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{{DISPLAYTITLE:Functional Logic : Inquiry and Analogy}}
{{DISPLAYTITLE:Functional Logic : Inquiry and Analogy}}
'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''
==Abstract==
==Abstract==
This report discusses C.S. Peirce's treatment of analogy, placing it in relation to his overall theory of inquiry. The first order of business is to introduce the three fundamental types of reasoning that Peirce adopted from classical logic. In Peirce's analysis both inquiry and analogy are complex programs of reasoning that develop through stages of these three types, although normally in different orders.
This report discusses C.S. Peirce's treatment of analogy, placing it in relation to his overall theory of inquiry. The first order of business is to introduce the three fundamental types of reasoning that Peirce adopted from classical logic. In Peirce's analysis both inquiry and analogy are complex programs of reasoning that develop through stages of these three types, although normally in different orders.
'''Note on notation.''' The discussion that follows uses [[minimal negations]], expressed as bracketed tuples of the form <math>\texttt{(} e_1 \texttt{,} \ldots \texttt{,} e_k \texttt{)},</math> and logical conjunctions, expressed as concatenated tuples of the form <math>e_1 ~\ldots~ e_k,</math> as the sole expression-forming operations of a calculus for [[boolean-valued functions]] or ''propositions''. The expressions of this calculus parse into data structures whose underlying graphs are called ''cacti'' by graph theorists. Hence the name ''[[cactus language]]'' for this dialect of propositional calculus.
'''Note on notation.''' The discussion that follows uses [[minimal negations]], expressed as bracketed tuples of the form <math>\texttt{(} e_1 \texttt{,} \ldots \texttt{,} e_k \texttt{)},\!</math> and logical conjunctions, expressed as concatenated tuples of the form <math>e_1 ~\ldots~ e_k,\!</math> as the sole expression-forming operations of a calculus for [[boolean-valued functions]] or ''propositions''. The expressions of this calculus parse into data structures whose underlying graphs are called ''cacti'' by graph theorists. Hence the name ''[[cactus language]]'' for this dialect of propositional calculus.
<div class="nonumtoc">__TOC__</div>
<div class="nonumtoc">__TOC__</div>
===1.1. Types of Reasoning in Aristotle===
===1.1. Types of Reasoning in Aristotle===
{| align="center" cellpadding="10"
{| align="center" style="text-align:center"
| [[Image:Types of Reasoning in Aristotle.jpg|600px]]
| [[Image:Types of Reasoning in Aristotle.jpg|600px]]
|-
| <math>\text{Figure 1. Types of Reasoning in Aristotle}\!</math>
|}
|}
| We have then three different kinds of inference:
| We have then three different kinds of inference:
|-
|-
| Deduction or inference ''à priori'',
| Deduction or inference ''à priori'',
|-
|-
| Induction or inference ''à particularis'',
| Induction or inference ''à particularis'',
|-
|-
| Hypothesis or inference ''à posteriori''.
| Hypothesis or inference ''à posteriori''.
|-
|-
| align="right" | (Peirce, CE 1, p. 267).
| align="right" | (Peirce, CE 1, p. 267).
{| align="center" width="90%"
{| align="center" width="90%"
| If I reason that certain conduct is wise because it has a character which belongs ''only'' to wise things, I reason ''à priori''.
| If I reason that certain conduct is wise because it has a character which belongs ''only'' to wise things, I reason ''à priori''.
|-
|-
| If I think it is wise because it once turned out to be wise, that is, if I infer that it is wise on this occasion because it was wise on that occasion, I reason inductively [''à particularis''].
| If I think it is wise because it once turned out to be wise, that is, if I infer that it is wise on this occasion because it was wise on that occasion, I reason inductively [''à particularis''].
|-
|-
| But if I think it is wise because a wise man does it, I then make the pure hypothesis that he does it because he is wise, and I reason ''à posteriori''.
| But if I think it is wise because a wise man does it, I then make the pure hypothesis that he does it because he is wise, and I reason ''à posteriori''.
|-
|-
| align="right" | (Peirce, CE 1, p. 180).
| align="right" | (Peirce, CE 1, p. 180).
| [[Image:Triply Wise Act.jpg|400px]]
| [[Image:Triply Wise Act.jpg|400px]]
|-
|-
| <math>\text{Figure 2. A Triply Wise Act}</math>
| <math>\text{Figure 2. A Triply Wise Act}\!</math>
|}
|}
===1.3. Comparison of the Analyses===
===1.3. Comparison of the Analyses===
{| align="center" style="text-align:center"
| [[Image:Types of Reasoning in Transition.jpg|600px]]
|-
| <math>\text{Figure 3. Types of Reasoning in Transition}\!</math>
|}
<br>
Figure 4. Types of Reasoning in Peirce
{| align="center" style="text-align:center"
| [[Image:Types of Reasoning in Peirce.jpg|600px]]
|-
| <math>\text{Figure 4. Types of Reasoning in Peirce}\!</math>
|}
===1.4. Aristotle's “Apagogy” : Abductive Reasoning as Problem Reduction===
===1.4. Aristotle's “Apagogy” : Abductive Reasoning as Problem Reduction===
<p> We have an Example (παραδειγμα, or analogy) when the major extreme is shown to be applicable to the middle term by means of a term similar to the third. It must be known both that the middle applies to the third term and that the first applies to the term similar to the third.</p>
<p> We have an Example (παραδειγμα, or analogy) when the major extreme is shown to be applicable to the middle term by means of a term similar to the third. It must be known both that the middle applies to the third term and that the first applies to the term similar to the third.</p>
<p> ''E.g.'', let A be “bad”, B “to make war on neighbors”, C “Athens against Thebes”, and D “Thebes against Phocis”. Then if we require to prove that war against Thebes is bad, we must be satisfied that war against neighbors is bad. Evidence of this can be drawn from similar examples, ''e.g.'', that war by Thebes against Phocis is bad. Then since war against neighbors is bad, and war against Thebes is against neighbors, it is evident that war against Thebes is bad.
<p> ''E.g.'', let A be “bad”, B “to make war on neighbors”, C “Athens against Thebes”, and D “Thebes against Phocis”. Then if we require to prove that war against Thebes is bad, we must be satisfied that war against neighbors is bad. Evidence of this can be drawn from similar examples, ''e.g.'', that war by Thebes against Phocis is bad. Then since war against neighbors is bad, and war against Thebes is against neighbors, it is evident that war against Thebes is bad.</p>
|-
|-
| align="right" | (Aristotle, “Prior Analytics” 2.24)
| align="right" | (Aristotle, “Prior Analytics” 2.24)
| [[Image:Aristotle's Paradigm.jpg|600px]]
| [[Image:Aristotle's Paradigm.jpg|600px]]
|-
|-
| <math>\operatorname{Figure~6.~Aristotle's~Paradigm}</math>
| <math>\operatorname{Figure~6.~Aristotle's} ~ {}^{\backprime\backprime}\text{Paradigm}{}^{\prime\prime}\!</math>
|}
|}
{| align="center" width="90%"
{| align="center" width="90%"
|
|
<p>The formula of analogy is as follows: <math>S^{\prime}, S^{\prime\prime}, ~\operatorname{and}~ S^{\prime\prime\prime}</math> are taken at random from such a class that their characters at random are such as <math>P^{\prime}, P^{\prime\prime}, P^{\prime\prime\prime}.</math></p>
<p>The formula of analogy is as follows: <math>S^{\prime}, S^{\prime\prime}, ~\operatorname{and}~ S^{\prime\prime\prime}\!</math> are taken at random from such a class that their characters at random are such as <math>{P^{\prime}, P^{\prime\prime}, P^{\prime\prime\prime}}.\!</math></p>
|-
|-
|
|
<center>
<center>
<math>\begin{matrix}
<math>\begin{matrix}
T ~\operatorname{is}~ P^{\prime}, P^{\prime\prime}, P^{\prime\prime\prime},
\\[4pt]
\\[4pt]
S^{\prime}, S^{\prime\prime}, ~\operatorname{and}~ S^{\prime\prime\prime} ~\operatorname{are}~ q;
S^{\prime}, S^{\prime\prime}, S^{\prime\prime\prime} ~\operatorname{are}~ Q;
\\[4pt]
\\[4pt]
\therefore t ~\operatorname{is}~ q.
\therefore T ~\operatorname{is}~ Q.
\end{matrix}</math>
\end{matrix}</math>
</center>
</center>
|-
|-
| <p>Such an argument is double. It combines the two following:
| <p>Such an argument is double. It combines the two following:</p>
|-
|-
|
|
S^{\prime}, S^{\prime\prime}, S^{\prime\prime\prime} ~\operatorname{are~taken~as~being}~ P^{\prime}, P^{\prime\prime}, P^{\prime\prime\prime},
S^{\prime}, S^{\prime\prime}, S^{\prime\prime\prime} ~\operatorname{are~taken~as~being}~ P^{\prime}, P^{\prime\prime}, P^{\prime\prime\prime},
\\[4pt]
\\[4pt]
S^{\prime}, S^{\prime\prime}, S^{\prime\prime\prime} ~\operatorname{are}~ q;
S^{\prime}, S^{\prime\prime}, S^{\prime\prime\prime} ~\operatorname{are}~ Q;
\\[4pt]
\\[4pt]
\therefore ~(\operatorname{By~induction})~ P^{\prime}, P^{\prime\prime}, P^{\prime\prime\prime} ~\operatorname{is}~ q,
\therefore ~(\operatorname{By~induction})~ P^{\prime}, P^{\prime\prime}, P^{\prime\prime\prime} ~\operatorname{is}~ Q,
\\[4pt]
\\[4pt]
T ~\operatorname{is}~ P^{\prime}, P^{\prime\prime}, P^{\prime\prime\prime};
\\[4pt]
\\[4pt]
\therefore ~(\operatorname{Deductively})~ t ~\operatorname{is}~ q.
\therefore ~(\operatorname{Deductively})~ T ~\operatorname{is}~ Q.
\end{matrix}</math>
\end{matrix}</math>
</center>
</center>
S^{\prime}, S^{\prime\prime}, S^{\prime\prime\prime} ~\operatorname{are,~for~instance,}~ P^{\prime}, P^{\prime\prime}, P^{\prime\prime\prime},
S^{\prime}, S^{\prime\prime}, S^{\prime\prime\prime} ~\operatorname{are,~for~instance,}~ P^{\prime}, P^{\prime\prime}, P^{\prime\prime\prime},
\\[4pt]
\\[4pt]
T ~\operatorname{is}~ P^{\prime}, P^{\prime\prime}, P^{\prime\prime\prime};
\\[4pt]
\\[4pt]
\therefore ~(\operatorname{By~hypothesis})~ t ~\operatorname{has~the~common~characters~of}~ S^{\prime}, S^{\prime\prime}, S^{\prime\prime\prime},
\therefore ~(\operatorname{By~hypothesis})~ T ~\operatorname{has~the~common~characters~of}~ S^{\prime}, S^{\prime\prime}, S^{\prime\prime\prime},
\\[4pt]
\\[4pt]
S^{\prime}, S^{\prime\prime}, S^{\prime\prime\prime} ~\operatorname{are}~ q;
S^{\prime}, S^{\prime\prime}, S^{\prime\prime\prime} ~\operatorname{are}~ Q;
\\[4pt]
\\[4pt]
\therefore ~(\operatorname{Deductively})~ t ~\operatorname{is}~ q.
\therefore ~(\operatorname{Deductively})~ T ~\operatorname{is}~ Q.
\end{matrix}</math>
\end{matrix}</math>
</center>
</center>
The form of this analysis is illustrated in Figure 7.
The form of this analysis is illustrated in Figure 7.
{| align="center" style="text-align:center"
| [[Image:Peirce's Formulation of Analogy (Version 1).jpg|600px]]
|-
| <math>\operatorname{Figure~7.~Peirce's~Formulation~of~Analogy~(Version~1)}\!</math>
|}
====Version 2. “A Theory of Probable Inference” (1883)====
====Version 2. “A Theory of Probable Inference” (1883)====
{| align="center" width="90%"
{| align="center" width="90%"
|
|
<p>The formula of the analogical inference presents, therefore, three premisses, thus: <math>S^{\prime}, S^{\prime\prime}, S^{\prime\prime\prime},</math> are a random sample of some undefined class, <math>X,\!</math> of whose characters <math>P^{\prime}, P^{\prime\prime}, P^{\prime\prime\prime},</math> are samples,</p>
<p>The formula of the analogical inference presents, therefore, three premisses, thus: <math>S^{\prime}, S^{\prime\prime}, S^{\prime\prime\prime},\!</math> are a random sample of some undefined class, <math>X,\!</math> of whose characters <math>P^{\prime}, P^{\prime\prime}, P^{\prime\prime\prime},\!</math> are samples,</p>
|-
|-
|
|
<math>\begin{array}{l}
<center>
<math>\begin{matrix}
T ~\text{is}~ P^{\prime}, P^{\prime\prime}, P^{\prime\prime\prime};
\\[4pt]
\\[4pt]
S^{\prime}, S^{\prime\prime}, S^{\prime\prime\prime}, ~\text{are}~ R\operatorname{'s};
S^{\prime}, S^{\prime\prime}, S^{\prime\prime\prime}, ~\text{are}~ Q\operatorname{'s};
\\[4pt]
\\[4pt]
\text{Hence,}~ Q ~\text{is an}~ R.
\text{Hence,}~ T ~\text{is a}~ Q.
\end{array}</math>
\end{matrix}</math>
</center>
|-
|-
|
|
|-
|-
|
|
<center>
<math>\begin{array}{l|l}
<math>\begin{array}{l|l}
\text{Every}~ X ~\text{is, for example,}~ P^{\prime}, P^{\prime\prime}, P^{\prime\prime\prime}, ~\text{etc.},
\text{Every}~ X ~\text{is, for example,}~ P^{\prime}, P^{\prime\prime}, P^{\prime\prime\prime}, ~\text{etc.},
S^{\prime}, S^{\prime\prime}, S^{\prime\prime\prime}, ~\text{etc., are samples of the}~ X\operatorname{'s},
S^{\prime}, S^{\prime\prime}, S^{\prime\prime\prime}, ~\text{etc., are samples of the}~ X\operatorname{'s},
\\[4pt]
\\[4pt]
T ~\text{is found to be}~ P^{\prime}, P^{\prime\prime}, P^{\prime\prime\prime}, ~\text{etc.};
&
&
S^{\prime}, S^{\prime\prime}, S^{\prime\prime\prime}, ~\text{etc., are found to be}~ R\operatorname{'s};
S^{\prime}, S^{\prime\prime}, S^{\prime\prime\prime}, ~\text{etc., are found to be}~ Q\operatorname{'s};
\\[4pt]
\\[4pt]
\text{Hence, hypothetically,}~ Q ~\text{is an}~ X.
\text{Hence, hypothetically,}~ T ~\text{is an}~ X.
&
&
\text{Hence, inductively, every}~ X ~\text{is an}~ R.
\text{Hence, inductively, every}~ X ~\text{is a}~ Q.
\end{array}</math>
\end{array}</math>
</center>
|-
|-
|
|
<center>
<center>
<math>\text{Hence, deductively,}~ Q ~\text{is an}~ R.</math>
<math>\text{Hence, deductively,}~ T ~\text{is a}~ Q.\!</math>
</center>
</center>
|-
|-
The form of this analysis is illustrated in Figure 8.
The form of this analysis is illustrated in Figure 8.
{| align="center" style="text-align:center"
| [[Image:Peirce's Formulation of Analogy (Version 2).jpg|600px]]
|-
| <math>\operatorname{Figure~8.~Peirce's~Formulation~of~Analogy~(Version~2)}\!</math>
|}
===1.7. Dewey's “Sign of Rain” : An Example of Inquiry===
===1.7. Dewey's “Sign of Rain” : An Example of Inquiry===
Figure 9 illustrates Dewey's “Sign of Rain” example, tracing the structure and function of the sign relation as it informs the activity of inquiry, including both the movements of surprise explanation and intentional action. The dyadic faces of the sign relation are labeled with just a few of the loosest terms that apply, indicating the “significance” of signs for eventual occurrences and the “correspondence&rdqu; of ideas with external orientations. Nothing essential is meant by these dyadic role distinctions, since it is only in special or degenerate cases that their shadowy projections can maintain enough information to determine the original sign relation.
Figure 9 illustrates Dewey's “Sign of Rain” example, tracing the structure and function of the sign relation as it informs the activity of inquiry, including both the movements of surprise explanation and intentional action. The dyadic faces of the sign relation are labeled with just a few of the loosest terms that apply, indicating the “significance” of signs for eventual occurrences and the “correspondence&rdqu; of ideas with external orientations. Nothing essential is meant by these dyadic role distinctions, since it is only in special or degenerate cases that their shadowy projections can maintain enough information to determine the original sign relation.
{| align="center" cellpadding="4" style="text-align:center"
{| align="center" style="text-align:center"
| [[Image:Signs And Inquiry In Dewey.gif|600px]]
| [[Image:Dewey's Sign of Rain Example.jpg|600px]]
|-
|-
| <math>\text{Figure 9. Signs and Inquiry in Dewey}</math>
| <math>\operatorname{Figure~9.~Dewey's} ~ {}^{\backprime\backprime}\text{Sign of Rain}{}^{\prime\prime} ~ \text{Example}\!</math>
|}
|}
Figure 10 charts the progress of inquiry in this example according to the three stages of reasoning identified by Peirce.
Figure 10 charts the progress of inquiry in this example according to the three stages of reasoning identified by Peirce.
{| align="center" cellpadding="4" style="text-align:center"
{| align="center" style="text-align:center"
| [[Image:Cycle of Inquiry.jpg|600px]]
| [[Image:Cycle of Inquiry.jpg|600px]]
|-
|-
| <math>\text{Figure 10. The Cycle of Inquiry}</math>
| <math>\text{Figure 10. The Cycle of Inquiry}\!</math>
|}
|}
# Abduction. The first, faltering step into the cycle of inquiry is taken through the flexion of abductive reasoning. The fact AC, the coolness of the air in the pedestrian's current situation, brings into play from his wordly experience (or from other kinds of background knowledge) the rule AB, that a chill in the air is a feature of situations that betoken rain. This fact and this rule, working in tandem, precipitate a plausible explanation for the observed phenomena. The hiker abduces the case BC, that bodes for rain in the current situation.
# Abduction. The first, faltering step into the cycle of inquiry is taken through the flexion of abductive reasoning. The fact ''C'' ⇒ ''A'', the coolness of the air in the pedestrian's current situation, brings into play from his worldly experience (or from other kinds of background knowledge) the rule ''B'' ⇒ ''A'', that a chill in the air is a feature of situations that betoken rain. This fact and this rule, working in tandem, precipitate a plausible explanation for the observed phenomena. The hiker abduces the case ''C'' ⇒ ''B'', that bodes for rain in the current situation.
# Deduction. …
# Deduction. …
# Induction. …
# Induction. …
In this analysis of the first steps of Inquiry, we have a complex or a mixed form of inference that can be seen as taking place in two steps:
Up till now quantification theory has been based on the assumption of individual variables ranging over universal collections of perfectly determinate elements. Merely to write down quantified formulas like <math>\forall_{x \in X} f(x)</math> and <math>\exists_{x \in X} f(x)</math> involves a subscription to such notions, as shown by the membership relations invoked in their indices. Reflected on pragmatic and constructive principles, however, these ideas begin to appear as problematic hypotheses whose warrants are not beyond question, projects of exhaustive determination that overreach the powers of finite information and control to manage. Therefore, it is worth considering how we might shift the scene of quantification theory closer to familiar ground, toward the predicates themselves that represent our continuing acquaintance with phenomena.
1. The first step is an Abduction that abstracts a Case from the consideration of a Fact and a Rule.
{|
| width=36 |
| Fact: || ''C'' ⇒ ''A'', || In the Current situation the Air is cool.
|-
|
| Rule: || ''B'' ⇒ ''A'', || Just Before it rains, the Air is cool.
|-
|
| Case: || ''C'' ⇒ ''B'', || The Current situation is just Before it rains.
|}
2. The final step is a Deduction that admits this Case to another Rule and so arrives at a novel Fact.
{|
| width=36 |
| Case: || ''C'' ⇒ ''B'', || The Current situation is just Before it rains.
|-
|
| Rule: || ''B'' ⇒ ''D'', || Just Before it rains, a Dark cloud will appear.
|-
|
| Fact: || ''C'' ⇒ ''D'', || In the Current situation, a Dark cloud will appear.
|}
This is nowhere near a complete analysis of the Rainy Day inquiry, even insofar as it might be carried out within the constraints of the syllogistic framework, and it covers only the first two steps of the relevant inquiry process, but maybe it will do for a start.
==2. Functional Conception of Quantification Theory==
Up till now quantification theory has been based on the assumption of individual variables ranging over universal collections of perfectly determinate elements. Merely to write down quantified formulas like <math>\forall_{x \in X} f(x)\!</math> and <math>\exists_{x \in X} f(x)\!</math> involves a subscription to such notions, as shown by the membership relations invoked in their indices. Reflected on pragmatic and constructive principles, however, these ideas begin to appear as problematic hypotheses whose warrants are not beyond question, projects of exhaustive determination that overreach the powers of finite information and control to manage. Therefore, it is worth considering how we might shift the scene of quantification theory closer to familiar ground, toward the predicates themselves that represent our continuing acquaintance with phenomena.
===Higher Order Propositional Expressions===
===Higher Order Propositional Expressions===
====Higher Order Propositions and Logical Operators (''n'' = 1)====
====Higher Order Propositions and Logical Operators (''n'' = 1)====
A ''higher order proposition'' is, very roughly speaking, a proposition about propositions. If the original order of propositions is a class of indicator functions <math>f : X \to \mathbb{B},</math> then the next higher order of propositions consists of maps of the type <math>m : (X \to \mathbb{B}) \to \mathbb{B}.</math>
A ''higher order proposition'' is, very roughly speaking, a proposition about propositions. If the original order of propositions is a class of indicator functions <math>{f : X \to \mathbb{B}},\!</math> then the next higher order of propositions consists of maps of the type <math>{m : (X \to \mathbb{B}) \to \mathbb{B}}.\!</math>
For example, consider the case where <math>X = \mathbb{B}.</math> Then there are exactly four propositions <math>f : \mathbb{B} \to \mathbb{B},</math> and exactly sixteen higher order propositions that are based on this set, all bearing the type <math>m : (\mathbb{B} \to \mathbb{B}) \to \mathbb{B}.</math>
For example, consider the case where <math>{X = \mathbb{B}}.\!</math> Then there are exactly four propositions <math>{f : \mathbb{B} \to \mathbb{B}},\!</math> and exactly sixteen higher order propositions that are based on this set, all bearing the type <math>{m : (\mathbb{B} \to \mathbb{B}) \to \mathbb{B}}.\!</math>
Table 10 lists the sixteen higher order propositions about propositions on one boolean variable, organized in the following fashion: Columns 1 and 2 form a truth table for the four <math>f : \mathbb{B} \to \mathbb{B},</math> turned on its side from the way that one is most likely accustomed to see truth tables, with the row leaders in Column 1 displaying the names of the functions <math>f_i,\!</math> for <math>i\!</math> = 1 to 4, while the entries in Column 2 give the values of each function for the argument values that are listed in the corresponding column head. Column 3 displays one of the more usual expressions for the proposition in question. The last sixteen columns are topped by a collection of conventional names for the higher order propositions, also known as the ''measures'' <math>m_j,\!</math> for <math>j\!</math> = 0 to 15, where the entries in the body of the Table record the values that each <math>m_j\!</math> assigns to each <math>f_i.\!</math>
Table 11 lists the sixteen higher order propositions about propositions on one boolean variable, organized in the following fashion: Columns 1 and 2 form a truth table for the four <math>{f : \mathbb{B} \to \mathbb{B}},\!</math> turned on its side from the way that one is most likely accustomed to see truth tables, with the row leaders in Column 1 displaying the names of the functions <math>{f_i},\!</math> for <math>{i}\!</math> = 1 to 4, while the entries in Column 2 give the values of each function for the argument values that are listed in the corresponding column head. Column 3 displays one of the more usual expressions for the proposition in question. The last sixteen columns are topped by a collection of conventional names for the higher order propositions, also known as the ''measures'' <math>{m_j},\!</math> for <math>{j}\!</math> = 0 to 15, where the entries in the body of the Table record the values that each <math>{m_j}\!</math> assigns to each <math>{f_i}.\!</math>
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:white; color:black; font-weight:bold; text-align:center; width:90%"
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:white; color:black; font-weight:bold; text-align:center; width:90%"
|+ '''Table 10. Higher Order Propositions (''n'' = 1)'''
|+ style="height:25px" |
<math>\text{Table 11. Higher Order Propositions} ~~ (n = 1)\!</math>
|- style="background:ghostwhite"
|- style="background:ghostwhite"
| align="right" | <math>x:</math>
| align="right" | <math>x:\!</math>
| 1 0
| 1 0
| <math>f\!</math>
| <math>f\!</math>
| <math>m_0</math>
| <math>m_0\!</math>
| <math>m_1</math>
| <math>m_1\!</math>
| <math>m_2</math>
| <math>m_2\!</math>
| <math>m_3</math>
| <math>m_3\!</math>
| <math>m_4</math>
| <math>m_4\!</math>
| <math>m_5</math>
| <math>m_5\!</math>
| <math>m_6</math>
| <math>m_6\!</math>
| <math>m_7</math>
| <math>m_7\!</math>
| <math>m_8</math>
| <math>m_8\!</math>
| <math>m_9</math>
| <math>m_9\!</math>
| <math>m_{10}</math>
| <math>m_{10}\!</math>
| <math>m_{11}</math>
| <math>m_{11}\!</math>
| <math>m_{12}</math>
| <math>m_{12}\!</math>
| <math>m_{13}</math>
| <math>m_{13}\!</math>
| <math>m_{14}</math>
| <math>m_{14}\!</math>
| <math>m_{15}</math>
| <math>m_{15}\!</math>
|-
|-
| <math>f_0</math>
| <math>f_0\!</math>
| 0 0
| 0 0
| <math>0\!</math>
| <math>0\!</math>
| 0 || style="background:black; color:white" | 1
| 0 || style="background:black; color:white" | 1
| 0 || style="background:black; color:white" | 1
| 0 || style="background:black; color:white" | 1
| 0 || style="background:black; color:white" | 1
| 0 || style="background:black; color:white" | 1
| 0 || style="background:black; color:white" | 1
| 0 || style="background:black; color:white" | 1
| 0 || style="background:black; color:white" | 1
| 0 || style="background:black; color:white" | 1
| 0 || style="background:black; color:white" | 1
| 0 || style="background:black; color:white" | 1
| 0 || style="background:black; color:white" | 1
| 0 || style="background:black; color:white" | 1
|-
|-
| <math>f_1</math>
| <math>f_1\!</math>
| 0 1
| 0 1
| <math>\texttt{(} x \texttt{)}</math>
| <math>\texttt{(} x \texttt{)}\!</math>
| 0 || 0
| 0 || 0
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|-
|-
| <math>f_2</math>
| <math>f_2\!</math>
| 1 0
| 1 0
| <math>x\!</math>
| <math>x\!</math>
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|-
|-
| <math>f_3</math>
| <math>f_3\!</math>
| 1 1
| 1 1
| <math>1\!</math>
| <math>1\!</math>
<br>
<br>
I am going to put off explaining Table 11, that presents a sample of what I call ''interpretive categories'' for higher order propositions, until after we get beyond the 1-dimensional case, since these lower dimensional cases tend to be a bit ''condensed'' or ''degenerate'' in their structures, and a lot of what is going on here will almost automatically become clearer as soon as we get even two logical variables into the mix.
I am going to put off explaining Table 12, that presents a sample of what I call ''interpretive categories'' for higher order propositions, until after we get beyond the 1-dimensional case, since these lower dimensional cases tend to be a bit ''condensed'' or ''degenerate'' in their structures, and a lot of what is going on here will almost automatically become clearer as soon as we get even two logical variables into the mix.
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"
|+ '''Table 11. Interpretive Categories for Higher Order Propositions (''n'' = 1)'''
|+ style="height:25px" |
<math>\text{Table 12. Interpretive Categories for Higher Order Propositions} ~~ (n = 1)~\!</math>
|- style="background:ghostwhite"
|- style="background:ghostwhite"
| Measure
| Measure
====Higher Order Propositions and Logical Operators (''n'' = 2)====
====Higher Order Propositions and Logical Operators (''n'' = 2)====
By way of reviewing notation and preparing to extend it to higher order universes of discourse, let us first consider the universe of discourse <math>X^\circ = [\mathcal{X}] = [x_1, x_2] = [u, v],</math> based on two logical features or boolean variables <math>u\!</math> and <math>v.\!</math>
By way of reviewing notation and preparing to extend it to higher order universes of discourse, let us first consider the universe of discourse <math>X^\bullet = [\mathcal{X}] = [x_1, x_2] = [u, v],\!</math> based on two logical features or boolean variables <math>{u}\!</math> and <math>{v}.\!</math>
The universe of discourse <math>X^\circ\!</math> consists of two parts, a set of ''points'' and a set of ''propositions''.
The universe of discourse <math>{X^\bullet}\!</math> consists of two parts, a set of ''points'' and a set of ''propositions''.
The points of <math>X^\circ</math> form the space:
The points of <math>{X^\bullet}\!</math> form the space:
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| <math>X \quad = \quad \langle \mathcal{X} \rangle \quad = \quad \langle u, v \rangle \quad = \quad \{ (u, v) \} \quad \cong \quad \mathbb{B}^2.</math>
| <math>{X \quad = \quad \langle \mathcal{X} \rangle \quad = \quad \langle u, v \rangle \quad = \quad \{ (u, v) \} \quad \cong \quad \mathbb{B}^2}.~\!</math>
|}
|}
Each point in <math>X\!</math> may be indicated by means of a ''singular proposition'', that is, a proposition that describes it uniquely. This form of representation leads to the following enumeration of points:
Each point in <math>{X}\!</math> may be indicated by means of a ''singular proposition'', that is, a proposition that describes it uniquely. This form of representation leads to the following enumeration of points:
{| align="center" cellpadding="8" style="text-align:center"
{| align="center" cellpadding="8" style="text-align:center"
| <math>X\!</math>
| <math>X\!</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>\{ ~ \texttt{(} u \texttt{)(} v \texttt{)} ~,~ \texttt{(} u \texttt{)} ~ v ~,~ u ~ \texttt{(} v \texttt{)} ~,~ u ~ v ~ \}</math>
| <math>\{ ~ \texttt{(} u \texttt{)(} v \texttt{)} ~,~ \texttt{(} u \texttt{)} ~ v ~,~ u ~ \texttt{(} v \texttt{)} ~,~ u ~ v ~ \}\!</math>
| <math>\cong\!</math>
| <math>\cong\!</math>
| <math>\mathbb{B}^2,</math>
| <math>\mathbb{B}^2,\!</math>
|}
|}
Each point in <math>X\!</math> may also be described by means of its ''coordinates'', that is, by the ordered pair of values in <math>\mathbb{B}</math> that the coordinate propositions <math>u\!</math> and <math>v\!</math> take on that point. This form of representation leads to the following enumeration of points:
Each point in <math>X\!</math> may also be described by means of its ''coordinates'', that is, by the ordered pair of values in <math>\mathbb{B}\!</math> that the coordinate propositions <math>u\!</math> and <math>v\!</math> take on that point. This form of representation leads to the following enumeration of points:
{| align="center" cellpadding="8" style="text-align:center"
{| align="center" cellpadding="8" style="text-align:center"
| <math>X\!</math>
| <math>X\!</math>
| <math>=\!</math>
| <math>=\!</math>
| <math>\{\ (0, 0),\ (0, 1),\ (1, 0),\ (1, 1)\ \}</math>
| <math>\{\ (0, 0),\ (0, 1),\ (1, 0),\ (1, 1)\ \}\!</math>
| <math>\cong\!</math>
| <math>\cong\!</math>
| <math>\mathbb{B}^2.</math>
| <math>\mathbb{B}^2.\!</math>
|}
|}
The propositions of <math>X^\circ</math> form the space:
The propositions of <math>{X^\bullet}\!</math> form the space:
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| <math>X^\uparrow \quad = \quad (X \to \mathbb{B}) \quad = \quad \{ f : X \to \mathbb{B} \} \quad \cong \quad (\mathbb{B}^2 \to \mathbb{B}).</math>
| <math>X^\uparrow \quad = \quad (X \to \mathbb{B}) \quad = \quad \{ f : X \to \mathbb{B} \} \quad \cong \quad (\mathbb{B}^2 \to \mathbb{B}).\!</math>
|}
|}
As always, it is frequently convenient to omit a few of the finer markings of distinctions among isomorphic structures, so long as one is aware of their presence and knows when it is crucial to call upon them again.
As always, it is frequently convenient to omit a few of the finer markings of distinctions among isomorphic structures, so long as one is aware of their presence and knows when it is crucial to call upon them again.
The next higher order universe of discourse that is built on <math>X^\circ</math> is <math>X^{\circ 2} = [X^\circ] = [[u, v]],</math> which may be developed in the following way. The propositions of <math>X^\circ</math> become the points of <math>X^{\circ 2},</math> and the mappings of the type <math>m : (X \to \mathbb{B}) \to \mathbb{B}</math> become the propositions of <math>X^{\circ 2}.</math> In addition, it is convenient to equip the discussion with a selected set of higher order operators on propositions, all of which have the form <math>w : (\mathbb{B}^2 \to \mathbb{B})^k \to \mathbb{B}.</math>
The next higher order universe of discourse that is built on <math>{X^\bullet}\!</math> is <math>X^{\bullet 2} = [X^\bullet] = [[u, v]],\!</math> which may be developed in the following way. The propositions of <math>{X^\bullet}\!</math> become the points of <math>X^{\bullet 2},\!</math> and the mappings of the type <math>m : (X \to \mathbb{B}) \to \mathbb{B}\!</math> become the propositions of <math>X^{\bullet 2}.\!</math> In addition, it is convenient to equip the discussion with a selected set of higher order operators on propositions, all of which have the form <math>w : (\mathbb{B}^2 \to \mathbb{B})^k \to \mathbb{B}.\!</math>
To save a few words in the remainder of this discussion, I will use the terms ''measure'' and ''qualifier'' to refer to all types of higher order propositions and operators. To describe the present setting in picturesque terms, the propositions of <math>[u, v]\!</math> may be regarded as a gallery of sixteen venn diagrams, while the measures <math>m : (X \to \mathbb{B}) \to \mathbb{B}</math> are analogous to a body of judges or a panel of critical viewers, each of whom evaluates each of the pictures as a whole and reports the ones that find favor or not. In this way, each judge <math>m_j\!</math> partitions the gallery of pictures into two aesthetic portions, the pictures <math>m_j^{-1}(1)\!</math> that <math>m_j\!</math> likes and the pictures <math>m_j^{-1}(0)\!</math> that <math>m_j\!</math> dislikes.
To save a few words in the remainder of this discussion, I will use the terms ''measure'' and ''qualifier'' to refer to all types of higher order propositions and operators. To describe the present setting in picturesque terms, the propositions of <math>[u, v]\!</math> may be regarded as a gallery of sixteen venn diagrams, while the measures <math>m : (X \to \mathbb{B}) \to \mathbb{B}\!</math> are analogous to a body of judges or a panel of critical viewers, each of whom evaluates each of the pictures as a whole and reports the ones that find favor or not. In this way, each judge <math>m_j\!</math> partitions the gallery of pictures into two aesthetic portions, the pictures <math>m_j^{-1}(1)\!</math> that <math>m_j\!</math> likes and the pictures <math>m_j^{-1}(0)\!</math> that <math>m_j\!</math> dislikes.
There are <math>2^{16} = 65536\!</math> measures of the type <math>m : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}.</math> Table 12 introduces the first 24 of these measures in the fashion of the higher order truth table that I used before. The column headed <math>m_j\!</math> shows the values of the measure <math>m_j\!</math> on each of the propositions <math>f_i : \mathbb{B}^2 \to \mathbb{B},</math> for <math>i\!</math> = 0 to 23, with blank entries in the Table being optional for values of zero. The arrangement of measures that continues according to the plan indicated here is referred to as the ''standard ordering'' of these measures. In this scheme of things, the index <math>j\!</math> of the measure <math>m_j\!</math> is the decimal equivalent of the bit string that is associated with <math>m_j\!</math>'s functional values, which can be obtained in turn by reading the <math>j^\mathrm{th}\!</math> column of binary digits in the Table as the corresponding range of boolean values, taking them up in the order from bottom to top.
There are <math>2^{16} = 65536\!</math> measures of the type <math>m : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}.\!</math> Table 13 introduces the first 24 of these measures in the fashion of the higher order truth table that I used before. The column headed <math>m_j\!</math> shows the values of the measure <math>m_j\!</math> on each of the propositions <math>f_i : \mathbb{B}^2 \to \mathbb{B},\!</math> for <math>i\!</math> = 0 to 23, with blank entries in the Table being optional for values of zero. The arrangement of measures that continues according to the plan indicated here is referred to as the ''standard ordering'' of these measures. In this scheme of things, the index <math>j\!</math> of the measure <math>m_j\!</math> is the decimal equivalent of the bit string that is associated with <math>m_j\!</math>'s functional values, which can be obtained in turn by reading the <math>j^\mathrm{th}\!</math> column of binary digits in the Table as the corresponding range of boolean values, taking them up in the order from bottom to top.
<br>
{| align="center" style="background:white; color:black; text-align:center; width:90%"
{| align="center" style="background:white; color:black; text-align:center; width:90%"
|+ '''Table 12. Higher Order Propositions (''n'' = 2)'''
|+ style="height:25px" |
<math>\text{Table 13. Higher Order Propositions} ~~ (n = 2)\!</math>
|- style="background:ghostwhite"
|- style="background:ghostwhite"
| align="right" | <math>\begin{matrix}u\!:\\v\!:\end{matrix}</math>
| align="right" | <math>\begin{matrix}u\!:\\v\!:\end{matrix}</math>
| <math>\begin{matrix}1100\\1010\end{matrix}</math>
| <math>\begin{matrix}1100\\1010\end{matrix}</math>
| <math>f\!</math>
| <math>f\!</math>
| <math>\underset{0}{m}</math>
| <math>{\underset{0}{m}}\!</math>
| <math>\underset{1}{m}</math>
| <math>{\underset{1}{m}}\!</math>
| <math>\underset{2}{m}</math>
| <math>{\underset{2}{m}}\!</math>
| <math>\underset{3}{m}</math>
| <math>{\underset{3}{m}}\!</math>
| <math>\underset{4}{m}</math>
| <math>{\underset{4}{m}}\!</math>
| <math>\underset{5}{m}</math>
| <math>{\underset{5}{m}}\!</math>
| <math>\underset{6}{m}</math>
| <math>{\underset{6}{m}}\!</math>
| <math>\underset{7}{m}</math>
| <math>{\underset{7}{m}}\!</math>
| <math>\underset{8}{m}</math>
| <math>{\underset{8}{m}}\!</math>
| <math>\underset{9}{m}</math>
| <math>{\underset{9}{m}}\!</math>
| <math>\underset{10}{m}</math>
| <math>{\underset{10}{m}}\!</math>
| <math>\underset{11}{m}</math>
| <math>{\underset{11}{m}}\!</math>
| <math>\underset{12}{m}</math>
| <math>{\underset{12}{m}}\!</math>
| <math>\underset{13}{m}</math>
| <math>{\underset{13}{m}}\!</math>
| <math>\underset{14}{m}</math>
| <math>{\underset{14}{m}}\!</math>
| <math>\underset{15}{m}</math>
| <math>{\underset{15}{m}}\!</math>
| <math>\underset{16}{m}</math>
| <math>{\underset{16}{m}}\!</math>
| <math>\underset{17}{m}</math>
| <math>{\underset{17}{m}}\!</math>
| <math>\underset{18}{m}</math>
| <math>{\underset{18}{m}}\!</math>
| <math>\underset{19}{m}</math>
| <math>{\underset{19}{m}}\!</math>
| <math>\underset{20}{m}</math>
| <math>{\underset{20}{m}}\!</math>
| <math>\underset{21}{m}</math>
| <math>{\underset{21}{m}}\!</math>
| <math>\underset{22}{m}</math>
| <math>{\underset{22}{m}}\!</math>
| <math>\underset{23}{m}</math>
| <math>{\underset{23}{m}}\!</math>
|-
|-
| <math>f_0</math>
| <math>f_0\!</math>
| <math>0000</math>
| <math>0000\!</math>
| <math>\texttt{(~)}</math>
| <math>\texttt{(~)}\!</math>
| 0 || style="background:black; color:white" | 1
| 0 || style="background:black; color:white" | 1
| 0 || style="background:black; color:white" | 1
| 0 || style="background:black; color:white" | 1
| 0 || style="background:black; color:white" | 1
| 0 || style="background:black; color:white" | 1
|-
|-
| <math>f_1</math>
| <math>f_1\!</math>
| <math>0001</math>
| <math>0001\!</math>
| <math>\texttt{(u)(v)}</math>
| <math>\texttt{(u)(v)}\!</math>
| 0 || 0
| 0 || 0
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|-
|-
| <math>f_2</math>
| <math>f_2\!</math>
| <math>0010</math>
| <math>{0010}\!</math>
| <math>\texttt{(u) v}</math>
| <math>\texttt{(u) v}\!</math>
| 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|-
|-
| <math>f_3</math>
| <math>f_3\!</math>
| <math>0011</math>
| <math>0011\!</math>
| <math>\texttt{(u)}</math>
| <math>\texttt{(u)}\!</math>
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| <math>f_4</math>
| <math>f_4\!</math>
| <math>0100</math>
| <math>0100\!</math>
| <math>\texttt{u (v)}</math>
| <math>\texttt{u (v)}\!</math>
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|-
|-
| <math>f_5</math>
| <math>f_5\!</math>
| <math>0101</math>
| <math>{0101}\!</math>
| <math>\texttt{(v)}</math>
| <math>\texttt{(v)}\!</math>
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| <math>f_6</math>
| <math>f_6\!</math>
| <math>0110</math>
| <math>0110\!</math>
| <math>\texttt{(u, v)}</math>
| <math>\texttt{(u, v)}\!</math>
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| <math>f_7</math>
| <math>f_7\!</math>
| <math>0111</math>
| <math>0111\!</math>
| <math>\texttt{(u v)}</math>
| <math>\texttt{(u v)}\!</math>
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| <math>f_8</math>
| <math>f_8\!</math>
| <math>1000</math>
| <math>1000\!</math>
| <math>\texttt{u v}</math>
| <math>\texttt{u v}\!</math>
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| <math>f_9</math>
| <math>f_9\!</math>
| <math>1001</math>
| <math>1001\!</math>
| <math>\texttt{((u, v))}</math>
| <math>\texttt{((u, v))}\!</math>
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| <math>f_{10}</math>
| <math>f_{10}\!</math>
| <math>1010</math>
| <math>1010\!</math>
| <math>\texttt{v}</math>
| <math>\texttt{v}\!</math>
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| <math>f_{11}</math>
| <math>f_{11}\!</math>
| <math>1011</math>
| <math>1011\!</math>
| <math>\texttt{(u (v))}</math>
| <math>\texttt{(u (v))}\!</math>
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| <math>f_{12}</math>
| <math>f_{12}\!</math>
| <math>1100</math>
| <math>1100\!</math>
| <math>\texttt{u}</math>
| <math>\texttt{u}\!</math>
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| <math>f_{13}</math>
| <math>f_{13}\!</math>
| <math>1101</math>
| <math>1101\!</math>
| <math>\texttt{((u) v)}</math>
| <math>\texttt{((u) v)}\!</math>
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| <math>f_{14}</math>
| <math>f_{14}\!</math>
| <math>1110</math>
| <math>1110\!</math>
| <math>\texttt{((u)(v))}</math>
| <math>\texttt{((u)(v))}\!</math>
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
|-
| <math>f_{15}</math>
| <math>f_{15}\!</math>
| <math>1111</math>
| <math>1111\!</math>
| <math>\texttt{((~))}</math>
| <math>\texttt{((~))}\!</math>
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
===Umpire Operators===
===Umpire Operators===
We now examine measures at the high end of the standard ordering. Instrumental to this purpose we define a couple of higher order operators, <math>\Upsilon_1 : (\langle u, v \rangle \to \mathbb{B}) \to \mathbb{B}</math> and <math>\Upsilon : (\langle u, v \rangle \to \mathbb{B})^2 \to \mathbb{B},</math> both symbolized by cursive upsilon characters and referred to as the absolute and relative ''umpire operators'', respectively. If either one of these operators is defined in terms of more primitive notions then the remaining operator can be defined in terms of the one first established.
We now examine measures at the high end of the standard ordering. Instrumental to this purpose we define a couple of higher order operators, <math>\Upsilon_1 : (\langle u, v \rangle \to \mathbb{B}) \to \mathbb{B}\!</math> and <math>\Upsilon : (\langle u, v \rangle \to \mathbb{B})^2 \to \mathbb{B},\!</math> both symbolized by cursive upsilon characters and referred to as the absolute and relative ''umpire operators'', respectively. If either one of these operators is defined in terms of more primitive notions then the remaining operator can be defined in terms of the one first established.
Given an ordered pair of propositions <math>e, f : \langle u, v \rangle \to \mathbb{B}</math> as arguments, the relative operator reports the value <math>1\!</math> if the first implies the second, otherwise <math>0.\!</math>
Given an ordered pair of propositions <math>e, f : \langle u, v \rangle \to \mathbb{B}\!</math> as arguments, the relative operator reports the value <math>1\!</math> if the first implies the second, otherwise <math>0.\!</math>
{| align="center" cellpadding="8" style="text-align:center"
{| align="center" cellpadding="8" style="text-align:center"
| <math>\Upsilon (e, f) = 1\!</math>
| <math>\Upsilon (e, f) = 1\!</math>
| <math>\operatorname{if~and~only~if}</math>
| <math>\operatorname{if~and~only~if}\!</math>
| <math>e \Rightarrow f.\!</math>
| <math>e \Rightarrow f.\!</math>
|}
|}
{| align="center" cellpadding="8" style="text-align:center"
{| align="center" cellpadding="8" style="text-align:center"
| <math>\Upsilon (e, f) = 1\!</math>
| <math>\Upsilon (e, f) = 1\!</math>
| <math>\iff</math>
| <math>\iff\!</math>
| <math>\texttt{(} e \texttt{(} f \texttt{))} = 1</math>
| <math>\texttt{(} e \texttt{(} f \texttt{))} = 1\!</math>
|}
|}
{| align="center" cellpadding="8" style="text-align:center"
{| align="center" cellpadding="8" style="text-align:center"
| <math>\Upsilon (e, f) = 1 \in \mathbb{B}</math>
| <math>\Upsilon (e, f) = 1 \in \mathbb{B}\!</math>
| <math>\iff</math>
| <math>\iff\!</math>
| <math>\texttt{(} e \texttt{(} f \texttt{))} = 1 : \langle u, v \rangle \to \mathbb{B}</math>
| <math>\texttt{(} e \texttt{(} f \texttt{))} = 1 : \langle u, v \rangle \to \mathbb{B}\!</math>
|}
|}
Writing types as subscripts and using the fact that <math>X = \langle u, v \rangle,</math> it is possible to express this a little more succinctly as follows:
Writing types as subscripts and using the fact that <math>X = \langle u, v \rangle,\!</math> it is possible to express this a little more succinctly as follows:
{| align="center" cellpadding="8" style="text-align:center"
{| align="center" cellpadding="8" style="text-align:center"
| <math>\Upsilon (e, f) = 1_\mathbb{B}</math>
| <math>\Upsilon (e, f) = 1_\mathbb{B}\!</math>
| <math>\iff</math>
| <math>\iff\!</math>
| <math>\texttt{(} e \texttt{(} f \texttt{))} = 1_{X \to \mathbb{B}}</math>
| <math>\texttt{(} e \texttt{(} f \texttt{))} = 1_{X \to \mathbb{B}}\!</math>
|}
|}
Finally, it is often convenient to write the first argument as a subscript, hence <math>\Upsilon_e (f) = \Upsilon (e, f).\!</math>
Finally, it is often convenient to write the first argument as a subscript, hence <math>\Upsilon_e (f) = \Upsilon (e, f).\!</math>
As a special application of this operator, we next define the absolute umpire operator, also called the ''umpire measure''. This is a higher order proposition <math>\Upsilon_1 : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}</math> which is given by the relation <math>\Upsilon_1 (f) = \Upsilon (1, f).\!</math> Here, the subscript <math>1\!</math> on the left and the argument <math>1\!</math> on the right both refer to the constant proposition <math>1 : \mathbb{B}^2 \to \mathbb{B}.</math> In most contexts where <math>\Upsilon_1\!</math> is actually applied the subscript <math>1\!</math> is safely omitted, since the number of arguments indicates which type of operator is intended. Thus, we have the following identities and equivalents:
As a special application of this operator, we next define the absolute umpire operator, also called the ''umpire measure''. This is a higher order proposition <math>\Upsilon_1 : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}\!</math> which is given by the relation <math>\Upsilon_1 (f) = \Upsilon (1, f).\!</math> Here, the subscript <math>1\!</math> on the left and the argument <math>1\!</math> on the right both refer to the constant proposition <math>1 : \mathbb{B}^2 \to \mathbb{B}.\!</math> In most contexts where <math>\Upsilon_1\!</math> is actually applied the subscript <math>1\!</math> is safely omitted, since the number of arguments indicates which type of operator is intended. Thus, we have the following identities and equivalents:
{| align="center" cellpadding="10" style="text-align:center"
{| align="center" cellpadding="10" style="text-align:center"
| <math>\Upsilon f = \Upsilon_1 (f) = 1 \in \mathbb{B}</math>
| <math>\Upsilon f = \Upsilon_1 (f) = 1 \in \mathbb{B}\!</math>
| <math>\iff</math>
| <math>\iff\!</math>
| <math>\texttt{(} 1 ~ \texttt{(} f \texttt{))} = 1</math>
| <math>\texttt{(} 1 ~ \texttt{(} f \texttt{))} = 1\!</math>
| <math>\iff</math>
| <math>\iff\!</math>
| <math>f = 1 : \mathbb{B}^2 \to \mathbb{B}</math>
| <math>f = 1 : \mathbb{B}^2 \to \mathbb{B}\!</math>
|}
|}
The umpire measure is defined at the level of truth functions, but can also be understood in terms of its implied judgments at the syntactic level. Interpreted this way, <math>\Upsilon_1\!</math> recognizes theorems of the propositional calculus over <math>[u, v],\!</math> giving a score of <math>1\!</math> to tautologies and a score of <math>0\!</math> to everything else, regarding all contingent statements as no better than falsehoods.
The umpire measure is defined at the level of truth functions, but can also be understood in terms of its implied judgments at the syntactic level. Interpreted this way, <math>\Upsilon_1\!</math> recognizes theorems of the propositional calculus over <math>[u, v],\!</math> giving a score of <math>1\!</math> to tautologies and a score of <math>0\!</math> to everything else, regarding all contingent statements as no better than falsehoods.
One remark in passing for those who might prefer an alternative definition. If we had originally taken <math>\Upsilon\!</math> to mean the absolute measure, then the relative version could have been defined as <math>\Upsilon_e f = \Upsilon \texttt{(} e \texttt{(} f \texttt{))}.</math>
One remark in passing for those who might prefer an alternative definition. If we had originally taken <math>\Upsilon\!</math> to mean the absolute measure, then the relative version could have been defined as <math>\Upsilon_e f = \Upsilon \texttt{(} e \texttt{(} f \texttt{))}.~\!</math>
===Measure for Measure===
===Measure for Measure===
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| <math>\alpha_i, \beta_i : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}, i = 0 \ldots 15,</math>
| <math>\alpha_i, \beta_i : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}, i = 0 \ldots 15,\!</math>
|}
|}
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| <math>\alpha_i f = \Upsilon (f_i, f) = \Upsilon (f_i \Rightarrow f),</math>
| <math>\alpha_i f = \Upsilon (f_i, f) = \Upsilon (f_i \Rightarrow f),\!</math>
|-
|-
| <math>\beta_i f = \Upsilon (f, f_i) = \Upsilon (f \Rightarrow f_i).</math>
| <math>\beta_i f = \Upsilon (f, f_i) = \Upsilon (f \Rightarrow f_i).\!</math>
|}
|}
The values of the sixteen <math>\alpha_i\!</math> on each of the sixteen boolean functions <math>f : \mathbb{B}^2 \to \mathbb{B}</math> are shown in Table 13. Expressed in terms of the implication ordering on the sixteen functions, <math>\alpha_i f = 1\!</math> says that <math>f\!</math> is ''above or identical to'' <math>f_i\!</math> in the implication lattice, that is, <math>\ge f_i\!</math> in the implication ordering.
<br>
The values of the sixteen <math>\alpha_i\!</math> on each of the sixteen boolean functions <math>f : \mathbb{B}^2 \to \mathbb{B}\!</math> are shown in Table 14. Expressed in terms of the implication ordering on the sixteen functions, <math>\alpha_i f = 1\!</math> says that <math>f\!</math> is ''above or identical to'' <math>f_i\!</math> in the implication lattice, that is, <math>\ge f_i\!</math> in the implication ordering.
<br>
{| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
{| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|+ '''Table 13. Qualifiers of Implication Ordering: <math>\alpha_i f = \Upsilon (f_i, f) = \Upsilon (f_i \Rightarrow f)</math>'''
|+ style="height:25px" |
<math>\text{Table 14. Qualifiers of Implication Ordering:} ~~ \alpha_i f = \Upsilon (f_i, f) = \Upsilon (f_i \Rightarrow f)\!</math>
|- style="background:ghostwhite"
|- style="background:ghostwhite"
| align="right" | <math>u:</math><br><math>v:</math>
| align="right" | <math>u:\!</math><br><math>v:\!</math>
| 1100<br>1010
| 1100<br>1010
| <math>f\!</math>
| <math>f\!</math>
| <math>\alpha_0</math>
| <math>\alpha_0~\!</math>
| <math>\alpha_1</math>
| <math>\alpha_1\!</math>
| <math>\alpha_2</math>
| <math>\alpha_2\!</math>
| <math>\alpha_3</math>
| <math>\alpha_3\!</math>
| <math>\alpha_4</math>
| <math>\alpha_4\!</math>
| <math>\alpha_5</math>
| <math>\alpha_5\!</math>
| <math>\alpha_6</math>
| <math>\alpha_6\!</math>
| <math>\alpha_7</math>
| <math>\alpha_7\!</math>
| <math>\alpha_8</math>
| <math>\alpha_8\!</math>
| <math>\alpha_9</math>
| <math>\alpha_9\!</math>
| <math>\alpha_{10}</math>
| <math>\alpha_{10}\!</math>
| <math>\alpha_{11}</math>
| <math>\alpha_{11}\!</math>
| <math>\alpha_{12}</math>
| <math>\alpha_{12}\!</math>
| <math>\alpha_{13}</math>
| <math>\alpha_{13}~\!</math>
| <math>\alpha_{14}</math>
| <math>\alpha_{14}\!</math>
| <math>\alpha_{15}</math>
| <math>\alpha_{15}\!</math>
|-
|-
| <math>f_0</math>
| <math>f_0\!</math>
| 0000
| 0000
| <math>\texttt{(~)}</math>
| <math>\texttt{(~)}\!</math>
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
|-
|-
| <math>f_1</math>
| <math>f_1\!</math>
| 0001
| 0001
| <math>\texttt{(} u \texttt{)(} v \texttt{)}</math>
| <math>\texttt{(} u \texttt{)(} v \texttt{)}\!</math>
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
|-
|-
| <math>f_2</math>
| <math>f_2\!</math>
| 0010
| 0010
| <math>\texttt{(} u \texttt{)} ~ v</math>
| <math>\texttt{(} u \texttt{)} ~ v\!</math>
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
|-
|-
| <math>f_3</math>
| <math>f_3\!</math>
| 0011
| 0011
| <math>\texttt{(} u \texttt{)}</math>
| <math>\texttt{(} u \texttt{)}~\!</math>
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
|-
|-
| <math>f_4</math>
| <math>f_4\!</math>
| 0100
| 0100
| <math>u ~ \texttt{(} v \texttt{)}</math>
| <math>u ~ \texttt{(} v \texttt{)}\!</math>
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
|-
|-
| <math>f_5</math>
| <math>f_5\!</math>
| 0101
| 0101
| <math>\texttt{(} v \texttt{)}</math>
| <math>\texttt{(} v \texttt{)}\!</math>
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
|-
|-
| <math>f_6</math>
| <math>f_6\!</math>
| 0110
| 0110
| <math>\texttt{(} u ~ \texttt{,} ~ v \texttt{)}</math>
| <math>\texttt{(} u ~ \texttt{,} ~ v \texttt{)}\!</math>
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
|-
|-
| <math>f_7</math>
| <math>f_7\!</math>
| 0111
| 0111
| <math>\texttt{(} u ~ v \texttt{)}</math>
| <math>\texttt{(} u ~ v \texttt{)}\!</math>
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
|-
|-
| <math>f_8</math>
| <math>f_8\!</math>
| 1000
| 1000
| <math>u ~ v</math>
| <math>u ~ v\!</math>
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
|-
|-
| <math>f_9</math>
| <math>f_9\!</math>
| 1001
| 1001
| <math>\texttt{((} u ~ \texttt{,} ~ v \texttt{))}</math>
| <math>\texttt{((} u ~ \texttt{,} ~ v \texttt{))}\!</math>
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
|-
|-
| <math>f_{10}</math>
| <math>f_{10}\!</math>
| 1010
| 1010
| <math>v\!</math>
| <math>v\!</math>
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
|-
|-
| <math>f_{11}</math>
| <math>f_{11}\!</math>
| 1011
| 1011
| <math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math>
| <math>\texttt{(} u ~ \texttt{(} v \texttt{))}\!</math>
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
|-
|-
| <math>f_{12}</math>
| <math>f_{12}\!</math>
| 1100
| 1100
| <math>u\!</math>
| <math>u\!</math>
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
|-
|-
| <math>f_{13}</math>
| <math>f_{13}\!</math>
| 1101
| 1101
| <math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math>
| <math>\texttt{((} u \texttt{)} ~ v \texttt{)}\!</math>
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
|-
|-
| <math>f_{14}</math>
| <math>f_{14}\!</math>
| 1110
| 1110
| <math>\texttt{((} u \texttt{)(} v \texttt{))}</math>
| <math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math>
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
|-
|-
| <math>f_{15}</math>
| <math>f_{15}\!</math>
| 1111
| 1111
| <math>\texttt{((~))}</math>
| <math>\texttt{((~))}\!</math>
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|}<br>
|}
<br>
The values of the sixteen <math>{\beta_i}\!</math> on each of the sixteen boolean functions <math>{f : \mathbb{B}^2 \to \mathbb{B}}\!</math> are shown in Table 15. Expressed in terms of the implication ordering on the sixteen functions, <math>{\beta_i f = 1}\!</math> says that <math>{f}\!</math> is ''below or identical to'' <math>{f_i}\!</math> in the implication lattice, that is, <math>{\le f_i}\!</math> in the implication ordering.
<br>
{| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
{| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|+ '''Table 14. Qualifiers of Implication Ordering: <math>\beta_i f = \Upsilon (f, f_i) = \Upsilon (f \Rightarrow f_i)</math>'''
|+ style="height:25px" |
<math>\text{Table 15. Qualifiers of Implication Ordering:} ~~ \beta_i f = \Upsilon (f, f_i) = \Upsilon (f \Rightarrow f_i)\!</math>
|- style="background:ghostwhite"
|- style="background:ghostwhite"
| align="right" | <math>u:</math><br><math>v:</math>
| align="right" | <math>u:\!</math><br><math>v:\!</math>
| 1100<br>1010
| 1100<br>1010
| <math>f\!</math>
| <math>f\!</math>
| <math>\beta_0</math>
| <math>\beta_0\!</math>
| <math>\beta_1</math>
| <math>\beta_1\!</math>
| <math>\beta_2</math>
| <math>\beta_2\!</math>
| <math>\beta_3</math>
| <math>\beta_3\!</math>
| <math>\beta_4</math>
| <math>\beta_4\!</math>
| <math>\beta_5</math>
| <math>\beta_5\!</math>
| <math>\beta_6</math>
| <math>\beta_6\!</math>
| <math>\beta_7</math>
| <math>\beta_7\!</math>
| <math>\beta_8</math>
| <math>\beta_8\!</math>
| <math>\beta_9</math>
| <math>\beta_9\!</math>
| <math>\beta_{10}</math>
| <math>\beta_{10}\!</math>
| <math>\beta_{11}</math>
| <math>\beta_{11}\!</math>
| <math>\beta_{12}</math>
| <math>\beta_{12}\!</math>
| <math>\beta_{13}</math>
| <math>\beta_{13}\!</math>
| <math>\beta_{14}</math>
| <math>\beta_{14}\!</math>
| <math>\beta_{15}</math>
| <math>\beta_{15}\!</math>
|-
|-
| <math>f_0</math>
| <math>f_0\!</math>
| 0000
| 0000
| <math>\texttt{(~)}</math>
| <math>\texttt{(~)}\!</math>
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|-
|-
| <math>f_1</math>
| <math>f_1\!</math>
| 0001
| 0001
| <math>\texttt{(} u \texttt{)(} v \texttt{)}</math>
| <math>\texttt{(} u \texttt{)(} v \texttt{)}\!</math>
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|-
|-
| <math>f_2</math>
| <math>f_2\!</math>
| 0010
| 0010
| <math>\texttt{(} u \texttt{)} ~ v</math>
| <math>\texttt{(} u \texttt{)} ~ v\!</math>
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|-
|-
| <math>f_3</math>
| <math>f_3\!</math>
| 0011
| 0011
| <math>\texttt{(} u \texttt{)}</math>
| <math>\texttt{(} u \texttt{)}~\!</math>
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|-
|-
| <math>f_4</math>
| <math>f_4\!</math>
| 0100
| 0100
| <math>u ~ \texttt{(} v \texttt{)}</math>
| <math>u ~ \texttt{(} v \texttt{)}\!</math>
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|-
|-
| <math>f_5</math>
| <math>f_5\!</math>
| 0101
| 0101
| <math>\texttt{(} v \texttt{)}</math>
| <math>\texttt{(} v \texttt{)}\!</math>
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|-
|-
| <math>f_6</math>
| <math>f_6\!</math>
| 0110
| 0110
| <math>\texttt{(} u ~ \texttt{,} ~ v \texttt{)}</math>
| <math>\texttt{(} u ~ \texttt{,} ~ v \texttt{)}\!</math>
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|-
|-
| <math>f_7</math>
| <math>f_7\!</math>
| 0111
| 0111
| <math>\texttt{(} u ~ v \texttt{)}</math>
| <math>\texttt{(} u ~ v \texttt{)}\!</math>
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|-
|-
| <math>f_8</math>
| <math>f_8\!</math>
| 1000
| 1000
| <math>u ~ v</math>
| <math>u ~ v\!</math>
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|-
|-
| <math>f_9</math>
| <math>f_9\!</math>
| 1001
| 1001
| <math>\texttt{((} u ~ \texttt{,} ~ v \texttt{))}</math>
| <math>\texttt{((} u ~ \texttt{,} ~ v \texttt{))}\!</math>
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|-
|-
| <math>f_{10}</math>
| <math>f_{10}\!</math>
| 1010
| 1010
| <math>v\!</math>
| <math>v\!</math>
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|-
|-
| <math>f_{11}</math>
| <math>f_{11}\!</math>
| 1011
| 1011
| <math>\texttt{(} u ~ \texttt{(} v \texttt{))}</math>
| <math>\texttt{(} u ~ \texttt{(} v \texttt{))}\!</math>
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|-
|-
| <math>f_{12}</math>
| <math>f_{12}\!</math>
| 1100
| 1100
| <math>u\!</math>
| <math>u\!</math>
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|-
|-
| <math>f_{13}</math>
| <math>f_{13}\!</math>
| 1101
| 1101
| <math>\texttt{((} u \texttt{)} ~ v \texttt{)}</math>
| <math>\texttt{((} u \texttt{)} ~ v \texttt{)}\!</math>
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|-
|-
| <math>f_{14}</math>
| <math>f_{14}\!</math>
| 1110
| 1110
| <math>\texttt{((} u \texttt{)(} v \texttt{))}</math>
| <math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math>
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|-
|-
| <math>f_{15}</math>
| <math>f_{15}\!</math>
| 1111
| 1111
| <math>\texttt{((~))}</math>
| <math>\texttt{((~))}\!</math>
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|}<br>
|}
<br>
Applied to a given proposition <math>f,\!</math> the qualifiers <math>\alpha_i\!</math> and <math>\beta_i\!</math> tell whether <math>f\!</math> rests <math>\operatorname{above}\ f_i</math> or <math>\operatorname{below}\ f_i,</math> respectively, in the implication ordering. By way of example, let us trace the effects of several such measures, namely, those that occupy the limiting positions of the Tables.
Applied to a given proposition <math>f,\!</math> the qualifiers <math>\alpha_i\!</math> and <math>\beta_i\!</math> tell whether <math>f\!</math> rests <math>\operatorname{above}\ f_i\!</math> or <math>\operatorname{below}\ f_i,\!</math> respectively, in the implication ordering. By way of example, let us trace the effects of several such measures, namely, those that occupy the limiting positions of the Tables.
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
|}
|}
Thus, <math>\alpha_0 = \beta_{15}\!</math> is a totally indiscriminate measure, one that accepts all propositions <math>f : \mathbb{B}^2 \to \mathbb{B},</math> whereas <math>\alpha_{15}\!</math> and <math>\beta_0\!</math> are measures that value the constant propositions <math>1 : \mathbb{B}^2 \to \mathbb{B}</math> and <math>0 : \mathbb{B}^2 \to \mathbb{B},</math> respectively, above all others.
Thus, <math>\alpha_0 = \beta_{15}\!</math> is a totally indiscriminate measure, one that accepts all propositions <math>{f : \mathbb{B}^2 \to \mathbb{B}},\!</math> whereas <math>\alpha_{15}\!</math> and <math>\beta_0\!</math> are measures that value the constant propositions <math>1 : \mathbb{B}^2 \to \mathbb{B}\!</math> and <math>0 : \mathbb{B}^2 \to \mathbb{B},\!</math> respectively, above all others.
Finally, in conformity with the use of the fiber notation to indicate sets of models, it is natural to use notations like:
Finally, in conformity with the use of the fiber notation to indicate sets of models, it is natural to use notations like:
Previously I introduced a calculus for propositional logic, fixing its meaning according to what C.S. Peirce called the ''existential interpretation''. As far as it concerns propositional calculus this interpretation settles the meanings that are associated with merely the most basic symbols and logical connectives. Now we must extend and refine the existential interpretation to comprehend the analysis of ''quantifications'', that is, quantified propositions. In doing so we recognize two additional aspects of logic that need to be developed, over and above the material of propositional logic. At the formal extreme there is the aspect of higher order functional types, into which we have already ventured a little above. At the level of the fundamental content of the available propositions we have to introduce a different interpretation for what we may call ''elemental'' or ''singular'' propositions.
Previously I introduced a calculus for propositional logic, fixing its meaning according to what C.S. Peirce called the ''existential interpretation''. As far as it concerns propositional calculus this interpretation settles the meanings that are associated with merely the most basic symbols and logical connectives. Now we must extend and refine the existential interpretation to comprehend the analysis of ''quantifications'', that is, quantified propositions. In doing so we recognize two additional aspects of logic that need to be developed, over and above the material of propositional logic. At the formal extreme there is the aspect of higher order functional types, into which we have already ventured a little above. At the level of the fundamental content of the available propositions we have to introduce a different interpretation for what we may call ''elemental'' or ''singular'' propositions.
Let us return to the 2-dimensional case <math>X^\circ = [u, v].</math> In order to provide a bridge between propositions and quantifications it serves to define a set of qualifiers <math>\ell_{ij} : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}</math> that have the following characters:
Let us return to the 2-dimensional case <math>X^\bullet = [u, v].\!</math> In order to provide a bridge between propositions and quantifications it serves to define a set of qualifiers <math>\ell_{ij} : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}\!</math> that have the following characters:
<center><math>\begin{array}{llllll}
<center><math>\begin{array}{llllll}
\end{array}</math></center>
\end{array}</math></center>
Intuitively, the <math>\ell_{ij}\!</math> operators may be thought of as qualifying propositions according to the elements of the universe of discourse that each proposition positively values. Taken together, these measures provide us with the means to express many useful observations about the propositions in <math>X^\circ = [u, v],</math> and so they mediate a subtext <math>[\ell_{00}, \ell_{01}, \ell_{10}, \ell_{11}]\!</math> that takes place within the higher order universe of discourse <math>X^{\circ 2} = [X^\circ] = [[u, v]].\!</math> Figure 15 summarizes the action of the <math>\ell_{ij}\!</math> operators on the <math>f_i\!</math> within <math>X^{\circ 2}.\!</math>
Intuitively, the <math>\ell_{ij}\!</math> operators may be thought of as qualifying propositions according to the elements of the universe of discourse that each proposition positively values. Taken together, these measures provide us with the means to express many useful observations about the propositions in <math>X^\bullet = [u, v],\!</math> and so they mediate a subtext <math>[\ell_{00}, \ell_{01}, \ell_{10}, \ell_{11}]\!</math> that takes place within the higher order universe of discourse <math>X^{\bullet 2} = [X^\bullet] = [[u, v]].\!</math> Figure 16 summarizes the action of the <math>\ell_{ij}\!</math> operators on the <math>f_i\!</math> within <math>X^{\bullet 2}.\!</math>
{| align="center" cellpadding="10" style="text-align:center"
{| align="center" cellpadding="10" style="text-align:center"
| [[Image:Venn Diagram 4 Dimensions UV Cacti 8 Inch.png]]
| [[Image:Venn Diagram 4 Dimensions UV Cacti 8 Inch.jpg]]
|-
|-
| <math>\text{Figure 6.} ~~ \text{Higher Order Universe of Discourse} ~ \left[ \ell_{00}, \ell_{01}, \ell_{10}, \ell_{11} \right] \subseteq \left[\left[ u, v \right]\right]</math>
| <math>\text{Figure 16.} ~~ \text{Higher Order Universe of Discourse} ~ \left[ \ell_{00}, \ell_{01}, \ell_{10}, \ell_{11} \right] \subseteq \left[\left[ u, v \right]\right]\!</math>
|}
|}
Our excursion into the vastening landscape of higher order propositions has finally come round to the stage where we can bring its returns to bear on opening up new perspectives for quantificational logic.
Our excursion into the vastening landscape of higher order propositions has finally come round to the stage where we can bring its returns to bear on opening up new perspectives for quantificational logic.
There is a question arising next that is still experimental in my mind. Whether it makes much difference from a purely formal point of view is not a question I can answer yet, but it does seem to aid the intuition to invent a slightly different interpretation for the two-valued space that we use as the target of our basic indicator functions. Therefore, let us declare a type of ''existential-valued'' functions <math>f : \mathbb{B}^k \to \mathbb{E},</math> where <math>\mathbb{E} = \{ -e, +e \} = \{ \operatorname{empty}, \operatorname{exist} \}</math> is a couple of values that we interpret as indicating whether of not anything exists in the cells of the underlying universe of discourse, venn diagram, or other domain. As usual, let us not be too strict about the coding of these functions, reverting to binary codes whenever the interpretation is clear enough.
There is a question arising next that is still experimental in my mind. Whether it makes much difference from a purely formal point of view is not a question I can answer yet, but it does seem to aid the intuition to invent a slightly different interpretation for the two-valued space that we use as the target of our basic indicator functions. Therefore, let us declare a type of ''existential-valued'' functions <math>f : \mathbb{B}^k \to \mathbb{E},\!</math> where <math>\mathbb{E} = \{ -e, +e \} = \{ \operatorname{empty}, \operatorname{exist} \}\!</math> is a couple of values that we interpret as indicating whether of not anything exists in the cells of the underlying universe of discourse, venn diagram, or other domain. As usual, let us not be too strict about the coding of these functions, reverting to binary codes whenever the interpretation is clear enough.
With this interpretation in mind we note the following correspondences between classical quantifications and higher order indicator functions:
With this interpretation in mind we note the following correspondences between classical quantifications and higher order indicator functions:
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|+ '''Table 16. Syllogistic Premisses as Higher Order Indicator Functions'''
|+ style="height:25px" |
<math>\text{Table 17. Syllogistic Premisses as Higher Order Indicator Functions}\!</math>
|
|
<math>\begin{array}{clcl}
<math>\begin{array}{clcl}
\mathrm{Indicator~of}\ u (v) = 1 \\
\mathrm{Indicator~of}\ u (v) = 1 \\
\end{array}</math>
\end{array}</math>
|}<br>
|}
<br>
The following Tables develop these ideas in more detail.
The following Tables develop these ideas in more detail.
<br>
{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|+ '''Table 17. Simple Qualifiers of Propositions (Version 1)'''
|+ style="height:25px" |
<math>\text{Table 18. Simple Qualifiers of Propositions (Version 1)}\!</math>
|- style="background:ghostwhite"
|- style="background:ghostwhite"
| align="right" | <math>u:</math><br><math>v:</math>
| align="right" | <math>u:\!</math><br><math>v:\!</math>
| 1100<br>1010
| 1100<br>1010
| <math>f\!</math>
| <math>f\!</math>
| <math>(\ell_{11})</math><br><math>\text{No } u </math><br><math>\text{is } v </math>
| <math>(\ell_{11})\!</math><br><math>\text{No } u \!</math><br><math>\text{is } v \!</math>
| <math>(\ell_{10})</math><br><math>\text{No } u </math><br><math>\text{is }(v)</math>
| <math>(\ell_{10})\!</math><br><math>\text{No } u \!</math><br><math>\text{is }(v)\!</math>
| <math>(\ell_{01})</math><br><math>\text{No }(u)</math><br><math>\text{is } v </math>
| <math>(\ell_{01})\!</math><br><math>\text{No }(u)\!</math><br><math>\text{is } v \!</math>
| <math>(\ell_{00})</math><br><math>\text{No }(u)</math><br><math>\text{is }(v)</math>
| <math>(\ell_{00})\!</math><br><math>\text{No }(u)\!</math><br><math>\text{is }(v)\!</math>
| <math> \ell_{00} </math><br><math>\text{Some }(u)</math><br><math>\text{is }(v)</math>
| <math> \ell_{00} \!</math><br><math>\text{Some }(u)\!</math><br><math>\text{is }(v)\!</math>
| <math> \ell_{01} </math><br><math>\text{Some }(u)</math><br><math>\text{is } v </math>
| <math> \ell_{01} \!</math><br><math>\text{Some }(u)\!</math><br><math>\text{is } v \!</math>
| <math> \ell_{10} </math><br><math>\text{Some } u </math><br><math>\text{is }(v)</math>
| <math> \ell_{10}~\!</math><br><math>\text{Some } u \!</math><br><math>\text{is }(v)\!</math>
| <math> \ell_{11} </math><br><math>\text{Some } u </math><br><math>\text{is } v </math>
| <math> \ell_{11} \!</math><br><math>\text{Some } u \!</math><br><math>\text{is } v \!</math>
|-
|-
| <math>f_0</math>
| <math>f_0\!</math>
| 0000
| 0000
| <math>(~)</math>
| <math>(~)\!</math>
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
|-
|-
| <math>f_1</math>
| <math>f_1\!</math>
| 0001
| 0001
| <math>(u)(v)\!</math>
| <math>(u)(v)\!</math>
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
|-
|-
| <math>f_2</math>
| <math>f_2\!</math>
| 0010
| 0010
| <math>(u) v\!</math>
| <math>(u) v\!</math>
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
|-
|-
| <math>f_3</math>
| <math>f_3\!</math>
| 0011
| 0011
| <math>(u)\!</math>
| <math>(u)~\!</math>
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
|-
|-
| <math>f_4</math>
| <math>f_4\!</math>
| 0100
| 0100
| <math>u (v)\!</math>
| <math>u (v)\!</math>
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
|-
|-
| <math>f_5</math>
| <math>f_5\!</math>
| 0101
| 0101
| <math>(v)\!</math>
| <math>(v)\!</math>
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
|-
|-
| <math>f_6</math>
| <math>f_6\!</math>
| 0110
| 0110
| <math>(u, v)\!</math>
| <math>(u, v)\!</math>
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
|-
|-
| <math>f_7</math>
| <math>f_7\!</math>
| 0111
| 0111
| <math>(u v)\!</math>
| <math>(u v)\!</math>
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
|-
|-
| <math>f_8</math>
| <math>f_8\!</math>
| 1000
| 1000
| <math>u v\!</math>
| <math>u v\!</math>
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|-
|-
| <math>f_9</math>
| <math>f_9\!</math>
| 1001
| 1001
| <math>((u, v))\!</math>
| <math>((u, v))\!</math>
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|-
|-
| <math>f_{10}</math>
| <math>f_{10}\!</math>
| 1010
| 1010
| <math>v\!</math>
| <math>v\!</math>
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|-
|-
| <math>f_{11}</math>
| <math>f_{11}\!</math>
| 1011
| 1011
| <math>(u (v))\!</math>
| <math>(u (v))\!</math>
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|-
|-
| <math>f_{12}</math>
| <math>f_{12}\!</math>
| 1100
| 1100
| <math>u\!</math>
| <math>u\!</math>
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|-
|-
| <math>f_{13}</math>
| <math>f_{13}\!</math>
| 1101
| 1101
| <math>((u) v)\!</math>
| <math>((u) v)\!</math>
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|-
|-
| <math>f_{14}</math>
| <math>f_{14}\!</math>
| 1110
| 1110
| <math>((u)(v))\!</math>
| <math>((u)(v))~\!</math>
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|-
|-
| <math>f_{15}</math>
| <math>f_{15}\!</math>
| 1111
| 1111
| <math>((~))</math>
| <math>((~))\!</math>
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|}<br>
|}
<br>
{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|+ '''Table 18. Simple Qualifiers of Propositions (Version 2)'''
|+ style="height:25px" |
<math>\text{Table 19. Simple Qualifiers of Propositions (Version 2)}\!</math>
|- style="background:ghostwhite"
|- style="background:ghostwhite"
| align="right" | <math>u:</math><br><math>v:</math>
| align="right" | <math>u:\!</math><br><math>v:\!</math>
| 1100<br>1010
| 1100<br>1010
| <math>f\!</math>
| <math>f\!</math>
| <math>(\ell_{11})</math><br><math>\text{No } u </math><br><math>\text{is } v </math>
| <math>(\ell_{11})\!</math><br><math>\text{No } u \!</math><br><math>\text{is } v \!</math>
| <math>(\ell_{10})</math><br><math>\text{No } u </math><br><math>\text{is }(v)</math>
| <math>(\ell_{10})\!</math><br><math>\text{No } u \!</math><br><math>\text{is }(v)\!</math>
| <math>(\ell_{01})</math><br><math>\text{No }(u)</math><br><math>\text{is } v </math>
| <math>(\ell_{01})\!</math><br><math>\text{No }(u)\!</math><br><math>\text{is } v \!</math>
| <math>(\ell_{00})</math><br><math>\text{No }(u)</math><br><math>\text{is }(v)</math>
| <math>(\ell_{00})\!</math><br><math>\text{No }(u)\!</math><br><math>\text{is }(v)\!</math>
| <math> \ell_{00} </math><br><math>\text{Some }(u)</math><br><math>\text{is }(v)</math>
| <math> \ell_{00} \!</math><br><math>\text{Some }(u)\!</math><br><math>\text{is }(v)\!</math>
| <math> \ell_{01} </math><br><math>\text{Some }(u)</math><br><math>\text{is } v </math>
| <math> \ell_{01} \!</math><br><math>\text{Some }(u)\!</math><br><math>\text{is } v \!</math>
| <math> \ell_{10} </math><br><math>\text{Some } u </math><br><math>\text{is }(v)</math>
| <math> \ell_{10}~\!</math><br><math>\text{Some } u \!</math><br><math>\text{is }(v)\!</math>
| <math> \ell_{11} </math><br><math>\text{Some } u </math><br><math>\text{is } v </math>
| <math> \ell_{11} \!</math><br><math>\text{Some } u \!</math><br><math>\text{is } v \!</math>
|-
|-
| <math>f_0</math>
| <math>f_0\!</math>
| 0000
| 0000
| <math>(~)</math>
| <math>(~)\!</math>
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
|-
|-
| <math>f_1</math>
| <math>f_1\!</math>
| 0001
| 0001
| <math>(u)(v)\!</math>
| <math>(u)(v)\!</math>
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
|-
|-
| <math>f_2</math>
| <math>f_2\!</math>
| 0010
| 0010
| <math>(u) v\!</math>
| <math>(u) v\!</math>
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
|-
|-
| <math>f_4</math>
| <math>f_4\!</math>
| 0100
| 0100
| <math>u (v)\!</math>
| <math>u (v)\!</math>
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
|-
|-
| <math>f_8</math>
| <math>f_8\!</math>
| 1000
| 1000
| <math>u v\!</math>
| <math>u v\!</math>
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|-
|-
| <math>f_3</math>
| <math>f_3\!</math>
| 0011
| 0011
| <math>(u)\!</math>
| <math>(u)~\!</math>
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
|-
|-
| <math>f_{12}</math>
| <math>f_{12}\!</math>
| 1100
| 1100
| <math>u\!</math>
| <math>u\!</math>
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|-
|-
| <math>f_6</math>
| <math>f_6\!</math>
| 0110
| 0110
| <math>(u, v)\!</math>
| <math>(u, v)\!</math>
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
|-
|-
| <math>f_9</math>
| <math>f_9\!</math>
| 1001
| 1001
| <math>((u, v))\!</math>
| <math>((u, v))\!</math>
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|-
|-
| <math>f_5</math>
| <math>f_5\!</math>
| 0101
| 0101
| <math>(v)\!</math>
| <math>(v)\!</math>
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
|-
|-
| <math>f_{10}</math>
| <math>f_{10}\!</math>
| 1010
| 1010
| <math>v\!</math>
| <math>v\!</math>
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|-
|-
| <math>f_7</math>
| <math>f_7\!</math>
| 0111
| 0111
| <math>(u v)\!</math>
| <math>(u v)\!</math>
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
|-
|-
| <math>f_{11}</math>
| <math>f_{11}\!</math>
| 1011
| 1011
| <math>(u (v))\!</math>
| <math>(u (v))\!</math>
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|-
|-
| <math>f_{13}</math>
| <math>f_{13}\!</math>
| 1101
| 1101
| <math>((u) v)\!</math>
| <math>((u) v)\!</math>
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|-
|-
| <math>f_{14}</math>
| <math>f_{14}\!</math>
| 1110
| 1110
| <math>((u)(v))\!</math>
| <math>((u)(v))~\!</math>
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|-
|-
| <math>f_{15}</math>
| <math>f_{15}\!</math>
| 1111
| 1111
| <math>((~))</math>
| <math>((~))\!</math>
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:white; color:black" | 0
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
| style="background:black; color:white" | 1
|}<br>
|}
<br>
{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|+ '''Table 19. Relation of Quantifiers to Higher Order Propositions'''
|+ style="height:25px" |
<math>\text{Table 20. Relation of Quantifiers to Higher Order Propositions}\!</math>
|- style="background:ghostwhite"
|- style="background:ghostwhite"
| <math>\text{Mnemonic}</math>
| <math>\text{Mnemonic}\!</math>
| <math>\text{Category}</math>
| <math>\text{Category}\!</math>
| <math>\text{Classical Form}</math>
| <math>\text{Classical Form}\!</math>
| <math>\text{Alternate Form}</math>
| <math>\text{Alternate Form}\!</math>
| <math>\text{Symmetric Form}</math>
| <math>\text{Symmetric Form}\!</math>
| <math>\text{Operator}</math>
| <math>\text{Operator}\!</math>
|-
|-
| <math>\text{E}\!</math><br><math>\text{Exclusive}</math>
| <math>\text{E}\!</math><br><math>\text{Exclusive}\!</math>
| <math>\text{Universal}</math><br><math>\text{Negative}</math>
| <math>\text{Universal}\!</math><br><math>\text{Negative}\!</math>
| <math>\text{All}\ u\ \text{is}\ (v)</math>
| <math>\text{All}~ u ~\text{is}~ (v)\!</math>
|
|
| <math>\text{No}\ u\ \text{is}\ v </math>
| <math>\text{No}~ u ~\text{is}~ v \!</math>
| <math>(\ell_{11})</math>
| <math>{(\ell_{11})}\!</math>
|-
|-
| <math>\text{A}\!</math><br><math>\text{Absolute}</math>
| <math>\text{A}\!</math><br><math>\text{Absolute}~\!</math>
| <math>\text{Universal}</math><br><math>\text{Affirmative}</math>
| <math>\text{Universal}\!</math><br><math>\text{Affirmative}\!</math>
| <math>\text{All}\ u\ \text{is}\ v </math>
| <math>\text{All}~ u ~\text{is}~ v \!</math>
|
|
| <math>\text{No}\ u\ \text{is}\ (v)</math>
| <math>\text{No}~ u ~\text{is}~ (v)\!</math>
| <math>(\ell_{10})</math>
| <math>{(\ell_{10})}\!</math>
|-
|-
|
|
|
|
| <math>\text{All}\ v\ \text{is}\ u </math>
| <math>\text{All}~ v ~\text{is}~ u \!</math>
| <math>\text{No}\ v\ \text{is}\ (u)</math>
| <math>\text{No}~ v ~\text{is}~ (u)\!</math>
| <math>\text{No}\ (u)\ \text{is}\ v </math>
| <math>\text{No}~ (u) ~\text{is}~ v \!</math>
| <math>(\ell_{01})</math>
| <math>{(\ell_{01})}\!</math>
|-
|-
|
|
|
|
| <math>\text{All}\ (v)\ \text{is}\ u </math>
| <math>\text{All}~ (v) ~\text{is}~ u \!</math>
| <math>\text{No}\ (v)\ \text{is}\ (u)</math>
| <math>\text{No}~ (v) ~\text{is}~ (u)\!</math>
| <math>\text{No}\ (u)\ \text{is}\ (v)</math>
| <math>\text{No}~ (u) ~\text{is}~ (v)\!</math>
| <math>(\ell_{00})</math>
| <math>{(\ell_{00})}\!</math>
|-
|-
|
|
|
|
| <math>\text{Some}\ (u)\ \text{is}\ (v)</math>
| <math>\text{Some}~ (u) ~\text{is}~ (v)\!</math>
|
|
| <math>\text{Some}\ (u)\ \text{is}\ (v)</math>
| <math>\text{Some}~ (u) ~\text{is}~ (v)\!</math>
| <math>\ell_{00}\!</math>
| <math>{\ell_{00}}\!</math>
|-
|-
|
|
|
|
| <math>\text{Some}\ (u)\ \text{is}\ v</math>
| <math>\text{Some}~ (u) ~\text{is}~ v\!</math>
|
|
| <math>\text{Some}\ (u)\ \text{is}\ v</math>
| <math>\text{Some}~ (u) ~\text{is}~ v\!</math>
| <math>\ell_{01}\!</math>
| <math>{\ell_{01}}\!</math>
|-
|-
| <math>\text{O}\!</math><br><math>\text{Obtrusive}</math>
| <math>\text{O}\!</math><br><math>\text{Obtrusive}\!</math>
| <math>\text{Particular}</math><br><math>\text{Negative}</math>
| <math>\text{Particular}\!</math><br><math>\text{Negative}\!</math>
| <math>\text{Some}\ u\ \text{is}\ (v)</math>
| <math>\text{Some}~ u ~\text{is}~ (v)\!</math>
|
|
| <math>\text{Some}\ u\ \text{is}\ (v)</math>
| <math>\text{Some}~ u ~\text{is}~ (v)\!</math>
| <math>\ell_{10}\!</math>
| <math>{\ell_{10}}\!</math>
|-
|-
| <math>\text{I}\!</math><br><math>\text{Indefinite}</math>
| <math>\text{I}\!</math><br><math>\text{Indefinite}\!</math>
| <math>\text{Particular}</math><br><math>\text{Affirmative}</math>
| <math>\text{Particular}\!</math><br><math>\text{Affirmative}\!</math>
| <math>\text{Some}\ u\ \text{is}\ v</math>
| <math>\text{Some}~ u ~\text{is}~ v\!</math>
|
|
| <math>\text{Some}\ u\ \text{is}\ v</math>
| <math>\text{Some}~ u ~\text{is}~ v\!</math>
| <math>\ell_{11}\!</math>
| <math>{\ell_{11}}\!</math>
|}<br>
|}
<br>
==Appendix : Generalized Umpire Operators==
==Appendix : Generalized Umpire Operators==
{| align="center" cellpadding="8" style="text-align:center"
{| align="center" cellpadding="8" style="text-align:center"
| <math>\Upsilon : \bigcup_{\ell = 1, 2, 3} ((\mathbb{B}^k \to \mathbb{B})^\ell \to \mathbb{B}).</math>
| <math>\Upsilon : \bigcup_{\ell = 1, 2, 3} ((\mathbb{B}^k \to \mathbb{B})^\ell \to \mathbb{B}).\!</math>
|}
|}
| <math>\Upsilon_p^r q = \Upsilon (p, q, r)\!</math>
| <math>\Upsilon_p^r q = \Upsilon (p, q, r)\!</math>
|-
|-
| <math>\Upsilon_p^r : (\mathbb{B}^k \to \mathbb{B}) \to \mathbb{B}</math>
| <math>\Upsilon_p^r : (\mathbb{B}^k \to \mathbb{B}) \to \mathbb{B}\!</math>
|}
|}
The intention of this operator is that we evaluate the proposition <math>q\!</math> on each model of the proposition <math>p\!</math> and combine the results according to the method indicated by the connective parameter <math>r.\!</math> In principle, the index <math>r\!</math> might specify any connective on as many as <math>2^k\!</math> arguments, but usually we have in mind a much simpler form of combination, most often either collective products or collective sums. By convention, each of the accessory indices <math>p, r\!</math> is assigned a default value that is understood to be in force when the corresponding argument place is left blank, specifically, the constant proposition <math>1 : \mathbb{B}^k \to \mathbb{B}</math> for the lower index <math>p,\!</math> and the continued conjunction or continued product operation <math>\textstyle\prod</math> for the upper index <math>r.\!</math> Taking the upper default value gives license to the following readings:
The intention of this operator is that we evaluate the proposition <math>q\!</math> on each model of the proposition <math>p\!</math> and combine the results according to the method indicated by the connective parameter <math>r.\!</math> In principle, the index <math>r\!</math> might specify any connective on as many as <math>2^k\!</math> arguments, but usually we have in mind a much simpler form of combination, most often either collective products or collective sums. By convention, each of the accessory indices <math>p, r\!</math> is assigned a default value that is understood to be in force when the corresponding argument place is left blank, specifically, the constant proposition <math>1 : \mathbb{B}^k \to \mathbb{B}\!</math> for the lower index <math>p,\!</math> and the continued conjunction or continued product operation <math>\textstyle\prod\!</math> for the upper index <math>r.\!</math> Taking the upper default value gives license to the following readings:
{| align="center" cellpadding="8" style="text-align:center"
{| align="center" cellpadding="8" style="text-align:center"
| <math>\Upsilon_p (q) = \Upsilon (p, q) = \Upsilon (p, q, \textstyle\prod).</math>
| <math>\Upsilon_p (q) = \Upsilon (p, q) = \Upsilon (p, q, \textstyle\prod).\!</math>
|-
|-
| <math>\Upsilon_p = \Upsilon (p, \underline{~~}, \textstyle\prod) : (\mathbb{B}^k \to \mathbb{B}) \to \mathbb{B}.</math>
| <math>\Upsilon_p = \Upsilon (p, \underline{~~}, \textstyle\prod) : (\mathbb{B}^k \to \mathbb{B}) \to \mathbb{B}.\!</math>
|}
|}
This means that <math>\Upsilon_p (q) = 1\!</math> if and only if <math>q\!</math> holds for all models of <math>p.\!</math> In propositional terms, this is tantamount to the assertion that <math>p \Rightarrow q,</math> or that <math>\underline{(p (q))} = \underline{1}.</math>
This means that <math>\Upsilon_p (q) = 1\!</math> if and only if <math>q\!</math> holds for all models of <math>p.\!</math> In propositional terms, this is tantamount to the assertion that <math>p \Rightarrow q,\!</math> or that <math>\underline{(p (q))} = \underline{1}.\!</math>
Throwing in the lower default value permits the following abbreviations:
Throwing in the lower default value permits the following abbreviations:
{| align="center" cellpadding="8" style="text-align:center"
{| align="center" cellpadding="8" style="text-align:center"
| <math>\Upsilon q = \Upsilon (q) = \Upsilon_1 (q) = \Upsilon (1, q, \textstyle\prod).</math>
| <math>\Upsilon q = \Upsilon (q) = \Upsilon_1 (q) = \Upsilon (1, q, \textstyle\prod).\!</math>
|-
|-
| <math>\Upsilon = \Upsilon (1, \underline{~~}, \textstyle\prod)) : (\mathbb{B}^k\ \to \mathbb{B}) \to \mathbb{B}.</math>
| <math>\Upsilon = \Upsilon (1, \underline{~~}, \textstyle\prod)) : (\mathbb{B}^k\ \to \mathbb{B}) \to \mathbb{B}.\!</math>
|}
|}
This means that <math>\Upsilon q = 1\!</math> if and only if <math>q\!</math> holds for the whole universe of discourse in question, that is, if and only <math>q\!</math> is the constantly true proposition <math>1 : \mathbb{B}^k \to \mathbb{B}.</math> The ambiguities of this usage are not a problem so long as we distinguish the context of definition from the context of application and restrict all shorthand notations to the latter.
This means that <math>\Upsilon q = 1\!</math> if and only if <math>q\!</math> holds for the whole universe of discourse in question, that is, if and only <math>q\!</math> is the constantly true proposition <math>1 : \mathbb{B}^k \to \mathbb{B}.\!</math> The ambiguities of this usage are not a problem so long as we distinguish the context of definition from the context of application and restrict all shorthand notations to the latter.
==References==
==References==
| align="left" | Course:
| align="left" | Course:
| align="center" | Engineering 690, Graduate Project
| align="center" | Engineering 690, Graduate Project
| align="right" | Cont'd from Winter 1995
| align="right" | Winter Term, January 1995
|-
|-
| align="left" | Supervisors:
| align="left" | Supervisors:
| align="center" | F. Mili & M.A. Zohdy
| align="center" | M.A. Zohdy and F. Mili
| align="right" | Oakland University
| align="right" | Oakland University
|}
|}
===Functional Logic===
===Functional Logic===
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[[Category:Adaptive Systems]]
[[Category:Adaptive Systems]]
[[Category:Artificial Intelligence]]
[[Category:Artificial Intelligence]]
[[Category:Charles Sanders Peirce]]
[[Category:Combinatorics]]
[[Category:Combinatorics]]
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[[Category:Computer Science]]
