2.9 The Semantics of “Definitely”
2.9.1 A First, Crude Pass
The operator “D” is to be read as “it is definitely the case that.” We add it to our language with the following syntactic clause.
Definition 9.1 Ifφis a formula of our language, thenDφis a formula of our language.
With the introduction ofD into our language, we may now also introduce the operatorI, read as “it is indefinite whether” and given the following, standard definition.
Definition 9.2 Iφ=df ¬Dφ∧ ¬D¬φ
Now, the intuitive meanings of D and I place certain restrictions on the semantic definition we offer forD. φshould be T ifDφis T, and ifφis not T, thenDφshould not be T either. Given our definition of “truth,” we might offer the following semantics forD as a first pass.
Definition 9.3 Let σ ⊆W ×OP be an information state. [[Dφ]](σ) = σ if
[[φ]](σ) =σ. Otherwise, [[Dφ]](σ) =∅.
That this definition is sufficient for capturing the intuitive relationship between the truth ofDφandφmay be seen via the following proposition.
Proposition 9.4 Letσ⊆W ×OP be an information state. Dφis T atσiff
φis T atσ. Dφis F atσotherwise.
proof: By Definition 6.1 and Definition 6.2,Dφis T atσiff [[Dφ]](σ) =σ, iff, by Definition 9.3, [[φ]](σ) =σ, iff, by Definition 6.1 and 6.2,φis T atσ.
Now suppose thatφ is not T atσ. By Definition 6.1 and 6.2, [[φ]](σ)6=σ, and so by Definition 9.3, [[Dφ]](σ) =∅. Finally, by Definition 6.1 and 6.2,Dφis F atσ.
With this primitively dynamic semantics forD, however, we must alter slightly our present interpretation of the sentence connectives, adopting instead a prim- itively dynamic interpretation for them as well.
Definition 9.5 Ifφis a sentence of the form¬ψ, then [[φ]](σ) ={hw, δi ∈σ:
hw, δi∈/[[ψ]](σ)}. Ifφis a sentence of the formψ∧χ, then [[φ]](σ) = [[χ]]([[ψ]](σ)). Proposition 9.4 clearly entails thatφis T atσifDφis, andDφis F atσif φis F. Another pleasant result of this semantics is thatIφis T at a state σif and only ifφis I atσ.
Proposition 9.6 Iφis T atσif and only ifφis I atσ.
proof: Suppose thatφis I atσ. Then, there are sequencess∈σwhich model φ and ones which do not. Thus, [[φ]](σ)6=σ and [[¬φ]](σ)6=σ, and so Dφis F atσas well asD¬φ. By Definition 9.5, then,¬Dφand¬D¬φare both T atσ, and so by Definitions 9.2 and 9.5,Iφis T atσ.
Now suppose thatφ is not I atσ. Thus,φ is either T or F at σ. Suppose that φ is T at σ. Then, by Definition 9.3 and Definition 9.5, [[¬Dφ]](σ) = ∅. Moreover, by Definition 9.3 and 9.5, [[¬D¬φ]](∅) =∅. Thus, by Definition 9.2 and 9.5, [[Iφ]](σ) =∅and so Iφis not T atσ. Suppose that φis F at σ. Then, by Definition 9.3 and 9.5, [[¬Dφ]](σ) =σ. However, by Definition 9.3 and 9.5, [[¬D¬φ]](σ) =∅, which entails by Defintion 9.2 and 9.5, that [[Iφ]](σ) =∅. Thus,
Iφis not T atσ.
Our semantics for D thus captures an intuitive relationship between the truth value of a sentence φand the truth values of the sentences Dφ and Iφ. Moreover, it also correctly relates the meaning of “it is definitely the case thatφ” to a speaker’s knowing thatφ. For example, if one says that Dave is “definitely tall,” one means that there is no possible doubt concerning Dave’s tallness. Formally, we would represent this situation as an information stateσat which tall(dave) is supported, and so our semantics stands as a plausible analysis of these natural language expressions.
Happily, the addition of D andI to our language does not seriously affect the nice results obtained in the last section. For example, we have not lost the deduction theorem for our language.
Proposition 9.7 Letφ,ψbe any two sentences of our extended language. ψ is a consequence ofφiffφ→ψis valid.
proof: Recall thatφ→ψis defined in section 8.1 as ¬(¬¬φ∧ ¬ψ), which by Definition 9.5 is equivalent to¬(φ∧ ¬ψ).
First, let us suppose that ψ is a consequence of φ, and let σ ⊆ W ×OP
be an arbitrary information state. At σ, φ is either T, F or I. Now, suppose that φ is T at σ. Let us consider the value of [[¬(φ∧ ¬ψ)]](σ) = {hw, δi ∈
σ : hw, δi ∈/ [[¬ψ]]([[φ]](σ))}. Since, by assumption, [[φ]](σ) = σ, [[ψ]](σ) = σ. Therefore,{hw, δi ∈σ:hw, δi∈/ [[¬ψ]]([[φ]](σ))}={hw, δi ∈σ:hw, δi∈ ∅}/ =σ. Thus, φ→ ψ is T atσ. Now, suppose that φis F. Then, [[φ]](σ) = ∅, and so
{hw, δi ∈ σ : hw, δi ∈/ [[¬ψ]]([[φ]](σ))} ={hw, δi ∈ σ : hw, δi∈ ∅}/ = σ. Thus, φ→ψis T atσ. Finally, suppose thatφis I. Then [[φ]](σ) =σ06=∅. Now, since [[φ]](σ0) =σ0, then by assumption [[ψ]](σ0) =σ0. Therefore,{hw, δi ∈σ:hw, δi∈/ [[¬ψ]]([[φ]](σ))}={hw, δi ∈σ:hw, δi∈/ [[¬ψ]](σ0)}={hw, δi ∈σ:hw, δi∈ ∅}/ =
σ. Thus,φ→ψis T atσ. Sinceσwas arbitrary, φ→ψis valid.
Now, let us suppose that φ →ψ is valid, and let σ⊆W ×OP be a state
at which φ is T. Now, since φ → ψ is valid, then {hw, δi ∈ σ : hw, δi ∈/ [[¬ψ]]([[φ]](σ))} = σ. Moreover, by definition of σ, {hw, δi ∈ σ : hw, δi ∈/ [[¬ψ]](σ)} = σ. Thus, σ∩[[¬ψ]](σ) = ∅. However, by the nature of updates, [[¬ψ]](σ)⊆ σ. Therefore, [[¬ψ]](σ) = ∅, and soψ(σ) = σ. Thus, ψ is T at σ. Sinceσwas arbitrary,ψ is a consequence ofσ.
Given the tight connection between the truth ofφand that ofDφdemonstrated in Proposition 9.4, the relation of consequence between sentences φ, ψ within our extended language can also be characterized by the formulaDφ→ψ.
Proposition 9.8 (Quasi-Deduction Theorem for Vague Language) Let P be a vague predicate, and letφ, ψ be sentences either of which containP. ψ is a consequence ofφiffDφ→ψis valid.
proof: First, suppose that ψ is a consequence of φ, and let σ ⊆ W ×OP
be an arbitrary information state. φ is either supported or not supported by σ. Suppose that φ is supported by σ. Consider the value of [[Dφ→ ψ]](σ) = [[¬(Dφ∧ ¬ψ)]](σ) ={hw, δi ∈σ:hw, δi∈/ [[¬ψ]]([[Dφ]](σ))}. Sinceφis supported by σ, then [[Dφ]](σ) = σ, and so {hw, δi ∈ σ : hw, δi ∈/ [[¬ψ]]([[Dφ]](σ))} =
{hw, δi ∈σ:hw, δi∈/ [[¬ψ]](σ)}. Moreover, sinceψ is a consequence ofφ, then [[ψ]](σ) =σ. Thus,{hw, δi ∈σ:hw, δi∈/ [[¬ψ]](σ)}={hw, δi ∈σ:hw, δi∈ ∅}/ = σ. Therefore, Dφ→ψ is T at σ. Now, suppose that that φ is not supported by σ. Thus, {hw, δi ∈ σ : hw, δi ∈/ [[¬ψ]]([[Dφ]](σ))} = {hw, δi ∈ σ : hw, δi ∈/ [[¬ψ]](∅)}={hw, δi ∈σ:hw, δi∈ ∅}/ =σ. Therefore,Dφ→ψis T atσ. Since σwas arbitrary, Dφ→ψis valid.
Now, suppose thatDφ→ψis valid, and letσ⊆W×OP be an information
state at which φ is T. Thus, [[φ]](σ) = σ, and by assumption, {hw, δi ∈ σ :
hw, δi∈/ [[¬ψ]]([[Dφ]](σ))} =σ. Therefore, {hw, δi ∈σ: hw, δi∈/ [[¬ψ]](σ)}=σ. By reasoning similar to that in the proof of Proposition 9.7, we conclude that [[ψ]](σ) =σ, and soψis T atσ. Sinceσwas arbitrary,ψ is a consequence ofφ. With this semantics forDandI, our extended language can express a limited range of statements about the information which speakers have at a context. We are immediately faced with a problem, however, if we wish our semantics forDto capture the widespread intuition that there exists a difference between something being “definitely” the case and it being “definitely definitely” the case. That is, we would like for there to be states in which Dφ is T but not DDφ. Proposition 9.4 already shows that this is out of the question. The source of this difficulty is that at any state, sentences of the form “Dφ” will have only either the value T or F. But, it seems that it could be indeterminate whether the proposition “it is definitely the case that φ” is true, and so it should be possible forDφto be assigned the value I. An immediate nasty consequence of this limitation on the truth values ofDφis revealed in the following proposition.
Proposition 9.9 IIφcannot be T at any information state.
proof: Letσ⊆W×OP be an information state. Now, atσ, the sentenceφis
either T, F or I. Suppose that φis T atσ. Then, by the reasoning laid out in the proof of Proposition 9.6, [[Iφ]](σ) =∅and so Iφis F atσ. Now, sinceIφis not I at σ, by Proposition 9.6, IIφ cannot be T atσ. Suppose that φ is F at σ. Again, by the reasoning laid out in the proof of Proposition 9.6, [[Iφ]](σ) =∅
and so Iφis F at σ. Again, it follows from Proposition 9.6 thatIIφcannot be T at σ. Finally, suppose thatφis I atσ. Then, by Proposition 9.6, Iφis T at σ, and so by Proposition 9.6,IIφ cannot be T atσ.
In section 3.2.2, we will see that Proposition 9.9 reveals a critical failure of our present semantics for D and I. In the next subsection, we therefore introduce a more sophisticated semantics which resolves these tensions.