One may hope to induce a plausibility relation on worlds from a plausibility relation on propositions, in an analogous way to how the plausibility ordering used to describe (part of) an agent’s doxastic state on ipms could be given qualitatively via icdms. To this end
we briefly explore the definability of binary plausibility operators between propositions inIPL.
This allows us to look for representation of such terms as ‘agentaholds' to be at least as plausible as .’ As the plausibility relation on ipms is between worlds the binary operators lifted from this relation holds between declaratives, for these are the formu- las which characterise what does and does not hold at any given world. These binary operators defined with respect to declaratives can then be lifted to interrogatives via the ability to characterise interrogatives in terms of their resolutions. We make suggest some avenues to pursue at the end of this section, but leave a complete development to later work.
Our presentation follows Girard (2008, Ch. 4).
Definition 7.1.8(Binary plausibility operators).The following seven binary plausibility operators given below are defined following Girard (2008, p. 60).3
M; w˛99a ˇ iff 9v;9uWM; v˛&M; uˇ&vwa u
M; w˛89a ˇ iff 8v;9uWM; v˛)M; uˇ&vwa u M; w˛ <99a ˇ iff 9v;9uWM; v˛&M; uˇ&v <wa u
M; w˛ <89a ˇ iff 8v;9uWM; v˛)M; uˇ&v <wa u
M; w˛ <88a ˇiff8v;8uWM; v˛&M; uˇ)v <wa u M; w˛88a ˇiff8v;8uWM; v˛&M; uˇ)vwa u
M; w˛98a ˇ iff 9v;8uWM; v˛&M; uˇ)uwa v
To aid in establishing the definability of these operators inIPLwe establish a number of lemmas.
Lemma 7.1.9. M; whEai'iff9v2a.w/; M; v'
Proof. From left to right supposeM; w hEai'. Expanding, this reads: M; w
:Ea:'. From this it follows thatM; w²Ea:', and so,9t 2˙a.w/; M; t²:'. By proposition 1.2.15 this entails that for somev2t; M; v²:', whenceM; v'. And, asv2t andt 2˙a.w/, it is immediate thatv2a.w/.
From right to left the reasoning is analogous in the converse direction. Lemma 7.1.10. M; whEai'iff9vWvwa wandM; v'
Proof. The proof is analogous to lemma 7.1.9.
Lemma 7.1.11. M; whEa—i'iff9vWw <wa vandM; v'
Proof. The proof is analogous to lemma 7.1.9, with the aid of proposition 7.1.3 to inter- changev—w
a wandw <wa v.
Proposition 7.1.12. The plausibility operators of definition 7.1.8 can be defined inIPL.
M; w˛99a ˇ ´ hEai.ˇ^ hEai˛/ M; w˛89a ˇ ´Ea.˛! hEaiˇ/ M; w˛ <99a ˇ ´ hEai.hEai˛^ hEa—iˇ/ M; w˛ <89a ˇ ´Ea.˛! hEa—iˇ/ M; w˛ <88a ˇ´Ea.˛!Ea:ˇ/ M; w˛88a ˇ´Ea.ˇ!Ea—:˛/ M; w˛98a ˇ ´ hEai.˛^Ea—:ˇ/
3Note, Girard defines eight binary operators. Here, we omit the operator˛ >98
a ˇ, defined by
9v;8uWM; v˛&M; uˇ)u <w
Proof.
M; whEai.ˇ^ hEai˛/iff9v;9uWM; v˛&M; uˇ&vwa u
From left to right supposeM; whEai.ˇ^hEai˛/. Therefore,9u2a.w/; M; u
ˇ^ hEai˛, by lemma 7.1.9. Moreover,M; uˇ^ hEai˛iffM; uˇandM; u
hEai˛. So, by lemma 7.1.10 and the latter conjunct,9v ua u; M; v ˛. Asu 2
a.w/we infer by the condition of introspection 2. on ipms that9vwa u; M; v ˛.
So, we have shown9v;9uWM; v˛&M; uˇ&vw a u.
From right to left the reasoning is analogous in the converse direction.
M; wEa.˛! hEa—iˇ/iff8v2a.w/;9uWM; u˛)M; uˇ&vwa u
From left to right, supposeM; w Ea.˛ ! hEaiˇ/. So,8t 2 ˙a.w/; M; t
˛ ! hEaiˇ. Therefore, by persistence we know that8v 2 a.w/; M; v ˛ !
hEaiˇ. So, as worlds behave classically, ifM; v ˛thenM; v hEa—iˇ, whence
by lemma 7.1.11 we have that9uW M; u ˇandv va u. As before, we knowv 2 a.w/and so by introspection 2. on ipms this entails that9uWM; uˇandvwa u.
Therefore, ifM; v˛then9uWM; uˇandvwa u.
From right to left the reasoning is analogous in the converse direction.
M; whEai.hEai˛^ hEa—iˇ/iff9v;9uWM; v˛&M; uˇ&v <w a u
From left to right supposeM; w hEai.hEai˛^ hEa—iˇ/. So, by lemma 7.1.9 we
know that8x2a.w/thatM; xhEai˛^ hEa—iˇ. Letx2a.w/be arbitrary. As
M; xhEai˛^ hEa—iˇwe knowM; xhEai˛andM; xhEa—iˇ.
By the former conjunct and lemma 7.1.10 we know that9vWvxa xand by the latter conjunct and lemma 7.1.11 we know that9uWx <xa uandM; u ˇ. So, we can infer thatv <xa u. And, asx 2 a.w/this implies thatv <xa uby the condition of
introspection 2. on ipms.
From right to left suppose9v;9uW M; v ˛&M; u ˇ &v <wa u. Letv; u
instantiate their respective quantifiers. Then,v <wa u. From this we know thatv 2
a.w/and therefore by the condition of introspection 2. on ipms we knowv <va u.
So, asvvavandM; v˛we know by lemma 7.1.10 thatM; vhEai˛. Similarly
by the fact thatv <va uandM; uˇwe know by lemma 7.1.11 thatM; vhEa—iˇ.
So,M; v hEai˛^ hEa—iˇ. And, asv 2 a.w/we know by lemma 7.1.9 that
M; whEai.hEai˛^ hEa—iˇ/.
M; wEa.˛! hEa—iˇ/iff8v;9uWM; v˛)M; uˇ&v <w a u
From left to right supposeM; w Ea.˛ ! hEa—iˇ/. Letv 2 a.w/be arbitrary,
and supposeM; v ˛. So, asM; w Ea.˛ ! hEa—iˇ/we know that8t 2
˙a.w/; M; t ˛ ! hEa—iˇ, and by persistence this entailsM; v ˛ ! hEa—iˇ,
whence we infer thatM; vhEa—iˇ. So, by lemma 7.1.11 we know that9u; v <va u
andM; u ˇ. As before, we use the fact thatv 2 a.w/to observe via the condi-
tion of introspection 2. on ipms and the previous fact thatv <w
a u, whence the result
follows.
M; wEa.˛!Ea:ˇ/iff8v;8uWM; v˛&M; uˇ)v <wa u
From left to right supposeM; w Ea.˛ !Ea:ˇ/. So,8t 2˙a.w/; M; t ˛!
Ea:ˇ.
Now, suppose for an arbitraryv; u 2 a.w/; M; v ˛andM; u ˇ. From the
above observation and our assumption thatM; v ˛we inferM; v Ea:ˇ. So,
8t02˙a.v/; M; t0:ˇ.
Now, supposeuw
a v. Using the condition of introspection 2. on ipms and the fact
thatv2a.w/we inferuva v. Therefore,fug 2˙a.v/, whenceM; u:ˇ. This
contradicts our inference thatM; uˇ, and therefore we infer thatu—wa v. However,
asv; u2a.w/we know by this and proposition 7.1.3 thatvwa u, whencev <wa u.
From right to left supposeM; w ²Ea.˛ !Ea:ˇ/. Then9t 2 ˙a.w/; M; t ˛ whileM; t²Ea:ˇ. So, as worlds behave classically it must be the case that for some
v2t; M; vhEaiˇSo, by lemma 7.1.10 we have someuva vsuch thatM; uˇ.
By the fact thatv2a.w/and the condition of introspection 2. on ipms it follows that
uwa v, whencev6<wa u.
M; wEa.ˇ!Ea—:˛/iff8v;8uWM; v˛&M; uˇ)vwa u
From left to right supposeM; w Ea.ˇ!Ea—:˛/, so8t 2˙a.w/; M; t ˇ!
Ea—:˛. So, suppose for arbitraryv; u 2 a.w/; M; v ˛andM; u ˇ. Then
M; u Ea—:˛, whencev… —a.u/, and sov ua u. Asu2 a.w/this entails, via
the condition of introspection 2. on ipms, thatvw a u.
From right to left suppose8v;8uW.M; v˛&M; uˇ/)vwa ubutM; w²
Ea.ˇ ! Ea—:˛/. ThenM; w hEai.ˇ^ hEa—i˛/, and so, by lemma 7.1.9, for
someu 2 a.w/; M; u ˇ^ hEa—i˛, whence by lemma 7.1.11, for somevWu <ua
v; M; v ˛, but thenu wa v, via the fact thatu 2 a.w/and the condition of
introspection 2. on ipms, and we have a contradiction.
M; whEai.'^Ea—: /iff9v;8uWM; v˛&M; uˇ)uwa v
From left to right supposeM; whEai.˛^Ea—:ˇ/. Then by lemma 7.1.9, for some
v 2 a.w/; M; v ˛^Ea—:ˇ. For an arbitraryu 2 a.w/supposeu ˇ, then
u…—a.v/on pain of contradiction.
Asu…—a.v/we know thatuva v, and by the fact thatv2a.w/and the condition
of introspection 2. this entailsuwa v.
From right to left supposeM; w ² hEai.˛^Ea—:ˇ/. Therefore, as worlds behave
classically we knowM; w Ea:.˛^Ea—:ˇ/So, for allt 2˙a.w/; M; t :.˛^
Ea—:ˇ/.
Now, suppose9v;8uW M; v ˛&.M; u ˇ ) u wa v/. Letv instantiate
the existential quantifier. Then, asv 2 a.w/we know via persistence thatM; v
:.˛^Ea—:ˇ/. So, asM; v˛we knowM; v²Ea—:ˇ. From this we inferM; v
:Ea—:ˇ, which can be rewritten asM; vhEa—iˇ. From the latter observation and
lemma 7.1.11 we know that for someuW v <va u; M; u ˇ. So, by the fact that
v2a.w/and the condition of introspection 2. on ipms we infer thatv <wa u. This
Using the preceding definitions binary plausibility operators can be defined with respect to interrogatives in a number of ways. For example, we may identify thatis a more plausible interrogative toby observing that for every resolution tothere is a more plausible resolution to. To capture this we may say <98a iff8ˇ2R./9˛2
R./W˛ <88a ˇ, which is captured by the formula
V
ˇ2R./
W
˛2R./.˛ <88a ˇ/, etc.