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)}
Simple Qualifiers of Propositions (Version 2)

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

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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)}
Simple Qualifiers of Propositions (Version 1)

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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}
Syllogistic Premisses as Higher Order Indicator Functions

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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}.

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

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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)
Qualifiers of the Implication Ordering α

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)
Qualifiers of the Implication Ordering β

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}

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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{))}.

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Functional Logic • Inquiry and Analogy • 13

Posted on July 10, 2023 by Jon Awbrey

Inquiry and Analogy • Higher Order Propositional Expressions

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’s first consider the universe of discourse X^\bullet = [\mathcal{X}] = [x_1, x_2] = [u, v], based on two logical features or boolean variables u and v.

The universe of discourse X^\bullet consists of two parts, a set of points and a set of propositions.

The points of X^\bullet form the space:

\begin{matrix} X & = & \langle \mathcal{X} \rangle & = & \langle u, v \rangle & = & \{ (u, v) \} & \cong & \mathbb{B}^2. \end{matrix}

Each point in X may be indicated by means of a singular proposition, that is, a proposition which describes it uniquely.  This form of representation leads to the following enumeration of points.

\begin{matrix} X & = & \{ ~ \texttt{(} u \texttt{)(} v \texttt{)} ~,~ \texttt{(} u \texttt{)} ~ v ~,~ u ~ \texttt{(} v \texttt{)} ~,~ u ~ v ~ \} & \cong & \mathbb{B}^2. \end{matrix}

Each point in X may also be described by means of its coordinates, that is, by the ordered pair of values in \mathbb{B} which the coordinate propositions u and v take on that point.  This form of representation leads to the following enumeration of points.

\begin{matrix} X & = & \{\ (0, 0),\ (0, 1),\ (1, 0),\ (1, 1)\ \} & \cong & \mathbb{B}^2. \end{matrix}

The propositions of X^\bullet form the space:

\begin{matrix} X^\uparrow & = & (X \to \mathbb{B}) & = & \{ f : X \to \mathbb{B} \} & \cong & (\mathbb{B}^2 \to \mathbb{B}). \end{matrix}

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 on them again.

The next higher order universe of discourse built on X^\bullet is X^{\bullet 2} = [X^\bullet] = [[u, v]], which may be developed in the following way.  The propositions of X^\bullet become the points of X^{\bullet 2}, and the mappings of the type m : (X \to \mathbb{B}) \to \mathbb{B} become the propositions of X^{\bullet 2}.  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 w : (\mathbb{B}^2 \to \mathbb{B})^k \to \mathbb{B}.

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 [u, v] may be regarded as a gallery of sixteen venn diagrams, while the measures m : (X \to \mathbb{B}) \to \mathbb{B} 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 m_j partitions the gallery of pictures into two aesthetic portions, the pictures m_j^{-1}(1) that m_j likes and the pictures m_j^{-1}(0) that m_j dislikes.

There are 2^{16} = 65536 measures of the form m : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}.  Table 13 shows the first 24 of their number in the style of higher order truth table I used before.  The column headed m_j shows the value of the measure m_j on each of the propositions f_i : \mathbb{B}^2 \to \mathbb{B} for i = 0 to 15.  The arrangement of measures in the order indicated will be referred to as their standard ordering.  In this scheme of things, the index j of the measure m_j is the decimal equivalent of the bit string in the corresponding column of the Table, reading the binary digits in order from bottom to top.

\text{Table 13. Higher Order Propositions}~ (n = 2)
Higher Order Propositions (n = 2)

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Functional Logic • Inquiry and Analogy • 12

Posted on July 9, 2023 by Jon Awbrey

Inquiry and Analogy • Higher Order Propositional Expressions

Interpretive Categories for Higher Order Propositions (n = 1)

Table 12 presents a series of interpretive categories for the higher order propositions in Table 11.  I’ll leave those for now to the reader’s contemplation and discuss them when we get two variables into the mix.  The lower dimensional cases tend to exhibit condensed or degenerate structures and their full significance will become clearer once we get beyond the 1‑dimensional case.

\text{Table 12. Interpretive Categories for Higher Order Propositions}~ (n = 1)
Interpretive Categories for Higher Order Propositions (n = 1)

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Functional Logic • Inquiry and Analogy • 11

Posted on July 8, 2023 by Jon Awbrey

Inquiry and Analogy • Higher Order Propositional Expressions

Higher Order Propositions and Logical Operators (n = 1)

A higher order proposition is a proposition about propositions.  If the original order of propositions is a set of indicator functions f : X \to \mathbb{B} then the next higher order of propositions consists of maps of type m : (X \to \mathbb{B}) \to \mathbb{B}.

For example, consider the case where X = \mathbb{B}.  There are exactly four propositions one can make about the elements of X.  Each proposition has the concrete type f: X \to \mathbb{B} and the abstract type f : \mathbb{B} \to \mathbb{B}.  From that beginning there are exactly sixteen higher order propositions one can make about the initial set of four propositions.  Each higher order proposition has the abstract type m : (\mathbb{B} \to \mathbb{B}) \to \mathbb{B}.

Table 11 lists the sixteen higher order propositions about propositions on one boolean variable, organized in the following fashion.

  • Columns 1 and 2 taken together present a form of truth table for the four propositions f : \mathbb{B} \to \mathbb{B}.  Column 1 displays the names of the propositions f_i, for i = 1 to 4, while the entries in Column 2 show the value each proposition takes on the argument value 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 headed by a series of conventional names for the higher order propositions, also known as the measures m_j, for j = 0 to 15.  The entries in the body of the Table show the value each measure assigns to each proposition f_i.

\text{Table 11. Higher Order Propositions}~ (n = 1)
Higher Order Propositions (n = 1)

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Functional Logic • Inquiry and Analogy • 10

Posted on July 7, 2023 by Jon Awbrey

Inquiry and Analogy • 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.  The mere act of writing quantified formulas like \forall_{x \in X} f(x) and \exists_{x \in X} f(x) involves a subscription to such notions, as shown by the membership relations invoked in their indices.

As we reflect more critically on the conventional assumptions in the light of pragmatic and constructive principles, however, they begin to appear as problematic hypotheses whose warrants are not beyond question, as projects of exhaustive determination overreaching the powers of finite information and control to manage.

Thus it is worth considering how the scene of quantification theory might be shifted nearer to familiar ground, toward the predicates themselves which represent our continuing acquaintance with phenomena.

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