Contextual Semantics for EQL-formulae

The formulae ofQL ⊂EQL received their official semantics in 4.3.1. This sets the stage for the semantics of explanatory formulae.

(57) a. For all formulae of the form φ

χ?ψ∈EQL,

Jφ

χ?ψKw,K,g = 1 iff

(i) w(φ

χ?ψ) = 1 (Veracity Condition); and

(ii) RKh_Jψ_KK,g,hJφKK,g,Jχ?KK,gii(Relevance Condition)

OtherwiseJφ χ?ψKw,K,g = 0. b. Forφ χ?ψas above,Jφχ?ψKK,g = {w:wan admissible world s.t. _Jφ χ?ψKw,K,g = 1}.

c. For all formulae of the form ¬(φ

χ?ψ)∈EQL, J¬(φ χ?ψ)Kw,K,g = 1−Jφχ?ψKw,K,g. d. For¬(φ χ?ψ) as above, J ¬(φ χ?ψ)KK,g = {w:wan admissible world s.t. _J¬(φ χ?ψ)Kw,K,g = 1 }. e. For all formulae of the form φ∧ψ∈EQL,

(i) Jφ∧ψKw,K,g =JφKw,K,g·JψKw,K,g and

(ii) _Jφ∧ψ_KK,g =JφKK,g∩JψKK,g

Both of the constraints in (57-a) arise naturally from the preceding discus- sion. (57-a-i) is the condition that φ ψ is intersubjectively acceptable in contextK only if it is not ruled out by the world in question. (57-a-ii) adds to this the condition that ψ be contextually relevant to φ relative to its contrast class χ?. Whether the relevance condition holds is determined by the intersection of the output of the CPs’ worldviews.

(58) Fact: RKh_Jψ_KK,hJφKK,Jχ?KKii iff, for each CP Pi, Vi(K) = R

K i

such that RK_i h_JψKK,hJφKK,Jχ?KKii

Of course, (58) follows trivially from the definitions of worldviews and con- textual relevance relations in (55) and (56), respectively.

Further, (59) shows that the standard condition that an explanans be true (better: intersubjectively acceptable) holds.

(59) Fact: If JφψKw,K,g = 1, then JφKw,K,g =JψKw,K,g = 1.

This fact follows trivially from the definition of admissible worlds in (46) together with(57-a-i).

The desired facts about the relations between topics, contrast-classes, and answers also hold as a result of earlier definitions.

(60) Fact: _Jφ

χ?ψKw,K,g is defined only if JφKK,g

∈_Jχ?_KK,g.

(60), which guarantees thatφis a member of the contrast-class χ?, trivially follows from (53) and the definition of the language in (45). (61) guarantees that all of φ’s contrasts are false.

(61) Fact: If _Jφ

χ?ψKw,K,g = 1, then for all ξ s.t. JξKK,g ∈ Jχ?KK,g,

JξKK,g =JφKK,g orJξKw,K,g = 0.

It follows from the definitions of_Jχ?_Kw,K,g andJχ?KK,g in (50-c) and (50-d), respectively. Additionally, (62) guarantees that there is a deductively strongest explanation available at each world-context pair.

(62) Fact: There is some θ such that _Jφ

χ?θKw,K,g = 1 and, for all ψ

such that _Jφ

χ?ψKw,K,g = 1,θ|=f.o.l. ψ.

This fact, which is of the utmost importance for the possibility of giving complete answers in a partition semantics for explanatory interrogatives, follows from (46-c), (54-a), the assumption that only a finite number of propositions are relevant toφ relative to χ?, and the use of intersection in (56).91

91

Note that the fact would also hold if a definition of the contextual relevance relation using union like (41) had been preferred.

Incidentally, since the only wff of the form φ ψ are such that φ and ψare sentences, (63) also holds.

(63) Fact: For any two assignment functions from variables xi to the

fixed domain D,g, g0,_Jφψ_Kw,K,g =JφψKw,K,g0.

Having presented the main facts about the semantics for explanatory indica- tives, the semantics for explanatory interrogatives is given in (64).

(64) a. For all formulae of the form ?(φ

χ?ψ)∈EQL, admissible

worldsw, admissible contexts K, and assignment functionsg J?(φ χ?ψ)Kw,K,g = {v:van admissible world s.t. _Jφ χ?ψKw,K,g =Jφχ?ψKv,K,g}. b. For ?(φ χ?ψ), K, g as above,J?(φχ?ψ)Kw,K = {_J?(φ χ?ψ)Kw,K,g :w an admissible world }.

c. For all formulae of the form ?χ?φ∈EQL,w, K, g as above,

J?χ?φKw,K,g ={v:v an admissible world s.t., for all sentences ψ∈P L,_Jφ

χ?ψKw,K,g =Jφχ?ψKv,K,g}.

d. For ?χ?φ, K, g as above,J?χ?φKK,g =

{_J?χ?φKw,K,g :w an admissible world}.

(64-a) and (64-b) give the obvious definitions for the extension and intension of the yes-no question ?(φψ). (64-c) mirrors the definition of the exten- sion of the non-explanatory interrogatives ofEQL, with universal quantifi- cation over sentences replacing the quantification over n-tuples of elements of the fixed domainD, and (64-d) defines the corresponding intension. Note that one element of the intension is

V :={v:v an admissible world s.t., for all sentencesψ∈P L,_Jφ

χ?ψKw,K,g = 0}

For these worlds, in context K, the why-question cannot be answered and should be rejected. But something related is known about the worlds inV.

(65) Fact: Where Ψ ={ψ1, . . . , ψn:RhJψiKK,hJφKK,Jχ?KKii},v∈ V as defined above iff J

n

W

i=1

ψiKv,K = 0.

(65) follows from the semantics for ?χ?φin (64-c), the definition ofV, and the

semantics forφ

χ?ψ in (57-a). This fact accounts for van Fraassen (1980)’s

identification of the third presupposition of why-questions, discussed in§3.1 above. Recall that the presupposition was not that a why-question is known to have a contextually relevant answer, but that it was not known not to have one. The presence of a block of the partition populated by worlds where the why-question is to be rejected is thus important for the formalization of van Fraassen’s theory.