Inquiry Into Inquiry

Survey of Pragmatic Semiotic Information • 7

Posted on July 23, 2023 by Jon Awbrey

This is a Survey of blog and wiki posts on a theory of information which grows out of pragmatic semiotic ideas. All my projects are exploratory in character but this line of inquiry is more open‑ended than most. The question is —

What is information and how does it impact the spectrum of activities answering to the name of inquiry?

Setting out on what would become his lifelong quest to explore and explain the “Logic of Science”, C.S. Peirce pierced the veil of historical confusions obscuring the issue and fixed on what he called the “laws of information” as the key to solving the puzzle.

The first hints of the Information Revolution in our understanding of scientific inquiry may be traced to Peirce’s lectures of 1865–1866 at Harvard University and the Lowell Institute. There Peirce took up “the puzzle of the validity of scientific inference” and claimed it was “entirely removed by a consideration of the laws of information”.

Fast forward to the present and I see the Big Question as follows. Having gone through the exercise of comparing and contrasting Peirce’s theory of information, however much it yet remains in a rough‑hewn state, with Shannon’s paradigm so pervasively informing the ongoing revolution in our understanding and use of information, I have reason to believe Peirce’s idea is root and branch more general and has the potential, with due development, to resolve many mysteries still bedeviling our grasp of inference, information, and inquiry.

Inference, Information, Inquiry

OEIS Wiki • Information = Comprehension × Extension

Blog Series • { Information = Comprehension × Extension }

Selections • (1) • (2) • (3) • (4) • (5) • (6)

Comments • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8)

Discussions • (1) • (2) • (3) • (4) • (5) • (6) • (7)

Blog Series • { Information = Comprehension × Extension } • Revisited

Selections • (1) • (2) • (3) • (4) • (5) • (6)

Comments • (1) • (2) • (3) • (4) • (5)

Discussions • (8) • (9) • (10) • (11) • (12) • (13) • (14) • (15) • (16) • (17)

Pragmatic Semiotic Information

Pragmatic Semiotic Information • Ψ

Discussions • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12) • (13) • (14) • (15) • (16) • (17) • (18) • (19) • (20) • (21)

Semiotics, Semiosis, Sign Relations

Semeiotic • OEIS Wiki • Wikiversity
Blog Series • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9)

Sign Relations, Triadic Relations, Relation Theory

Blog Series • (1)

Discusssions • (1) • (2)

Excursions

Blog Dialogs

Semiotic Theory Of Information • (1) • (2) • (3) • (4) • (5) • (6)
Icon Index Symbol • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12) • (13) • (14) • (15) • (16) • (17)
Sign Relations • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12)
Sign Relations, Triadic Relations, Relations • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11)

References

Peirce, C.S. (1867), “Upon Logical Comprehension and Extension”. Online.
Awbrey, J.L., and Awbrey, S.M. (1992), “Interpretation as Action : The Risk of Inquiry”, The Eleventh International Human Science Research Conference, Oakland University, Rochester, Michigan.
Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), pp. 40–52. Archive. Journal. Online.

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cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Posted in Abduction, C.S. Peirce, Communication, Control, Cybernetics, Deduction, Determination, Discovery, Doubt, Epistemology, Fixation of Belief, Induction, Information, Information = Comprehension × Extension, Information Theory, Inquiry, Inquiry Driven Systems, Inquiry Into Inquiry, Interpretation, Invention, Knowledge, Learning Theory, Logic, Logic of Relatives, Logic of Science, Mathematics, Peirce, Philosophy, Philosophy of Science, Pragmatic Information, Probable Reasoning, Process Thinking, Relation Theory, Scientific Inquiry, Scientific Method, Semeiosis, Semiosis, Semiotic Information, Semiotics, Sign Relational Manifolds, Sign Relations, Surveys, Triadic Relations, Uncertainty | Tagged Abduction, C.S. Peirce, Communication, Control, Cybernetics, Deduction, Determination, Discovery, Doubt, Epistemology, Fixation of Belief, Induction, Information, Information = Comprehension × Extension, Information Theory, Inquiry, Inquiry Driven Systems, Inquiry Into Inquiry, Interpretation, Invention, Knowledge, Learning Theory, Logic, Logic of Relatives, Logic of Science, Mathematics, Peirce, Philosophy, Philosophy of Science, Pragmatic Information, Probable Reasoning, Process Thinking, Relation Theory, Scientific Inquiry, Scientific Method, Semeiosis, Semiosis, Semiotic Information, Semiotics, Sign Relational Manifolds, Sign Relations, Surveys, Triadic Relations, Uncertainty | 3 Comments

Pragmatic Semiotic Information • Ψ

Posted on July 22, 2023 by Jon Awbrey

I remember it was back in ’76 when I began to notice a subtle shift of focus in the computer science journals I was reading, from discussing X to discussing Information About X, a transformation I noted mentally as $X \to \mathrm{Info}(X)$ whenever I ran across it. I suppose that small arc of revolution had been building for years but it struck me as crossing a threshold to a more explicit, self‑conscious stage about that time.

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cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Posted in Abduction, C.S. Peirce, Comprehension, Deduction, Definition, Determination, Extension, Hypothesis, Icon Index Symbol, Induction, Inference, Information, Information = Comprehension × Extension, Inquiry, Intension, Logic, Logic of Science, Peirce, Pragmatic Semiotic Information, Pragmatism, Scientific Method, Semiotics, Sign Relations | Tagged Abduction, C.S. Peirce, Comprehension, Deduction, Definition, Determination, Extension, Hypothesis, Icon Index Symbol, Induction, Inference, Information, Information = Comprehension × Extension, Inquiry, Intension, Logic, Logic of Science, Peirce, Pragmatic Semiotic Information, Pragmatism, Scientific Method, Semiotics, Sign Relations | 3 Comments

Functional Logic • Inquiry and Analogy • 21

Posted on July 19, 2023 by Jon Awbrey

Inquiry and Analogy • Generalized Umpire Operators

To get a better handle on the space of higher order propositions and continue developing our functional approach to quantification theory, we’ll need a number of specialized tools. To begin, we define a higher order operator $\Upsilon,$ called the umpire operator, which takes 1, 2, or 3 propositions as arguments and returns a single truth value as the result. Operators with optional numbers of arguments are called multigrade operators, typically defined as unions over function types. Expressing $\Upsilon$ in that form gives the following formula.

In contexts of application, that is, where a multigrade operator is actually being applied to arguments, the number of arguments in the argument list tells which of the optional types is “operative”. In the case of $\Upsilon,$ the first and last arguments appear as indices, the one in the middle serving as the main argument while the other two arguments serve to modify the sense of the operation in question. Thus, we have the following forms.

The operation $\Upsilon_p^r q$ evaluates the proposition $q$ on each model of the proposition $p$ and combines the results according to the method indicated by the connective parameter $r.$ In principle, the index $r$ may specify any logical connective on as many as $2^k$ arguments but in practice we usually have a much simpler form of combination in mind, typically either products or sums. By convention, each of the accessory indices $p, r$ is assigned a default value understood to be in force when the corresponding argument place is left blank, specifically, the constant proposition $1 : \mathbb{B}^k \to \mathbb{B}$ for the lower index $p$ and the continued conjunction or continued product operation $\textstyle\prod$ for the upper index $r.$ Taking the upper default value gives license to the following readings.

This means $\Upsilon_p (q) = 1$ if and only if $q$ holds for all models of $p.$ In propositional terms, this is tantamount to the assertion that $p \Rightarrow q,$ or that $\texttt{(} p \texttt{(} q \texttt{))} = 1.$

Throwing in the lower default value permits the following abbreviations.

This means $\Upsilon q = 1$ if and only if $q$ holds for the whole universe of discourse in question, that is, if and only $q$ is the constantly true proposition $1 : \mathbb{B}^k \to \mathbb{B}.$ 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.

Resources

Logic Syllabus
Boolean Function
Boolean-Valued Function
Logical Conjunction
Multigrade Operator
Parametric Operator
Minimal Negation Operator
Introduction to Inquiry Driven Systems
Functional Logic • Part 1 • Part 2 • Part 3
Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

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cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Posted in Abduction, Analogy, Argument, Aristotle, C.S. Peirce, Constraint, Deduction, Determination, Diagrammatic Reasoning, Diagrams, Differential Logic, Functional Logic, Hypothesis, Indication, Induction, Inference, Information, Inquiry, Logic, Logic of Science, Mathematics, Pragmatic Semiotic Information, Probable Reasoning, Propositional Calculus, Propositions, Reasoning, Retroduction, Semiotics, Sign Relations, Syllogism, Triadic Relations, Visualization | Tagged Abduction, Analogy, Argument, Aristotle, C.S. Peirce, Constraint, Deduction, Determination, Diagrammatic Reasoning, Diagrams, Differential Logic, Functional Logic, Hypothesis, Indication, Induction, Inference, Information, Inquiry, Logic, Logic of Science, Mathematics, Pragmatic Semiotic Information, Probable Reasoning, Propositional Calculus, Propositions, Reasoning, Retroduction, Semiotics, Sign Relations, Syllogism, Triadic Relations, Visualization | 4 Comments

Functional Logic • Inquiry and Analogy • 20

Posted on July 18, 2023 by Jon Awbrey

Inquiry and Analogy • Application of Higher Order Propositions to Quantification Theory

Table 21 provides a thumbnail sketch of the relationships discussed in this section.

$\text{Table 21. Relation of Quantifiers to Higher Order Propositions}$

Resources

Logic Syllabus
Boolean Function
Boolean-Valued Function
Logical Conjunction
Minimal Negation Operator
Introduction to Inquiry Driven Systems
Functional Logic • Part 1 • Part 2 • Part 3
Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Functional Logic • Inquiry and Analogy • 19

Posted on July 17, 2023 by Jon Awbrey

Inquiry and Analogy • Application of Higher Order Propositions to Quantification Theory

Reflection is turning a topic over in various aspects and in various lights so that nothing significant about it shall be overlooked — almost as one might turn a stone over to see what its hidden side is like or what is covered by it.

John Dewey • How We Think

Tables 19 and 20 present the same information as Table 18, sorting the rows in different orders to reveal other symmetries in the arrays.

$\text{Table 19. Simple Qualifiers of Propositions (Version 2)}$

$\text{Table 20. Simple Qualifiers of Propositions (Version 3)}$

Resources

Logic Syllabus
Boolean Function
Boolean-Valued Function
Logical Conjunction
Minimal Negation Operator
Introduction to Inquiry Driven Systems
Functional Logic • Part 1 • Part 2 • Part 3
Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Functional Logic • Inquiry and Analogy • 18

Posted on July 16, 2023 by Jon Awbrey

Inquiry and Analogy • Application of Higher Order Propositions to Quantification Theory

Last time we took up a fourfold scheme of quantified propositional forms traditionally known as a “Square of Opposition”, relating it to a quartet of higher order propositions which, depending on context, are also known as measures, qualifiers, or higher order indicator functions.

Table 18 develops the above ideas in further detail, expressing a larger set of quantified propositional forms by means of propositions about propositions.

$\text{Table 18. Simple Qualifiers of Propositions (Version 1)}$

Resources

Logic Syllabus
Boolean Function
Boolean-Valued Function
Logical Conjunction
Minimal Negation Operator
Introduction to Inquiry Driven Systems
Functional Logic • Part 1 • Part 2 • Part 3
Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Functional Logic • Inquiry and Analogy • 17

Posted on July 15, 2023 by Jon Awbrey

Inquiry and Analogy • Application of Higher Order Propositions to Quantification Theory

Our excursion into the expanding landscape of higher order propositions has come round to the point where we can begin to open up new perspectives on quantificational logic.

Though it may be all the same from a purely formal point of view, it does serve intuition to adopt a slightly different interpretation for the two‑valued space we take as the target of our basic indicator functions. In that spirit we declare a novel type of existence-valued functions $f : \mathbb{B}^k \to \mathbb{E}$ where $\mathbb{E} = \{ -e, +e \} = \{ \mathrm{empty}, \mathrm{existent} \}$ is a pair of values indicating whether anything exists in the cells of the underlying universe of discourse. As usual, we won’t be too picky about the coding of those functions, reverting to binary codes whenever the intended interpretation is clear enough.

With that interpretation in mind we observe the following correspondence between classical quantifications and higher order indicator functions.

$\text{Table 17. Syllogistic Premisses as Higher Order Indicator Functions}$

Resources

Logic Syllabus
Boolean Function
Boolean-Valued Function
Logical Conjunction
Minimal Negation Operator
Introduction to Inquiry Driven Systems
Functional Logic • Part 1 • Part 2 • Part 3
Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Functional Logic • Inquiry and Analogy • 16

Posted on July 13, 2023 by Jon Awbrey

Inquiry and Analogy • Extending the Existential Interpretation to Quantificational Logic

One of the resources we have for our investigation is a formal calculus based on C.S. Peirce’s logical graphs. For the present we’ll adopt the existential interpretation of that calculus, fixing the meanings of logical constants and connectives at the core level of propositional logic. To build on that core we’ll need to extend the existential interpretation to encompass the analysis of quantified propositions, or quantifications. That in turn will take developing two further capacities of our calculus. On the formal side we’ll need to consider higher order functional types, continuing our earlier venture above. In terms of content we’ll need to consider new species of elemental or singular propositions.

Let us return to the 2‑dimensional universe $X^\bullet = [u, v].$ A bridge between propositions and quantifications is afforded by a set of measures or qualifiers $\ell_{ij} : (\mathbb{B} \times \mathbb{B} \to \mathbb{B}) \to \mathbb{B}$ defined by the following equations.

$\begin{array}{*{11}{l}} \ell_{00} f & = & \ell_{\texttt{(} u \texttt{)(} v \texttt{)}} f & = & \alpha_1 f & = & \Upsilon_{\texttt{(} u \texttt{)(} v \texttt{)}} f & = & \Upsilon_{\texttt{(} u \texttt{)(} v \texttt{)} \,\Rightarrow\, f} & = & f ~\text{likes}~ \texttt{(} u \texttt{)(} v \texttt{)} \\ \ell_{01} f & = & \ell_{\texttt{(} u \texttt{)} v} f & = & \alpha_2 f & = & \Upsilon_{\texttt{(} u \texttt{)} v} f & = & \Upsilon_{\texttt{(} u \texttt{)} v \,\Rightarrow\, f} & = & f ~\text{likes}~ \texttt{(} u \texttt{)} v \\ \ell_{10} f & = & \ell_{u \texttt{(} v \texttt{)}} f & = & \alpha_4 f & = & \Upsilon_{u \texttt{(} v \texttt{)}} f & = & \Upsilon_{u \texttt{(} v \texttt{)} \,\Rightarrow\, f} & = & f ~\text{likes}~ u \texttt{(} v \texttt{)} \\ \ell_{11} f & = & \ell_{u \, v} f & = & \alpha_8 f & = & \Upsilon_{u \, v} f & = & \Upsilon_{u \, v \,\Rightarrow\, f} & = & f ~\text{likes}~ u \, v \end{array}$

A higher order proposition $\ell_{ij} : (\mathbb{B} \times \mathbb{B} \to \mathbb{B}) \to \mathbb{B}$ tells us something about the proposition $f :\mathbb{B} \times \mathbb{B} \to \mathbb{B},$ namely, which elements in the space of type $\mathbb{B} \times \mathbb{B}$ are assigned a positive value by $f.$ Taken together, the $\ell_{ij}$ operators give us a way to express many useful observations about the propositions in $X^\bullet = [u, v].$ Figure 16 summarizes the action of the $\ell_{ij}$ operators on the propositions of type $f :\mathbb{B} \times \mathbb{B} \to \mathbb{B}.$

$\text{Figure 16. Higher Order Universe of Discourse}~ [ \ell_{00}, \ell_{01}, \ell_{10}, \ell_{11} ] \subseteq [[ u, v ]]$

Resources

Logic Syllabus
Boolean Function
Boolean-Valued Function
Logical Conjunction
Minimal Negation Operator
Introduction to Inquiry Driven Systems
Functional Logic • Part 1 • Part 2 • Part 3
Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Functional Logic • Inquiry and Analogy • 15

Posted on July 12, 2023 by Jon Awbrey

Inquiry and Analogy • Measure for Measure

Let us define two families of measures,

$\alpha_i, \beta_i : (\mathbb{B} \times \mathbb{B} \to \mathbb{B}) \to \mathbb{B} ~\text{for}~ i = 0 ~\text{to}~ 15,$

by means of the following equations:

$\begin{matrix} \alpha_i f & = & \Upsilon (f_i, f) & = & \Upsilon (f_i \Rightarrow f), \\[6pt] \beta_i f & = & \Upsilon (f, f_i) & = & \Upsilon (f \Rightarrow f_i). \end{matrix}$

Table 14 shows the value of each $\alpha_i$ on each of the 16 boolean functions $f: \mathbb{B} \times \mathbb{B} \to \mathbb{B}.$ In terms of the implication ordering on the 16 functions, $\alpha_i f = 1$ says that $f$ is above or identical to $f_i$ in the implication lattice, that is, $f \ge f_i$ in the implication ordering.

$\text{Table 14. Qualifiers of the Implication Ordering}~ \alpha_i f = \Upsilon (f_i, f)$

Table 15 shows the value of each $\beta_i$ on each of the 16 boolean functions $f: \mathbb{B} \times \mathbb{B} \to \mathbb{B}.$ In terms of the implication ordering on the 16 functions, $\beta_i f = 1$ says that $f$ is below or identical to $f_i$ in the implication lattice, that is, $f \le f_i$ in the implication ordering.

$\text{Table 15. Qualifiers of the Implication Ordering}~ \beta_i f = \Upsilon (f, f_i)$

Applied to a given proposition $f,$ the qualifiers $\alpha_i$ and $\beta_i$ tell whether $f$ is above $f_i$ or below $f_i,$ respectively, in the implication ordering. By way of example, let us trace the effects of several such measures, namely, those which occupy the limiting positions in the Tables.

$\begin{array}{*{8}{r}} \alpha_{0} f = 1 & \mathrm{iff} & f_{0} \Rightarrow f & \mathrm{iff} & 0 \Rightarrow f, & \mathrm{hence} & \alpha_{0} f = 1 & \mathrm{for~all} ~ f. \\[4pt] \alpha_{15} f = 1 & \mathrm{iff} & f_{15} \Rightarrow f & \mathrm{iff} & 1 \Rightarrow f, & \mathrm{hence} & \alpha_{15} f = 1 & \mathrm{iff} ~ f = 1. \\[4pt] \beta_{0} f = 1 & \mathrm{iff} & f \Rightarrow f_{0} & \mathrm{iff} & f \Rightarrow 0, & \mathrm{hence} & \beta_{0} f = 1 & \mathrm{iff} ~ f = 0. \\[4pt] \beta_{15} f = 1 & \mathrm{iff} & f \Rightarrow f_{15} & \mathrm{iff} & f \Rightarrow 1, & \mathrm{hence} & \beta_{15} f = 1 & \mathrm{for~all} ~ f. \end{array}$

Expressed in terms of the propositional forms they value positively, $\alpha_{0} = \beta_{15}$ is a wholly indifferent or indiscriminate measure, accepting every proposition $f : \mathbb{B} \times \mathbb{B} \to \mathbb{B},$ whereas the measures $\alpha_{15}$ and $\beta_{0}$ value the constant propositions $1 : \mathbb{B} \times \mathbb{B} \to \mathbb{B}$ and $0 : \mathbb{B} \times \mathbb{B} \to \mathbb{B},$ respectively, above all others.

Finally, in conformity with the use of fiber notation to indicate sets of models, it is natural to use notations like the following to denote sets of propositions satisfying the umpires in question.

$\begin{matrix} [| \alpha_i |] & = & \alpha_i^{-1}(1), \\[6pt] [| \beta_i |] & = & \beta_i^{-1}(1), \\[6pt] [| \Upsilon_p |] & = & \Upsilon_p^{-1}(1). \end{matrix}$

Resources

Logic Syllabus
Boolean Function
Boolean-Valued Function
Logical Conjunction
Minimal Negation Operator
Introduction to Inquiry Driven Systems
Functional Logic • Part 1 • Part 2 • Part 3
Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Functional Logic • Inquiry and Analogy • 14

Posted on July 11, 2023 by Jon Awbrey

Inquiry and Analogy • Umpire Operators

The $2^{16}$ measures of type $(\mathbb{B} \times \mathbb{B} \to \mathbb{B}) \to \mathbb{B}$ present a formidable array of propositions about propositions about 2‑dimensional universes of discourse. The early entries in their standard ordering define universes too amorphous to detain us for long on a first pass but as we turn toward the high end of the ordering we begin to recognize familiar structures worth examining from new angles.

Instrumental to our study we define a couple of higher order operators,

$\begin{matrix} \Upsilon : (\mathbb{B} \times \mathbb{B} \to \mathbb{B})^2 \to \mathbb{B} && \text{and} && \Upsilon_1 : (\mathbb{B} \times \mathbb{B} \to \mathbb{B}) \to \mathbb{B}, \end{matrix}$

referred to as the relative and absolute umpire operators, respectively. If either operator is defined in terms of more primitive notions then the remaining operator can be defined in terms of the one first established.

Let $X = \langle u, v \rangle$ be a two‑dimensional boolean space, $X \cong \mathbb{B} \times \mathbb{B},$ generated by two boolean variables or logical features $u$ and $v.$

Given an ordered pair of propositions $e, f : \langle u, v \rangle \to \mathbb{B}$ as arguments, the relative umpire operator reports the value $1$ if the first implies the second, otherwise it reports the value $0.$

$\begin{matrix} \Upsilon (e, f) = 1 && \text{if and only if} && e \Rightarrow f \end{matrix}$

Expressing it another way:

$\begin{matrix} \Upsilon (e, f) = 1 && \iff && \texttt{(} e \texttt{(} f \texttt{))} = 1 \end{matrix}$

In writing this, however, it is important to observe that the $1$ appearing on the left side and the $1$ appearing on the right side of the logical equivalence have different meanings. Filling in the details, we have the following.

$\begin{matrix} \Upsilon (e, f) = 1 \in \mathbb{B} && \iff && \texttt{(} e \texttt{(} f \texttt{))} = 1 : \langle u, v \rangle \to \mathbb{B} \end{matrix}$

Writing types as subscripts and using the fact that $X = \langle u, v \rangle,$ it is possible to express this more succinctly as follows.

$\begin{matrix} \Upsilon (e, f) = 1_\mathbb{B} && \iff && \texttt{(} e \texttt{(} f \texttt{))} = 1_{X \to \mathbb{B}} \end{matrix}$

Finally, it is often convenient to write the first argument as a subscript. Thus we have the following equation.

$\begin{matrix} \Upsilon_e (f) & = & \Upsilon (e, f). \end{matrix}$

The absolute umpire operator, also known as the umpire measure, is a higher order proposition $\Upsilon_1 : (\mathbb{B} \times \mathbb{B} \to \mathbb{B}) \to \mathbb{B}$ defined by the equation $\Upsilon_1 (f) = \Upsilon (1, f).$ In this case the subscript $1$ on the left and the argument $1$ on the right both refer to the constant proposition $1 : \mathbb{B} \times \mathbb{B} \to \mathbb{B}.$ In most settings where $\Upsilon_1$ is applied to arguments it is safe to omit the subscript $1$ since the number of arguments indicates which type of operator is meant. Thus, we have the following identities and equivalents.

$\begin{matrix} \Upsilon f = \Upsilon_1 (f) = 1_\mathbb{B} & \iff & \texttt{(} 1 \texttt{(} f \texttt{))} = \mathbf{1} & \iff & f = 1_{\mathbb{B} \times \mathbb{B} \to \mathbb{B}} \end{matrix}$

The umpire measure $\Upsilon_1$ is defined on boolean functions regarded as mathematical objects but can also be understood in terms of the judgments it induces on the syntactic level. In that interpretation $\Upsilon_1$ recognizes theorems of the propositional calculus over $[u, v],$ giving a score of $1$ to tautologies and a score of $0$ to everything else, counting 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 $\Upsilon$ to mean the absolute measure then the relative measure could have been defined as $\Upsilon_e f = \Upsilon \texttt{(} e \texttt{(} f \texttt{))}.$

Resources

Logic Syllabus
Boolean Function
Boolean-Valued Function
Logical Conjunction
Minimal Negation Operator
Introduction to Inquiry Driven Systems
Functional Logic • Part 1 • Part 2 • Part 3
Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

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