Functional Logic • Inquiry and Analogy • 16

Inquiry and Analogy • Extending the Existential Interpretation to Quantificational Logic

One of the resources we have for this work is a formal calculus based on C.S. Peirce’s logical graphs. For now 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: Conceptual Graphs • Cybernetics • Laws of Form • Ontolog Forum
cc: FB | Peirce Matters • Structural Modeling • Systems Science

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