We are in a position to establish completeness for ipnms. The method taken to establish completeness of IPLwith respect to ipnms follows that of ICDLwith respect to icdms, from chapter 6.
7.4.1 Preliminaries
We will not restate the preliminaries for completeness forICDLfrom chapter 6, as these straightforwardly carry over toIPL. We note only the adjustment required to adapt the definition of the finite fragment of ICDLto that ofIPL, arising from a difference in the modal operators between the two languages, in the definition of a subformula.
Definition 7.4.1(Subformulas).LetF be a set of formulas, we definesub.F/to be the smallest set satisfying the following conditions:
1. If'ı 2F then'; 2sub.F/forı 2 f^;!g. 2. If‹f˛1; : : : ; ˛ng 2F, then˛1; : : : ; ˛n2sub.F/.
3. IfE
a '2F, then'2sub.F/, for2 f;—;g.
As inICDLto establish completeness forIPLwe begin by taking a finite set of formulas F fromIPLwhich then close under successive operations.
As definition 6.2.10 makes no reference toICDLwe apply the same method to define the finite fragment ofIPL,F, via successive closures on a finite set of formulasF.
The fragment ofIPL,F, is then used as a basis for the construction of nuclei and then atoms of the canonical model, as inICDL.
As the syntactic characterisation of atoms, states, and propositions for fragments of
ICDLmakes no specific reference to the language ofICDL, nor toF, the definitions and lemmas established in section 6.2.1 carry over to the fragment ofIPL, as do the relevant lemmas of section 6.3.
7.4.2
Constructing the Canonical Model
Definition 7.4.2(Canonical model forIPLoverF).The canonical model forIPLis the tuple:MF D hW;f˙
aga2A;f˙—aga2Af˙aga2A; Vi, defined as follows: – W is the set of complete theories of declaratives ofIPL.
– V .A/D fp2Atjp2Ag
– For2 f;—; g,˙a.A/is the set of statesS W defined by:
S 2˙
a.A/()
\
S `'wheneverA`E
a '
7.4.3
The Support Lemma
Lemma 7.4.3. ForA2At.F/,' 2F, ifA°E
a'then9S 2˙a.A/W
T
S °', for
2 f;—;g.
Proof. Analogous to lemma 6.3.3.
Lemma 7.4.4(Support lemma for the canonical model overF). For a set of formulas
F and the canonical model overF,MF, for anyS At.F/and any' 2F;
MF; S'()\S `'
Proof. Proof of the support lemma forIPLdiffers from the the support lemma forICDL
only with respect to modal formulas, therefore we consider only these steps. I4)´Ea'
From left to right supposeMF; S
Ea'. Then8A2S;8T 2˙a.A/; MF; T ', whence by the induction hypothesis we have8A2S;8T 2˙a.A/;TT `'. So, by
lemma 7.4.3 we have8A2S; A`Ea', whenceTS `Ea'.
From right to left supposeTS ` Ea , so8A 2 S; A ` Ea . Therefore, asT 2
˙a.A/()TT `wheneverEa2Awe can infer by the induction hypothesis
that8T 2˙a.A/; M; T , whenceM; AEa for allA2S, whenceM; S
Ea .
I5)´Ea'
Analogous to the case for the entertains modality. I6)´Ea—'
Analogous to the case for the entertains modality.
It is certainly not clear that the canonical model forIPLis an ipnm, as it appears plausible that for someA; B2At.F/,B2a.A/\—a.A/—it is certainly possible for
some formula'thatA` hEai'^h—ai', whence it seems thatA` hEaiγB^h—aiγB
cannot be excluded. We therefore define a transformation of any given canonical ipnm into a regular ipnm.
Definition 7.4.5(RegularIPLmodel overF).Given the canonicalIPLmodel overF
M D hW;f˙aga2A;f˙—aga2Af˙aga2A; Vi we define the regular model forIPLas the tuple:
such that for everyB2
a.A/\—a.A/we create two copies,BandB—distinguished
by adding some set-theoretic element outside ofIPLtoBto obtainBand omitting it fromB—.
We omitBfromW0in place ofB; B—, which are then defined to belong to
a.A/
and—a.A/respectively, for anyAsuch thatB 2 a.A/\—a.A/and with˙a.A/,
˙a.A/, and˙—a.A/modified accordingly. We then define˙a.B/as˙a.B/with
Bin place ofB, and˙—a.B—/as˙—a.B/. With bothBandB—in place ofBwe
then revise the valuation functionV0accordingly, following the definition ofV from the canonical model.
The rest of the canonical model remains unchanged.
Lemma 7.4.6(Support lemma for the regular model overF). For a set of formulasF
and the regular model overF,MF, for anySAt.F/and any' 2F;
MF; S'()
\
S `'
Proof. Inherited from the canonical model. For, it is straightforward to see that the changes made to the canonical modal are purely cosmetic—duplicating certain atoms, which the fragment ofIPLcannot distinguish between, to ensure the intersection of cer- tain sets are indeed empty.
Corollary 7.4.7. For all'2F, anyS At.F/, and2 f;—;g:
\
S`E
a 'iffMF; SEa'
Proof. IfSD ;the result is trivial, so we assumeSD fA1; : : : ; AngforAi2At.F/.
From left to right supposeTS `E
a '. Then, asS D fA1; : : : ; Angwe know that
T
S Aiforin, whenceAi `Ea'for alli n.
Therefore, by definition ofE
a 'we know thatT 2˙a.Ai/iffTT `'. Therefore,
as' 2 Fwe know by the support lemma thatTT ` ' iff MF; T
', whence we inferMF; Ai Ea'. AsAi was arbitrary this holds for alli n, and so via
proposition 1.2.15 we can infer thatMF; SE
a'.
As each inference appealed to used one direction of an equivalence the left to right direction follows easily from the same reasoning in reverse.
Lemma 7.4.8. The regular model is an ipnm. Proof.
1 ifB2a.A/andT 2˙a.B/, thenT 2˙a.A/
SupposeB 2
a.A/andT 2 ˙a.B/. AsT 2 ˙a.B/we know that
T
T `
wheneverB`E
a . And, asB2a.A/, whencefBg 2˙a.A/, which means that
B` wheneverA`E
a . Our task is to show that
T
T `wheneverA`E
a,
so supposeA `Ea. By axiom 4 it follows thatA` EaEa, whenceB `Ea, and soTT `.
2 ifB2a—.A/andT 2˙a—.B/, thenT 2˙a—.A/
3A2a.A/
SupposeA…a.A/. ThenMF; AEa:γA.
For, if˙a.A/D f;gthenMF; AEa'for any'. And, if˙a.A/¤ f;gthen, as
we know thatA…a.A/, andγA 2BiffBDAby lemma 6.2.26, it must be the case
thatMF; B :γAfor allB2a.A/. So, by proposition 1.2.15MF; S:γAfor
allS 2 ˙a.A/, whenceMF; A Ea:γA. Therefore, by corollary 7.4.7 we know
thatA`E
a:γA.
So, by axiom 9 we have`E
a:γA ! :γA. Therefore, asA`Ea:γAwe have that
A` :γAby modus ponens. However, we know byIPLcounterpart to lemma 6.2.26
and proposition 6.2.22 thatA`γA. A contradiction.
4a.A/\—a.A/D ;
By construction.
5a.A/[—a.A/Da.A/
From left to right assumeB2a.A/[—a.A/. Then it is either the case thatB`
wheneverA`E
a orB` wheneverA`Ea— . Without loss of generality let us
suppose the latter is the case. To showB 2a.A/it is sufficient to show thatB `
wheneverA`Ea.
By axiom 7,` Ea !.Ea^Ea—/, whenceA ` Ea ! .Ea^Ea—/. So,
ifA ` EathenA ` Ea—, whenceB ` . Therefore, it is the case thatB `
wheneverA`Ea.
From right to left supposeB 2 a.A/, whileB … a.A/[—a.A/. So,B …a.A/
andB…—a.A/, whencefBg …˙a.A/andfBg …˙—a.A/.
So,B` wheneverA`Ea , but for some1; 2,A`Ea1andA`Ea—2, yet
B°1andB°2.
However, by axiom 8 it follows thatA `Ea.W˛2R.1/ W
ˇ2R.2/.˛_ˇ//, whence
B`.W˛2R.1/Wˇ2R.2/.˛_ˇ//. AsBis an atom it has the disjunction property by proposition 6.2.23. So,B`˛_ˇ, for some˛2R.χ1/andˇ2R.χ2/. Appealing
again to the fact thatB has the disjunction property we know thatB ` ˛for˛ 2
R.χ1/orB ` ˇforˇ 2 R.χ2/. Therefore, by theorem 6.2.7 we know that either
B`χ1orB`χ2, a contradiction.
6 ifB2a.A/, then˙a.A/D˙a.B/
This comes via the introspection axioms for the entertains modality, paralleling the proof for introspection with respect to considers on icdms.
7˙a.A/D.}.a.A//\˙a.A//
From left to right, ifT 2˙
a.A/thenT a.A/, and soT 2}.a.A//. Further-
more,TT ` wheneverA ` E
a. SupposeA ` Ea then by axiom 7 and the
elimination rule for conjunction,A ` E
a , whence
T
T. So,TT ` whenever
A`Ea , and soT 2˙a.A/. Therefore,T 2}.a.A//\˙a.A/.
From right to left, letT 2}.a.A//\˙a.A/be arbitrary.
We begin by establishing thatA`Ea—:γT. For, asT 2 ˙a.A/we knowTT `
condition 4 on ipnms, proved to be satisfied byMF above, we knowT\—
a.A/D ;,
whenceT …}.—a.A//. So, as˙—a.A/}.—a.A//it follows thatT …˙—a.A/.
AsB ` γT iffB 2 T by lemma 6.2.29, this means that8B 2 —a.A/; B ° γT,
whenceB ` :γT, asBis a maximally consistent set of declaratives. By the support
lemma, then, as:γT 2D, we know that for allB 2—a.A/; MF; B :γT. So, by
lemma 1.2.15 we know that for allT 2 ˙—a.A/; MF; T :γT, and soMF; A
Ea—:γT. Therefore, by the support lemma we knowA`Ea—:γT.
We now show that ifA`Ea'thenTT `'. For, supposeA`Ea'for an arbitrary
'2F. By the previously established fact thatA`Ea—:γTand the introduction rule
for conjunction we haveA`Ea'^Ea—:γT So, as an instantiation of axiom 10 we
haveA`.Ea'^Ea—:γT/!Ea.γT !'/, from which we inferA`Ea.γT !'/.
Therefore, ifS2˙a.A/thenTS `γT !'.
As we knowTT 2 ˙a.A/this means thatTT `γT !', whence asTT ` γT
by lemma 6.2.32 we know thatTT `'. Therefore, we have show that ifA`E
a'
thenT 2 ˙
a.A/. And, asT was an arbitrary element of}.a.A//\˙a.A/this
establishes}.a.A//\˙a.A/˙a.A/.
8˙—a.A/D.}.—a.A//\˙a.A//
Proof is analogous to the previous condition using axiom 11.
Theorem 7.4.9(Completeness ofIPLwrt. ipnms.). IPLis weakly complete with respect to ipnms.
Proof. As with completeness ofICDLwith respect to icdms, theorem 6.4.1.