User:Jon Awbrey/Cactus Language Semantics Work

Cactus Language • Semantics • Last Part of Wiki Version

If one takes the point of view that PARCs and PARCEs amount to a pair of intertranslatable languages for the same domain of objects, then denotation brackets of the form \(\downharpoonleft \ldots \downharpoonright\) can be used to indicate the logical denotation \(\downharpoonleft C_j \downharpoonright\) of a cactus \(C_j\) or the logical denotation \(\downharpoonleft s_j \downharpoonright\) of a sentence \(s_j.\)

Tables 14 and 15 summarize the relations that serve to connect the formal language of sentences with the logical language of propositions.  Between these two realms of expression there is a family of graphical data structures that arise in parsing the sentences and that serve to facilitate the performance of computations on the indicator functions.  The graphical language supplies an intermediate form of representation between the formal sentences and the indicator functions, and the form of mediation that it provides is very useful in rendering the possible connections between the other two languages conceivable in fact, not to mention in carrying out the necessary translations on a practical basis.  These Tables include this intermediate domain in their Central Columns.  Between their First and Middle Columns they illustrate the mechanics of parsing the abstract sentences of the cactus language into the graphical data structures of the corresponding species.  Between their Middle and Final Columns they summarize the semantics of interpreting the graphical forms of representation for the purposes of reasoning with propositions.

\(\text{Table 14. Semantic Translation : Functional Form}\)

\(\mathrm{Sentence}\) \(\xrightarrow[\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}]{\mathrm{Parse}}\) \(\mathrm{Graph}\) \(\xrightarrow[\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}]{\mathrm{Denotation}}\) \(\mathrm{Proposition}\)

\(s_j\) \(\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}\) \(C_j\) \(\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}\) \(q_j\)

\(\mathrm{Conc}^0\) \(\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}\) \(\mathrm{Node}^0\) \(\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}\) \(1\) \(\mathrm{Conc}^k_j s_j\) \(\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}\) \(\mathrm{Node}^k_j C_j\) \(\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}\) \(\mathrm{Conj}^k_j q_j\)

\(\mathrm{Surc}^0\) \(\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}\) \(\mathrm{Lobe}^0\) \(\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}\) \(0\) \(\mathrm{Surc}^k_j s_j\) \(\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}\) \(\mathrm{Lobe}^k_j C_j\) \(\xrightarrow{\mathrm{~ ~ ~ ~ ~ ~ ~ ~ ~ ~}}\) \(\mathrm{Surj}^k_j q_j\)

\(\text{Table 15. Semantic Translation : Equational Form}\)

\(\downharpoonleft \mathrm{Sentence} \downharpoonright\) \(\stackrel{\mathrm{Parse}}{=}\) \(\downharpoonleft \mathrm{Graph} \downharpoonright\) \(\stackrel{\mathrm{Denotation}}{=}\) \(\mathrm{Proposition}\)

\(\downharpoonleft s_j \downharpoonright\) \(=\) \(\downharpoonleft C_j \downharpoonright\) \(=\) \(q_j\)

\(\downharpoonleft \mathrm{Conc}^0 \downharpoonright\) \(=\) \(\downharpoonleft \mathrm{Node}^0 \downharpoonright\) \(=\) \(1\) \(\downharpoonleft \mathrm{Conc}^k_j s_j \downharpoonright\) \(=\) \(\downharpoonleft \mathrm{Node}^k_j C_j \downharpoonright\) \(=\) \(\mathrm{Conj}^k_j q_j\)

\(\downharpoonleft \mathrm{Surc}^0 \downharpoonright\) \(=\) \(\downharpoonleft \mathrm{Lobe}^0 \downharpoonright\) \(=\) \(0\) \(\downharpoonleft \mathrm{Surc}^k_j s_j \downharpoonright\) \(=\) \(\downharpoonleft \mathrm{Lobe}^k_j C_j \downharpoonright\) \(=\) \(\mathrm{Surj}^k_j q_j\)

Aside from their common topic, the two Tables present slightly different ways of conceptualizing the operations that go to establish their maps.  Table 14 records the functional associations that connect each domain with the next, taking the triplings of a sentence \(s_j,\) a cactus \(C_j,\) and a proposition \(q_j\) as basic data, and fixing the rest by recursion on these.  Table 15 records these associations in the form of equations, treating sentences and graphs as alternative kinds of signs, and generalizing the denotation bracket operator to indicate the proposition that either denotes.  It should be clear at this point that either scheme of translation puts the sentences, the graphs, and the propositions that it associates with each other roughly in the roles of the signs, the interpretants, and the objects, respectively, whose triples define an appropriate sign relation.  Indeed, the "roughly" can be made "exactly" as soon as the domains of a suitable sign relation are specified precisely.

A good way to illustrate the action of the conjunction and surjunction operators is to demonstrate how they can be used to construct the boolean functions on any finite number of variables.  Let us begin by doing this for the first three cases, \(k = 0, 1, 2.\)

A boolean function \(F^{(0)}\) on \(0\) variables is just an element of the boolean domain \(\underline\mathbb{B} = \{ 0, 1 \}.\)  Table 16 shows several different ways of referring to these elements, just for the sake of consistency using the same format that will be used in subsequent Tables, no matter how degenerate it tends to appear in the initial case.

\(\text{Table 16. Boolean Functions on Zero Variables}\)

\(F\)	\(F\)	\(F()\)	\(F\)
\(0\)	\(F_0^{(0)}\)	\(0\)	\(\texttt{(~)}\)
\(1\)	\(F_1^{(0)}\)	\(1\)	\(\texttt{((~))}\)

Column 1 lists each boolean element or boolean function under its ordinary constant name or under a succinct nickname, respectively.

Column 2 lists each boolean function in a style of function name \(F_j^{(k)}\) that is constructed as follows:  The superscript \((k)\) gives the dimension of the functional domain, that is, the number of its functional variables, and the subscript \(j\) is a binary string that recapitulates the functional values, using the obvious translation of boolean values into binary values.

Column 3 lists the functional values for each boolean function, or possibly a boolean element appearing in the guise of a function, for each combination of its domain values.

Column 4 shows the usual expressions of these elements in the cactus language, conforming to the practice of omitting the underlines in display formats.  Here I illustrate also the convention of using the expression \(\text{“} ((~)) \text{”}\) as a visible stand-in for the expression of the logical value \(\mathrm{true},\) a value that is minimally represented by a blank expression that tends to elude our giving it much notice in the context of more demonstrative texts.

Table 17 presents the boolean functions on one variable, \(F^{(1)} : \underline\mathbb{B} \to \underline\mathbb{B},\) of which there are precisely four.

\(\text{Table 17. Boolean Functions on One Variable}\)

\(F\)	\(F\)	\(F(x)\)		\(F\)
		\(F(1)\)	\(F(0)\)
\(F_0^{(1)}\)	\(F_{00}^{(1)}\)	\(0\)	\(0\)	\(\texttt{(~)}\)
\(F_1^{(1)}\)	\(F_{01}^{(1)}\)	\(0\)	\(1\)	\(\texttt{(} x \texttt{)}\)
\(F_2^{(1)}\)	\(F_{10}^{(1)}\)	\(1\)	\(0\)	\(x\)
\(F_3^{(1)}\)	\(F_{11}^{(1)}\)	\(1\)	\(1\)	\(\texttt{((~))}\)

Here, Column 1 codes the contents of Column 2 in a more concise form, compressing the lists of boolean values, recorded as bits in the subscript string, into their decimal equivalents.  Naturally, the boolean constants reprise themselves in this new setting as constant functions on one variable.  Thus, one has the synonymous expressions for constant functions that are expressed in the next two chains of equations:

\(\begin{matrix} F_0^{(1)} & = & F_{00}^{(1)} & = & 0 ~:~ \underline\mathbb{B} \to \underline\mathbb{B} \\ \\ F_3^{(1)} & = & F_{11}^{(1)} & = & 1 ~:~ \underline\mathbb{B} \to \underline\mathbb{B} \end{matrix}\)

As for the rest, the other two functions are easily recognized as corresponding to the one-place logical connectives, or the monadic operators on \(\underline\mathbb{B}.\)  Thus, the function \(F_1^{(1)} = F_{01}^{(1)}\) is recognizable as the negation operation, and the function \(F_2^{(1)} = F_{10}^{(1)}\) is obviously the identity operation.

Table 18 presents the boolean functions on two variables, \(F^{(2)} : \underline\mathbb{B}^2 \to \underline\mathbb{B},\) of which there are precisely sixteen.

\(\text{Table 18. Boolean Functions on Two Variables}\)

\(F\)	\(F\)	\(F(x, y)\)				\(F\)
		\(F(1, 1)\)	\(F(1, 0)\)	\(F(0, 1)\)	\(F(0, 0)\)
\(F_{0}^{(2)}\)	\(F_{0000}^{(2)}\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\texttt{(~)}\)
\(F_{1}^{(2)}\)	\(F_{0001}^{(2)}\)	\(0\)	\(0\)	\(0\)	\(1\)	\(\texttt{(} x \texttt{)(} y \texttt{)}\)
\(F_{2}^{(2)}\)	\(F_{0010}^{(2)}\)	\(0\)	\(0\)	\(1\)	\(0\)	\(\texttt{(} x \texttt{)} y\)
\(F_{3}^{(2)}\)	\(F_{0011}^{(2)}\)	\(0\)	\(0\)	\(1\)	\(1\)	\(\texttt{(} x \texttt{)}\)
\(F_{4}^{(2)}\)	\(F_{0100}^{(2)}\)	\(0\)	\(1\)	\(0\)	\(0\)	\(x \texttt{(} y \texttt{)}\)
\(F_{5}^{(2)}\)	\(F_{0101}^{(2)}\)	\(0\)	\(1\)	\(0\)	\(1\)	\(\texttt{(} y \texttt{)}\)
\(F_{6}^{(2)}\)	\(F_{0110}^{(2)}\)	\(0\)	\(1\)	\(1\)	\(0\)	\(\texttt{(} x \texttt{,} y \texttt{)}\)
\(F_{7}^{(2)}\)	\(F_{0111}^{(2)}\)	\(0\)	\(1\)	\(1\)	\(1\)	\(\texttt{(} x y \texttt{)}\)
\(F_{8}^{(2)}\)	\(F_{1000}^{(2)}\)	\(1\)	\(0\)	\(0\)	\(0\)	\(x y\)
\(F_{9}^{(2)}\)	\(F_{1001}^{(2)}\)	\(1\)	\(0\)	\(0\)	\(1\)	\(\texttt{((} x \texttt{,} y \texttt{))}\)
\(F_{10}^{(2)}\)	\(F_{1010}^{(2)}\)	\(1\)	\(0\)	\(1\)	\(0\)	\(y\)
\(F_{11}^{(2)}\)	\(F_{1011}^{(2)}\)	\(1\)	\(0\)	\(1\)	\(1\)	\(\texttt{(} x \texttt{(} y \texttt{))}\)
\(F_{12}^{(2)}\)	\(F_{1100}^{(2)}\)	\(1\)	\(1\)	\(0\)	\(0\)	\(x\)
\(F_{13}^{(2)}\)	\(F_{1101}^{(2)}\)	\(1\)	\(1\)	\(0\)	\(1\)	\(\texttt{((} x \texttt{)} y \texttt{)}\)
\(F_{14}^{(2)}\)	\(F_{1110}^{(2)}\)	\(1\)	\(1\)	\(1\)	\(0\)	\(\texttt{((} x \texttt{)(} y \texttt{))}\)
\(F_{15}^{(2)}\)	\(F_{1111}^{(2)}\)	\(1\)	\(1\)	\(1\)	\(1\)	\(\texttt{((~))}\)

As before, all of the boolean functions of fewer variables are subsumed in this Table, though under a set of alternative names and possibly different interpretations.  Just to acknowledge a few of the more notable pseudonyms:

The constant function \(0 ~:~ \underline\mathbb{B}^2 \to \underline\mathbb{B}\) appears under the name \(F_{0}^{(2)}.\)

The constant function \(1 ~:~ \underline\mathbb{B}^2 \to \underline\mathbb{B}\) appears under the name \(F_{15}^{(2)}.\)

The negation and identity of the first variable are \(F_{3}^{(2)}\) and \(F_{12}^{(2)},\) respectively.

The negation and identity of the second variable are \(F_{5}^{(2)}\) and \(F_{10}^{(2)},\) respectively.

The logical conjunction is given by the function \(F_{8}^{(2)} (x, y) = x \cdot y.\)

The logical disjunction is given by the function \(F_{14}^{(2)} (x, y) = \underline{((} ~x~ \underline{)(} ~y~ \underline{))}.\)

Functions expressing the conditionals, implications, or if-then statements are given in the following ways:

\[[x \Rightarrow y] = F_{11}^{(2)} (x, y) = \underline{(} ~x~ \underline{(} ~y~ \underline{))} = [\mathrm{not}~ x ~\mathrm{without}~ y].\]

\[[x \Leftarrow y] = F_{13}^{(2)} (x, y) = \underline{((} ~x~ \underline{)} ~y~ \underline{)} = [\mathrm{not}~ y ~\mathrm{without}~ x].\]

The function that corresponds to the biconditional, the equivalence, or the if and only statement is exhibited in the following fashion:

\[[x \Leftrightarrow y] = [x = y] = F_{9}^{(2)} (x, y) = \underline{((} ~x~,~y~ \underline{))}.\]

Finally, there is a boolean function that is logically associated with the exclusive disjunction, inequivalence, or not equals statement, algebraically associated with the binary sum operation, and geometrically associated with the symmetric difference of sets.  This function is given by:

\[[x \neq y] = [x + y] = F_{6}^{(2)} (x, y) = \underline{(} ~x~,~y~ \underline{)}.\]

Let me now address one last question that may have occurred to some.  What has happened, in this suggested scheme of functional reasoning, to the distinction that is quite pointedly made by careful logicians between (1) the connectives called conditionals and symbolized by the signs \((\rightarrow)\) and \((\leftarrow),\) and (2) the assertions called implications and symbolized by the signs \((\Rightarrow)\) and \((\Leftarrow)\), and, in a related question:  What has happened to the distinction that is equally insistently made between (3) the connective called the biconditional and signified by the sign \((\leftrightarrow)\) and (4) the assertion that is called an equivalence and signified by the sign \((\Leftrightarrow)\)?  My answer is this:  For my part, I am deliberately avoiding making these distinctions at the level of syntax, preferring to treat them instead as distinctions in the use of boolean functions, turning on whether the function is mentioned directly and used to compute values on arguments, or whether its inverse is being invoked to indicate the fibers of truth or untruth under the propositional function in question.