Two methods to find truth value gaps and their application to the projection problem of homogeneity

Presupposition, vagueness, and oddness can lead some sentences to fail to have a clear truth value. The homogeneity property of plural predication with definite descriptions may also create truth-value gaps: The books are written in Dutch is true if all relevant books are in Dutch, false if none of them is, and neither true nor false if, say, half of the books are written in Dutch. We study the projection property of homogeneity by deploying methods of general interest to identify truth-value gaps. Method A consists in collecting both truth-value judgments (completely true vs. not completely true) and, independently, false value judgments (completely false vs. not completely false). The second method, employed in experiment series B and C, is based on one-shot ternary judgments: completely true vs. completely false vs. neither. After a calibration of these methods, we use them to demonstrate that homogeneity projects out of negation, the scope of universal sentences and the scope of non-monotonic quantifiers such as exactly two, to some extent (i.e., in two out of three conceivable kinds of gap situations). We assess our results in light of different theoretical approaches to homogeneity — approaches based on presuppositions, scalar implicatures, and something like supervaluations, respectively. We identify free parameters in these theories and assess various variants of them based on our results. Our experimental paradigms may be of broader significance insofar as they can be applied to other phenomena which result in the failure of a sentence to have a definite truth value. 1 Homogeneity and truth-value gaps The question of whether a sentence is true or false in a given situation has been part of the foundation of formal semantics since the beginning (Frege 1892; Tarski 1935, 1956). Various phenomena, however, may prevent a sentence from being definitely true or false: vagueness, presupposition, oddness, or suboptimal phrasing of one kind or another can all give rise to truth-value judgments that are at least uncomfortable, if not outright impossible. We will describe two methods to identify truth-value gaps empirically. We will do so with a focus on a phenomenon known as homogeneity, which arises with plural definite descriptions. Our primary concern is its projection behaviour from the scope of quantifiers, which may be seen as an analogue of the well-known projection problem for presuppositions (Karttunen and Peters 1979, Heim 1983, and many others since).