Propositions, representation, and truth

Abstract

Theories of propositions as sets of truth-supporting circumstances are committed to the thesis that sentences or other representations true in all and only the same circumstances express the same proposition. Theories of propositions as complex, structured entities are not committed to this thesis. As a result, structured propositions can play a role in our theories of language and thought that sets of truth-supporting circumstances cannot play. To illustrate this difference, I sketch a theory of transparent, non-deflationary truth consistent with some theories of structured propositions, but inconsistent with any theory of propositions as sets of truth-supporting circumstances.

Appendix A: Soames’s fineness of grain argument and Ripley’s response

Following  Barwise and Perry (1985, p. 153), I distinguish between Soames’s argument and Soames’s derivation. Soames’s derivation shows that any circumstantialist theory committed to certain natural semantic assumptions, including that names, indexicals, and variables are directly referential, predicts that (5) entails (6):

(5)

(6)

I will not reproduce all of Soames’s derivation here. To illustrate the role of the Representation Thesis, I will focus on one fragment that reveals Soames’s argument to be a fineness of grain argument. This fragment relies on the semantic theses Compositionality, Conjunction, Direct Reference, and Existential Quantification:

Compositionality

If S$_{1}$ and S$_{2}$ are non-intensional sentences/formulas with the same grammatical structure, which differ only in the substitution of constituents with the same semantic contents (relative to their respective contexts and assignments), then the semantic contents of S$_{1}$ and S$_{2}$ will be the same (relative to those contexts and assignments).

Conjunction

A sentence is true at a circumstance w, relative to a context c and assignment function f, if and only if both A and B are true at w relative to c and f.

Direct Reference

Proper names, indexicals (relative to contexts), and variables (relative to assignments) are directly referential.

Existential Quantification

A sentence is true at a circumstance w, relative to a context c and assignment f, if and only if $\phi$ is true at w relative to c and some v-variant of f (where a v-variant of f is a function that differs from f at most in what it assigns to v).

Given these assumptions, the fragment comprises the following lemmas:

Lemma 1

(11) is true at every circumstance at which (10) is true:

$$\begin{aligned}&\mathrm{`Hesperus'}~\mathrm{refers}~\mathrm{to}~\mathrm{Hesperus}, \mathrm{and}~\mathrm{`Phosphorus'}~\mathrm{refers}~\mathrm{to}~\mathrm{Hesperus.}\end{aligned}$$

(10)

$$\begin{aligned}&\exists x \mathrm{(`Hesperus'}~\mathrm{refers to}~x~\mathrm{and}~\mathrm{`Phosphorus'}~\mathrm{refers to }~x). \end{aligned}$$

(11)

Proof

Assume that (10) is true at some circumstance w relative to some context c and assignment f. By Direct Reference, both the name ‘Hesperus’ and the variable ‘x’ relative to the ‘x’-variant of f that assigns Venus to ‘x’ directly refer to Venus. So via Compositionality, the truth of (10) at w relative to c and f guarantees the truth at w of the open formula

$$\begin{aligned} \hbox {`Hesperus' refers to } x \hbox { and `Phosphorus' refers to } x \end{aligned}$$

relative to c and the ‘x’-variant of f that assigns Venus to ‘x’, because the formulas have the same content (relative to c and the respective assignments). So by the right-to-left direction of Existential Quantification, (11) is true at w relative to c and f. $\square$

Lemma 2

The conjunction is true at a circumstance w iff (10) is true at w.

Proof

Left to right: assume that is true at a circumstance w (relative to c and f—omitted henceforth). Then by Conjunction, (10) is true at w. Right to left: assume that (10) is true at w. Then by Lemma 1, (11) is true at w. Hence both (10) and (11) are true at w, and so by Conjunction is . $\square$

The set of circumstances at which a sentence is true is the truth-conditional content of the sentence. Thus Lemma 2 entails that (10) and are a representational pair. Given Circumstantialism (and hence the Representation Thesis), it follows that (10) and express the same proposition.

It is at this stage that Soames’s argument becomes a fineness of grain argument. Soames’s derivation as a whole goes through only if the result above—that (10) and express the same proposition—holds. Distinguishing between the propositions expressed by (10) and would block the problematic derivation, and Soames argues that other attempts to block the derivation face independent problems. If these further arguments are correct, we have independent grounds for distinguishing propositions that Circumstantialism identifies.

Ripley (2012), building on Priest’s (2005) semantics for impossible and open worlds, shows how to reject Soames’s argument.³³ The account of quantification in Priest illustrates this. Ripley adopts Priest’s notion of a matrix:

Call a formula a matrix, if all its free terms are variables, no free variable has multiple occurrences and—for the sake of definiteness—the free variables that occur in it are the least variables greater than all the variables bound in the formula, in some canonical ordering, in ascending order from left to right. (Priest 2005, p. 17)

Priest and Ripley adopt the following notational convention: where C is any matrix containing the exactly the variables $v_{i} \ldots v_{j}$ free, and $t_{i}, \ldots , t_{j}$ a sequence of terms (some of which may be variables), $C(t_{i},\ldots , t_{j})$ is the unique formula that results from substituting $t_{i}$ for $v_{i}$, ..., and $t_{j}$ for $v_{j}$. Given this convention, every formula is the result of substituting a unique sequence of terms in a unique matrix. The unique matrix from which a formula A results via the appropriate substitution of terms is called the matrix, $\overline{A}$, of A. So $\overline{(11)}$ is (12) (assuming a natural alphabetic ordering of the variables), and (13) is a notational variant of (11) given the convention above:

$$\begin{aligned}&\exists x (y \hbox { refers to }x\hbox { and }z\hbox { refers to }x.)\end{aligned}$$

(12)

$$\begin{aligned}&\exists x (y \hbox { refers to }x\hbox { and }z\hbox { refers to }x\hbox {)(` `Hesperus' ',` `Phosphorus' ')} \end{aligned}$$

(13)

At logically impossible circumstances, Priest and Ripley treat matrices as n-place predicates that are assigned arbitrary extensions. For atomic formulas, Ripley introduces a general purpose denotation function, or $[\![ \, \, ]\!]$, that maps terms to objects or individuals and maps predicates (and matrices) to intensions (functions from circumstances to extensions). Each denotation function $[\![ \, \, ]\!]$ also determines a unique assignment f of values to variables. We can now state the rule for impossible quantification from Ripley (2012, p. 110):

Impossible Quantification

For a quantified sentence $A = \overline{A}(t_{1},t_{2},\ldots ,t_{n})$:

A is true at a circumstance w (relative to a context c—but I’ll follow Ripley in ignoring this while discussing his argument) if and only if $\left\langle [\![ \,t_{1}\, ]\!], [\![ \,t_{2}\, ]\!],\ldots , [\![ \,t_{n}\, ]\!] \right\rangle \in [\![ \,\overline{A}\, ]\!](w)$

To maintain the proper behavior of existential quantification at logically possible circumstances, we must restrict the denotation of the matrix in various ways. Assume that we have done so. (Alternatively, assume that Impossible Quantification only applies at logically impossible circumstances, and that at all other circumstances, quantifiers are governed by rules like Existential Quantification above.) Now let w be a circumstance at which (10) is true, but at which the extension of the matrix (12) of (11) does not include the pair $\left\langle \hbox {`Hesperus'}, \hbox {`Phosphorus'} \right\rangle$. Then (13) is not true at w. But since (13) is a notational variant of (11), (11) is also not true at w. Thus w is both logically impossible and open. Given circumstances such as w, the proof of Lemma 1 fails. This blocks Soames’s derivation.