Witnesses

Abstract

The meaning of definite descriptions is a central topic in philosophy and linguistics. Indefinites have been relatively neglected by philosophers, under the Russellian assumption that they are simply existential quantifiers. However, a robust set of patterns suggest that this assumption is wrong. In this paper I develop a new approach to (in)definites which aims to capture these patterns. On my theory, truth-conditions are classical. But in addition to truth-conditions, meanings comprise a second dimension, which I call bounds. It is at the level of bounds, not truth-conditions, that the characteristic coordination between indefinites and definites takes place. My system has a classical logic, thus avoiding serious problems which face the most plausible extant account of these patterns, namely, dynamic semantics. More generally, my approach yields a new perspective on the relation between truth-conditions and dynamic effects in natural language.

Appendix: Semantics

For ease of reference, I summarize the semantics given in the text. We specify the truth-conditions for our language, relative to pairs of a world and a variable assignment: where g is a variable assignment, w a world, and p a sentence, $\llbracket {p}\rrbracket ^{g,w}$ is the bivalent truth-value of p at $\left\langle {g,w} \right\rangle$. We simultaneously specify the bounds of sentences relative to a context (a set of assignment-world pairs), an assignment, and a world. Bounds never influence truth-conditions, so contexts play no role in determining the latter. For any context c and sentence p, $c^p$ is the set of world-assignment pairs in c where p is true and satt relative to c. $\mathfrak {I}$ is an atomic valuation taking n-ary atoms and worlds to sets of n-tuples.

• Atoms:

  $\llbracket {A(x_1, x_2, \dots x_n)}\rrbracket ^{g,w}$$=1$ iff $g(x_1), \dots g(x_n)$ are all defined and $\left\langle {g(x_1), \dots g(x_n)} \right\rangle \in \mathfrak {I}(A,w)$, 0 otherwise

  $A(x_1, x_2, \dots x_n)$ is satt at $\left\langle {c,g,w} \right\rangle$ iff $g(x_1), \dots g(x_n)$ are all defined

• Conjunction:

  $\llbracket {p \& q}\rrbracket ^{g,w}$$=1$ iff $\llbracket {p}\rrbracket ^{g,w}$$=\llbracket {q}\rrbracket ^{g,w}=1$

  $p \& q$ is satt at $\left\langle {c,g,w} \right\rangle$ iff p is satt at $\left\langle {c,g,w} \right\rangle$ and q is satt at $\left\langle {c^{p},g,w} \right\rangle$

• Disjunction:

  $\llbracket {p\vee q}\rrbracket ^{g,w}$$=1$ iff $\llbracket {p}\rrbracket ^{g,w}$$=1$ or $\llbracket {q}\rrbracket ^{g,w}=1$

  $p\vee q$ is satt at $\left\langle {c,g,w} \right\rangle$ iff p is satt at $\left\langle {c,g,w} \right\rangle$ and q is satt at $\left\langle {c^{\lnot p},g,w} \right\rangle$

• Negation:

  $\llbracket {\lnot p}\rrbracket ^{g,w}$$=1$ iff $\llbracket {p}\rrbracket ^{g,w}=0$

  $\lnot p$ is satt at $\left\langle {c,g,w} \right\rangle$ iff p is satt at $\left\langle {c,g,w} \right\rangle$

• Indefinites:

  iff $\llbracket {\exists x( p \& q)}\rrbracket ^{ g,w}=1$ iff $\exists a: \llbracket {p \& q}\rrbracket ^{g_{[x\rightarrow a]},w}=1$

  is satt at $\left\langle {c,g,w} \right\rangle$ iff $\exists g': p \& q$ is satt at $\left\langle {c,g',w} \right\rangle$ and $\llbracket {\exists x(p \& q)}\rrbracket ^{ g,w}=1\rightarrow p \& q$ is true and satt at $\left\langle {c,g,w} \right\rangle$

• Definites:

  $\llbracket {\iota x (p,q)}\rrbracket ^{g,w}$$=\llbracket {p \& q}\rrbracket ^{g,w}$

  $\iota x (p,q)$ is satt at $\left\langle {c,g,w} \right\rangle$ iff $\forall \left\langle {g',w'} \right\rangle \in c: p$ is true and satt at $\left\langle {c,g',w'} \right\rangle$ and, if p is true at $\left\langle {c,g,w} \right\rangle$, then q is satt at $\left\langle {c^p,g,w} \right\rangle$

• Quantifiers:

  $\llbracket {\textsc {every} x_\delta (p,q)}\rrbracket ^{g,w}$$=1$ iff $\forall \left\langle {a,g'} \right\rangle \in g(\delta ): \llbracket {p}\rrbracket ^{g'_{[x\rightarrow a]},w}=1 \rightarrow \llbracket {p \& q}\rrbracket ^{g'_{[x\rightarrow a]},w}=1$

  $\textsc {every} x_\delta (p,q)$ is satt at $\left\langle {c,g,w} \right\rangle$ iff

  • $\forall a: \exists ! \left\langle {a',g'} \right\rangle \in g(\delta ) : a'=a;$

  • $\forall \left\langle {a,g'} \right\rangle \in g(\delta ): p$ and $p \& q$ are satt at $\left\langle {c,g'_{[x\rightarrow a]},w} \right\rangle$; and

  • $\forall \left\langle {a,g'} \right\rangle \in g(\delta ) : g'\sim _{c} g$, where $g'\sim _c g$ iff $g'$ agrees with g on the values of all variables which are not novel in c.

  $\llbracket {\textsc {most} x_\delta (p,q)}\rrbracket ^{g,w}$$=1$ iff $\frac{|\{\left\langle {a,g'} \right\rangle \in g(\delta ): \llbracket p \& q\rrbracket ^{ g'_{[x\rightarrow a]},w}=1\}|}{|\{\left\langle {a,g'} \right\rangle \in g(\delta ): \llbracket p\rrbracket ^{g'_{[x\rightarrow a]},w}=1\}|}> .5$

  $\textsc {most} x_\delta (p,q)$ is satt at $\left\langle {c,g,w} \right\rangle$ iff

  • $\forall a: \exists ! \left\langle {a',g'} \right\rangle \in g(\delta ) : a'=a;$

  • $\forall \left\langle {a,g'} \right\rangle \in g(\delta ): p$ and $p \& q$ are satt at $\left\langle {c,g'_{[x\rightarrow a]},w} \right\rangle$; and

  • $\forall \left\langle {a,g'} \right\rangle \in g(\delta ) : g'\sim _{c} g$.