To find our way into a reasonable dynamic semantics for vague predicates, let us first recall what is standardly said about the semantics of gradeable adjectives used as predicates. The following is an abstraction and simplification of ideas found in the works of Cresswell 1976, Kennedy 1997, Seuren 1978, and von Stechow 1984, among others.
When used as predicates, gradable adjectives are undoubtedly vague. How- ever, in many works on the semantics of gradable adjectives, the author begins by “bracketing” the issue of their vagueness. That is, the vagueness of these predicates is typically considered beyond the scope of such studies 3, and the simplifying assumption is made that gradable adjectives are not vague. I will thus follow suit, but the reader can be assured that their vagueness will re-enter the arena soon.
What can we say about the semantics of the sentence “John is tall”, given this simplified picture? The analysis typically offered would seek to capture the standard-sensitive context dependency of this predicate by proposing the exis- tence of a “tallness scale”, a linearly ordered infinite set of “tallness degrees.” These tallness degrees and their ordering relation are usually taken as primi- tive, though some analyses attempt to “construct” these entities from classes of objects or contexts4. It is then hypothesized that “John is tall” is true if and only if the degree on the tallness scale assigned to John is greater than the “standard” degree of tallness provided by the context5.
Getting down to the formalism, we assign to “John is tall” the following truth conditions.
w, δtall|= tall(john) iff ftall(w)(john)>tallδtall
The analysis we adopt, then, is that “John is tall” is true at a worldwand some adopted standardδtalliff John’s tallness atwis greater thanδtall. John’s tallness
at w is represented by the formula ftall(w)(john), where ftall is a function
from worlds and individuals to tallness degrees. Intuitively, this function takes an individual to its degree of tallness at w. Given this truth definition, it is obvious how we should treat the proposition that John is tall. ∩tall(john) is simply the set of all pairshw, δtallisuch thatw, δtall|= tall(john).
Although analyses of this form are usually aimed at gradable adjectives, there is no reason why this approach could not be generalized to all predicates with standard-sensitive context dependency. That is, we propose that for any
3The studies I have in mind are primarily working towards a compositional semantics for
the comparative.
4If the reader is skeptical of our taking such “degrees” as primitive, we will in the next
chapter discuss some points in its favor.
5It should be pointed out that this semantics is neutral on the issue of whether the “stan-
dards” required for the interpretation of standard-sensitive predicates are “comparison classes” or something distinct from the implicit comparison class. The postulation of this contextually determined degree, for example, could be understood as abstracting from all the details going into the calculation of what might be expected from a member of the implicit comparison class.
standard-sensitive predicateP, there exists an associated infinite set of degrees OP, linearly ordered without beginning or end point by a relation <
P, and an
associated function fP from pairs of worlds and objects to OP 6 7. This scale and function are then used to interpretP in the expected way. Whereδ∈OP
andαis some object in the domain ofw, we offer the following truth conditions. w, δ|=P(α) ifffP(w)(α)>
P δ
Similarly, the proposition ∩P(α) is simply the set {hw, δi : w, δ |=P(α)}, or equivalently,{hw, δi:fP(w)(α)>
P δ}.
The reader may worry that this definition assumes our language to include only one standard-sensitive context dependent predicate. How should we in- terpret the sentenceP(α)∧Q(β), where bothP andQare standard-sensitive? One solution is obvious 8, but throughout this thesis we will make the simpli- fying assumption that our language has only one standard-sensitive predicate, usually written P. Due to the relationship between standard-sensitive context dependency and vagueness, this will mean that we likewise assume throughout that P is our language’s only vague predicate.
The reader may also be reluctant about extending the notion of a linearly ordered scale of degrees to other standard-sensitive predicates besides grad- able adjectives, such as nouns. Indeed, some theorists have expressed remorse over even extending this analysis to “multi-dimensional” gradable adjectives like “nice.” If this reluctance arises purely from our straightforward appeal to the “degrees” of a predicate, the reader is advised to check his reluctance until hav- ing read our discussion in Section 3.5 supporting such an approach. However, if the reader accepts degrees as representations of the standards used in interpret- ing gradable adjectives, it is unclear why he should be in principle unwilling to accept degrees as representations of the standards used in interpreting nouns.
The source of the reader’s reluctance here could also be the fact that we assume the degrees of a predicate to be in alinear ordering. Are the degrees of redness or of fatherliness to be found linearly ordered? To mitigate this worry, it may be said that the standards of these predicates certainly appear to beordered
by some measure of magnitude. One speaks informally of a person’s “measure of niceness” or “level of fatherliness” being greater than another’s. Our assumption that the ordering is linear merely allows a simpler formal model of how the degree to which a predicate holds of an object interacts with the contextually
6As in many philosophical works on the analysis of vague predicates, the formal meta-
language which we employ throughout this thesis will not be explicitly defined. The reader may understand this language to include nothing more than the logical constants and syntax available from first order logic. Moreover, at every sequence within a state, unless a sentence is of the formP(α) wherePis a standard-sensitive predicate, a sentence is interpreted exactly as in a standard first order model structure.
7Regarding the lack of beginning or end point for <
P, this is intended to capture the
intuition that if an object has a property to any degree, then it is possible for that object to have the property to a greater or a lesser degree. This intuition, however, requires that objects entirely lacking a property cannot have that property to any degree.
8Extend the sequence of indices to include degrees for all the vague predicates in the
given standard for the predicate to determine the truth of an utterance. The reader will find that none of our core claims regarding the nature of vagueness rest on the ordering of these standards being linear. Our claimswill rest on the existence of such an ordering of standards, and on this ordering having certain other properties, but its linearity is for now only a matter of convenience. The reader may, if he likes, scoff at the very idea.
Finally, because of the correlation between standard-sensitive context depen- dency and vagueness, the classic semantics offered above for standard-sensitive predicates is thereby our proposed classic analysis for all vague predicates.