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{{DISPLAYTITLE:Propositional Equation Reasoning Systems}}
{{DISPLAYTITLE:Propositional Equation Reasoning Systems}}
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''.
'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''
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''.
==Formal development==
==Formal development==
The first order of business is to give the exact forms of the axioms that we use, devolving from Peirce's "[[Logical Graphs]]" 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.
The first order of business is to give the exact forms of the axioms that we use, devolving from Peirce's “[[Logical Graphs]]” 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.
===Axioms===
===Axioms===
{| align="center" cellpadding="10"
{| align="center" cellpadding="10"
| [[Image:PERS_Figure_01.jpg|500px]] || (α)
| [[Image:PERS_Figure_01.jpg|500px]] || (1)
|-
|-
| [[Image:PERS_Figure_02.jpg|500px]] || (β)
| [[Image:PERS_Figure_02.jpg|500px]] || (2)
|-
|-
| [[Image:PERS_Figure_03.jpg|500px]] || (γ)
| [[Image:PERS_Figure_03.jpg|500px]] || (3)
|-
|-
| [[Image:PERS_Figure_04.jpg|500px]] || (δ)
| [[Image:PERS_Figure_04.jpg|500px]] || (4)
|}
|}
One way of assigning logical meaning to the initial equations is known as the ''entitative interpretation'' (<math>\operatorname{En}</math>). Under <math>\operatorname{En},</math> the axioms read as follows:
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"
{| align="center" cellpadding="10"
I_1
I_1
& : &
& : &
\operatorname{true} ~\operatorname{or}~ \operatorname{true}
\mathrm{true} ~\mathrm{or}~ \mathrm{true}
& = &
& = &
\operatorname{true}
\mathrm{true}
\\
\\
I_2
I_2
& : &
& : &
\operatorname{not}~ \operatorname{true}
\mathrm{not}~ \mathrm{true}
& = &
& = &
\operatorname{false}
\mathrm{false}
\\
\\
J_1
J_1
& : &
& : &
a ~\operatorname{or}~ \operatorname{not}~ a
a ~\mathrm{or}~ \mathrm{not}~ a
& = &
& = &
\operatorname{true}
\mathrm{true}
\\
\\
J_2
J_2
& : &
& : &
(a ~\operatorname{or}~ b) ~\operatorname{and}~ (a ~\operatorname{or}~ c)
(a ~\mathrm{or}~ b) ~\mathrm{and}~ (a ~\mathrm{or}~ c)
& = &
& = &
a ~\operatorname{or}~ (b ~\operatorname{and}~ c)
a ~\mathrm{or}~ (b ~\mathrm{and}~ c)
\end{matrix}</math>
\end{matrix}</math>
|}
|}
Another way of assigning logical meaning to the initial equations is known as the ''existential interpretation'' (<math>\operatorname{Ex}</math>). Under <math>\operatorname{Ex},</math> the axioms read as follows:
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:
{| align="center" cellpadding="10"
{| align="center" cellpadding="10"
I_1
I_1
& : &
& : &
\operatorname{false} ~\operatorname{and}~ \operatorname{false}
\mathrm{false} ~\mathrm{and}~ \mathrm{false}
& = &
& = &
\operatorname{false}
\mathrm{false}
\\
\\
I_2
I_2
& : &
& : &
\operatorname{not}~ \operatorname{false}
\mathrm{not}~ \mathrm{false}
& = &
& = &
\operatorname{true}
\mathrm{true}
\\
\\
J_1
J_1
& : &
& : &
a ~\operatorname{and}~ \operatorname{not}~ a
a ~\mathrm{and}~ \mathrm{not}~ a
& = &
& = &
\operatorname{false}
\mathrm{false}
\\
\\
J_2
J_2
& : &
& : &
(a ~\operatorname{and}~ b) ~\operatorname{or}~ (a ~\operatorname{and}~ c)
(a ~\mathrm{and}~ b) ~\mathrm{or}~ (a ~\mathrm{and}~ c)
& = &
& = &
a ~\operatorname{and}~ (b ~\operatorname{or}~ 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 <math>\overline{\underline{~~~~~~}}</math> 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.
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 ═════ 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.
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>\operatorname{En}</math> and <math>\operatorname{Ex},</math> respectively. The early CSP, as in his essay on "Qualitative Logic", and also GSB, emphasized the <math>\operatorname{En}</math> interpretation, while the later CSP developed mostly the <math>\operatorname{Ex}</math> interpretation.
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 “Qualitative Logic”, and also GSB, emphasized the <math>\mathrm{En}\!</math> interpretation, while the later CSP developed mostly the <math>\mathrm{Ex}\!</math> interpretation.
===Frequently used theorems===
===Frequently used theorems===
====C<sub>1</sub>. Double negation theorem====
====C<sub>1</sub>. Double negation====
The first theorem goes under the names of ''Consequence 1'' <math>(C_1)\!</math>, the ''double negation theorem'' (DNT), or ''Reflection''.
The first theorem goes under the names of ''Consequence 1'' <math>(C_1),\!</math> the ''double negation theorem'' (DNT), or ''Reflection''.
{| align="center" border="0" cellpadding="10" cellspacing="0"
{| align="center" cellpadding="10"
| [[Image:PERS_Figure_05.jpg|500px]] || (1.2.1.a)
| [[Image:Double Negation 1.0 Splash Page.png|500px]] || (5)
|}
|}
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.
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.
{| align="center" border="0" cellpadding="10" cellspacing="0"
{| align="center" cellpadding="8"
| [[Image:PERS_Figure_06.jpg|500px]] || (1.2.1.b)
|
{| 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"
|-
| [[Image:Double Negation 1.0 Marquee Title.png|500px]]
|-
| [[Image:Double Negation 1.0 Storyboard 1.png|500px]]
|-
| [[Image:Equational Inference I2 Elicit (( )).png|500px]]
|-
| [[Image:Double Negation 1.0 Storyboard 2.png|500px]]
|-
| [[Image:Equational Inference J1 Insert (a).png|500px]]
|-
| [[Image:Double Negation 1.0 Storyboard 3.png|500px]]
|-
| [[Image:Equational Inference J2 Distribute ((a)).png|500px]]
|-
| [[Image:Double Negation 1.0 Storyboard 4.png|500px]]
|-
| [[Image:Equational Inference J1 Delete (a).png|500px]]
|-
| [[Image:Double Negation 1.0 Storyboard 5.png|500px]]
|-
| [[Image:Equational Inference J1 Insert a.png|500px]]
|-
| [[Image:Double Negation 1.0 Storyboard 6.png|500px]]
|-
| [[Image:Equational Inference J2 Collect a.png|500px]]
|-
| [[Image:Double Negation 1.0 Storyboard 7.png|500px]]
|-
| [[Image:Equational Inference J1 Delete ((a)).png|500px]]
|-
| [[Image:Double Negation 1.0 Storyboard 8.png|500px]]
|-
| [[Image:Equational Inference I2 Cancel (( )).png|500px]]
|-
| [[Image:Double Negation 1.0 Storyboard 9.png|500px]]
|-
| [[Image:Equational Inference Marquee QED.png|500px]]
|}
|}
| (6)
|}
|}
| [[Image:Double Negation 2.0 Animation.gif]]
| [[Image:Double Negation 2.0 Animation.gif]]
|}
|}
| (1.2.1.d)
| (7)
|}
|}
One theorem of frequent use goes under the nickname of the ''weed and seed theorem'' (WAST). The proof is just an exercise in mathematical induction, once a suitable basis is laid down, and it will be left as an exercise for the reader. What the WAST says is that a label can be freely distributed or freely erased anywhere in a subtree whose root is labeled with that label. The second in our list of frequently used theorems is in fact the base case of this weed and seed theorem. In LOF, it goes by the names of ''Consequence 2'' <math>(C_2)\!</math> or ''Generation''.
One theorem of frequent use goes under the nickname of the ''weed and seed theorem'' (WAST). The proof is just an exercise in mathematical induction, once a suitable basis is laid down, and it will be left as an exercise for the reader. What the WAST says is that a label can be freely distributed or freely erased anywhere in a subtree whose root is labeled with that label. The second in our list of frequently used theorems is in fact the base case of this weed and seed theorem. In LOF, it goes by the names of ''Consequence 2'' <math>(C_2)\!</math> or ''Generation''.
{| align="center" border="0" cellpadding="10" cellspacing="0"
{| align="center" cellpadding="10"
| [[Image:PERS_Figure_08.jpg|500px]] || (1.2.2.a)
| [[Image:Generation Theorem 1.0 Splash Page.png|500px]] || (8)
|}
|}
Here is a proof of the Generation Theorem.
Here is a proof of the Generation Theorem.
{| align="center" border="0" cellpadding="10" cellspacing="0"
{| align="center" cellpadding="8"
| [[Image:PERS_Figure_09.jpg|500px]] || (1.2.2.b)
|
|}
{| 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"
|-
| [[Image:Generation Theorem 1.0 Marquee Title.png|500px]]
|-
| [[Image:Generation Theorem 1.0 Storyboard 1.png|500px]]
|-
| [[Image:Equational Inference C1 Reflect a(b).png|500px]]
|-
| [[Image:Generation Theorem 1.0 Storyboard 2.png|500px]]
|-
| [[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 third of the frequently used theorems of service to this survey is one that Spencer-Brown annotates as ''Consequence 3'' <math>(C_3)\!</math> or ''Integration''. A better mnemonic might be ''dominance and recession theorem'' (DART), but perhaps the brevity of ''dominant form theorem'' (DFT) is sufficient reminder of its double-edged role in proofs.
The steps of this proof are replayed in the following animation.
{| align="center" border="0" cellpadding="10" cellspacing="0"
{| align="center" cellpadding="8"
| [[Image:PERS_Figure_10.jpg|500px]] || (1.2.3.a)
|
{| 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"
|-
| [[Image:Generation Theorem 2.0 Animation.gif]]
|}
| (10)
|}
|}
====C<sub>3</sub>. Dominant form theorem====
The third of the frequently used theorems of service to this survey is one that Spencer-Brown annotates as ''Consequence 3'' <math>(C_3)\!</math> or ''Integration''. A better mnemonic might be ''dominance and recession theorem'' (DART), but perhaps the brevity of ''dominant form theorem'' (DFT) is sufficient reminder of its double-edged role in proofs.
{| align="center" border="0" cellpadding="10" cellspacing="0"
{| align="center" cellpadding="10"
| [[Image:PERS_Figure_11.jpg|500px]] || (1.2.3.b)
| [[Image:Dominant Form 1.0 Splash Page.png|500px]] || (11)
|}
|}
===Exemplary proofs===
Here is a proof of the Dominant Form Theorem.
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 theorems. A couple of all-time favorites are given next.
{| align="center" cellpadding="8"
|
{| 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"
|-
| [[Image:Dominant Form 1.0 Marquee Title.png|500px]]
|-
| [[Image:Dominant Form 1.0 Storyboard 1.png|500px]]
|-
| [[Image:Equational Inference C2 Regenerate a.png|500px]]
|-
| [[Image:Dominant Form 1.0 Storyboard 2.png|500px]]
|-
| [[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.
{| align="center" cellpadding="8"
|
{| 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"
|-
| [[Image:Dominant Form 2.0 Animation.gif]]
|}
| (13)
|}
===Exemplary proofs===
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 theorems. A couple of all-time favorites are given next.
====Peirce's law====
====Peirce's law====
{| align="center" cellpadding="10"
{| align="center" cellpadding="10"
| <math>((p \Rightarrow q) \Rightarrow p) \Rightarrow p</math>
| <math>((p \Rightarrow q) \Rightarrow p) \Rightarrow p\!</math>
|}
|}
The existential graph representation of Peirce's law is shown in Figure 12.
The existential graph representation of Peirce's law is shown below.
{| align="center" cellpadding="10"
{| align="center" cellpadding="10"
| [[Image:PERS_Figure_12.jpg|500px]] || (12)
| [[Image:Peirce's Law 1.0 Splash Page.png|500px]] || (14)
|}
|}
A graphical proof of Peirce's law is shown in Figure 13.
A graphical proof of Peirce's law is shown next.
{| align="center" cellpadding="10"
{| align="center" cellpadding="8"
| [[Image:PERS_Figure_13.jpg|500px]] || (13)
|
|}
{| 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"
|-
====Praeclarum theorema====
| [[Image:Peirce's Law 1.0 Marquee Title.png|500px]]
|-
| [[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]]
|-
| [[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]]
|-
| [[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]]
|}
| (15)
|}
The following animation replays the steps of the proof.
{| align="center" cellpadding="8"
|
{| 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"
|-
| [[Image:Peirce's Law 2.0 Animation.gif]]
|}
| (16)
|}
====Praeclarum theorema====
An illustrious example of a propositional theorem is the ''praeclarum theorema'', the ''admirable'', ''shining'', or ''splendid'' theorem of Leibniz.
An illustrious example of a propositional theorem is the ''praeclarum theorema'', the ''admirable'', ''shining'', or ''splendid'' theorem of Leibniz.
{| align="center" cellpadding="10"
{| align="center" cellpadding="10"
| [[Image:PERS_Figure_14.jpg|500px]] || (14)
| [[Image:Praeclarum Theorema 1.0 Splash Page.png|500px]] || (17)
|}
|}
And here's a neat proof of that nice theorem.
And here's a neat proof of that nice theorem.
{| align="center" cellpadding="10"
{| align="center" cellpadding="8"
| [[Image:PERS_Figure_15.jpg|500px]] || (15)
|
|}
{| 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"
|-
====Majority function example====
| [[Image:Praeclarum Theorema 1.0 Marquee Title.png|500px]]
|-
| [[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]]
|-
| [[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]]
|-
| [[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]]
|-
| [[Image:Praeclarum Theorema 1.0 Storyboard 7.png|500px]]
|-
| [[Image:Equational Inference Marquee QED.png|500px]]
|}
| (18)
|}
The steps of the proof are replayed in the following animation.
{| align="center" cellpadding="8"
|
{| 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"
|-
| [[Image:Praeclarum Theorema 2.0 Animation.gif]]
|}
| (19)
|}
====Two-thirds majority function====
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].
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].
{| align="center" cellpadding="10"
{| align="center" cellpadding="20"
|
|
<math>\begin{matrix}
<math>\begin{matrix}
a b + a c + b c
a b + a c + b c
\end{matrix}</math>
\end{matrix}</math>
| (20)
|}
|}
The required equation can be proven in the medium of logical graphs as shown in the following Figure.
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| [[Image:Majority Function Example 2.0 Proof 1 Frame 1.jpg|500px]]
| [[Image:Majority Function Example 2.0 Proof 1 Frame 1.jpg|500px]]
|-
|-
| [[Image:Equational Inference Reflect ab, ac, bc.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:Majority Function Example 2.0 Proof 1 Frame 2.jpg|500px]]
| [[Image:Majority Function Example 2.0 Proof 1 Frame 3.jpg|500px]]
| [[Image:Majority Function Example 2.0 Proof 1 Frame 3.jpg|500px]]
|-
|-
| [[Image:Equational Inference Collect ab, ac, bc.jpg|500px]]
| [[Image:Equational Inference Collect ab, ac, bc ISW.jpg|500px]]
|-
|-
| [[Image:Majority Function Example 2.0 Proof 1 Frame 4.jpg|500px]]
| [[Image:Majority Function Example 2.0 Proof 1 Frame 4.jpg|500px]]
| [[Image:Majority Function Example 2.0 Proof 1 Frame 7.jpg|500px]]
| [[Image:Majority Function Example 2.0 Proof 1 Frame 7.jpg|500px]]
|-
|-
| [[Image:Equational Inference Delete a, b, c.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:Majority Function Example 2.0 Proof 1 Frame 8.jpg|500px]]
| [[Image:Equational Inference QED.jpg|500px]]
| [[Image:Equational Inference QED.jpg|500px]]
|}
|}
| (16)
| (21)
|}
|}
Here's an animated recap of the graphical transformations that occur in the above proof:
Here's an animated recap of the graphical transformations that occur in the above proof:
{| 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="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"
|-
|-
| [[Image:Majority Function Example 2.0 Animation.gif]]
| [[Image:Two-Thirds Majority Function 500 x 250 Animation.gif]]
|}
|}
| (17)
| (22)
|}
|}
Let us now extend the CSP–GSB calculus in the following way:
Let us now extend the CSP–GSB calculus in the following way:
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, treating <math>\texttt{(\_)}</math> as the ''[[minimal negation operator]]'' of order 1, and adding another such operator for each integer parameter greater than 1. Taken in series, the minimal negation operators are symbolized by parenthesized argument lists of the following shapes: <math>\texttt{(\_)},</math> <math>\texttt{(\_, \_)},</math> <math>\texttt{(\_, \_, \_)},</math> and so on, where the number of argument slots is the order of the reflective negation operator in question.
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{(} \_ \texttt{)}\!</math> in a particular manner, treating <math>\texttt{(} \_ \texttt{)}\!</math> as the ''[[minimal negation operator]]'' of order 1 and adding another such operator for each order greater than 1. Taken in series, the minimal negation operators are symbolized by parenthesized argument lists of the following shapes: <math>\texttt{(} \_ \texttt{)},\!</math> <math>\texttt{(} \_ \texttt{,} \_ \texttt{)},\!</math> <math>\texttt{(} \_ \texttt{,} \_ \texttt{,} \_ \texttt{)},\!</math> and so on, where the number of argument slots is the order of the reflective negation operator in question.
===Fundamental evaluation rule===
The formal rule of evaluation for a <math>k\!</math>''-lobe'' or <math>k\!</math>-operator may be summarized as follows:
The formal rule of evaluation for a <math>k\!</math>''-lobe'' or <math>k\!</math>-operator may be summarized as follows:
{| align="center" cellpadding="10" style="text-align:center; width:90%"
{| align="center" border="0" cellpadding="10"
|
|
<pre>
<pre>
o-----------------------------------------------------------o
o-----------------------------------------------------------o
| Evaluation Rule |
| Fundamental Evaluation Rule |
o-----------------------------------------------------------o
o-----------------------------------------------------------o
| |
| |
o-----------------------------------------------------------o
o-----------------------------------------------------------o
</pre>
</pre>
| (18)
| (23)
|}
|}
These operators may be interpreted for logic as assertions about the values of their listed arguments, resulting in the following pair of dual interpretations.
{| align="center" cellpadding="10" style="text-align:center; width:90%"
{| align="center" border="0" cellpadding="10"
|
|
<pre>
<pre>
o-----------------------------------------------------------o
o-----------------------------------------------------------o
</pre>
</pre>
| (19)
| (24)
|}
|}
==Case analysis-synthesis theorem==
===Additional evaluation rules===
===Discusssion===
Three immediate corollaries of the fundamental evaluation rule are listed below for future reference.
{| align="center" cellpadding="10" style="text-align:center; width:90%"
{| align="center" border="0" cellpadding="10"
|
|
<pre>
<pre>
o-----------------------------------------------------------o
o-----------------------------------------------------------o
| Case Analysis-Synthesis Theorem (CAST) |
| Emptiness Rule |
o-----------------------------------------------------------o
o-----------------------------------------------------------o
| |
| |
| x |
| o o---o o-o-o |
| | \ / \ / |
| @ = @ = @ = ... |
| Q \ / |
| @ = @ |
| |
| |
o-----------------------------------------------------------o
o-----------------------------------------------------------o
</pre>
</pre>
| (20)
| (23a)
|}
|}
{| align="center" border="0" cellpadding="10"
{| align="center" cellpadding="10" style="text-align:center; width:90%"
|
|
<pre>
<pre>
o-----------------------------------------------------------o
o-----------------------------------------------------------o
| Precept L_1. Indifference |
| Indistinctness Rule |
o-----------------------------------------------------------o
o-----------------------------------------------------------o
| |
| |
| a |
| a a a a a |
| o |
| o o---o o-o-o |
| | \ / \ / |
| @ = @ = @ = ... |
| \ / |
| @ = @ |
| |
| |
o-----------------------------------------------------------o
o-----------------------------------------------------------o
</pre>
</pre>
| (21)
| (23b)
|-
|}
{| align="center" border="0" cellpadding="10"
|
|
<pre>
<pre>
o-----------------------------------------------------------o
o-----------------------------------------------------------o
| Precept L_2. Equality. The Following Are Equivalent: |
| Setup/Spike Evaluation Rule |
o-----------------------------------------------------------o
o-----------------------------------------------------------o
| |
| |
| b a b a |
| o |
| | x_2 ... x_k |
| a | \ / | b |
| o---o-...-o---o |
| o---o o o---o |
| \ / |
| \ / | \ / |
| \ / |
| @ = @ = @ |
| \ / |
| \ / |
| \ / |
| \ / x_2 ... x_k |
| @ = @ |
| |
| |
o-----------------------------------------------------------o
o-----------------------------------------------------------o
| |
| |
| (a, (b)) = ((a , b)) = ((a), b) |
| ((), x_2, ..., x_k) = x_2 ... x_k |
| |
| |
o-----------------------------------------------------------o
| Setup <---- | ----> Spike |
o-----------------------------------------------------------o
o-----------------------------------------------------------o
</pre>
</pre>
| (22)
| (23c)
|-
|}
|
<pre>
==Case analysis-synthesis theorem==
o-----------------------------------------------------------o
| Precept L_3. Dispersion |
===Discussion===
The task at hand is build a bridge between model-theoretic and proof-theoretic perspectives on logical procedure, though for now we join them at a point so close to their common source that it may not seem worth the candle at all. The substance of this principle was known to Boole in the 1850's, tantamount to what we now call the ''boolean expansion'' of a propositional expression. The only novelty here resides in a certain manner of presentation, in which we will prove the basic principle from the axioms given before. One name for this rule is the ''Case Analysis-Synthesis Theorem'' (CAST).
I am going to revert to my customarily sloppy workshop manners and refer to propositions and proposition expressions on rough analogy with functions and function expressions, which implies that a proposition will be regarded as the chief formal object of discussion, enjoying many proposition expressions, formulas, or sentences that express it, but worst of all I will probably just go ahead and use any and all of these terms as loosely as I see fit, taking a bit of extra care only when I see the need.
Let <math>Q\!</math> be a propositional expression with an unspecified, but context-appropriate number of variables, say, none, or <math>x,\!</math> or <math>x_1, \ldots, x_k,\!</math> as the case may be.
:* Strings and graphs that have no labels are called ''bare''.
:* A bare terminal node, symbolized by a small circle <math>{}^{\backprime\backprime} \circ {}^{\prime\prime}\!</math> in text, is known as a ''stone''.
:* A bare terminal edge, symbolized by a vertical bar <math>{}^{\backprime\backprime} \vert {}^{\prime\prime}\!</math> in text, is known as a ''stick''.
Let the ''replacement expression'' of the form <math>Q[\circ /x]\!</math> denote the proposition that results from <math>Q\!</math> by replacing every token of the variable <math>x\!</math> with a blank, that is to say, by erasing <math>x.\!</math>
Let the ''replacement expression'' of the form <math>Q[\,\vert /x]\!</math> denote the proposition that results from <math>Q\!</math> by replacing every token of the variable <math>x\!</math> with a stick stemming from the site of <math>x.\!</math>
In the case of a propositional expression <math>Q\!</math> that has no token of the designated variable <math>x,\!</math> let it be stipulated that <math>Q[\circ /x] = Q = Q[\,\vert /x].\!</math>
I think that I am at long last ready to state the following:
{| align="center" border="0" cellpadding="10"
|
<pre>
o-----------------------------------------------------------o
| Case Analysis-Synthesis Theorem (CAST) |
o-----------------------------------------------------------o
o-----------------------------------------------------------o
| |
| |
| For k > 1, the following equation holds: |
| x |
| o |
| x | |
| Q[o/x] o---o Q[|/x] |
| Q \ / |
| @ = @ |
| |
o-----------------------------------------------------------o
| |
| |
| y_1 y_2 ... y_k x y_1 x y_2 ... x y_k |
| Q = ( Q[o/x] x , Q[|/x] (x) ) |
| |
| |
o-----------------------------------------------------------o
o-----------------------------------------------------------o
</pre>
</pre>
| (23)
| (25)
|}
|}
In order to think of tackling even the roughest sketch toward a proof of this theorem, we need to add a number of axioms and axiom schemata. Because I abandoned proof-theoretic purity somewhere in the middle of grinding this calculus into computational form, I never got around to finding the most elegant and minimal, or anything near a complete set of axioms for the ''cactus language'', so what I list here are just the slimmest rudiments of the hodge-podge of ''rules of thumb'' that I have found over time to be necessary and useful in most working settings. Some of these special precepts are probably provable from genuine axioms, but I have yet to go looking for a more proper formulation.
{| align="center" cellpadding="10" style="text-align:center; width:90%"
{| align="center" border="0" cellpadding="10"
|
|
<pre>
<pre>
o-----------------------------------------------------------o
o-----------------------------------------------------------o
| Case Analysis-Synthesis Theorem. Proof Sketch. |
| Precept L_1. Indifference |
o-----------------------------------------------------------o
o-----------------------------------------------------------o
| |
| |
| Q |
| a |
| @ |
| o |
| a | |
| o---o |
| \ / |
| @ = @ |
| |
o-----------------------------------------------------------o
| |
| |
| (a, (a)) = |
| |
| |
| x |
o-----------------------------------------------------------o
| Split <---- | ----> Merge |
| x | |
o-----------------------------------------------------------o
</pre>
| \ / |
| (26)
|-
|
<pre>
o-----------------------------------------------------------o
| Precept L_2. Equality. The Following Are Equivalent: |
o-----------------------------------------------------------o
| |
| |
o=============================< L3. Disperse "Q" >==========o
| b a b a |
| o o---o o |
| a | \ / | b |
| o---o o o---o |
| \ / | \ / |
| @ = @ = @ |
| |
| |
o-----------------------------------------------------------o
| |
| |
| (a, (b)) = ((a , b)) = ((a), b) |
| |
| |
| x |
o-----------------------------------------------------------o
</pre>
| (27)
|-
| \ / |
|
<pre>
o-----------------------------------------------------------o
| Precept L_3. Dispersion |
o-----------------------------------------------------------o
| |
| |
| For k > 1, the following equation holds: |
| |
| |
| x |
| y_1 y_2 ... y_k x y_1 x y_2 ... x y_k |
| o |
| o------o-...-o------o o------o-...-o------o |
| x | |
| \ / \ / |
| Q[o/x] o---o Q[|/x] |
| \ / \ / |
| \ / |
| \ / \ / |
| @ |
| \ / \ / |
| |
| \ / \ / |
o=============================< QES >=======================o
| \ / \ / |
</pre>
| \ / \ / |
| (24)
| \ / \ / |
|}
| \ / \ / |
| x @ = @ |
| |
| x (y_1, ..., y_k) = (x y_1, ..., x y_k) |
| |
o-----------------------------------------------------------o
| Distill <---- | ----> Disperse |
o-----------------------------------------------------------o
</pre>
| (28)
|}
To see why the ''Dispersion Rule'' holds, look at it this way: If <math>x\!</math> is true, then the presence of <math>x\!</math> makes no difference on either side of the equation, but if <math>x\!</math> is false, then both sides of the equation are false.
Here is a proof sketch for the ''Case Analysis-Synthesis Theorem'' (CAST):
{| align="center" cellpadding="8"
{| align="center" border="0" cellpadding="10"
|
|
<pre>
|-
o-----------------------------------------------------------o
| Case Analysis-Synthesis Theorem. Proof Sketch. |
|-
o-----------------------------------------------------------o
| [[Image:Proof Praeclarum Theorema CAST 01.jpg|500px]]
| |
|-
| Q |
| [[Image:Equational Inference Bar -- Cast A.jpg|500px]]
| @ |
|-
| |
| [[Image:Proof Praeclarum Theorema CAST 02.jpg|500px]]
o=============================< L1. Split " " >=============o
|-
| |
| [[Image:Equational Inference Bar -- Domination.jpg|500px]]
| x |
|-
| o |
| [[Image:Proof Praeclarum Theorema CAST 03.jpg|500px]]
| x | |
|-
| o---o |
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
| \ / |
|-
| Q @ |
| [[Image:Proof Praeclarum Theorema CAST 04.jpg|500px]]
| |
|-
o=============================< L3. Disperse "Q" >==========o
| [[Image:Equational Inference Bar -- Domination.jpg|500px]]
| |
|-
| x |
| [[Image:Proof Praeclarum Theorema CAST 05.jpg|500px]]
| o |
|-
| x | |
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
| Q o---o Q |
|-
| \ / |
| [[Image:Proof Praeclarum Theorema CAST 06.jpg|500px]]
| @ |
|-
| |
| [[Image:Equational Inference Bar -- Cast D.jpg|500px]]
o=============================< C1. Reflect "x" >===========o
|-
| |
| [[Image:Proof Praeclarum Theorema CAST 07.jpg|500px]]
| x |
|-
| o |
| [[Image:Equational Inference Bar -- Domination.jpg|500px]]
| x | |
|-
| Q o---o Q[((x))/x] |
| [[Image:Proof Praeclarum Theorema CAST 08.jpg|500px]]
| \ / |
|-
| @ |
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
| |
|-
o=============================< C2. Weed "x", "(x)" >=======o
| [[Image:Proof Praeclarum Theorema CAST 09.jpg|500px]]
| |
| x |
| o |
|-
| x | |
| [[Image:Proof Praeclarum Theorema CAST 10.jpg|500px]]
| Q[o/x] o---o Q[|/x] |
| \ / |
| @ |
| |
o=============================< QES >=======================o
</pre>
| (29)
|}
===Praeclarum theorema : Proof by CAST===
Some of the jobs that the CAST can be put to work on are proving propositional theorems and establishing equations between propositions. Once again, let us turn to the example of Leibniz's ''Praeclarum Theorema'' as a way of illustrating how.
{| align="center" cellpadding="8"
|
{| 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"
|-
| [[Image:Proof Praeclarum Theorema CAST 00.jpg|500px]]
|-
| [[Image:Proof Praeclarum Theorema CAST 01.jpg|500px]]
|-
|-
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
| [[Image:Equational Inference Bar -- Cast A.jpg|500px]]
|-
|-
| [[Image:Proof Praeclarum Theorema CAST 11.jpg|500px]]
| [[Image:Proof Praeclarum Theorema CAST 02.jpg|500px]]
|-
|-
| [[Image:Equational Inference Bar -- Cast B.jpg|500px]]
| [[Image:Equational Inference Bar -- Domination ISW.jpg|500px]]
|-
|-
| [[Image:Proof Praeclarum Theorema CAST 12.jpg|500px]]
| [[Image:Proof Praeclarum Theorema CAST 03.jpg|500px]]
|-
|-
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
|-
|-
| [[Image:Proof Praeclarum Theorema CAST 13.jpg|500px]]
| [[Image:Proof Praeclarum Theorema CAST 04.jpg|500px]]
|-
|-
| [[Image:Equational Inference Bar -- Domination.jpg|500px]]
| [[Image:Equational Inference Bar -- Domination ISW.jpg|500px]]
|-
|-
| [[Image:Proof Praeclarum Theorema CAST 14.jpg|500px]]
| [[Image:Proof Praeclarum Theorema CAST 05.jpg|500px]]
|-
|-
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
|-
|-
| [[Image:Proof Praeclarum Theorema CAST 15.jpg|500px]]
| [[Image:Proof Praeclarum Theorema CAST 06.jpg|500px]]
|-
| [[Image:Equational Inference Bar -- Cast D.jpg|500px]]
|-
| [[Image:Proof Praeclarum Theorema CAST 07.jpg|500px]]
|-
|-
| [[Image:Equational Inference Bar -- Cast C.jpg|500px]]
| [[Image:Equational Inference Bar -- Domination ISW.jpg|500px]]
|-
|-
| [[Image:Proof Praeclarum Theorema CAST 16.jpg|500px]]
| [[Image:Proof Praeclarum Theorema CAST 08.jpg|500px]]
|-
|-
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
|-
|-
| [[Image:Proof Praeclarum Theorema CAST 17.jpg|500px]]
| [[Image:Proof Praeclarum Theorema CAST 09.jpg|500px]]
|-
| [[Image:Equational Inference Bar -- Domination ISW.jpg|500px]]
|-
| [[Image:Proof Praeclarum Theorema CAST 10.jpg|500px]]
|-
|-
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
|-
|-
| [[Image:Proof Praeclarum Theorema CAST 18.jpg|500px]]
| [[Image:Proof Praeclarum Theorema CAST 11.jpg|500px]]
|-
| [[Image:Equational Inference Bar -- Cast B.jpg|500px]]
|-
|-
| [[Image:Equational Inference Bar -- QED.jpg|500px]]
| [[Image:Proof Praeclarum Theorema CAST 12.jpg|500px]]
|-
|-
| [[Image:Praeclarum Theorema CAST 2.0 Animation.gif]]
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
|}
|-
| (26)
| [[Image:Proof Praeclarum Theorema CAST 13.jpg|500px]]
|}
|-
| [[Image:Equational Inference Bar -- Domination ISW.jpg|500px]]
|-
| [[Image:Proof Praeclarum Theorema CAST 14.jpg|500px]]
|-
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
|-
| [[Image:Proof Praeclarum Theorema CAST 15.jpg|500px]]
|-
| [[Image:Equational Inference Bar -- Cast C ISW.jpg|500px]]
|-
| [[Image:Proof Praeclarum Theorema CAST 16.jpg|500px]]
|-
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
|-
| [[Image:Proof Praeclarum Theorema CAST 17.jpg|500px]]
|-
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
|-
| [[Image:Proof Praeclarum Theorema CAST 18.jpg|500px]]
|-
| [[Image:Equational Inference Bar -- QED.jpg|500px]]
|}
| (30)
|}
The logical graph that concludes this proof is a variant type of ''disjunctive normal form'' (DNF) for the logical graph that was to be demonstrated.
The following Figure provides an animated recap of the graphical transformations that occur in the above proof:
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| [[Image:Praeclarum Theorema DNF.jpg|500px]]
|
| (27)
{| 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"
|-
| [[Image:Praeclarum Theorema CAST 500 x 389 Animation.gif]]
|}
| (31)
|}
|}
Remembering that a blank node is the graphical equivalent of a logical value <math>\operatorname{true},</math> the resulting DNF may be read as follows:
The logical graph that concludes this proof is a variant type of ''disjunctive normal form'' (DNF) for the logical graph that was to be demonstrated.
{| align="center" cellpadding="8"
| [[Image:Praeclarum Theorema DNF.jpg|500px]]
| (32)
|}
Remembering that a blank node is the graphical equivalent of a logical value <math>{\mathrm{true}},\!</math> the resulting DNF may be read as follows:
{| align="center" cellpadding="10" style="text-align:center; width:90%"
{| align="center" border="0" cellpadding="10"
|
|
<pre>
<pre>
o-----------------------------------------------------------o
o-----------------------------------------------------------o
</pre>
</pre>
| (28)
| (33)
|}
|}
==Logic as sign transformation==
==Logic as sign transformation==
We have been looking at various ways of transforming propositional expressions, expressed in the parallel formats of character strings and graphical structures, all the while preserving certain aspects of their "meaning" — and here I risk using that vaguest of all possible words, but only as a promissory note, hopefully to be cached out in a more meaningful species of currency as the discussion develops.
We have been looking at various ways of transforming propositional expressions, expressed in the parallel formats of character strings and graphical structures, all the while preserving certain aspects of their “meaning” — and here I risk using that vaguest of all possible words, but only as a promissory note, hopefully to be cached out in a more meaningful species of currency as the discussion develops.
I cannot pretend to be acquainted with or to comprehend every form of intension that others might find of interest in a given form of expression, nor can I speak for every form of meaning that another might find in a given form of syntax. The best that I can hope to do is to specify what my object is in using these expressions, and to say what aspects of their syntax are meant to serve this object, lending these properties the interest I have in preserving them as I put the expressions through the paces of their transformations.
I cannot pretend to be acquainted with or to comprehend every form of intension that others might find of interest in a given form of expression, nor can I speak for every form of meaning that another might find in a given form of syntax. The best that I can hope to do is to specify what my object is in using these expressions, and to say what aspects of their syntax are meant to serve this object, lending these properties the interest I have in preserving them as I put the expressions through the paces of their transformations.
{| align="center" cellpadding="10"
{| align="center" cellpadding="10"
| [[Image:Praeclarum Theorema.jpg|500px]] || (29)
| [[Image:Praeclarum Theorema.jpg|500px]] || (34)
|}
|}
{| align="center" cellpadding="10" width="90%"
{| align="center" cellpadding="10" width="90%"
| [[Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems#Praeclarum_theorema|The first way of transforming the expression]] that appears on the left hand side of the equation can be described as ''proof-theoretic'' in character.
| [[#Praeclarum theorema|The first way of transforming the expression]] that appears on the left hand side of the equation can be described as ''proof-theoretic'' in character.
|-
|-
| [[Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems#Example|The second way of transforming the expression]] that appears on the left hand side of the equation can be described as ''model-theoretic'' in character.
| [[#Praeclarum theorema : Proof by CAST|The second way of transforming the expression]] that appears on the left hand side of the equation can be described as ''model-theoretic'' in character.
|}
|}
My broader interest lies in the theory of inquiry as a special application or a special case of the theory of signs. Another name for the theory of inquiry is ''logic'' and another name for the theory of signs is ''semiotics''. So I might as well have said that I am interested in logic as a special application or a special case of semiotics. But what sort of a special application? What sort of a special case? Well, I think of logic as ''formal semiotics'' — though, of course, I am not the first to have said such a thing — and by ''formal'' we say, in our etymological way, that logic is concerned with the ''form'', indeed, with the ''animate beauty'' and the very ''life force'' of signs and sign actions. Yes, perhaps that is far too Latin a way of understanding logic, but it's all I've got.
My broader interest lies in the theory of inquiry as a special application or a special case of the theory of signs. Another name for the theory of inquiry is ''logic'' and another name for the theory of signs is ''semiotics''. So I might as well have said that I am interested in logic as a special application or a special case of semiotics. But what sort of a special application? What sort of a special case? Well, I think of logic as ''formal semiotics'' — though, of course, I am not the first to have said such a thing — and by ''formal'' we say, in our etymological way, that logic is concerned with the ''form'', indeed, with the ''animate beauty'' and the very ''life force'' of signs and sign actions. Yes, perhaps that is far too Latin a way of understanding logic, but it's all I've got.
Now, if you think about these things just a little more, I know that you will find them just a little suspicious, for what besides logic would I use to do this theory of signs that I would apply to this theory of inquiry that I'm also calling ''logic''? But that is precisely one of the things signified by the word ''formal'', for what I'd be required to use would have to be some brand of logic, that is, some sort of innate or inured skill at inquiry, but a style of logic that is casual, catch-as-catch-can, formative, incipient, inchoate, unformalized, a work in progress, partially built into our natural language and partially more primitive than our most artless language. In so far as I use it more than mention it, mention it more than describe it, and describe it more than fully formalize it, then to that extent it must be consigned to the realm of unformalized and unreflective logic, where some say "there be oracles", but I don't know.
Now, if you think about these things just a little more, I know that you will find them just a little suspicious, for what besides logic would I use to do this theory of signs that I would apply to this theory of inquiry that I'm also calling ''logic''? But that is precisely one of the things signified by the word ''formal'', for what I'd be required to use would have to be some brand of logic, that is, some sort of innate or inured skill at inquiry, but a style of logic that is casual, catch-as-catch-can, formative, incipient, inchoate, unformalized, a work in progress, partially built into our natural language and partially more primitive than our most artless language. In so far as I use it more than mention it, mention it more than describe it, and describe it more than fully formalize it, then to that extent it must be consigned to the realm of unformalized and unreflective logic, where some say “there be oracles”, but I don't know.
Still, one of the aims of formalizing what acts of reasoning that we can is to draw them into an arena where we can examine them more carefully, perhaps to get better at their performance than we can unreflectively, and thus to live, to formalize again another day. Formalization is not the be-all end-all of human life, not by a long shot, but it has its uses on that behalf.
Still, one of the aims of formalizing what acts of reasoning that we can is to draw them into an arena where we can examine them more carefully, perhaps to get better at their performance than we can unreflectively, and thus to live, to formalize again another day. Formalization is not the be-all end-all of human life, not by a long shot, but it has its uses on that behalf.
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| [[Image:Logical Graph (P (Q)) (P (R)).jpg|500px]] || (30)
| [[Image:Logical Graph (P (Q)) (P (R)).jpg|500px]] || (35)
|}
|}
For the sake of simplicity in discussing this example, let's stick with the existential interpretation (<math>\operatorname{Ex}</math>) of logical graphs and their corresponding parse strings. Under <math>\operatorname{Ex}</math> the formal expression <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}</math> translates into the vernacular expression <math>{}^{\backprime\backprime} p ~\operatorname{implies}~ q ~\operatorname{and}~ p ~\operatorname{implies}~ r {}^{\prime\prime},</math> in symbols, <math>(p \Rightarrow q) \land (p \Rightarrow r),</math> so this is the reading that we'll want to keep in mind for the present. Where brevity is required, we may refer to the propositional expression <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}</math> under the name <math>f\!</math> by making use of the following definition:
For the sake of simplicity in discussing this example, let's stick with the existential interpretation <math>(\mathrm{Ex})\!</math> of logical graphs and their corresponding parse strings. Under <math>\mathrm{Ex}\!</math> the formal expression <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}\!</math> translates into the vernacular expression <math>{}^{\backprime\backprime} p ~\mathrm{implies}~ q ~\mathrm{and}~ p ~\mathrm{implies}~ r {}^{\prime\prime},\!</math> in symbols, <math>(p \Rightarrow q) \land (p \Rightarrow r),\!</math> so this is the reading that we'll want to keep in mind for the present. Where brevity is required, we may refer to the propositional expression <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}\!</math> under the name <math>f\!</math> by making use of the following definition:
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| <math>f ~=~ \texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}</math>
| <math>f ~=~ \texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}\!</math>
|}
|}
Since the expression <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}</math> involves just three variables, it may be worth the trouble to draw a venn diagram of the situation. There are in fact two different ways to execute the picture.
Since the expression <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}\!</math> involves just three variables, it may be worth the trouble to draw a venn diagram of the situation. There are in fact two different ways to execute the picture.
Figure 31 indicates the points of the universe of discourse <math>X\!</math> for which the proposition <math>f : X \to \mathbb{B}</math> has the value 1, here interpreted as the logical value <math>\operatorname{true}.</math> In this ''paint by numbers'' style of picture, one simply paints over the cells of a generic template for the universe <math>X,\!</math> going according to some previously adopted convention, for instance: Let the cells that get the value 0 under <math>f\!</math> remain untinted and let the cells that get the value 1 under <math>f\!</math> be painted or shaded. In doing this, it may be good to remind ourselves that the value of the picture as a whole is not in the ''paints'', in other words, the <math>0, 1\!</math> in <math>\mathbb{B},</math> but in the pattern of regions that they indicate.
Figure 2 indicates the points of the universe of discourse <math>X\!</math> for which the proposition <math>f : X \to \mathbb{B}\!</math> has the value 1, here interpreted as the logical value <math>\mathrm{true}.\!</math> In this ''paint by numbers'' style of picture, one simply paints over the cells of a generic template for the universe <math>X,\!</math> going according to some previously adopted convention, for instance: Let the cells that get the value 0 under <math>f\!</math> remain untinted and let the cells that get the value 1 under <math>f\!</math> be painted or shaded. In doing this, it may be good to remind ourselves that the value of the picture as a whole is not in the ''paints'', in other words, the <math>0, 1\!</math> in <math>\mathbb{B},\!</math> but in the pattern of regions that they indicate.
{| align="center" cellpadding="8" style="text-align:center"
{| align="center" cellpadding="8" style="text-align:center"
| [[Image:Venn Diagram (P (Q)) (P (R)).jpg|500px]] || (31)
| [[Image:Venn Diagram (P (Q)) (P (R)).jpg|500px]] || (36)
|-
|-
| <math>\text{Venn Diagram for}~ \texttt{(} p \texttt{~(} q \texttt{))~(} p \texttt{~(} r \texttt{))}</math>
| <math>\text{Venn Diagram for}~ \texttt{(} p \texttt{ (} q \texttt{)) (} p \texttt{ (} r \texttt{))}\!</math>
|}
|}
There are a number of standard ways in mathematics and statistics for talking about the subset <math>W\!</math> of the functional domain <math>X\!</math> that gets painted with the value <math>z \in \mathbb{B}</math> by the indicator function <math>f : X \to \mathbb{B}.</math> The region <math>W \subseteq X</math> is called by a variety of names in different settings, for example, the ''antecedent'', the ''fiber'', the ''inverse image'', the ''level set'', or the ''pre-image'' in <math>X\!</math> of <math>z\!</math> under <math>f.\!</math> It is notated and defined as <math>W = f^{-1}(z).\!</math> Here, <math>f^{-1}\!</math> is called the ''converse relation'' or the ''inverse relation'' — it is not in general an inverse function — corresponding to the function <math>f.\!</math> Whenever possible in simple examples, we use lower case letters for functions <math>f : X \to \mathbb{B},</math> and it is sometimes useful to employ capital letters for subsets of <math>X,\!</math> if possible, in such a way that <math>F\!</math> is the fiber of 1 under <math>f,\!</math> in other words, <math>F = f^{-1}(1).\!</math>
There are a number of standard ways in mathematics and statistics for talking about the subset <math>W\!</math> of the functional domain <math>X\!</math> that gets painted with the value <math>z \in \mathbb{B}\!</math> by the indicator function <math>f : X \to \mathbb{B}.\!</math> The region <math>W \subseteq X\!</math> is called by a variety of names in different settings, for example, the ''antecedent'', the ''fiber'', the ''inverse image'', the ''level set'', or the ''pre-image'' in <math>X\!</math> of <math>z\!</math> under <math>f.\!</math> It is notated and defined as <math>W = f^{-1}(z).\!</math> Here, <math>f^{-1}\!</math> is called the ''converse relation'' or the ''inverse relation'' — it is not in general an inverse function — corresponding to the function <math>f.\!</math> Whenever possible in simple examples, we use lower case letters for functions <math>f : X \to \mathbb{B},\!</math> and it is sometimes useful to employ capital letters for subsets of <math>X,\!</math> if possible, in such a way that <math>F\!</math> is the fiber of 1 under <math>f,\!</math> in other words, <math>F = f^{-1}(1).\!</math>
The easiest way to see the sense of the venn diagram is to notice that the expression <math>\texttt{(} p \texttt{(} q \texttt{))},</math> read as <math>p \Rightarrow q,</math> can also be read as <math>{}^{\backprime\backprime} \operatorname{not}~ p ~\operatorname{without}~ q {}^{\prime\prime}.</math> Its assertion effectively excludes any tincture of truth from the region of <math>P\!</math> that lies outside the region <math>Q.\!</math> In a similar manner, the expression <math>\texttt{(} p \texttt{(} r \texttt{))},</math> read as <math>p \Rightarrow r,</math> can also be read as <math>{}^{\backprime\backprime} \operatorname{not}~ p ~\operatorname{without}~ r {}^{\prime\prime}.</math> Asserting it effectively excludes any tincture of truth from the region of <math>P\!</math> that lies outside the region <math>R.\!</math>
The easiest way to see the sense of the venn diagram is to notice that the expression <math>\texttt{(} p \texttt{(} q \texttt{))},\!</math> read as <math>p \Rightarrow q,\!</math> can also be read as <math>{}^{\backprime\backprime} \mathrm{not}~ p ~\mathrm{without}~ q {}^{\prime\prime}.\!</math> Its assertion effectively excludes any tincture of truth from the region of <math>P\!</math> that lies outside the region <math>Q.\!</math> In a similar manner, the expression <math>\texttt{(} p \texttt{(} r \texttt{))},\!</math> read as <math>p \Rightarrow r,\!</math> can also be read as <math>{}^{\backprime\backprime} \mathrm{not}~ p ~\mathrm{without}~ r {}^{\prime\prime}.\!</math> Asserting it effectively excludes any tincture of truth from the region of <math>P\!</math> that lies outside the region <math>R.\!</math>
Figure 32 shows the other standard way of drawing a venn diagram for such a proposition. In this ''punctured soap film'' style of picture — others may elect to give it the more dignified title of a ''logical quotient topology'' — one begins with Figure 31 and then proceeds to collapse the fiber of 0 under <math>X\!</math> down to the point of vanishing utterly from the realm of active contemplation, arriving at the following picture:
Figure 3 shows the other standard way of drawing a venn diagram for such a proposition. In this ''punctured soap film'' style of picture — others may elect to give it the more dignified title of a ''logical quotient topology'' — one begins with Figure 31 and then proceeds to collapse the fiber of 0 under <math>X\!</math> down to the point of vanishing utterly from the realm of active contemplation, arriving at the following picture:
{| align="center" cellpadding="8" style="text-align:center"
{| align="center" cellpadding="8" style="text-align:center"
| [[Image:Venn Diagram (P (Q R)).jpg|500px]] || (32)
| [[Image:Venn Diagram (P (Q R)).jpg|500px]] || (37)
|-
|-
| <math>\text{Venn Diagram for}~ \texttt{(} p \texttt{~(} q ~ r \texttt{))}</math>
| <math>\text{Venn Diagram for}~ \texttt{(} p \texttt{~(} q ~ r \texttt{))}\!</math>
|}
|}
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))},</math>
| <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))},\!</math>
|}
|}
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| <math>(p \Rightarrow q) \land (p \Rightarrow r),</math>
| <math>(p \Rightarrow q) \land (p \Rightarrow r),\!</math>
|}
|}
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| <math>\texttt{(} p \texttt{(} q r \texttt{))},</math>
| <math>\texttt{(} p \texttt{(} q r \texttt{))},\!</math>
|}
|}
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| <math>p \Rightarrow (q \land r).</math>
| <math>p \Rightarrow (q \land r).\!</math>
|}
|}
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| [[Image:Logical Graph (P (Q)) (P (R)) = (P (Q R)).jpg|500px]] || (33)
| [[Image:Logical Graph (P (Q)) (P (R)) = (P (Q R)).jpg|500px]] || (38)
|}
|}
While we go through each of these ways let us keep one eye out for the character and the conduct of each type of proceeding as a semiotic process, that is, as an orbit, in this case discrete, through a locus of signs, in this case propositional expressions, and as it happens in this case, a sequence of transformations that perseveres in the denotative objective of each expression, that is, in the abstract proposition that it expresses, while it preserves the informed constraint on the universe of discourse that gives us one viable candidate for the informational content of each expression in the interpretive chain of sign metamorphoses.
While we go through each of these ways let us keep one eye out for the character and the conduct of each type of proceeding as a semiotic process, that is, as an orbit, in this case discrete, through a locus of signs, in this case propositional expressions, and as it happens in this case, a sequence of transformations that perseveres in the denotative objective of each expression, that is, in the abstract proposition that it expresses, while it preserves the informed constraint on the universe of discourse that gives us one viable candidate for the informational content of each expression in the interpretive chain of sign metamorphoses.
A ''sign relation'' <math>L\!</math> is a subset of a cartesian product <math>O \times S \times I,</math> where <math>O, S, I\!</math> are sets known as the ''object'', ''sign'', and ''interpretant sign'' domains, respectively. These facts are symbolized by writing <math>L \subseteq O \times S \times I.</math> Accordingly, a sign relation <math>L\!</math> consists of ordered triples of the form <math>(o, s, i),\!</math> where <math>o, s, i\!</math> belong to the domains <math>O, S, I,\!</math> respectively. An ordered triple of the form <math>(o, s, i) \in L</math> is referred to as a ''sign triple'' or an ''elementary sign relation''.
A ''sign relation'' <math>L\!</math> is a subset of a cartesian product <math>O \times S \times I,\!</math> where <math>O, S, I\!</math> are sets known as the ''object'', ''sign'', and ''interpretant sign'' domains, respectively. These facts are symbolized by writing <math>L \subseteq O \times S \times I.\!</math> Accordingly, a sign relation <math>L\!</math> consists of ordered triples of the form <math>(o, s, i),\!</math> where <math>{o, s, i}\!</math> belong to the domains <math>{O, S, I},\!</math> respectively. An ordered triple of the form <math>(o, s, i) \in L\!</math> is referred to as a ''sign triple'' or an ''elementary sign relation''.
The ''syntactic domain'' of a sign relation <math>L \subseteq O \times S \times I</math> is defined as the set-theoretic union <math>S \cup I</math> of its sign domain <math>S\!</math> and its interpretant domain <math>I.\!</math> It is not uncommon, especially in formal examples, for the sign domain and the interpretant domain to be equal as sets, in short, to have <math>S = I.\!</math>
The ''syntactic domain'' of a sign relation <math>L \subseteq O \times S \times I\!</math> is defined as the set-theoretic union <math>S \cup I\!</math> of its sign domain <math>S\!</math> and its interpretant domain <math>I.\!</math> It is not uncommon, especially in formal examples, for the sign domain and the interpretant domain to be equal as sets, in short, to have <math>S = I.\!</math>
Sign relations may contain any number of sign triples, finite or infinite. Finite sign relations do arise in applications and can be very instructive as expository examples, but most of the sign relations of significance in logic have infinite sign and interpretant domains, and usually infinite object domains, in the long run, at least, though one frequently works up to infinite domains by a series of finite approximations and gradual stages.
Sign relations may contain any number of sign triples, finite or infinite. Finite sign relations do arise in applications and can be very instructive as expository examples, but most of the sign relations of significance in logic have infinite sign and interpretant domains, and usually infinite object domains, in the long run, at least, though one frequently works up to infinite domains by a series of finite approximations and gradual stages.
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| [[Image:Logical Graph (P (Q)) (P (R)) = (P (Q R)) Proof 1.jpg|500px]]
| [[Image:Logical Graph (P (Q)) (P (R)) = (P (Q R)) Proof 1.jpg|500px]]
| (34)
| (39)
|}
|}
For some reason I always think of this as the way that our DNA would prove it.
For some reason I always think of this as the way that our DNA would prove it.
We are in the process of examining various proofs of the propositional equation <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))} = \texttt{(} p \texttt{(} q r \texttt{))},</math> and viewing these proofs in the light of their character as semiotic processes, in essence, as sign-theoretic transformations.
We are in the process of examining various proofs of the propositional equation <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))} = \texttt{(} p \texttt{(} q r \texttt{))},\!</math> and viewing these proofs in the light of their character as semiotic processes, in essence, as sign-theoretic transformations.
The second way of establishing the truth of this equation is one that I see, rather loosely, as ''model-theoretic'', for no better reason than the sense of its ending with a pattern of expression, a variant of the ''disjunctive normal form'' (DNF), that is commonly recognized to be the form that one extracts from a truth table by pulling out the rows of the table that evaluate to true and constructing the disjunctive expression that sums up the senses of the corresponding interpretations.
The second way of establishing the truth of this equation is one that I see, rather loosely, as ''model-theoretic'', for no better reason than the sense of its ending with a pattern of expression, a variant of the ''disjunctive normal form'' (DNF), that is commonly recognized to be the form that one extracts from a truth table by pulling out the rows of the table that evaluate to true and constructing the disjunctive expression that sums up the senses of the corresponding interpretations.
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-2.jpg|500px]]
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-2.jpg|500px]]
|-
|-
| [[Image:Equational Inference Bar -- Domination.jpg|500px]]
| [[Image:Equational Inference Bar -- Domination ISW.jpg|500px]]
|-
|-
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-3.jpg|500px]]
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-3.jpg|500px]]
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-6.jpg|500px]]
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-6.jpg|500px]]
|-
|-
| [[Image:Equational Inference Bar -- Domination.jpg|500px]]
| [[Image:Equational Inference Bar -- Domination ISW.jpg|500px]]
|-
|-
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-7.jpg|500px]]
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-7.jpg|500px]]
| [[Image:Equational Inference Bar -- DNF.jpg|500px]]
| [[Image:Equational Inference Bar -- DNF.jpg|500px]]
|}
|}
| (35)
| (40)
|}
|}
The final graph in the sequence of equivalents is a disjunctive normal form (DNF) for the proposition on the left hand side of the equation <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))} = \texttt{(} p \texttt{(} q r \texttt{))}.</math>
The final graph in the sequence of equivalents is a disjunctive normal form (DNF) for the proposition on the left hand side of the equation <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))} = \texttt{(} p \texttt{(} q r \texttt{))}.\!</math>
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| [[Image:Logical Graph (P (Q)) (P (R)) DNF.jpg|500px]]
| [[Image:Logical Graph (P (Q)) (P (R)) DNF.jpg|500px]]
| (36)
| (41)
|}
|}
Remembering that a blank node is the graphical equivalent of a logical value <math>\operatorname{true},</math> the resulting DNF may be read as follows:
Remembering that a blank node is the graphical equivalent of a logical value <math>\mathrm{true},\!</math> the resulting DNF may be read as follows:
{| align="center" cellpadding="8" style="text-align:center; width:90%"
{| align="center" border="0" cellpadding="10"
|
|
<pre>
<pre>
o-----------------------------------------------------------o
o-----------------------------------------------------------o
</pre>
</pre>
| (37)
| (42)
|}
|}
It remains to show that the right hand side of the equation <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))} = \texttt{(} p \texttt{(} q r \texttt{))}</math> is logically equivalent to the DNF just obtained. The needed chain of equations is as follows:
It remains to show that the right hand side of the equation <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))} = \texttt{(} p \texttt{(} q r \texttt{))}\!</math> is logically equivalent to the DNF just obtained. The needed chain of equations is as follows:
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| [[Image:Equational Inference Bar -- Cast P.jpg|500px]]
| [[Image:Equational Inference Bar -- Cast P.jpg|500px]]
|-
|-
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-2.jpg|500px]]
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-2 ISW.jpg|500px]]
|-
|-
| [[Image:Equational Inference Bar -- Domination.jpg|500px]]
| [[Image:Equational Inference Bar -- Domination ISW.jpg|500px]]
|-
|-
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-3.jpg|500px]]
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-3.jpg|500px]]
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-5.jpg|500px]]
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-5.jpg|500px]]
|-
|-
| [[Image:Equational Inference Bar -- Domination.jpg|500px]]
| [[Image:Equational Inference Bar -- Domination ISW.jpg|500px]]
|-
|-
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-6.jpg|500px]]
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-6 ISW.jpg|500px]]
|-
|-
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
| [[Image:Equational Inference Bar -- DNF.jpg|500px]]
| [[Image:Equational Inference Bar -- DNF.jpg|500px]]
|}
|}
| (38)
| (43)
|}
|}
This is not only a logically equivalent DNF but exactly the same DNF expression that we obtained before, so we have established the given equation <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))} = \texttt{(} p \texttt{(} q r \texttt{))}.</math> Incidentally, one may wish to note that this DNF expression quickly folds into the following form:
This is not only a logically equivalent DNF but exactly the same DNF expression that we obtained before, so we have established the given equation <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))} = \texttt{(} p \texttt{(} q r \texttt{))}.\!</math> Incidentally, one may wish to note that this DNF expression quickly folds into the following form:
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| [[Image:Logical Graph (P Q R , (P)).jpg|500px]] || (39)
| [[Image:Logical Graph (P Q R , (P)).jpg|500px]] || (44)
|}
|}
This can be read to say <math>{}^{\backprime\backprime} \operatorname{either}~ p q r ~\operatorname{or}~ \operatorname{not}~ p {}^{\prime\prime},</math> which gives us yet another equivalent for the expression <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}</math> and the expression <math>\texttt{(} p \texttt{(} q r \texttt{))}.</math> Still another way of writing the same thing would be as follows:
This can be read to say <math>{}^{\backprime\backprime} \mathrm{either}~ p q r ~\mathrm{or}~ \mathrm{not}~ p {}^{\prime\prime},\!</math> which gives us yet another equivalent for the expression <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}\!</math> and the expression <math>\texttt{(} p \texttt{(} q r \texttt{))}.\!</math> Still another way of writing the same thing would be as follows:
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| [[Image:Logical Graph ((P , P Q R)).jpg|500px]] || (40)
| [[Image:Logical Graph ((P , P Q R)).jpg|500px]] || (45)
|}
|}
In other words, <math>{}^{\backprime\backprime} p ~\operatorname{is~equivalent~to}~ p ~\operatorname{and}~ q ~\operatorname{and}~ r {}^{\prime\prime}.</math>
In other words, <math>{}^{\backprime\backprime} p ~\mathrm{is~equivalent~to}~ p ~\mathrm{and}~ q ~\mathrm{and}~ r {}^{\prime\prime}.\!</math>
One lemma that suggests itself at this point is a principle that may be canonized as the ''Emptiness Rule''. It says that a bare lobe expression like <math>\texttt{( \_, \_, \ldots )},</math> with any number of places for arguments but nothing but blanks as filler, is logically tantamount to the proto-typical expression of its type, namely, the constant expression <math>\texttt{(~)}</math> that <math>\operatorname{Ex}</math> interprets as denoting the logical value <math>\operatorname{false}.</math> To depict the rule in graphical form, we have the continuing sequence of equations:
One lemma that suggests itself at this point is a principle that may be canonized as the ''Emptiness Rule''. It says that a bare lobe expression like <math>\texttt{(} \_ \texttt{,} \_ \texttt{,} \ldots \texttt{)},\!</math> with any number of places for arguments but nothing but blanks as filler, is logically tantamount to the proto-typical expression of its type, namely, the constant expression <math>\texttt{(} ~ \texttt{)}\!</math> that <math>\mathrm{Ex}~\!</math> interprets as denoting the logical value <math>\mathrm{false}.~\!</math> To depict the rule in graphical form, we have the continuing sequence of equations:
{| align="center" cellpadding="8" style="text-align:center; width:90%"
{| align="center" border="0" cellpadding="10"
|
|
<pre>
<pre>
o-----------------------------------------------------------o
o-----------------------------------------------------------o
</pre>
</pre>
| (41)
| (46)
|}
|}
Yet another rule that we'll need is the following:
Yet another rule that we'll need is the following:
{| align="center" cellpadding="8" style="text-align:center; width:90%"
{| align="center" border="0" cellpadding="10"
|
|
<pre>
<pre>
o-----------------------------------------------------------o
o-----------------------------------------------------------o
</pre>
</pre>
| (42)
| (47)
|}
|}
This one is easy enough to derive from rules that are already known, but just for the sake of ready reference it is useful to canonize it as the ''Indistinctness Rule''. Finally, let me introduce a rule-of-thumb that is a bit more suited to routine computation, and that serves to replace the indistinctness rule in many cases where we actually have to call on it. This is actually just a special case of the evaluation rule listed above:
This one is easy enough to derive from rules that are already known, but just for the sake of ready reference it is useful to canonize it as the ''Indistinctness Rule''. Finally, let me introduce a rule-of-thumb that is a bit more suited to routine computation, and that serves to replace the indistinctness rule in many cases where we actually have to call on it. This is actually just a special case of the evaluation rule listed above:
{| align="center" cellpadding="8" style="text-align:center; width:90%"
{| align="center" border="0" cellpadding="10"
|
|
<pre>
<pre>
o-----------------------------------------------------------o
o-----------------------------------------------------------o
</pre>
</pre>
| (43)
| (48)
|}
|}
To continue with the beating of this still-kicking horse in the form of the equation <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))} = \texttt{(} p \texttt{(} q r \texttt{))},</math> let's now take up the third way that I mentioned for examining propositional equations, even if it is literally a third way only at the very outset, almost immediately breaking up according to whether one proceeds by way of the more routine model-theoretic path or else by way of the more strategic proof-theoretic path.
To continue with the beating of this still-kicking horse in the form of the equation <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))} = \texttt{(} p \texttt{(} q r \texttt{))},\!</math> let's now take up the third way that I mentioned for examining propositional equations, even if it is literally a third way only at the very outset, almost immediately breaking up according to whether one proceeds by way of the more routine model-theoretic path or else by way of the more strategic proof-theoretic path.
Let's convert the equation between propositions:
Let's convert the equation between propositions:
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| [[Image:Logical Graph (( (P (Q)) (P (R)) , (P (Q R)) )).jpg|500px]]
| [[Image:Logical Graph (( (P (Q)) (P (R)) , (P (Q R)) )).jpg|500px]]
| (44)
| (49)
|}
|}
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-02.jpg|500px]]
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-02.jpg|500px]]
|-
|-
| [[Image:Equational Inference Bar -- Domination.jpg|500px]]
| [[Image:Equational Inference Bar -- Domination ISW.jpg|500px]]
|-
|-
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-03.jpg|500px]]
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-03.jpg|500px]]
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-08.jpg|500px]]
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-08.jpg|500px]]
|-
|-
| [[Image:Equational Inference Bar -- Domination.jpg|500px]]
| [[Image:Equational Inference Bar -- Domination ISW.jpg|500px]]
|-
|-
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-09.jpg|500px]]
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-09.jpg|500px]]
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
| [[Image:Equational Inference Bar -- Cancellation.jpg|500px]]
|-
|-
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-14.jpg|500px]]
| [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-14 ISW.jpg|500px]]
|-
|-
| [[Image:Equational Inference Bar -- Emptiness.jpg|500px]]
| [[Image:Equational Inference Bar -- Emptiness.jpg|500px]]
| [[Image:Equational Inference Bar -- QED.jpg|500px]]
| [[Image:Equational Inference Bar -- QED.jpg|500px]]
|}
|}
| (45)
| (50)
|}
|}
e_0 & = &
e_0 & = &
{}^{\backprime\backprime}
{}^{\backprime\backprime}
\texttt{(~)}
\texttt{( )}
{}^{\prime\prime}
{}^{\prime\prime}
\\[4pt]
\\[4pt]
e_1 & = &
e_1 & = &
{}^{\backprime\backprime}
{}^{\backprime\backprime}
\texttt{~}
\texttt{ }
{}^{\prime\prime}
{}^{\prime\prime}
\\[4pt]
\\[4pt]
e_2 & = &
e_2 & = &
{}^{\backprime\backprime}
{}^{\backprime\backprime}
\texttt{(} p \texttt{~(} q \texttt{))~(} p \texttt{~(} r \texttt{))}
\texttt{(} p \texttt{ (} q \texttt{)) (} p \texttt{ (} r \texttt{))}
{}^{\prime\prime}
{}^{\prime\prime}
\\[4pt]
\\[4pt]
e_3 & = &
e_3 & = &
{}^{\backprime\backprime}
{}^{\backprime\backprime}
\texttt{(} p \texttt{~(} q~r \texttt{))}
\texttt{(} p \texttt{ (} q r \texttt{))}
{}^{\prime\prime}
{}^{\prime\prime}
\\[4pt]
\\[4pt]
e_4 & = &
e_4 & = &
{}^{\backprime\backprime}
{}^{\backprime\backprime}
\texttt{(} p~q~r \texttt{~,~(} p \texttt{))}
\texttt{(} p q r \texttt{ , (} p \texttt{))}
{}^{\prime\prime}
{}^{\prime\prime}
\\[4pt]
\\[4pt]
e_5 & = &
e_5 & = &
{}^{\backprime\backprime}
{}^{\backprime\backprime}
\texttt{((~(} p \texttt{~(} q \texttt{))~(} p \texttt{~(} r \texttt{))~,~(} p \texttt{~(} q~r \texttt{))~))}
\texttt{(( (} p \texttt{ (} q \texttt{)) (} p \texttt{ (} r \texttt{)) , (} p \texttt{ (} q r \texttt{)) ))}
{}^{\prime\prime}
{}^{\prime\prime}
\end{array}</math>
\end{array}\!</math>
|}
|}
Under <math>\operatorname{Ex}</math> we have the following interpretations:
Under <math>\mathrm{Ex}\!</math> we have the following interpretations:
{| align="center" cellpadding="8" width="90%"
{| align="center" cellpadding="8" width="90%"
| <math>e_0 = {}^{\backprime\backprime} \texttt{(~)} {}^{\prime\prime}</math> expresses the logical constant <math>\operatorname{false}.</math>
| <math>e_0 = {}^{\backprime\backprime} \texttt{( )} {}^{\prime\prime}\!</math> expresses the logical constant <math>\mathrm{false}.\!</math>
|-
|-
| <math>e_1 = {}^{\backprime\backprime} \texttt{~} {}^{\prime\prime}</math> expresses the logical constant <math>\operatorname{true}.</math>
| <math>e_1 = {}^{\backprime\backprime} \texttt{ } {}^{\prime\prime}\!</math> expresses the logical constant <math>\mathrm{true}.\!</math>
|-
|-
| <math>e_2 = {}^{\backprime\backprime} \texttt{(} p \texttt{~(} q \texttt{))~(} p \texttt{~(} r \texttt{))} {}^{\prime\prime}</math> says <math>{}^{\backprime\backprime} \operatorname{not}~ p ~\operatorname{without}~ q,</math> <math>\operatorname{and~not}~ p ~\operatorname{without}~ r {}^{\prime\prime}.</math>
| <math>e_2 = {}^{\backprime\backprime} \texttt{(} p \texttt{ (} q \texttt{)) (} p \texttt{ (} r \texttt{))} {}^{\prime\prime}\!</math> says <math>{}^{\backprime\backprime} \mathrm{not}~ p ~\mathrm{without}~ q,\!</math> <math>\mathrm{and~not}~ p ~\mathrm{without}~ r {}^{\prime\prime}.\!</math>
|-
|-
| <math>e_3 = {}^{\backprime\backprime} \texttt{(} p \texttt{~(} q~r \texttt{))} {}^{\prime\prime}</math> says <math>{}^{\backprime\backprime} \operatorname{not}~ p ~\operatorname{without}~ q ~\operatorname{and}~ r {}^{\prime\prime}.</math>
| <math>e_3 = {}^{\backprime\backprime} \texttt{(} p \texttt{ (} q~r \texttt{))} {}^{\prime\prime}\!</math> says <math>{}^{\backprime\backprime} \mathrm{not}~ p ~\mathrm{without}~ q ~\mathrm{and}~ r {}^{\prime\prime}.\!</math>
|-
|-
| <math>e_4 = {}^{\backprime\backprime} \texttt{(} p~q~r \texttt{~,~(} p \texttt{))} {}^{\prime\prime}</math> says <math>{}^{\backprime\backprime} p ~\operatorname{and}~ q ~\operatorname{and}~ r,</math> <math>~\operatorname{or~else~not}~ p{}^{\prime\prime}.</math>
| <math>e_4 = {}^{\backprime\backprime} \texttt{(} p~q~r \texttt{ , (} p \texttt{))} {}^{\prime\prime}\!</math> says <math>{}^{\backprime\backprime} p ~\mathrm{and}~ q ~\mathrm{and}~ r,\!</math> <math>~\mathrm{or~else~not}~ p{}^{\prime\prime}.\!</math>
|-
|-
| <math>e_5 = {}^{\backprime\backprime} \texttt{((~(} p \texttt{~(} q \texttt{))~(} p \texttt{~(} r \texttt{))~,~(} p \texttt{~(} q~r \texttt{))~))} {}^{\prime\prime}</math> says that <math>e_2\!</math> and <math>e_3\!</math> say the same thing.
| <math>e_5 = {}^{\backprime\backprime} \texttt{(( (} p \texttt{ (} q \texttt{)) (} p \texttt{ (} r \texttt{)) , (} p \texttt{ (} q~r \texttt{)) ))} {}^{\prime\prime}\!</math> says that <math>e_2\!</math> and <math>e_3\!</math> say the same thing.
|}
|}
{| align="center" cellpadding="8" width="90%"
{| align="center" cellpadding="8" width="90%"
| <math>\texttt{(} p \texttt{~(} q \texttt{))(} p \texttt{~(} r \texttt{))} = \texttt{(} p \texttt{~(} q~r \texttt{))}.</math>
| <math>\texttt{(} p \texttt{ (} q \texttt{))(} p \texttt{ (} r \texttt{))} = \texttt{(} p \texttt{ (} q~r \texttt{))}.\!</math>
|}
|}
{| align="center" cellpadding="8" width="90%"
{| align="center" cellpadding="8" width="90%"
| <math>s_1, s_2, s_3, \ldots, s_n.</math>
| <math>s_1, s_2, s_3, \ldots, s_n.\!</math>
|}
|}
'''Proof 1''' proceeded by the ''straightforward approach'', starting with <math>e_2\!</math> as <math>s_1\!</math> and ending with <math>e_3\!</math> as <math>s_n\!.</math>
'''Proof 1''' proceeded by the ''straightforward approach'', starting with <math>e_2\!</math> as <math>s_1\!</math> and ending with <math>e_3\!</math> as <math>s_n\!.</math>
: That is, Proof 1 commenced from the sign <math>{}^{\backprime\backprime} \texttt{(} p \texttt{~(} q \texttt{))~(} p \texttt{~(} r \texttt{))} {}^{\prime\prime}</math> and ended up at the sign <math>{}^{\backprime\backprime} \texttt{(} p \texttt{~(} q~r \texttt{))} {}^{\prime\prime}</math> by legal moves.
: That is, Proof 1 commenced from the sign <math>{}^{\backprime\backprime} \texttt{(} p \texttt{ (} q \texttt{)) (} p \texttt{ (} r \texttt{))} {}^{\prime\prime}\!</math> and ended up at the sign <math>{}^{\backprime\backprime} \texttt{(} p \texttt{ (} q~r \texttt{))} {}^{\prime\prime}\!</math> by legal moves.
'''Proof 2''' lit on by ''burning the candle at both ends'', changing <math>e_2\!</math> into a normal form that reduced to <math>e_4,\!</math> and changing <math>e_3\!</math> into a normal form that also reduced to <math>e_4,\!</math> in this way tethering <math>e_2\!</math> and <math>e_3\!</math> to a common stake.
'''Proof 2''' lit on by ''burning the candle at both ends'', changing <math>e_2\!</math> into a normal form that reduced to <math>e_4,\!</math> and changing <math>e_3\!</math> into a normal form that also reduced to <math>e_4,\!</math> in this way tethering <math>e_2\!</math> and <math>e_3\!</math> to a common stake.
: Filling in the details, one route went from <math>{}^{\backprime\backprime} \texttt{(} p \texttt{~(} q \texttt{))~(} p \texttt{~(} r \texttt{))} {}^{\prime\prime}</math> to <math>{}^{\backprime\backprime} \texttt{(} p~q~r \texttt{~,~(} p \texttt{))} {}^{\prime\prime},</math> and another went from <math>{}^{\backprime\backprime} \texttt{(} p \texttt{~(} q~r \texttt{))} {}^{\prime\prime}</math> to <math>{}^{\backprime\backprime} \texttt{(} p~q~r \texttt{~,~(} p \texttt{))} {}^{\prime\prime},</math> thus equating the two points of departure.
: Filling in the details, one route went from <math>{}^{\backprime\backprime} \texttt{(} p \texttt{ (} q \texttt{)) (} p \texttt{ (} r \texttt{))} {}^{\prime\prime}\!</math> to <math>{}^{\backprime\backprime} \texttt{(} p~q~r \texttt{ , (} p \texttt{))} {}^{\prime\prime},\!</math> and another went from <math>{}^{\backprime\backprime} \texttt{(} p \texttt{ (} q~r \texttt{))} {}^{\prime\prime}\!</math> to <math>{}^{\backprime\backprime} \texttt{(} p~q~r \texttt{ , (} p \texttt{))} {}^{\prime\prime},\!</math> thus equating the two points of departure.
'''Proof 3''' took the path of reflection, expressing the meta-equation between <math>e_2\!</math> and <math>e_3\!</math> in the form of the naturalized equation <math>e_5,\!</math> then taking <math>e_5\!</math> as <math>s_1\!</math> and exchanging it by dint of value preserving steps for <math>e_1\!</math> as <math>s_n.\!</math>
'''Proof 3''' took the path of reflection, expressing the meta-equation between <math>e_2\!</math> and <math>e_3\!</math> in the form of the naturalized equation <math>e_5,\!</math> then taking <math>e_5\!</math> as <math>s_1\!</math> and exchanging it by dint of value preserving steps for <math>e_1\!</math> as <math>s_n.\!</math>
: This way of proceeding went from <math>e_5 = {}^{\backprime\backprime} \texttt{((~(} p \texttt{~(} q \texttt{))~(} p \texttt{~(} r \texttt{))~,~(} p \texttt{~(} q~r \texttt{))~))} {}^{\prime\prime}</math> to the blank expression that <math>\operatorname{Ex}</math> recognizes as the value <math>\operatorname{true}.</math>
: This way of proceeding went from <math>e_5 = {}^{\backprime\backprime} \texttt{(( (} p \texttt{ (} q \texttt{)) (} p \texttt{ (} r \texttt{)) , (} p \texttt{ (} q~r \texttt{)) ))} {}^{\prime\prime}\!</math> to the blank expression that <math>\mathrm{Ex}\!</math> recognizes as the value <math>{\mathrm{true}}.\!</math>
==Computation and inference as semiosis==
==Computation and inference as semiosis==
~ p
~ p
\\
\\
\overline{~~~~~~~~~~~~~~~}
\overline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}
\\
\\
~ q
~ q
~ p
~ p
\\
\\
\overline{\underline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}
\\
\\
~ p ~ q
~ p ~ q
|
|
<math>\begin{array}{l}
<math>\begin{array}{l}
~ \textit{Expression~1}
~ \textit{Expression 1}
\\
\\
~ \textit{Expression~2}
~ \textit{Expression 2}
\\
\\
\overline{~~~~~~~~~~~~~~~~~~~~}
\overline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}
\\
\\
~ \textit{Expression~3}
~ \textit{Expression 3}
\end{array}</math>
\end{array}</math>
|}
|}
The expressions above the line are called ''premisses'' and the expression below the line is called a ''conclusion''. If the rule of inference is simple enough, the ''proof-theoretic turnstile symbol'' <math>{}^{\backprime\backprime} \vdash {}^{\prime\prime}</math> may be used to write the rule on a single line, as follows:
The expressions above the line are called ''premisses'' and the expression below the line is called a ''conclusion''. If the rule of inference is simple enough, the ''proof-theoretic turnstile symbol'' <math>{}^{\backprime\backprime} \vdash {}^{\prime\prime}\!</math> may be used to write the rule on a single line, as follows:
{| align="center" cellpadding="8" width="90%"
{| align="center" cellpadding="8" width="90%"
|
|
<math>\textit{Premiss~1}, \textit{Premiss~2} ~\vdash~ \textit{Conclusion}.</math>
<math>\textit{Premiss 1}, \textit{Premiss 2} ~\vdash~ \textit{Conclusion}.\!</math>
|}
|}
{| align="center" cellpadding="8" width="90%"
{| align="center" cellpadding="8" width="90%"
|
|
From <math>\textit{Expression~1}</math> and <math>\textit{Expression~2}</math> infer <math>\textit{Expression~3}.</math>
From <math>{\textit{Expression 1}}\!</math> and <math>{\textit{Expression 2}}\!</math> infer <math>{\textit{Expression 3}}.\!</math>
|}
|}
Looking to Example 1, the rule of inference known as ''modus ponens'' says the following: From the premiss <math>p \Rightarrow q</math> and the premiss <math>p\!</math> one may logically infer the conclusion <math>q.\!</math>
Looking to Example 1, the rule of inference known as ''modus ponens'' says the following: From the premiss <math>p \Rightarrow q\!</math> and the premiss <math>p\!</math> one may logically infer the conclusion <math>q.\!</math>
Modus ponens is an ''illative'' or ''implicational'' rule. Passage through its turnstile incurs the toll of some information loss, and thus from a fact of <math>q\!</math> alone one cannot infer with any degree of certainty that <math>p \Rightarrow q</math> and <math>p\!</math> are the reasons why <math>q\!</math> happens to be true.
Modus ponens is an ''illative'' or ''implicational'' rule. Passage through its turnstile incurs the toll of some information loss, and thus from a fact of <math>q\!</math> alone one cannot infer with any degree of certainty that <math>p \Rightarrow q\!</math> and <math>p\!</math> are the reasons why <math>q\!</math> happens to be true.
Further considerations along these lines may lead us to appreciate the difference between ''implicational rules of inference'' and ''equational rules of inference'', the latter indicated by an ''equational line of inference'' or a 2-way turnstile <math>{}^{\backprime\backprime} \Vdash {}^{\prime\prime}.</math>
Further considerations along these lines may lead us to appreciate the difference between ''implicational rules of inference'' and ''equational rules of inference'', the latter indicated by an ''equational line of inference'' or a 2-way turnstile <math>{}^{\backprime\backprime} \Vdash {}^{\prime\prime}.\!</math>
==Variations on a theme of transitivity==
==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.
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 <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>
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>
{| align="center" cellpadding="8" width="90%"
{| align="center" cellpadding="8" width="90%"
~ q \le r
~ q \le r
\\
\\
\overline{~~~~~~~~~~~~~~~}
\overline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}
\\
\\
~ p \le r
~ p \le r
~ q \le r
~ q \le r
\\
\\
\overline{\underline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}
\\
\\
~ p \le q \le r
~ p \le q \le r
|}
|}
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 ~\operatorname{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 — conveys more information — 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>
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 — conveys more information — 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>
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.
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.
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ <math>\text{Table 46.}~~\text{Composite and Compiled Order Relations}</math>
|+ <math>\text{Table 51.}~~\text{Composite and Compiled Order Relations}\!</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
|
|
<p><math>\mathcal{L}_1</math></p>
<p><math>\mathcal{L}_1\!</math></p>
<p><math>\text{Decimal}</math></p>
<p><math>\text{Decimal}\!</math></p>
|
|
<p><math>\mathcal{L}_2</math></p>
<p><math>\mathcal{L}_2\!</math></p>
<p><math>\text{Binary}</math></p>
<p><math>\text{Binary}\!</math></p>
|
|
<p><math>\mathcal{L}_3</math></p>
<p><math>\mathcal{L}_3\!</math></p>
<p><math>\text{Vector}</math></p>
<p><math>\text{Vector}\!</math></p>
|
|
<p><math>\mathcal{L}_4</math></p>
<p><math>\mathcal{L}_4\!</math></p>
<p><math>\text{Cactus}</math></p>
<p><math>\text{Cactus}\!</math></p>
|
|
<p><math>\mathcal{L}_5</math></p>
<p><math>\mathcal{L}_5\!</math></p>
<p><math>\text{Order}</math></p>
<p><math>\text{Order}\!</math></p>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
|
|
| align="right" | <math>p\colon\!</math>
| align="right" | <math>p\colon\!</math>
| <math>1~1~1~1~0~0~0~0</math>
| <math>1~1~1~1~0~0~0~0\!</math>
|
|
|
|
|
|
| align="right" | <math>q\colon\!</math>
| align="right" | <math>q\colon\!</math>
| <math>1~1~0~0~1~1~0~0</math>
| <math>1~1~0~0~1~1~0~0\!</math>
|
|
|
|
|
|
| align="right" | <math>r\colon\!</math>
| align="right" | <math>r\colon\!</math>
| <math>1~0~1~0~1~0~1~0</math>
| <math>1~0~1~0~1~0~1~0\!</math>
|
|
|
|
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\texttt{(} p \texttt{~(} q \texttt{))}
\texttt{(} p \texttt{ (} q \texttt{))}
\\[4pt]
\\[4pt]
\texttt{(} q \texttt{~(} r \texttt{))}
\texttt{(} q \texttt{ (} r \texttt{))}
\\[4pt]
\\[4pt]
\texttt{(} p \texttt{~(} r \texttt{))}
\texttt{(} p \texttt{ (} r \texttt{))}
\\[4pt]
\\[4pt]
\texttt{(} p \texttt{~(} q \texttt{))~(} q \texttt{~(} r \texttt{))}
\texttt{(} p \texttt{ (} q \texttt{)) (} q \texttt{ (} r \texttt{))}
\end{matrix}</math>
\end{matrix}</math>
|
|
{| align="center" cellpadding="6" style="text-align:center"
{| align="center" cellpadding="6" style="text-align:center"
| [[Image:Venn Diagram (P (Q)).jpg|500px]] || (47)
| [[Image:Venn Diagram (P (Q)).jpg|500px]] || (52)
|-
|-
| <math>f_{207}(p, q, r) ~=~ \texttt{(} p \texttt{~(} q \texttt{))}</math>
| <math>f_{207}(p, q, r) ~=~ \texttt{(} p \texttt{ (} q \texttt{))}\!</math>
|-
|-
|
|
|-
|-
| [[Image:Venn Diagram (Q (R)).jpg|500px]] || (48)
| [[Image:Venn Diagram (Q (R)).jpg|500px]] || (53)
|-
|-
| <math>f_{187}(p, q, r) ~=~ \texttt{(} q \texttt{~(} r \texttt{))}</math>
| <math>f_{187}(p, q, r) ~=~ \texttt{(} q \texttt{ (} r \texttt{))}\!</math>
|-
|-
|
|
|-
|-
| [[Image:Venn Diagram (P (R)).jpg|500px]] || (49)
| [[Image:Venn Diagram (P (R)).jpg|500px]] || (54)
|-
|-
| <math>f_{175}(p, q, r) ~=~ \texttt{(} p \texttt{~(} r \texttt{))}</math>
| <math>f_{175}(p, q, r) ~=~ \texttt{(} p \texttt{ (} r \texttt{))}\!</math>
|-
|-
|
|
|-
|-
| [[Image:Venn Diagram (P (Q)) (Q (R)).jpg|500px]] || (50)
| [[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>
| <math>f_{139}(p, q, r) ~=~ \texttt{(} p \texttt{ (} q \texttt{)) (} q \texttt{ (} r \texttt{))}\!</math>
|}
|}
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.
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.
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>
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>
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>
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>
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>
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>
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 2.
Thus we obtain the following four relational data tables for the propositions that we are looking at in Example 2.
{| 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%"
{| 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 51.} ~~ [| f_{207} |] ~=~ [| p \le q |]</math>
|+ style="height:30px" | <math>\text{Table 56.} ~~ [| f_{207} |] ~=~ [| p \le q |]\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| style="border-bottom:1px solid black" | <math>p\!</math>
| style="border-bottom:1px solid black" | <math>p\!</math>
{| 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%"
{| 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 52.} ~~ [| f_{187} |] ~=~ [| q \le r |]</math>
|+ style="height:30px" | <math>\text{Table 57.} ~~ [| f_{187} |] ~=~ [| q \le r |]\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| style="border-bottom:1px solid black" | <math>p\!</math>
| style="border-bottom:1px solid black" | <math>p\!</math>
{| 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%"
{| 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 53.} ~~ [| f_{175} |] ~=~ [| p \le r |]</math>
|+ style="height:30px" | <math>\text{Table 58.} ~~ [| f_{175} |] ~=~ [| p \le r |]\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| style="border-bottom:1px solid black" | <math>p\!</math>
| style="border-bottom:1px solid black" | <math>p\!</math>
{| 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%"
{| 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 54.} ~~ [| f_{139} |] ~=~ [| p \le q \le r |]</math>
|+ style="height:30px" | <math>\text{Table 59.} ~~ [| f_{139} |] ~=~ [| p \le q \le r |]\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| style="border-bottom:1px solid black" | <math>p\!</math>
| style="border-bottom:1px solid black" | <math>p\!</math>
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.
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 <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.
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.
Let us now consider the various rules of inference for transitivity in the light of their performance as information-developing actions.
Let us now consider the various rules of inference for transitivity in the light of their performance as information-developing actions.
~ q \le r
~ q \le r
\\
\\
\overline{~~~~~~~~~~~~~~~}
\overline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}
\\
\\
~ p \le r
~ p \le r
|-
|-
| valign="top" | <big>•</big>
| valign="top" | <big>•</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.
| 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.
|-
|-
| valign="top" | <big>•</big>
| valign="top" | <big>•</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.
| 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.
|-
|-
| valign="top" | <big>•</big>
| valign="top" | <big>•</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.
| 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.
|}
|}
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.
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="4" width="90%"
{| align="center" cellpadding="4" width="90%"
~ q \le r
~ q \le r
\\
\\
\overline{\underline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}
\\
\\
~ p \le q \le r
~ p \le q \le r
|-
|-
| valign="top" | <big>•</big>
| valign="top" | <big>•</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.
| 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.
|-
|-
| valign="top" | <big>•</big>
| valign="top" | <big>•</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.
| 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.
|}
|}
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.
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 <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>\operatorname{syll} : \mathbb{B}^3 \to \mathbb{B},</math> and let us refer to its fiber of truth <math>[| \operatorname{syll} |] = \operatorname{syll}^{-1}(1)</math> as the ''syllogism relation'', written as <math>\operatorname{Syll} \subseteq \mathbb{B}^3.</math> Table 51 shows <math>\operatorname{Syll}</math> as a relational dataset.
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 60 shows <math>\mathrm{Syll}\!</math> as a relational dataset.
<br>
<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%"
{| 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 55.} ~~ \text{Syllogism Relation}</math>
|+ style="height:30px" | <math>\text{Table 60.} ~~ \text{Syllogism Relation}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| style="border-bottom:1px solid black" | <math>p\!</math>
| style="border-bottom:1px solid black" | <math>p\!</math>
<br>
<br>
One of the first questions that we might ask about a 3-adic relation, in this case <math>\operatorname{Syll},</math> is whether it is ''determined by'' its 2-adic projections. I will illustrate what this means in the present case.
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 56 repeats the relation <math>\operatorname{Syll}</math> in the first column, listing its 3-tuples in bit-string form, followed by the 2-adic or ''planar'' projections of <math>\operatorname{Syll}</math> in the next three columns. For instance, <math>\operatorname{Syll}_{pq}</math> is the 2-adic projection of <math>\operatorname{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>\operatorname{Syll}_{pr}</math> is obtained by deleting the <math>q\!</math> column and <math>\operatorname{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.
Table 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>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table 56.} ~~ \text{Dyadic Projections of the Syllogism Relation}</math>
|+ style="height:30px" | <math>\text{Table 61.} ~~ \text{Dyadic Projections of the Syllogism Relation}\!</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| <math>\operatorname{Syll}</math>
| <math>\mathrm{Syll}\!</math>
| <math>\operatorname{Syll}_{pq}</math>
| <math>\mathrm{Syll}_{pq}\!</math>
| <math>\operatorname{Syll}_{pr}</math>
| <math>\mathrm{Syll}_{pr}\!</math>
| <math>\operatorname{Syll}_{qr}</math>
| <math>\mathrm{Syll}_{qr}\!</math>
|-
|-
|
|
\end{matrix}</math>
\end{matrix}</math>
|- style="height:40px; background:#f0f0ff"
|- style="height:40px; background:#f0f0ff"
| <math>p \le q \le r</math>
| <math>p \le q \le r\!</math>
| <math>\texttt{(} p \texttt{~(} q \texttt{))}</math>
| <math>\texttt{(} p \texttt{ (} q \texttt{))}\!</math>
| <math>\texttt{(} p \texttt{~(} r \texttt{))}</math>
| <math>\texttt{(} p \texttt{ (} r \texttt{))}\!</math>
| <math>\texttt{(} q \texttt{~(} r \texttt{))}</math>
| <math>\texttt{(} q \texttt{ (} r \texttt{))}\!</math>
|}
|}
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>
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>
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>\operatorname{syll} : \mathbb{B}^3 \to \mathbb{B}</math> and <math>\operatorname{Syll} \subseteq \mathbb{B}^3,</math> respectively.
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.
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.
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.
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 57 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".
Figure 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="10" style="text-align:center; width:90%"
{| align="center" border="0" cellpadding="10"
|
|
<pre>
<pre>
| |
| |
o-------------------------------------------------o
o-------------------------------------------------o
Figure 56. Boolean 3-Cube B^3
Figure 62. Boolean 3-Cube B^3
</pre>
</pre>
| (57)
| (62)
|}
|}
Table 58 shows the 3-adic relation <math>\operatorname{Syll} \subseteq \mathbb{B}^3</math> again, and Figure 59 shows it plotted on a 3-cube template.
Table 63 shows the 3-adic relation <math>\mathrm{Syll} \subseteq \mathbb{B}^3\!</math> again, and Figure 64 shows it plotted on a 3-cube template.
{| align="center" cellpadding="10" style="text-align:center; width:90%"
{| align="center" border="0" cellpadding="10"
|
|
<pre>
<pre>
Table 58. Syll c B^3
Table 63. Syll c B^3
o-----------------------o
o-----------------------o
| p q r |
| p q r |
o-----------------------o
o-----------------------o
</pre>
</pre>
| (58)
| (63)
|-
|}
{| align="center" border="0" cellpadding="10"
|
|
<pre>
<pre>
| |
| |
o-------------------------------------------------o
o-------------------------------------------------o
Figure 59. Triadic Relation Syll c B^3
Figure 64. Triadic Relation Syll c B^3
</pre>
</pre>
| (59)
| (64)
|}
|}
We return once more to the plane projections of <math>\operatorname{Syll} \subseteq \mathbb{B}^3.</math>
We return once more to the plane projections of <math>\mathrm{Syll} \subseteq \mathbb{B}^3.\!</math>
{| align="center" cellpadding="10" style="text-align:center; width:90%"
{| align="center" border="0" cellpadding="10"
|
|
<pre>
<pre>
Table 60. Syll c B^3
Table 65. Syll c B^3
o-----------------------o
o-----------------------o
| p q r |
| p q r |
o-----------------------o
o-----------------------o
</pre>
</pre>
| (60)
| (65)
|-
|}
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|
|
<pre>
<pre>
Table 61. Dyadic Projections of Syll
Table 66. Dyadic Projections of Syll
o-----------o o-----------o o-----------o
o-----------o o-----------o o-----------o
| Syll_12 | | Syll_13 | | Syll_23 |
| Syll_12 | | Syll_13 | | Syll_23 |
o-----------o o-----------o o-----------o
o-----------o o-----------o o-----------o
</pre>
</pre>
| (61)
| (66)
|}
|}
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.
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.
Figure 62 shows <math>\operatorname{Syll}</math> and its three 2-adic projections:
Figure 67 shows <math>\mathrm{Syll}\!</math> and its three 2-adic projections:
{| align="center" cellpadding="10" style="text-align:center; width:90%"
{| align="center" border="0" cellpadding="10"
|
|
<pre>
<pre>
| |
| |
o-------------------------------------------------o
o-------------------------------------------------o
Figure 62. Syll c B^3 and its Dyadic Projections
Figure 67. Syll c B^3 and its Dyadic Projections
</pre>
</pre>
| (62)
| (67)
|}
|}
We now compute the tacit extensions of the 2-adic projections of <math>\operatorname{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>
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>
{| align="center" cellpadding="10" style="text-align:center; width:90%"
{| align="center" border="0" cellpadding="10"
|
|
<pre>
<pre>
Table 63. Syll c B^3
Table 68. Syll c B^3
o-----------------------o
o-----------------------o
| p q r |
| p q r |
o-----------------------o
o-----------------------o
</pre>
</pre>
| (63)
| (68)
|-
|}
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|
|
<pre>
<pre>
Table 64. Dyadic Projections of Syll
Table 69. Dyadic Projections of Syll
o-----------o o-----------o o-----------o
o-----------o o-----------o o-----------o
| Syll_12 | | Syll_13 | | Syll_23 |
| Syll_12 | | Syll_13 | | Syll_23 |
o-----------o o-----------o o-----------o
o-----------o o-----------o o-----------o
</pre>
</pre>
| (64)
| (69)
|-
|}
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|
|
<pre>
<pre>
Table 65. Tacit Extensions of Projections of Syll
Table 70. Tacit Extensions of Projections of Syll
o---------------o o---------------o o---------------o
o---------------o o---------------o o---------------o
| te(Syll_12) | | te(Syll_13) | | te(Syll_23) |
| te(Syll_12) | | te(Syll_13) | | te(Syll_23) |
o---------------o o---------------o o---------------o
o---------------o o---------------o o---------------o
</pre>
</pre>
| (65)
| (70)
|-
|}
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|
|
<pre>
<pre>
| |
| |
o-------------------------------------------------o
o-------------------------------------------------o
Figure 66. Tacit Extension te_12_3 (Syll_12)
Figure 71. Tacit Extension te_12_3 (Syll_12)
</pre>
</pre>
| (66)
| (71)
|-
|}
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|
|
<pre>
<pre>
| |
| |
o-------------------------------------------------o
o-------------------------------------------------o
Figure 67. Tacit Extension te_13_2 (Syll_13)
Figure 72. Tacit Extension te_13_2 (Syll_13)
</pre>
</pre>
| (67)
| (72)
|-
|}
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|
|
<pre>
<pre>
| |
| |
o-------------------------------------------------o
o-------------------------------------------------o
Figure 68. Tacit Extension te_23_1 (Syll_23)
Figure 73. Tacit Extension te_23_1 (Syll_23)
</pre>
</pre>
| (68)
| (73)
|}
|}
The reader may wish to contemplate Figure 69 and use it to verify the following two facts:
The reader may wish to contemplate Figure 74 and use it to verify the following two facts:
{| align="center" cellpadding="8" width="90%"
{| align="center" cellpadding="8" width="90%"
|
|
<math>\begin{array}{lcc}
<math>\begin{array}{lcc}
\operatorname{Syll}
\mathrm{Syll}
& = &
& = &
\operatorname{te}(\operatorname{Syll}_{12})
\mathrm{te}(\mathrm{Syll}_{12})
\cap
\cap
\operatorname{te}(\operatorname{Syll}_{23})
\mathrm{te}(\mathrm{Syll}_{23})
\\[6pt]
\\[6pt]
\operatorname{Syll}_{13}
\mathrm{Syll}_{13}
& = &
& = &
\operatorname{Syll}_{12}
\mathrm{Syll}_{12}
\circ
\circ
\operatorname{Syll}_{23}
\mathrm{Syll}_{23}
\end{array}</math>
\end{array}</math>
|}
|}
{| align="center" cellpadding="10" style="text-align:center; width:90%"
{| align="center" border="0" cellpadding="10"
|
|
<pre>
<pre>
| |
| |
o-------------------------------------------------o
o-------------------------------------------------o
Figure 69. Syll = te(Syll_12) |^| te(Syll_23)
Figure 74. Syll = te(Syll_12) |^| te(Syll_23)
</pre>
</pre>
| (69)
| (74)
|}
|}
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.
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 70 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 — 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 <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.
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 <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 — 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 <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.
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|
<pre>
<pre>
o-------------------o o-------------------o
o-------------------o o-------------------o
o-------------------o
o-------------------o
Figure 70. Expressive Aspects of Transitive Inference
Figure 75. Expressive Aspects of Transitive Inference
</pre>
</pre>
|}
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.
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.
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''.
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 71 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:
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:
{| align="center" cellpadding="8" width="90%"
{| 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.
| 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.
|-
|-
| 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> <math>M \subseteq X:Z,</math> <math>N \subseteq Y:Z,</math> the forms <math>L \subseteq \mathbb{B}:\mathbb{B}:-,</math> <math>M \subseteq \mathbb{B}:-:\mathbb{B},</math> <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>
| 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> <math>{M \subseteq X:Z},\!</math> <math>{N \subseteq Y:Z},\!</math> the forms <math>{L \subseteq \mathbb{B}:\mathbb{B}:-},\!</math> <math>{M \subseteq \mathbb{B}:-:\mathbb{B}},\!</math> <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>
|}
|}
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|
<pre>
<pre>
o-------------------o o-------------------o
o-------------------o o-------------------o
o---------o---------o o---------o---------o
o---------o---------o o---------o---------o
Figure 71. Denotative Aspects of Transitive Inference
Figure 76. Denotative Aspects of Transitive Inference
</pre>
</pre>
|}
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.
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.
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.
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.
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.
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.
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.
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^\circ = [\mathcal{X}] = [p, q, r],</math> already getting sloppy about quotation marks to single out the signs.
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^\circ</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>
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:
Let us make the following definitions:
|}
|}
These are three sets of two abstract signs each, altogether staking out the qualitative dimensions of the universe of discourse <math>X^\circ</math>.
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>
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.
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.
To illustrate the use of concrete coordinates for points and concrete types for spaces and propositions, Figure 72 translates the contents of Figure 71 into the new language.
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.
{| align="center" border="0" cellpadding="10"
|
<pre>
<pre>
o-------------------o o-------------------o
o-------------------o o-------------------o
o---------o---------o o---------o---------o
o---------o---------o o---------o---------o
Figure 72. Denotative Aspects of Transitive Inference
Figure 77. Denotative Aspects of Transitive Inference
</pre>
</pre>
|}
==References==
==References==
* [[Gottfried Leibniz|Leibniz, G.W.]] (1679–1686 ?), "Addenda to the Specimen of the Universal Calculus", pp. 40–46 in Parkinson, G.H.R. (ed.), ''Leibniz : Logical Papers'', Oxford University Press, London, UK, 1966. (Cf. Gerhardt, 7, p. 223).
* Leibniz, G.W. (1679–1686 ?), “Addenda to the Specimen of the Universal Calculus”, pp. 40–46 in Parkinson, G.H.R. (ed.), ''Leibniz : Logical Papers'', Oxford University Press, London, UK, 1966. (Cf. Gerhardt, 7, p. 223).
* [[Charles Sanders Peirce (Bibliography)|Peirce, C.S., Bibliography]].
* [[Charles Sanders Peirce|Peirce, C.S.]] (1931–1935, 1958), ''Collected Papers of Charles Sanders Peirce'', vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA. Cited as CP volume.paragraph.
* Peirce, C.S. (1981–), ''Writings of Charles S. Peirce : A Chronological Edition'', Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianoplis, IN. Cited as CE volume, page.
* [[Charles Peirce (Bibliography)|Peirce, C.S., Bibliography]].
* Peirce, C.S. (1885), “On the Algebra of Logic : A Contribution to the Philosophy of Notation”, ''American Journal of Mathematics'' 7 (1885), 180–202. Reprinted as CP 3.359–403 and CE 5, 162–190.
* [[Charles Peirce|Peirce, C.S.]] (1931–1935, 1958), ''Collected Papers of Charles Sanders Peirce'', vols. 1–6, [[Charles Hartshorne]] and [[Paul Weiss (philosopher)|Paul Weiss]] (eds.), vols. 7–8, [[Arthur W. Burks]] (ed.), Harvard University Press, Cambridge, MA. Cited as CP volume.paragraph.
* Peirce, C.S. (c. 1886), “Qualitative Logic”, MS 736. Published as pp. 101–115 in Carolyn Eisele (ed., 1976), ''The New Elements of Mathematics by Charles S. Peirce, Volume 4, Mathematical Philosophy'', Mouton, The Hague.
* Peirce, C.S. (1981–), ''Writings of Charles S. Peirce: A Chronological Edition'', [[Peirce Edition Project]] (eds.), Indiana University Press, Bloomington and Indianoplis, IN. Cited as CE volume, page.
* Peirce, C.S. (1886 a), “Qualitative Logic”, MS 582. Published as pp. 323–371 in ''Writings of Charles S. Peirce : A Chronological Edition, Volume 5, 1884–1886'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1993.
* Peirce, C.S. (1885), "On the Algebra of Logic: A Contribution to the Philosophy of Notation", ''American Journal of Mathematics'' 7 (1885), 180–202. Reprinted as CP 3.359–403 and CE 5, 162–190.
* Peirce, C.S. (1886 b), “The Logic of Relatives : Qualitative and Quantitative”, MS 584. Published as pp. 372–378 in ''Writings of Charles S. Peirce : A Chronological Edition, Volume 5, 1884–1886'', Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1993.
* Peirce, C.S. (c. 1886), "Qualitative Logic", MS 736. Published as pp. 101–115 in Carolyn Eisele (ed., 1976), ''The New Elements of Mathematics by Charles S. Peirce, Volume 4, Mathematical Philosophy'', Mouton, The Hague.
* Spencer Brown, George (1969), ''[[Laws of Form]]'', George Allen and Unwin, London, UK.
==See also==
===Related essays and projects===
* [[Futures Of Logical Graphs]]
* [[Information = Comprehension × Extension]]
* [[Inquiry Driven Systems]]
* [[Introduction to Inquiry Driven Systems]]
* [[Peirce's Logic Of Information]]
* [[Futures Of Logical Graphs]]
* [[Semiotic Theory Of Information]]
* [[Information = Comprehension × Extension]]
* [[Inquiry Driven Systems]]
===Related concepts and topics===
* [[Introduction to Inquiry Driven Systems]]
* [[Peirce's Logic Of Information]]
{{col-begin}}
* [[Semiotic Theory Of Information]]
{{col-break}}
* [[Ampheck]]
===Related concepts and topics===
* [[Boolean domain]]
* [[Boolean function]]
{{col-begin}}
* [[Boolean-valued function]]
{{col-break}}
* [[Dynamics of inquiry]]
* [[Ampheck]]
* [[Laws of Form]]
* [[Logic of relatives]]
* [[Boolean domain]]
{{col-break}}
* [[Boolean function]]
* [[Logic of information]]
* [[Logical graph]]
* [[Boolean-valued function]]
* [[Logical matrix]]
* [[Dynamics of inquiry]]
* [[Logical NAND]]
* [[Entitative graph]]
* [[Logical NNOR]]
* [[Existential graph]]
* [[Minimal negation operator]]
{{col-break}}
* [[Multigrade operator]]
* [[Graph (mathematics)|Graph]]
* [[Logic of relatives]]
* [[Logic of information]]
* [[Logical graph]]
* [[Logical matrix]]
* [[Logical NAND]]
* [[Logical NNOR]]
{{col-break}}
{{col-break}}
* [[Parametric operator]]
* [[Parametric operator]]
* [[Peirce's law]]
* [[Peirce's law]]
===Standard Upper Ontology (Mar–Apr 2001)===
===Standard Upper Ontology (Mar–Apr 2001)===
* http://suo.ieee.org/email/thrd186.html#04187
* http://web.archive.org/web/20130109194711/http://suo.ieee.org/email/thrd187.html#04187
# http://suo.ieee.org/email/msg04187.html
# http://web.archive.org/web/20140423181000/http://suo.ieee.org/email/msg04187.html
# http://suo.ieee.org/email/msg04305.html
# http://web.archive.org/web/20070922193822/http://suo.ieee.org/email/msg04305.html
# http://suo.ieee.org/email/msg04413.html
# http://web.archive.org/web/20071007170752/http://suo.ieee.org/email/msg04413.html
# http://suo.ieee.org/email/msg04419.html
# http://web.archive.org/web/20070121063018/http://suo.ieee.org/email/msg04419.html
# http://suo.ieee.org/email/msg04422.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04422.html
# http://suo.ieee.org/email/msg04423.html
# http://web.archive.org/web/20070305132316/http://suo.ieee.org/email/msg04423.html
# http://suo.ieee.org/email/msg04432.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04432.html
# http://suo.ieee.org/email/msg04454.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04454.html
# http://suo.ieee.org/email/msg04455.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04455.html
# http://suo.ieee.org/email/msg04476.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04476.html
# http://suo.ieee.org/email/msg04510.html
# http://web.archive.org/web/20060718091105/http://suo.ieee.org/email/msg04510.html
# http://suo.ieee.org/email/msg04517.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04517.html
# http://suo.ieee.org/email/msg04525.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04525.html
# http://suo.ieee.org/email/msg04533.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04533.html
# http://suo.ieee.org/email/msg04536.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04536.html
# http://suo.ieee.org/email/msg04542.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04542.html
# http://suo.ieee.org/email/msg04546.html
# http://web.archive.org/web/20050824231950/http://suo.ieee.org/email/msg04546.html
===Ontology List (Mar–Apr 2001)===
===Ontology List (Mar–Apr 2001)===
* http://suo.ieee.org/ontology/thrd74.html#01779
* http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/thrd74.html#01779
# http://suo.ieee.org/ontology/msg01779.html
# http://web.archive.org/web/20070326233418/http://suo.ieee.org/ontology/msg01779.html
# http://suo.ieee.org/ontology/msg01897.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg01897.html
# http://suo.ieee.org/ontology/msg02005.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02005.html
# http://suo.ieee.org/ontology/msg02011.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02011.html
# http://suo.ieee.org/ontology/msg02014.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02014.html
# http://suo.ieee.org/ontology/msg02015.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02015.html
# http://suo.ieee.org/ontology/msg02024.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02024.html
# http://suo.ieee.org/ontology/msg02046.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02046.html
# http://suo.ieee.org/ontology/msg02047.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02047.html
# http://suo.ieee.org/ontology/msg02068.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02068.html
# http://suo.ieee.org/ontology/msg02102.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02102.html
# http://suo.ieee.org/ontology/msg02109.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02109.html
# http://suo.ieee.org/ontology/msg02117.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02117.html
# http://suo.ieee.org/ontology/msg02125.html
# http://web.archive.org/web/20040116001230/http://suo.ieee.org/ontology/msg02125.html
# http://suo.ieee.org/ontology/msg02128.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02128.html
# http://suo.ieee.org/ontology/msg02134.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02134.html
# http://suo.ieee.org/ontology/msg02138.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02138.html
===Arisbe List (Feb 2003)===
===Arisbe List (Feb 2003)===
* http://stderr.org/pipermail/arisbe/2003-February/thread.html#1541
* http://web.archive.org/web/20140619114718/http://stderr.org/pipermail/arisbe/2003-February/thread.html#1541
# http://stderr.org/pipermail/arisbe/2003-February/001541.html
# http://web.archive.org/web/20140619114651/http://stderr.org/pipermail/arisbe/2003-February/001541.html
# http://stderr.org/pipermail/arisbe/2003-February/001542.html
# http://web.archive.org/web/20140619114657/http://stderr.org/pipermail/arisbe/2003-February/001542.html
# http://stderr.org/pipermail/arisbe/2003-February/001543.html
# http://web.archive.org/web/20140619115440/http://stderr.org/pipermail/arisbe/2003-February/001543.html
# http://stderr.org/pipermail/arisbe/2003-February/001544.html
# http://web.archive.org/web/20140619114703/http://stderr.org/pipermail/arisbe/2003-February/001544.html
# http://stderr.org/pipermail/arisbe/2003-February/001545.html
# http://web.archive.org/web/20140619115220/http://stderr.org/pipermail/arisbe/2003-February/001545.html
# http://stderr.org/pipermail/arisbe/2003-February/001546.html
# http://web.archive.org/web/20140619115225/http://stderr.org/pipermail/arisbe/2003-February/001546.html
# http://stderr.org/pipermail/arisbe/2003-February/001547.html
# http://web.archive.org/web/20140619114707/http://stderr.org/pipermail/arisbe/2003-February/001547.html
# http://stderr.org/pipermail/arisbe/2003-February/001548.html
# http://web.archive.org/web/20140619114712/http://stderr.org/pipermail/arisbe/2003-February/001548.html
# http://stderr.org/pipermail/arisbe/2003-February/001549.html
# http://web.archive.org/web/20140619115446/http://stderr.org/pipermail/arisbe/2003-February/001549.html
# http://stderr.org/pipermail/arisbe/2003-February/001550.html
# http://web.archive.org/web/20140619114829/http://stderr.org/pipermail/arisbe/2003-February/001550.html
# http://stderr.org/pipermail/arisbe/2003-February/001552.html
# http://web.archive.org/web/20140619115451/http://stderr.org/pipermail/arisbe/2003-February/001552.html
# http://stderr.org/pipermail/arisbe/2003-February/001553.html
# http://web.archive.org/web/20140619115230/http://stderr.org/pipermail/arisbe/2003-February/001553.html
# http://stderr.org/pipermail/arisbe/2003-February/001554.html
# http://web.archive.org/web/20140619115456/http://stderr.org/pipermail/arisbe/2003-February/001554.html
# http://stderr.org/pipermail/arisbe/2003-February/001555.html
# http://web.archive.org/web/20140619114834/http://stderr.org/pipermail/arisbe/2003-February/001555.html
# http://stderr.org/pipermail/arisbe/2003-February/001557.html
# http://web.archive.org/web/20140619115501/http://stderr.org/pipermail/arisbe/2003-February/001557.html
# http://stderr.org/pipermail/arisbe/2003-February/001559.html
# http://web.archive.org/web/20140619115235/http://stderr.org/pipermail/arisbe/2003-February/001559.html
===Ontology List (Feb 2003)===
===Ontology List (Feb 2003)===
* http://suo.ieee.org/ontology/thrd19.html#04522
* http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/thrd19.html#04522
* http://suo.ieee.org/ontology/thrd19.html#04532
* http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/thrd19.html#04532
# http://suo.ieee.org/ontology/msg04522.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04522.html
# http://suo.ieee.org/ontology/msg04523.html
# http://web.archive.org/web/20070302152422/http://suo.ieee.org/ontology/msg04523.html
# http://suo.ieee.org/ontology/msg04524.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04524.html
# http://suo.ieee.org/ontology/msg04525.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04525.html
# http://suo.ieee.org/ontology/msg04526.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04526.html
# http://suo.ieee.org/ontology/msg04527.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04527.html
# http://suo.ieee.org/ontology/msg04528.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04528.html
# http://suo.ieee.org/ontology/msg04529.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04529.html
# http://suo.ieee.org/ontology/msg04530.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04530.html
# http://suo.ieee.org/ontology/msg04531.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04531.html
# http://suo.ieee.org/ontology/msg04532.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04532.html
# http://suo.ieee.org/ontology/msg04533.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04533.html
# http://suo.ieee.org/ontology/msg04534.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04534.html
# http://suo.ieee.org/ontology/msg04536.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04536.html
# http://suo.ieee.org/ontology/msg04537.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04537.html
# http://suo.ieee.org/ontology/msg04538.html
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04538.html
===Inquiry List (Mar 2003)===
===Inquiry List (Mar 2003)===
* http://stderr.org/pipermail/inquiry/2003-March/thread.html#126
* http://web.archive.org/web/20150224210000/http://stderr.org/pipermail/inquiry/2003-March/thread.html#126
# http://stderr.org/pipermail/inquiry/2003-March/000126.html
# http://web.archive.org/web/20150109150400/http://stderr.org/pipermail/inquiry/2003-March/000126.html
# http://stderr.org/pipermail/inquiry/2003-March/000127.html
# http://web.archive.org/web/20150109150401/http://stderr.org/pipermail/inquiry/2003-March/000127.html
# http://stderr.org/pipermail/inquiry/2003-March/000128.html
# http://web.archive.org/web/20140619114723/http://stderr.org/pipermail/inquiry/2003-March/000128.html
# http://stderr.org/pipermail/inquiry/2003-March/000129.html
# http://web.archive.org/web/20140619115507/http://stderr.org/pipermail/inquiry/2003-March/000129.html
# http://stderr.org/pipermail/inquiry/2003-March/000130.html
# http://web.archive.org/web/20140619115246/http://stderr.org/pipermail/inquiry/2003-March/000130.html
# http://stderr.org/pipermail/inquiry/2003-March/000131.html
# http://web.archive.org/web/20140619115251/http://stderr.org/pipermail/inquiry/2003-March/000131.html
# http://stderr.org/pipermail/inquiry/2003-March/000132.html
# http://web.archive.org/web/20140619114728/http://stderr.org/pipermail/inquiry/2003-March/000132.html
# http://stderr.org/pipermail/inquiry/2003-March/000133.html
# http://web.archive.org/web/20140619114733/http://stderr.org/pipermail/inquiry/2003-March/000133.html
# http://stderr.org/pipermail/inquiry/2003-March/000134.html
# http://web.archive.org/web/20140619115256/http://stderr.org/pipermail/inquiry/2003-March/000134.html
# http://stderr.org/pipermail/inquiry/2003-March/000135.html
# http://web.archive.org/web/20140619114844/http://stderr.org/pipermail/inquiry/2003-March/000135.html
# http://stderr.org/pipermail/inquiry/2003-March/000136.html
# http://web.archive.org/web/20140619114849/http://stderr.org/pipermail/inquiry/2003-March/000136.html
# http://stderr.org/pipermail/inquiry/2003-March/000137.html
# http://web.archive.org/web/20140619114738/http://stderr.org/pipermail/inquiry/2003-March/000137.html
# http://stderr.org/pipermail/inquiry/2003-March/000138.html
# http://web.archive.org/web/20140619115512/http://stderr.org/pipermail/inquiry/2003-March/000138.html
# http://stderr.org/pipermail/inquiry/2003-March/000139.html
# http://web.archive.org/web/20140619115301/http://stderr.org/pipermail/inquiry/2003-March/000139.html
# http://stderr.org/pipermail/inquiry/2003-March/000140.html
# http://web.archive.org/web/20140619114743/http://stderr.org/pipermail/inquiry/2003-March/000140.html
# http://stderr.org/pipermail/inquiry/2003-March/000141.html
# http://web.archive.org/web/20140619115306/http://stderr.org/pipermail/inquiry/2003-March/000141.html
# http://stderr.org/pipermail/inquiry/2003-March/000142.html
# http://web.archive.org/web/20140619114748/http://stderr.org/pipermail/inquiry/2003-March/000142.html
===NKS Forum (Apr–May 2004)===
===NKS Forum (Apr–May 2004)===
* http://forum.wolframscience.com/archive/topic/297-1.html
* http://web.archive.org/web/20141210150520/http://forum.wolframscience.com/archive/topic/297.html
* http://forum.wolframscience.com/printthread.php?threadid=297
* http://web.archive.org/web/20130929153121/http://forum.wolframscience.com/showthread.php?threadid=297&perpage=50
# http://web.archive.org/web/20140619112627/http://forum.wolframscience.com/showthread.php?postid=950#post950
# http://forum.wolframscience.com/showthread.php?postid=950#post950
# http://web.archive.org/web/20140619112742/http://forum.wolframscience.com/showthread.php?postid=952#post952
# http://forum.wolframscience.com/showthread.php?postid=952#post952
# http://web.archive.org/web/20140619112632/http://forum.wolframscience.com/showthread.php?postid=953#post953
# http://forum.wolframscience.com/showthread.php?postid=953#post953
# http://web.archive.org/web/20140619112642/http://forum.wolframscience.com/showthread.php?postid=954#post954
# http://forum.wolframscience.com/showthread.php?postid=954#post954
# http://web.archive.org/web/20140619112748/http://forum.wolframscience.com/showthread.php?postid=957#post957
# http://forum.wolframscience.com/showthread.php?postid=957#post957
# http://web.archive.org/web/20140619112648/http://forum.wolframscience.com/showthread.php?postid=958#post958
# http://forum.wolframscience.com/showthread.php?postid=958#post958
# http://web.archive.org/web/20140619112753/http://forum.wolframscience.com/showthread.php?postid=959#post959
# http://forum.wolframscience.com/showthread.php?postid=959#post959
# http://web.archive.org/web/20140619112758/http://forum.wolframscience.com/showthread.php?postid=961#post961
# http://forum.wolframscience.com/showthread.php?postid=961#post961
# http://web.archive.org/web/20140619112804/http://forum.wolframscience.com/showthread.php?postid=962#post962
# http://forum.wolframscience.com/showthread.php?postid=962#post962
# http://web.archive.org/web/20140619112653/http://forum.wolframscience.com/showthread.php?postid=964#post964
# http://forum.wolframscience.com/showthread.php?postid=964#post964
# http://web.archive.org/web/20140619112658/http://forum.wolframscience.com/showthread.php?postid=965#post965
# http://forum.wolframscience.com/showthread.php?postid=965#post965
# http://web.archive.org/web/20140619112808/http://forum.wolframscience.com/showthread.php?postid=967#post967
# http://forum.wolframscience.com/showthread.php?postid=967#post967
# http://web.archive.org/web/20140619112813/http://forum.wolframscience.com/showthread.php?postid=968#post968
# http://forum.wolframscience.com/showthread.php?postid=968#post968
# http://web.archive.org/web/20140619112703/http://forum.wolframscience.com/showthread.php?postid=973#post973
# http://forum.wolframscience.com/showthread.php?postid=973#post973
# http://web.archive.org/web/20140619112819/http://forum.wolframscience.com/showthread.php?postid=976#post976
# http://forum.wolframscience.com/showthread.php?postid=976#post976
# http://web.archive.org/web/20140619112708/http://forum.wolframscience.com/showthread.php?postid=977#post977
# http://forum.wolframscience.com/showthread.php?postid=977#post977
# http://web.archive.org/web/20140619112824/http://forum.wolframscience.com/showthread.php?postid=981#post981
# http://forum.wolframscience.com/showthread.php?postid=981#post981
# http://web.archive.org/web/20140619112829/http://forum.wolframscience.com/showthread.php?postid=988#post988
# http://forum.wolframscience.com/showthread.php?postid=988#post988
# http://web.archive.org/web/20140619112917/http://forum.wolframscience.com/showthread.php?postid=990#post990
# http://forum.wolframscience.com/showthread.php?postid=990#post990
# http://web.archive.org/web/20140619112922/http://forum.wolframscience.com/showthread.php?postid=994#post994
# http://forum.wolframscience.com/showthread.php?postid=994#post994
# http://web.archive.org/web/20140619112718/http://forum.wolframscience.com/showthread.php?postid=1003#post1003
# http://forum.wolframscience.com/showthread.php?postid=1003#post1003
# http://web.archive.org/web/20140619112535/http://forum.wolframscience.com/showthread.php?postid=1005#post1005
# http://forum.wolframscience.com/showthread.php?postid=1005#post1005
# http://web.archive.org/web/20140619112723/http://forum.wolframscience.com/showthread.php?postid=1015#post1015
# http://forum.wolframscience.com/showthread.php?postid=1015#post1015
# http://web.archive.org/web/20140619112540/http://forum.wolframscience.com/showthread.php?postid=1022#post1022
# http://forum.wolframscience.com/showthread.php?postid=1022#post1022
# http://web.archive.org/web/20140619112727/http://forum.wolframscience.com/showthread.php?postid=1025#post1025
# http://forum.wolframscience.com/showthread.php?postid=1025#post1025
# http://web.archive.org/web/20140619112545/http://forum.wolframscience.com/showthread.php?postid=1031#post1031
# http://forum.wolframscience.com/showthread.php?postid=1031#post1031
# http://web.archive.org/web/20140619112551/http://forum.wolframscience.com/showthread.php?postid=1220#post1220
# http://forum.wolframscience.com/showthread.php?postid=1220#post1220
# http://web.archive.org/web/20140619112733/http://forum.wolframscience.com/showthread.php?postid=1224#post1224
# http://forum.wolframscience.com/showthread.php?postid=1224#post1224
# http://web.archive.org/web/20140619112556/http://forum.wolframscience.com/showthread.php?postid=1227#post1227
# http://forum.wolframscience.com/showthread.php?postid=1227#post1227
# http://web.archive.org/web/20140619112601/http://forum.wolframscience.com/showthread.php?postid=1228#post1228
# http://forum.wolframscience.com/showthread.php?postid=1228#post1228
# http://web.archive.org/web/20140619112606/http://forum.wolframscience.com/showthread.php?postid=1232#post1232
# http://forum.wolframscience.com/showthread.php?postid=1232#post1232
# http://web.archive.org/web/20140619112611/http://forum.wolframscience.com/showthread.php?postid=1249#post1249
# http://forum.wolframscience.com/showthread.php?postid=1249#post1249
# http://web.archive.org/web/20140619112737/http://forum.wolframscience.com/showthread.php?postid=1262#post1262
# http://forum.wolframscience.com/showthread.php?postid=1262#post1262
# http://web.archive.org/web/20140619112616/http://forum.wolframscience.com/showthread.php?postid=1265#post1265
# http://forum.wolframscience.com/showthread.php?postid=1265#post1265
# http://web.archive.org/web/20140619112621/http://forum.wolframscience.com/showthread.php?postid=1273#post1273
# http://forum.wolframscience.com/showthread.php?postid=1273#post1273
===Inquiry List (2004–2006)===
===Inquiry List (2004–2006)===
* http://stderr.org/pipermail/inquiry/2004-April/thread.html#1341
* http://web.archive.org/web/20141210145032/http://stderr.org/pipermail/inquiry/2004-April/thread.html#1341
* http://stderr.org/pipermail/inquiry/2004-May/thread.html#1391
* http://web.archive.org/web/20141207162001/http://stderr.org/pipermail/inquiry/2004-May/thread.html#1391
* http://stderr.org/pipermail/inquiry/2006-January/thread.html#3364
* http://web.archive.org/web/20140619115359/http://stderr.org/pipermail/inquiry/2006-January/thread.html#3364
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[[Category:Artificial Intelligence]]
[[Category:Artificial Intelligence]]
[[Category:Boolean Functions]]
[[Category:Charles Sanders Peirce]]
[[Category:Combinatorics]]
[[Category:Combinatorics]]
[[Category:Computer Science]]
[[Category:Computer Science]]
[[Category:Mathematics]]
[[Category:Mathematics]]
[[Category:Philosophy]]
[[Category:Philosophy]]
[[Category:Propositional Calculus]]
[[Category:Semiotics]]
[[Category:Semiotics]]
[[Category:Visualization]]
[[Category:Visualization]]