Anatomy of a proposition

Abstract

This paper addresses the mereological problem of the unity of structured propositions. The problem is how to make multiple parts interact such that they form a whole that is ultimately related to truth and falsity. The solution I propose is based on a Platonist variant of procedural semantics. I think of procedures as abstract entities that detail a logical path from input to output. Procedures are modeled on a function/argument logic, but are not functions (mappings). Instead they are higher-order, fine-grained structures. I identify propositions with particular kinds of molecular procedures containing multiple sub-procedures as parts. Procedures are among the basic entities of my ontology, while propositions are derived entities. The core of a structured proposition is the procedure of predication, which is an instance of the procedure of functional application. The main thesis I defend is that procedurally conceived propositions are their own unifiers detailing how their parts interact so as to form a unit. They are not unified by one of their constituents, e.g., a relation or a sub-procedure, on pain of regress. The relevant procedural semantics is Transparent Intensional Logic, a hyperintensional, typed $\lambda$-calculus, whose $\lambda$-terms express four different kinds of procedures. While demonstrating how the theory works, I place my solution in a wider historical and systematic context.

Appendix

This appendix contains four definitions accompanied by remarks for clarification.

Above I have been using the term ‘procedure’. Tichý’s own term of art was ‘construction’. I prefer ‘procedure’ because the term seems more evocative than ‘construction’, which furthermore has misleading idealist connotations. Whichever the term, the inductive definition below lays down what TIL procedures are. They are embedded within a typed universe. The following three definitions constitute the core of TIL.

Definition 1

(types of order 1). Let B be a base, where a base is a collection of pair-wise disjoint, non-empty sets. Then:

• (i) Every member of B is an elementary type of order 1 over B.

• (ii) Let $\upalpha , \upbeta _{1}, \ldots ,\upbeta _{m} (m >0)$ be types of order 1 over B. Then the collection ($\upalpha \,\,\upbeta _{1 }\ldots \upbeta _{m})$ of all m-ary partial mappings from $\upbeta _{1} \times \cdots \times \upbeta _{m}$ into $\upalpha$ is a functional type of order 1 over B.

• (iii) Nothing is a type of order 1 over B unless it so follows from (i) and (ii).

$\square$

Remark 1

For the purposes of natural-language analysis, we are currently assuming the following base of ground types, which form part of the ontological commitments of TIL:

• $\hbox {o}$: the type of truth-values = {T, F}

• $\upiota$: the type of individuals (the universe of discourse)

• $\uptau$: the type of real numbers (doubling as discrete times)

• $\upomega$: the type of logically possible worlds (the logical space)

For the purposes of a fairly restricted arithmetic discourse, these may suffice as ground types:

• $\hbox {o}$: the type of truth-values = {T, F}

• $\upnu$: the type of natural numbers

Definition 2

(procedure).

• (i) The variable x is the procedure that produces an object Xof the respective type dependently on a valuation v; xv-producesX.

• (ii) Where Xis an object whatsoever (an extension, an intension or a procedure), Trivialization is the procedure ${}^{0}X$. ${}^{0}X$ produces X without any change of X.

• (iii) The Composition $[X Y_{1}{\ldots }Y_{m}]$ is the following procedure. If Xv-produces a function gof a type $(\upalpha \upbeta _{1}{\ldots }\upbeta _{m})$, and $Y_{1}, {\ldots }, Y_{m}$v-produce entities $\hbox {B}_{1}, {\ldots }, \hbox {B}_{m}$ of types $\upbeta _{1}$, ..., $\upbeta _{m}$, respectively, then the Composition $[X Y_{1}{\ldots }Y_{m}]$v-produces the value (an entity, if any, of type $\upalpha$) of g on the tuple argument $\langle \hbox {B}_{1}, {\ldots }, \hbox {B}_{m}\rangle$. Otherwise the Composition$[X Y_{1}{\ldots }Y_{m}]$ does not v-produce anything and so is v-improper.

• (iv) The Closure [$\lambda x_{1}{\ldots }x_{m} Y$] is the following procedure. Let $x_{1}, x_{2}, {\ldots }, x_{m}$ be pair-wise distinct variables v-producing entities of types $\upbeta _{1}, {\ldots }, \upbeta _{m}$ and Y a procedurev-producing an $\upalpha$-entity. Then $[\lambda x_{1}{\ldots } x_{m} Y]$ is the procedure$\lambda$-Closure. It v-produces the following function f of the type $(\upalpha \upbeta _{1}{\ldots }\upbeta _{m})$. Let $v(\hbox {B}_{1}/x_{1},{\ldots },\hbox {B}_{m}/x_{m})$ be a valuation identical with v at least up to assigning objects $\hbox {B}_{1}/\upbeta _{1}, {\ldots }, \hbox {B}_{m}/\upbeta _{m}$ to variables $x_{1}, {\ldots }, x_{m}$. If Y is $v(\hbox {B}_{1}/x_{1},{\ldots },\hbox {B}_{m}/x_{m})$-improper (see iii), then f is undefined on $\langle \hbox {B}_{1}, {\ldots }, \hbox {B}_{m}\rangle$. Otherwise the value of f on $\langle \hbox {B}_{1}, {\ldots }, \hbox {B}_{m}\rangle$ is the $\upalpha$-entity $v(\hbox {B}_{1}/x_{1},{\ldots },\hbox {B}_{m}/x_{m})$-produced by Y.

• (v) Nothing is a procedure, unless it so follows from (i) through (vi).$\square$

Remark 2

Def. 2 leaves out the procedures Single and Double Execution, ${}^{1}X$ and ${}^{2}X$, which we do not need for the present foundational study. I will just note that Double Execution is the dual of Trivialization. Thus, the effect of applying Double Execution to Trivialization is the annihilation of the effect of the latter. Hence ${}^{20}X$ is equivalent to X. Trivialization raises the type whereas Double Execution lowers it. See my (2015: pp. 328–329) for definitions and elucidation.

Remark 3

Since ${}^{1}X$ and ${}^{2}X$ are Single and Double Execution, respectively, it would be natural if ${}^{0}X$ was known as Zero Execution. In fact, whereas ‘Trivialization’ gives the wrong idea about Trivialization, which is anything but trivial, ‘Zero Execution’ sums up what ${}^{0}X$ is all about: ${}^{0}X$displaysX. If X is a procedure, then ${}^{0}X$ does not proceed to executing X. This is in fact the gist of Tichý’s original definition that ${}^{0}X$ produces Xwithout any change of X. In this paper, however, I will stick to the original term ‘Trivialization’ for continuity.

Remark 4

Our procedural reading of the $\lambda$-calculus affects how we count occurrences of variables. Syntactically, ‘$\lambda x$ [${}^{0}{\textit{Odd}}\,\,x$]’ contains two occurrences of ‘x’. Procedurally, $\lambda x\,\, [{}^{0}{\textit{Odd}}\,\,x]$ contains only one occurrence of x, namely in [${}^{0}{\textit{Odd}}\,\,x$]. $\lambda x$ is the procedure of abstracting over the values assigned to x, but the syntactic occurrence of ‘x’ serves only to connect to the x in [${}^{0}{\textit{Odd}}\,\,x$] in order to specify that the variable over whose values is to be abstracted is x, something which ‘$\lambda y\,\, [{}^{0}{\textit{Odd}}\,\,x]$’ would fail to express.

Definition 3

(ramified hierarchy of types).

$\hbox {T}_{1 }$(types of order 1). See Def. 1.

$\hbox {C}_{n}$ (procedures of order n)

• (i) Let x be a variable ranging over a type of order n. Then x is a procedure of order n over B.

• (ii) Let X be a member of a type of order n. Then ${}^{0}X$ is a procedure of order n over B.

• (iii) Let $X, X_{1},\ldots , X_{m} (m > 0)$ be procedures of order n over B. Then [$X X_{1}\ldots X_{m}$] is a procedure of order n over B.

• (iv) Let $x_{1},\ldots x_{m}, X (m > 0)$ be procedures of order n over B. Then [$\lambda x_{1}\ldots x_{m}\,X$] is a procedure of order n over B.

• (v) Nothing is a procedure of order n overB unless it so follows from $\hbox {C}_{n}$ (i)–(iv). $\hbox {T}_{n+1}$ (types of order n + 1). Let $*_{n}$ be the collection of all procedures of order n over B. Then:

  • (a) $*_{n}$ and every type of order n are types of order n + 1.

  • (b) If $0 < m$ and $\upalpha$, $\upbeta _{1},\ldots ,\upbeta _{m}$ are types of order $n + 1$ over B, then $(\upalpha \,\upbeta _{1} \ldots \upbeta _{m})$ (see $\hbox {T}_{1}$ ii)) is a type of order n + 1 over B.

  • (c) Nothing is a type of order n + 1 over B unless it so follows from T$_{n+1}$ (a) and (b). $\square$

The final definition is of displayed and executed procedures.

Definition 4

(displayed vs. executed procedure). Let C be a procedure and D a sub-procedure of C. Then:

• (i) If D is identical to C then the occurrence of D is executed in C.

• (ii) If C is identical to [$X_{1} X_{2} {\ldots }X_{m}$] and D is identical to one of the procedures$X_{1}, X_{2},{\ldots }, X_{m}$, then the occurrence of D is executed in C.

• (iii) If C is identical to [$\lambda x_{1} {\ldots }x_{m}\,\, X$] and D is identical to X, then the occurrence of D is executed in C.

• (iv) If an occurrence of D is executed in $C^\prime$ and this occurrence of $C^\prime$ is executed in C, then the occurrence of D is executed in C.

• (v) If an occurrence of a sub-procedureD of C is not executed in C then the occurrence of D is displayed in C.

• (vi) No occurrence of a sub-procedureD of C is executed/displayed in C unless it so follows from (i)–(vi).$\square$

Remark 5

Words come with the convention that they point to an object beyond themselves. Procedures come with the analogous convention that they point beyond themselves to a product. It is no accident that the default mode is the executed one. The primary purpose of a procedure is to be a path to a product so that the latter may be the argument of a function. But there are more ‘theoretical’ cases where a procedure becomes itself the object of study. One such case is hyperintensional attitude contexts. Another is the theoretical study of our apparatus, like when scrutinizing the unity and structure of propositions.

I have described in the main text two kinds of atomic propositions, while cursorily touching upon a third one. Since propositions are propositional unifiers in the particular sense I described above, we can individuate the three kinds of unifiers in terms of the sort of procedures they are.

• Where A is of type $\hbox {o}_{\uptau \upomega }$, the proposition ${}^{0}A$ is a Trivialization. It is a mereological atom, so it has unity in the minimal, if not vacuous sense that it is a structure whose zero proper parts are unified into a whole. ${}^{0}A$ is not a case of monadic predication; ${}^{0}A$ is a logical black box.

• Where F is of type ($\hbox {o}\upalpha$) and a of $\upalpha$, the non-empirical proposition [${}^{0}F\,\,{}^{0}a$] is a Composition. It is a mereological molecule with two proper parts, two sub-procedures, whose products are such that one is to be applied to the other. [${}^{0}F\,\,{}^{0}a$] is a fine-grained truth-condition that is typed to produce a truth-value.

• Where G is of type $(\hbox {o}\upbeta )_{\uptau \upomega }$ and b of $\upbeta$, the empirical proposition $\lambda w\lambda t$ [${}^{0}G_{wt}\,\,{}^{0}b$] is a Closure. It is a mereological molecule with eight proper parts, one of which is a Composition that is the predication of G of b. This predication is indexed to worlds and times. The twofold $\lambda$-abstraction is the procedure that the relevant characteristic function of $G_{wt}$ is applied to b at any world and at any time that are selected as points of evaluation.

By displaying the last two of these propositions, we are able to express within the theory itself that both of them unify sub-propositional procedures into the procedures that are the propositions in question. (Remember that if they occurred executed instead, we would be expressing something about their respective products.) ${}^{0}A$ is excluded because it does not unify one or more proper parts into a whole (nor, of course, does ${}^{0}A$ unify itself).

I will exemplify how this works by means of the minimal proposition [${}^{0}F\,\,{}^{0}a$]. The type of Unify is here ($o{}*_ 1{}*_1{}*_1)$, ${}*_{1}$ first-order procedures, where the two right-most types type the two input parts and the left-most higher-order type types the output whole. The result of applying the former to the latter is T if they are indeed the proper parts of the whole, and otherwise F. The unifier is displayed and can therefore occur as a functional argument. We need ramification to pull this off, the Trivialization ${}^{0}[{}^{0}{} \textit{Odd}\,\,{}^{0}1]$ being of type ${}*_2$, i.e., a second-order procedure.

$$\begin{aligned} \left[ {}^{0}{\textit{Unify}}\,\,{}^{0}\left[ {}^{0}{{\textit{Odd}}}\,\,{}^{0}1\right] \,\,{}^{00}{\textit{Odd}}\,\,{}^{00}1\right] \end{aligned}$$

The gloss is that what the Trivialization ${}^{0}[^{0}{\textit{Odd}}\,\,{}^{0}1]$ produces, namely the Composition $[{}^{0}{{\textit{Odd}}}\,\,{}^{0}1]$, unifies what ${}^{00}{{\textit{Odd}}}$ and ${}^{00}1$ produce, namely ${}^{0}{\textit{Odd}}$ and ${}^{0}1$ and in that order, into $[{}^{0}{\textit{Odd}}\,\,{}^{0}1]$: two Trivializations are unified into one Composition. We know what the ‘glue’ is between the parts, namely the procedure of predicating the product of ${}^{0}{\textit{Odd}}$ of the product of ${}^{0}1$. Propositional glue, in TIL, is not one item among several items, but instead a molecular procedure that is the form of a content.