Vague gradable adjectives
Adjectives such as tall, wide, expensive, and happy are often considered to be vague. The example in (1), taken from Kennedy (2007), appears to have somewhat uncertain truth conditions.
- The coffee in Rome is expensive.
It is true if the cost of coffee in Rome is as great as some salient threshold for costs—something associated with the adjective expensive—but this threshold intuitively remains uncertain when attempting to evaluate whether or not (1) might be true. It may range somewhere from 2 euros to 4 euros, for example, but an exact value appears very difficult to pin down.
A hallmark property of vague adjectives like expensive is that they exhibit certain unique inference patterns, such as borderline cases (Kennedy 2007). For example while the Mud Blend ($1.50/lb), might be considered not expensive, and Organic Kona ($20/lb) might be considered expensive, it’s harder to say which category the Swell Start ($9.25/lb) falls into.
Perhaps most famously, vague adjectives—and vague predicates in general—give rise to sorites paradoxes. Such paradoxes arise from considering arguments (known as sorites arguments) that go as follows.
- Premise 1: A $10 cup of coffee is expensive.
- Premise 2: If an expensive cup of coffee were 1 cent cheaper, it would still be expensive.
- Conclusion: Therefore, a free cup of coffee is expensive!
These kinds of inference profiles are notoriously tricky to analyze in terms of the classical notion of truth conditions. Specifically, borderline cases provide instances in which the property denoted by the adjective seems neither to apply nor not to apply to certain entities; but this conflicts with the theoretical requirement that truth conditions provide a definition when a sentence is true—they should bifurcate the space of possible situations into those in which it is true and those in which it is false. Such inference patterns thus at least suggest that traditional model theoretic tools might not be sufficient for studying these kinds of adjectives.
Meanwhile, the sorites paradox is troublesome because it appears to contravene the assumption that inferences should be closed under implication. For example, if we have the following two premises:
- $10 is expensive implies $9.99 is expensive
- $9.99 is expensive implies $9.98 is expensive
We should be able to draw the following conclusion:
- $10.00 is expensive implies $9.98 expensive
And so on, such that we should eventually be able to conclude that $10.00 being expensive (true) implies that $0.00 is expensive (false).