How might we capture these processes of information change within a dynamic semantics for vague predicates? Let P be a vague predicate and OP be its
associated set of degrees. Keeping our static semantics in mind, suppose that we take a state of total ignorance on the part of the listener to be the full set W ×OP. Now, given such a loathsome dearth of knowledge, we add to
the needy listener’s information by asserting sentences φ. The effect of this will be the intersection of our information state with the set of world-degree pairs constituting the proposition expressed by φ. Clearly, then, the dynamic
semantics we will propose is simply one in which the information state of a discourse employing the vague predicateP is taken to be a subset ofW ×OP,
and in which the propositions expressed by utterances of the formP(α) are as was proposed in Section 2.3. For example, the meaning of the sentence “John is tall” is the function [[tall(john)]] defined as follows.
[[tall(john)]] :℘(W×Otall)→℘(W×Otall)
[[tall(john)]] :σ7→σ∩ {hw, δtalli:hw, δtalli |= tall(john)}
Let us see how this proposal coheres with our data from the previous section. To neaten the appearance of our arguments, however, let us first introduce the following definitions.
Definition 5.1 Let σ ⊆ W ×OP be an information state. The degrees
available at σis the set{δ:∃w∈W hw, δi ∈σ}. Theworlds available at
σis the set{w:∃δ∈OP hw, δi ∈σ}.
1. Total Ignorance Suppose that we add the proposition∩tall(john) to our state of total ignoranceσ=W×Otall. The resulting state σ0 will be a proper subset of σ, and so in that sense the assertion was informative. However, σ0 is still such that for every w∈ W there is a pair hw, δtalli ∈ σ0 and for every
δtall∈Otall there is a pairhw, δtalli ∈σ0.
Proposition 5.2 For everyw∈W there is a pairhw, δtalli ∈σ0.
proof: Suppose w ∈W and ftall(w)(john) = δ0
tall. Adding ∩tall(john) to a
stateσremoves all pairshw, δtallisuch thatδtall0 ≤tallδtall. But, ifσisW×Otall,
then sincehOtall, <
tallihas no beginning point, there is a pairhw, δtalli ∈σsuch
that δ0
tall>tallδtall. So hw, δtalli survives the update and is an element of the
resulting stateσ0.
Proposition 5.3 For everyδtall∈Otall there is a pairhw, δtalli ∈σ0.
proof: Similar to that for the previous proposition.
Therefore, the listener still does not have any information about the identity of the height of John or of the contextual standard for “tall.” No possible worlds w or possible standards δtall have been ruled out by the assertion. All the
listener learns from the utterance is that whatever John’s height is, it will be greater than whatever the standard for “tall” turns out to be. Moreover, this information is reflected in the resulting stateσ0, since for every pairhw, δ
talli ∈
σ0,ftall(w)(john)>
2. Ignorance of Context, Awareness of Fact Let us now suppose that our initial knowledge stateσis one in which an upper bound on John’s height is known, but the listener lackscompletely any information about the contextual standard for “tall.” That is, this listener does not have any information as to the identity of this contextual parameter, nor any information linking its identity to factual states of affairs. σ, then, is the full productW0×Otall, whereW0⊂W and∃δtall∈Otall∀w∈W0 ftall(w)(john)≤tallδtall. If we add the information
that John is tall, the resulting state σ0 includes no new information about the world.
Proposition 5.4 The set of worlds available at σ is the same as the set of worlds available atσ0.
proof: Suppose thatwis in the set of worlds available atσandftall(w)(john) =
δ0
tall. Sinceσ= W0×Otall, andhOtall, <talliis without beginning point, there
is a pairhw, δtalli ∈σsuch thatδ0tall>tallδtall. Therefore,hw, δtalli ∈σ0 and so
wis a world available atσ0. Due to the nature of updates, the worlds available at σmust then be equal to the worlds available atσ0.
However, since an upper bound is known for John’s height, the assertion does supply information about the standard for “tall.”
Proposition 5.5 The set of degrees available atσ does not equal the set of degrees available atσ0.
proof: ∃δtall ∈ Otall ∀w ∈ W0 ftall(w)(john) ≤tall δtall. Suppose that
∀w∈W0 ftall(w)(john)≤
tallδ0tall. Consider an arbitrary pairhw, δtall0 i where
w ∈ W0. Since ftall(w)(john) ≤
tall δtall0 , ftall(w)(john) 6>tall δtall0 , and so
w, δ0
tall 6|= tall(john). Therefore, hw, δ0talli would not survive the update with
∩tall(john), and so hw, δ0
talli∈/ σ0. Sincewwas arbitrary,δ0tallis not in the set
of degrees available at σ0. By definition, however, δ0
tall ∈Otall; thus, δtall0 is in
the set of degrees available at σ.
The result is that the upper bound for John’s height now places an upper bound on the contextually given standard for tall.
Proposition 5.6 There exists aδ0
tall∈Otall such that for allhw, δtalli ∈σ0,
δtall<tallδtall0 .
proof: ∃δtall ∈ Otall ∀w ∈ W0 ftall(w)(john) ≤tall δtall. Suppose that
∀w ∈ W0 ftall(w)(john) ≤
tall δtall0 . Let hw, δtalli ∈ σ0. w, δtall |= tall(john).
Therefore, ftall(w)(john)>
tallδtall and soδtall<tallδ0tall. Sincehw, δtalliwas
Thus, the listener receives no information about the world, but he does obtain information regarding the standard employed for “tall”.
3. Total Awareness In our third scenario, the listener has information re- garding both John’s height and the standard used for “tall”. This case is ac- tually the most often occurring, it being a rather fanciful state of affairs where the listener bears no assumptions regarding the contextual standard. Typically, when a speaker asserts that John is tall, the listener accommodates the utter- ance so that their information state includes the information that the standard for “tall” is something plausible for human beings. The consequence is that the information that John is tall is never added to a state of complete ignorance re- garding what counts as tall, and so some information about the world is always attained.
Now, it was said that in this third scenario, it should be possible for a speaker to already believe that John is tall. We would with our adopted formalism represent a state of total awareness σ as some subset of W0 ×Otall
i , where
W0 ⊂W and Otall
i ⊂Otall. Clearly, one such σcould be a state in which for
everyhw, δtalli ∈σ, ftall(w)(john)>tallδtall. Since every pair in this putative
state models “tall(john)”, we thus correctly capture that there are some states of total awareness in which the speakers already believe that John is tall. Similarly, we predict that other states of total awareness may be ones in which speakers already believe that John is not tall.
Furthermore, we observed in Section 2.4 that a listener in a state of total awareness may be in a position where the information that John is tall is helpful, and informs him both of John’s height and of the standard for “tall.” Suppose that our listener knows that Sue is taller than John, that Sue is 5 feet tall and that Mary is not tall. If we add to this state the proposition ∩tall(john), the resulting stateσ0 contains both less worlds and less tallness degrees.
Proposition 5.7 Letσ⊆W0×Otall
i whereW0 ⊂W andOtalli ⊂Otall. Letσ
contain only the information that Sue is taller than John, that Sue is 5 feet tall and that Mary is not tall. The update ofσ with ∩tall(john) results in a state
σ0 such that the set of worlds available at σ does not equal the set of worlds available atσ0.
proof: Consider the worldwin the set of worlds available atσin which John is shorter than Mary; i.e., ftall(w)(john) <
tall ftall(w)(mary). Let hw, δtalli
be a pair inσ. Since hw, δtalli 6|= tall(mary),ftall(w)(mary)≤tallδtall, and so
ftall(w)(john)<
tall δtall, and thushw, δtalli 6|= tall(john). Therefore, hw, δtalli
does not survive the update with ∩tall(john) and so hw, δ
talli ∈/ σ0. Since
hw, δtalli was arbitrary, w does not survive in σ0, and so w is not in the set
of worlds available atσ0.
Proposition 5.8 Letσ be as is stated in Proposition 5.7. The update ofσ with∩tall(john) results in a stateσ0 such that the set of degrees available atσ
does not equal the set of degrees available atσ0.
proof: Consider the tallness degreeδtallin the set of degrees available atσsuch
that δtall= 5f t11. Lethw, δtallibe an element of σ. Sinceftall(w)(sue)>tall
ftall(w)(john) andftall(w)(sue) =δ
tall, ftall(w)(john)<tall δtall. Therefore,
hw, δtalli 6|= tall(john), and sohw, δtalli∈/ σ0. Sincehw, δtalliwas arbitrary, δtall
does not survive inσ0, and soδ
tallis not in the set of degrees available atσ0.
The reader is also invited to confirm that we may with this dynamic seman- tics predict that in the resulting state σ0 the speakers know that John is taller than Mary, that the standard for tallness is below five feet, that Sue is tall, and that Mary is below five feet in height.
4. Awareness of Context, Ignorance of Fact Finally, let us suppose that our initial knowledge state σ is one in which a lower bound on the contextual standard for “tall” is known, but the listener lacks completely any information regarding the vertical dimension of John. This listener, then, does not have any information regarding the height of John, nor any information linking his height to the identity of the standard for “tall.”σ, is therefore the full product W0×Otall
i where W0⊆W,Oitall⊂Otall,∃δtall0 ∈Otall∀δtall∈Otalli δtall>tall
δ0
talland∀δtall∈Otall,∃w∈W0 such thatftall(w)(john) =δtall. If we add the
information that John is tall, the resulting stateσ0 includes no new information about the standard for “tall.”
Proposition 5.9 The set of degrees available at σ is the same as the set of degrees available atσ0.
proof: Suppose that δtall is in the set of degrees available at σ. Since σ =
W0 ×Otall
i , there is a pair hw, δtalli ∈ σ such that ftall(w)(john) >tall δtall.
Therefore, hw, δtalli survives update with ∩tall(john) and so hw, δtalli ∈ σ0.
Thus,δtallis in the set of degrees available atσ0, and by the nature of update,
the degrees available atσand at σ0 must be the same.
However, since there exists a lower bound for the tallness standard, such an assertion does supply information about John’s height.
Proposition 5.10 The set of worlds available atσdoes not equal the set of worlds available atσ0.
proof: ∃δ0
tall∈Otall∀δtall∈Otalli δtall>tallδ0tall. Therefore,∃w∈W0∀δtall∈
Otall
i ftall(w)(john)<tallδtall. Suppose that∀δtall∈Oitallftall(w0)(john)<tall
δtall. Consider an arbitrary pair hw0, δtalli, where δtall is a degree available at
σ. Sinceftall(w0)(john)<
tallδtall, w0, δtall6|= tall(john). Therefore, hw0, δtalli
would not survive the update with ∩tall(john), and so hw0, δ
talli ∈/ σ0. Since
hw0, δ
talliwas arbitrary, w0 is not a world available at σ0. By definition, how-
ever,w0 is a world available atσ.
The result is that the lower bound for the standard of tallness now places a lower bound on John’s height.
Proposition 5.11 There exists aδ0
tall∈Otallsuch that for allhw, δtalli ∈σ0,
ftall(w)(john)>
tallδ0tall.
proof: Letδ0
tallbe such that ∀δtall∈Otalli δtall>tall δtall0 . Lethw, δtalli ∈σ0.
By definition, w, δtall|= tall(john). Therefore,ftall(w)(john)>tallδtall and so
ftall(w)(john)>
tallδtall0 . Since hw, δtalliwas arbitrary, for any hw, δtalli ∈σ0
ftall(w)(john)>
tallδ0tall.
Therefore, just as we informally observed in Section 2.4, although the listener receives no information about the standard for “tall”, he does obtain informa- tion regarding John’s height.
The proofs above demonstrate that the proposed dynamic semantics for vague predicates is at least a plausible way of representing the information con- tained in assertions employing those predicates and how such assertions affect the knowledge of a listener. Equipped with this semantics, we could, if we wished, pursue further this study of the conversational dynamics of vague lan- guage. Indeed, Kyburg & Morreau 2000 and Barker 2002 independently develop similar systems in their more focused investigations of the subtle interplay be- tween speakers using vague predicates. Although we certainly recommend that our semantic analysis be used for such research, in this thesis our aim is a bit to the right of that target. Our curiosity is focused on how this analysis might pressed into providing us some insight into the relationship between standard- sensitive context dependency, having borderline cases, and being susceptible to sorites paradoxes. In the following section, we will begin indulging this curiosity by connecting our proffered dynamic semantics to the fact that all standard- sensitive context dependent predicates have the potential for borderline cases.