | |   0   1   | |   0   1   |
|   1   1   | |   1   1   | |   1   1   |
o-----------o o-----------o o-----------o
|  (p (q))  | |  (p (r))  | |  (q (r))  |
o-----------o o-----------o o-----------o
(69)
Table 70.  Tacit Extensions of Projections of Syll
o---------------o o---------------o o---------------o
|  te(Syll_12)  | |  te(Syll_13)  | |  te(Syll_23)  |
o---------------o o---------------o o---------------o
|   p   q   r   | |   p   q   r   | |   p   q   r   |
o---------------o o---------------o o---------------o
|   0   0   0   | |   0   0   0   | |   0   0   0   |
|   0   0   1   | |   0   1   0   | |   1   0   0   |
|   0   1   0   | |   0   0   1   | |   0   0   1   |
|   0   1   1   | |   0   1   1   | |   1   0   1   |
|   1   1   0   | |   1   0   1   | |   0   1   1   |
|   1   1   1   | |   1   1   1   | |   1   1   1   |
o---------------o o---------------o o---------------o
| [| (p (q)) |] | | [| (p (r)) |] | | [| (q (r)) |] |
o---------------o o---------------o o---------------o
| [|  f_207  |] | | [|  f_175  |] | | [|  f_187  |] |
o---------------o o---------------o o---------------o
(70)
o-------------------------------------------------o
|                                                 |
|                       111                       |
|                       /|\                       |
|                      / | \                      |
|                     /  |  \                     |
|                    /   |   \                    |
|                   /    |    \                   |
|                  /     |     \                  |
|                 /      |      \                 |
|               110      o      011               |
|                |\     / \     /|                |
|                | \   /   \   / |                |
|                |  \ /     \ /  |                |
|                |   \       /   |                |
|                |  / \     / \  |                |
|                | /   \   /   \ |                |
|                |/     \ /     \|                |
|                o      010     001               |
|                 \      |      /                 |
|     11.          \     |     /                  |
|      |\           \    |    /                   |
|      | \           \   |   /                    |
|      |  \           \  |  /                     |
|      |   \           \ | /                      |
|      |    \           \|/                       |
|      |     \          000                       |
|      |      \                                   |
|      o      01.                                 |
|       \      |                                  |
|        \     |                                  |
|         \    |                                  |
|          \   |                                  |
|           \  |                                  |
|            \ |                                  |
|             \|                                  |
|             00.                                 |
|                                                 |
o-------------------------------------------------o
Figure 71.  Tacit Extension te_12_3 (Syll_12)
(71)
o-------------------------------------------------o
|                                                 |
|                       111                       |
|                       /|\                       |
|                      / | \                      |
|                     /  |  \                     |
|                    /   |   \                    |
|                   /    |    \                   |
|                  /     |     \                  |
|                 /      |      \                 |
|                o      101     011               |
|                |\     / \     /|                |
|                | \   /   \   / |                |
|                |  \ /     \ /  |                |
|                |   \       /   |                |
|                |  / \     / \  |                |
|                | /   \   /   \ |                |
|                |/     \ /     \|                |
|                o      010     001               |
|                 \      |      /                 |
|                  \     |     /                  |
|                   \    |    /                   |
|                    \   |   /                    |
|                     \  |  /                     |
|                      \ | /                      |
|                       \|/                       |
|                       000                       |
|                                                 |
|                                                 |
|                       1.1                       |
|                       / \                       |
|                      /   \                      |
|                     /     \                     |
|                    /       \                    |
|                   /         \                   |
|                  /           \                  |
|                 /             \                 |
|                o              0.1               |
|                 \             /                 |
|                  \           /                  |
|                   \         /                   |
|                    \       /                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                       0.0                       |
|                                                 |
o-------------------------------------------------o
Figure 72.  Tacit Extension te_13_2 (Syll_13)
(72)
o-------------------------------------------------o
|                                                 |
|                       111                       |
|                       /|\                       |
|                      / | \                      |
|                     /  |  \                     |
|                    /   |   \                    |
|                   /    |    \                   |
|                  /     |     \                  |
|                 /      |      \                 |
|                o      101     011               |
|                |\     / \     /|                |
|                | \   /   \   / |                |
|                |  \ /     \ /  |                |
|                |   \       /   |                |
|                |  / \     / \  |                |
|                | /   \   /   \ |                |
|                |/     \ /     \|                |
|               100      o      001               |
|                 \      |      /                 |
|                  \     |     /          .11     |
|                   \    |    /           /|      |
|                    \   |   /           / |      |
|                     \  |  /           /  |      |
|                      \ | /           /   |      |
|                       \|/           /    |      |
|                       000          /     |      |
|                                   /      |      |
|                                  o      .01     |
|                                  |      /       |
|                                  |     /        |
|                                  |    /         |
|                                  |   /          |
|                                  |  /           |
|                                  | /            |
|                                  |/             |
|                                 .00             |
|                                                 |
o-------------------------------------------------o
Figure 73.  Tacit Extension te_23_1 (Syll_23)
(73)

The reader may wish to contemplate Figure 74 and use it to verify the following two facts:

\(\begin{array}{lcc} \mathrm{Syll} & = & \mathrm{te}(\mathrm{Syll}_{12}) \cap \mathrm{te}(\mathrm{Syll}_{23}) \\[6pt] \mathrm{Syll}_{13} & = & \mathrm{Syll}_{12} \circ \mathrm{Syll}_{23} \end{array}\)

o-------------------------------------------------o
|                                                 |
|                        *                        |
|                       /|\                       |
|                      / | \                      |
|                     /  |  \                     |
|                    /   |   \                    |
|                   /    |    \                   |
|                  /     |     \                  |
|                 /      |      \                 |
|                o       o       *                |
|               /|\     / \     /|\               |
|              / | \   /   \   / | \              |
|             /  |  \ /     \ /  |  \             |
|            /   |   \       /   |   \            |
|           /    |  / \     / \  |    \           |
|          /     | /   \   /   \ |     \          |
|         /      |/     \ /     \|      \         |
|        /       o       o       *       \        |
|       /         \     /|      / \       \       |
|      *           \   / |     /   \       *      |
|      |\           \ /  |    /     \     /|      |
|      | \           /   |   /       \   / |      |
|      |  \         / \  |  /         \ /  |      |
|      |   \       /   \ | /           /   |      |
|      |    \     /     \|/           / \  |      |
|      |     \   /       *           /   \ |      |
|      |      \ /       / \         /     \|      |
|      o       *       /   \       o       *      |
|       \      |      /     \      |      /       |
|        \     |     /       \     |     /        |
|         \    |    /         \    |    /         |
|          \   |   /     *     \   |   /          |
|           \  |  /     / \     \  |  /           |
|            \ | /     /   \     \ | /            |
|             \|/     /     \     \|/             |
|              *     /       \     *              |
|                   /         \                   |
|                  /           \                  |
|                 /             \                 |
|                o               *                |
|                 \             /                 |
|                  \           /                  |
|                   \         /                   |
|                    \       /                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                        *                        |
|                                                 |
o-------------------------------------------------o
Figure 74.  Syll = te(Syll_12) |^| te(Syll_23)
(74)

I don't know about you, but I am still puzzled by all of thus stuff, that is to say, by the entanglements of composition and projection and their relationship to the information processing properties of logical inference rules. What I lack is a single picture that could show me all of the pieces and make the pattern of their informational relationships clear.

In accord with my experimental way, I will stick with the case of transitive inference until I have pinned it down thoroughly, but of course the real interest is much more general than that.

At first sight, the relationships seem easy enough to write out. Figure 75 shows how the various logical expressions are related to each other: The expressions \({}^{\backprime\backprime} \texttt{(} p \texttt{ (} q \texttt{))} {}^{\prime\prime}\!\) and \({}^{\backprime\backprime} \texttt{(} q \texttt{ (} r \texttt{))} {}^{\prime\prime}\!\) are conjoined in a purely syntactic fashion — much in the way that one might compile a theory from axioms without knowing what either the theory or the axioms were about — and the best way to sum up the state of information implicit in taking them together is just the expression \({}^{\backprime\backprime} \texttt{(} p \texttt{ (} q \texttt{)) (} q \texttt{ (} r \texttt{))}{}^{\prime\prime}\!\) that would the canonical result of an equational or reversible rule of inference. From that equational inference, one might arrive at the implicational inference \({}^{\backprime\backprime} \texttt{(} p \texttt{ (} r \texttt{))} {}^{\prime\prime}\!\) by the most conventional implication.

o-------------------o         o-------------------o
|                   |         |                   |
|         q         |         |         r         |
|         o         |         |         o         |
|         |         |         |         |         |
|       p o         |         |       q o         |
|         |         |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|      (p (q))      |         |      (q (r))      |
o-------------------o         o-------------------o
|       f_207       |         |       f_187       |
o---------o---------o         o---------o---------o
           \                           /           
            \       Conjunction       /            
             \                       /             
              v                     v              
               o-------------------o               
               |                   |               
               |                   |               
               |                   |               
               |       q   r       |               
               |       o   o       |               
               |       |   |       |               
               |     p o   o q     |               
               |        \ /        |               
               |         @         |               
               |                   |               
               o-------------------o               
               |  (p (q)) (q (r))  |               
               o-------------------o               
               |       f_139       |               
               o---------o---------o               
                         |                         
                    Implication                    
                         |                         
                         v                         
               o---------o---------o               
               |                   |               
               |         r         |               
               |         o         |               
               |         |         |               
               |       p o         |               
               |         |         |               
               |         @         |               
               |                   |               
               o-------------------o               
               |      (p (r))      |               
               o-------------------o               
               |       f_175       |               
               o-------------------o               
                                                   
Figure 75.  Expressive Aspects of Transitive Inference

Most of the customary names for this type of process have turned out to have misleading connotations, and so I will experiment with calling it the expressive aspect of the various rules for transitive inference, simply to emphasize the fact that rules can be given for it that operate solely on signs and expressions, without necessarily needing to look at the objects that are denoted by these signs and expressions.

In the way of many experiments, the word expressive does not seem to work for what I wanted to say here, since we too often use it to suggest something that expresses an object or a purpose, and I wanted it to imply what is purely a matter of expression, shorn of consideration for anything objective. Aside from coining a word like ennotative, some other options would be connotative, hermeneutic, semiotic, syntactic — each of which works in some range of interpretation but fails in others. Let's try formulaic.

Despite how simple the formulaic aspects of transitive inference might appear on the surface, there are problems that wait for us just beneath the syntactic surface, as we quickly discover if we turn to considering the kinds of objects, abstract and concrete, that these formulas are meant to denote, and all the more so if we try to do this in a context of computational implementations, where the "interpreters" to be addressed take nothing on faith. Thus we engage the denotative semantics or the model theory of these extremely simple programs that we call propositions.

Figure 76 is an attempt to outline the model-theoretic relationships that are involved in our study of transitive inference. A couple of alternative notations are introduced in this Table:

The forms \(X:Y:Z\!\) and \(x:y:z\!\) are used as alternative notations for the cartesian product \(X \times Y \times Z\!\) and the tuple \((x, y, z),\!\) respectively.
In situations where we have products like \(X:Y:Z\!\) with \(X = Y = Z = \mathbb{B},\!\) and relations like \({L \subseteq X:Y},\!\)   \({M \subseteq X:Z},\!\)   \({N \subseteq Y:Z},\!\) the forms \({L \subseteq \mathbb{B}:\mathbb{B}:-},\!\)   \({M \subseteq \mathbb{B}:-:\mathbb{B}},\!\)   \({N \subseteq -:\mathbb{B}:\mathbb{B}}\!\) are used to remind us that we are considering particular ways of situating \({L, M, N}\!\) within the product space \(X:Y:Z.\!\)
o-------------------o         o-------------------o
|                   |         |                   |
|       0:0:0       |         |       0:0:0       |
|       0:0:1       |         |       0:0:1       |
|       0:1:0       |         |       0:1:1       |
|       0:1:1       |         |       1:0:0       |
|       1:1:0       |         |       1:0:1       |
|       1:1:1       |         |       1:1:1       |
|                   |         |                   |
o-------------------o         o-------------------o
|te(Syll_12) c B:B:B|         |te(Syll_23) c B:B:B|
o-------------------o         o-------------------o
|    [| f_207 |]    |         |    [| f_187 |]    |
o----o---------o----o         o----o---------o----o
     ^          \                 /          ^     
     |           \ Intersection  /           |     
     |            \             /            |     
     |             v           v             |     
     |         o-------------------o         |     
     |         |                   |         |     
     |         |       0:0:0       |         |     
     |         |       0:0:1       |         |     
     |         |       0:1:1       |         |     
     |         |       1:1:1       |         |     
     |         |                   |         |     
     |         o-------------------o         |     
     |         |    Syll c B:B:B   |         |     
     |         o-------------------o         |     
     |         |    [| f_139 |]    |         |     
     |         o---------o---------o         |     
     |                   |                   |     
     |              Projection               |     
     |                   |                   |     
     |                   v                   |     
     |         o---------o---------o         |     
     |         |                   |         |     
     |         |        0:0        |         |     
     |         |        0:1        |         |     
     |         |        1:1        |         |     
     |         |                   |         |     
     |         o-------------------o         |     
     |         |  Syll_13 c B:~:B  |         |     
     |         o-------------------o         |     
     |         |   [| (p (r)) |]   |         |     
     |         o----o---------o----o         |     
     |             ^           ^             |     
     |            /             \            |     
     |           /  Composition  \           |     
     |          /                 \          |     
o----o---------o----o         o----o---------o----o
|                   |         |                   |
|        0:0        |         |        0:0        |
|        0:1        |         |        0:1        |
|        1:1        |         |        1:1        |
|                   |         |                   |
o-------------------o         o-------------------o
|  Syll_12 c B:B:~  |         |  Syll_23 c ~:B:B  |
o-------------------o         o-------------------o
|   [| (p (q)) |]   |         |   [| (q (r)) |]   |
o---------o---------o         o---------o---------o
                                                   
Figure 76.  Denotative Aspects of Transitive Inference

A piece of syntax like \({}^{\backprime\backprime} \texttt{(} p \texttt{(} q \texttt{))} {}^{\prime\prime}\!\) or \({}^{\backprime\backprime} p \Rightarrow q {}^{\prime\prime}\!\) is an abstract description, and abstraction is a process that loses information about the objects described. So when we go to reverse the abstraction, as we do when we look for models of that description, there is a degree of indefiniteness that comes into play.

For example, the proposition \(\texttt{(} p \texttt{(} q \texttt{))}\!\) is typically assigned the functional type \(\mathbb{B}^2 \to \mathbb{B},\!\) but that is only its canonical or its minimal abstract type. No sooner do we use it in a context that invokes additional variables, as we do when we next consider the proposition \(\texttt{(} q \texttt{(} r \texttt{))},\!\) than its type is tacitly adjusted to fit the new context, for instance, acquiring the extended type \({\mathbb{B}^3 \to \mathbb{B}}.\!\) This is one of those things that most people eventually learn to do without blinking an eye, that is to say, unreflectively, and this is precisely what makes the same facility so much trouble to implement properly in computational form.

Both the fibering operation, that takes us from the function \(\texttt{(} p \texttt{(} q \texttt{))}\!\) to the relation \([| \texttt{(} p \texttt{(} q \texttt{))} |],\!\) and the tacit extension operation, that takes us from the relation \([| \texttt{(} p \texttt{(} q \texttt{))} |] \subseteq \mathbb{B}:\mathbb{B}\!\) to the relation \([| q_{207} |] \subseteq \mathbb{B}:\mathbb{B}:\mathbb{B},\!\) have this same character of abstraction-undoing or modelling operations that require us to re-interpret the same pieces of syntax under different types. This accounts for a large part of the apparent ambiguities.

Up till now I've concentrated mostly on the abstract types of domains and propositions, things like \(\mathbb{B}^k\!\) and \(\mathbb{B}^k \to \mathbb{B},\!\) respectively. This is a little like trying to do physics all in dimensionless quantities without keeping track of the qualitative physical units. So much abstraction has its obvious limits, not to mention its hidden dangers.

To remedy this situation I will start to introduce the concrete types of domains and propositions, once again as they pertain to our current collection of examples.

We have been using the lower case letters \(p, q, r\!\) for the basic propositions of abstract type \(\mathbb{B}^3 \to \mathbb{B}\!\) and the upper case letters \(P, Q, R\!\) for the basic regions of the universe of discourse where \(p, q, r,\!\) respectively, hold true.

The set of signs \(\mathcal{X} = \{ {}^{\backprime\backprime} p {}^{\prime\prime}, {}^{\backprime\backprime} q {}^{\prime\prime}, {}^{\backprime\backprime} r {}^{\prime\prime} \}\!\) is the alphabet for the universe of discourse that is notated as \(X^\bullet = [\mathcal{X}] = [p, q, r],\!\) already getting sloppy about quotation marks to single out the signs.

The universe \({X^\bullet}\!\) is composed of two different spaces of objects. The first is the space of positions \(X = \langle p, q, r \rangle = \{ (p, q, r) \}.\!\) The second is the space of propositions \(X^\uparrow = (X \to \mathbb{B}).\!\)

Let us make the following definitions:

\(\begin{matrix} P^\ddagger & = & X_p & = & \{ \texttt{(} p \texttt{)}, p \}, \\[4pt] Q^\ddagger & = & X_q & = & \{ \texttt{(} q \texttt{)}, q \}, \\[4pt] R^\ddagger & = & X_r & = & \{ \texttt{(} r \texttt{)}, r \}. \end{matrix}\)

These are three sets of two abstract signs each, altogether staking out the qualitative dimensions of the universe of discourse \(X^\bullet.\!\)

Given this framework, the concrete type of the space \(X\!\) is \(P^\ddagger \times Q^\ddagger \times R^\ddagger ~\cong~ \mathbb{B}^3\!\) and the concrete type of each proposition in \(X^\uparrow = (X \to \mathbb{B})\!\) is \(P^\ddagger \times Q^\ddagger \times R^\ddagger \to \mathbb{B}.~\!\) Given the length of the type markers, we will often omit the cartesian product symbols and write just \(P^\ddagger Q^\ddagger R^\ddagger.\!\)

An abstract reference to a point of \(X\!\) is a triple in \(\mathbb{B}^3.\!\) A concrete reference to a point of \(X\!\) is a conjunction of signs from the dimensions \(P^\ddagger, Q^\ddagger, R^\ddagger,\!\) picking exactly one sign from each dimension.

To illustrate the use of concrete coordinates for points and concrete types for spaces and propositions, Figure 77 translates the contents of Figure 76 into the new language.

o-------------------o         o-------------------o
|                   |         |                   |
|     (p)(q)(r)     |         |     (p)(q)(r)     |
|     (p)(q) r      |         |     (p)(q) r      |
|     (p) q (r)     |         |     (p) q  r      |
|     (p) q  r      |         |      p (q)(r)     |
|      p  q (r)     |         |      p (q) r      |
|      p  q  r      |         |      p  q  r      |
|                   |         |                   |
o-------------------o         o-------------------o
|TE(Syll_12) c B:B:B|         |TE(Syll_23) c B:B:B|
o-------------------o         o-------------------o
|    [| f_207 |]    |         |    [| f_187 |]    |
o----o---------o----o         o----o---------o----o
     ^          \                 /          ^     
     |           \ Intersection  /           |     
     |            \             /            |     
     |             v           v             |     
     |         o-------------------o         |     
     |         |                   |         |     
     |         |     (p)(q)(r)     |         |     
     |         |     (p)(q) r      |         |     
     |         |     (p) q  r      |         |     
     |         |      p  q  r      |         |     
     |         |                   |         |     
     |         o-------------------o         |     
     |         |  Syll c P‡ Q‡ R‡  |         |     
     |         o-------------------o         |     
     |         |    [| f_139 |]    |         |     
     |         o---------o---------o         |     
     |                   |                   |     
     |              Projection               |     
     |                   |                   |     
     |                   v                   |     
     |         o---------o---------o         |     
     |         |                   |         |     
     |         |      (p) (r)      |         |     
     |         |      (p)  r       |         |     
     |         |       p   r       |         |     
     |         |                   |         |     
     |         o-------------------o         |     
     |         |  Syll_13 c P‡ R‡  |         |     
     |         o-------------------o         |     
     |         |   [| (p (r)) |]   |         |     
     |         o----o---------o----o         |     
     |             ^           ^             |     
     |            /             \            |     
     |           /  Composition  \           |     
     |          /                 \          |     
o----o---------o----o         o----o---------o----o
|                   |         |                   |
|      (p) (q)      |         |      (q) (r)      |
|      (p)  q       |         |      (q)  r       |
|       p   q       |         |       q   r       |
|                   |         |                   |
o-------------------o         o-------------------o
|  Syll_12 c P‡ Q‡  |         |  Syll_23 c Q‡ R‡  |
o-------------------o         o-------------------o
|   [| (p (q)) |]   |         |   [| (q (r)) |]   |
o---------o---------o         o---------o---------o
                                                    
Figure 77.  Denotative Aspects of Transitive Inference