One way to look at the problem we ended up with in the last section is that the formalization of the Gricean Principle given with |≡n is too strong. By |≡n
a speaker who wants to obey the principle has to give every bit of information about deontic accessible interpretations of the basic atoms that she has. Perhaps we can obtain a more natural notion of pragmatic entailment when we allow the speaker to withhold some of this information. The problem, then, becomes to find the right restriction that fits our intuitions.
To start with, we can ask ourselves which information about the deontic acces- sibility relation we can take to be not relevant for the order because it is accessible to the interpreter anyway. It turns out that if the speaker is competent on △, then which permissions the speaker believes to hold can be already concluded from taking her to convey all she knows about the valid obligations. If she is honest about this part of her beliefs, then if her utterance φ does not entail for some χ ∈ L0 that she believes ∇χ she cannot believe this obligation to be valid,
i.e. ¬∇χ holds. From her competence it follows that she has to believe that ¬χ is permitted. On the other hand, if for some χ ∈ L it holds that the speaker believes χ to be permitted, then, by competence, ¬∇¬χ is true and because we assume her to believe in her utterance φ, φ cannot entail ∇¬χ. Thus, a com- petent speaker believes some sentence χ ∈ L0 to be permitted if and only if her
about which permissions the speaker believes to be valid can be ignored by the order. It is enough to compare what a competent speaker believes to be a valid obligation.19 We obtain such an order when we delete condition (ii) from the
definition of n.20
2.3.7. Definition. (The Positive Information Order +)21
For s = hM, wi, s′ = hM′, w′i ∈ S we define s + s′ iff
def
∀v′ ∈ R′
♦[w′] ∃v ∈ R♦[w] :
(i) ∀p ∈ P : V (p)(v) = V′(p)(v′) &
(ii) ∀u′ ∈ R′
△[v′] ∃u ∈ R△[v] (∀p ∈ P : V (p)(u) = V (p)(u′)).
By substituting + in definition 2.3.1 we obtain a new notion of pragmatic
entailment: |≡S+, abbreviated |≡+S. It turns out that for |≡+C not only the free choice inferences for the epistemic modality are valid, but (D4) and ∇(p ∨ q) |≡+C ♦△p ∧ ♦△q as well. Parallel to the epistemic case the sentence ∇p ∨ ∇q is predicted to be dishonest when uttered by a competent speaker that obeys the Gricean Principle.
Let us discuss the validity of (D4). The argumentation we employ has exactly the same structure as in section 2.3.4.1. If for p, q ∈ L0 such that {p, q} is
satisfiable in C (D4): △(p ∨ q) |≡+C △p ∧ △q were not valid then there would be a state s ∈ C minimal with respect to + such that s |= △(p ∨ q) ∧ △(p ∨ q)
but not s |= △p ∧ △q. Now, we show that this cannot be the case: every state s ∈ C that semantically entails △(p ∨ q) ∧ △(p ∨ q) but where the consequence of (D4) is not true cannot be minimal with respect to +.
Assume that for s = hM, wi ∈ C we have s |= △(p ∨ q) ∧ △(p ∧ q), but s 6|= △p ∧ △q. Without loss of generality s 6|= △p. Let s∗ = hM∗, w∗i ∈ C be
19Of course, the same argument can be also used to show that the speaker does not have
to convey all she believes about valid obligations, as long as she is honest about her beliefs concerning permissions. However, minimizing beliefs on permissions does not result in a con- vincing notion of pragmatic entailment. For instance, this one wrongly predicts that sentences like △(p ∨ q) are dishonest. One would like to have some motivation for the choice of the order +besides the fact that it does the job, while some equally salient alternatives do not –
particularly, given that we formalize a theory of rational behavior. But so far I am not aware of any conclusive arguments.
20Also for this order an equivalent definition using a set of sentences can be given (for a close
discussion see Schulz (2004)).
2.3.6.Fact. Let L+⊆ L be language defined by the BNF-form χ+ ::= p(p ∈ L0)|χ+∧χ+|χ+∨ χ+|∇p(p ∈ L0). Then we have for s, s′∈ C:
s+s′
⇔ ∀χ ∈ L+: s |= χ ⇒ s′
|= χ.
21Again, +only compares beliefs about formulas {∇χ|χ ∈ L0}, but an extension to sentences
∇χ for χ ∈ L is easily possible (see Schulz (2004)). We use the simpler variant because the sentences we consider here are only of the former type.
the state that is like s except that from w∗ an additional world ˜v is △-accessible
where p is true.22 Thus s∗ |= △p. We show that (i) s∗△(p ∨ q) ∧ △(p ∧ q), (ii)
s∗ + s, and (iii) s 6+ s∗. Then s cannot be minimal because s∗ is smaller.
Ad (i) We have seen already that s∗ |= △p. It follows s∗ |= △(p ∨ q). Because s∗ is
an element of C we can conclude from this (by [C2]) that s∗ |= △(p ∨ q).
This shows (i).
Ad (ii) We have to show that for all v ∈ R♦[w] we can find a v∗ ∈ R∗♦[w∗] such that
(i) ∀p ∈ P(V (p)(v) = V′(p)(v′)) and (ii) ∀u′ ∈ R′
△[v′] ∃u ∈ R△[v] (∀p ∈ P :
V (p)(u) = V (p)(u′)). (i) is simple, let us go directly to the interesting case:
(ii). Because the difference between s∗ and s is that s∗ has one more △-
accessible world: ˜v, we have R△[w] ⊂ R∗△[w∗]. From s, s∗ ∈ C we conclude
∀v ∈ R♦[w] : R△[v] = R△[w] and ∀v∗ ∈ R∗♦[w∗] : R△∗[v∗] = R△[w∗].
Together, this gives: ∀v ∈ R♦[w] ∀v∗ ∈ R∗♦[w∗] : R△[v] ⊂ R∗△[v∗]. Because
by assumption s and s∗do not differ in the interpretation assigned in worlds
of R△[v] to elements of P this proves the claim.
Ad (iii) Finally, s 6+ s∗. Because s 6|= △p we obtain by [C
1] that s |= ¬△p.
Hence, for no v ∈ R♦[w] and no u ∈ R△[v] we have hM, ui |= p. But from
s∗ |= △p with [C
2] it follows s∗ |= △p, and, thus, ∀v∗ ∈ R∗♦[w∗]∃u∗ ∈
R∗
△[v∗] : hM∗, u∗i |= p. Because p ∈ L0 condition (ii) of the definition of
+ is violated for s +s∗.
Thus, we see that adopting |≡+ as a formalization of the Gricean Principle and applying it to the set of states C where the speaker is competent accounts for the free choice inferences.23
22Again, Schulz (2004) provides a formally precise version of this proof, including a construc-
tive description of s∗. s∗ is obtained from s by first adding a world to the model where p is
true – this is possible if p is satisfiable in C – then making this world △-accessible from w, and, finally, close the resulting accessibility relations R′
♦and R ′
△under the axioms [4], [5], [D], [C1],
and [C2] to obtain a state that belongs to C. 23There is another way to repair |≡n
C such that one can account for the deontic free choice
inferences. Instead of weakening the order and thereby be less strict on what a speaker has to convey with her utterance, we can also take her to be less competent. It turns out that the competence axiom we have to drop is [C2]: we weaken C to the set of states C+ where
[D], [4], [5], and [C1] are valid. In this case, the speaker knows all valid obligations, but she
may be not aware of certain permissions. While this accounts for the free choice inferences, other predictions made by |≡nC+ are less convincing than what is predicted by |≡
+
C. For a more
elaborate discussion the reader is referred to Schulz (2004).
Finally, it is interesting to note, that also the combination of |≡+ with C+, hence, the
combination of weakening the order and weakening the notion of entailment allows us to derive the free choice inferences. Also this combination of a concept of competence with a formalization of the Gricean Principle does not work as well as |≡+C.