We now turn to establishing soundness and completeness of IPLwith respect to ipms. This follows the same strategy as chapter 5, by defining transformations between the class of ipnms and the class of ipms that preserve the interpretation ofIPL.
7.5.1 From ipnms to ipms
Theorem 7.5.1(From ipnms to ipms). Any finite ipnm can be transformed into an ipm preserving the interpretation ofIPL.
Definition 7.5.2(Map from ipnms to ipms). Given an arbitrary ipnm,M D hW;f˙
ag;f˙a—g;f˙ag; Viwe define a mapM 7!
M], whereM]D hW];fw
aga2A;w2W];f˙aga2A; V]iis constructed in the following way:
1. W]´W
2. vwa uifv; u2a.w/andv2a.u/
3. ˙a]´˙a
4. V]´V
Proof. Note in the followinga].w/´S˙a].w/.
Factivity:w2a].w/; for allw2W
We know thatw2a.w/, by condition 3 on ipnms. Furthermore, from this it follows
thatw2a.w/, by condition 5. So, asw2a.w/andw2a.w/we havewwa w,
by definition ofwa, whencew2 ] a.w/. Introspection 1: ifv2a].w/; then˙ ] a.w/D˙ ] a.v/
Supposev2a].w/. Thenvwa ufor someu. So, it must be the case thatv2a.w/
andv2a.u/. And, asv2a.w/we know that˙a.w/D˙a.v/by condition 6 on
ipnms, but then˙a].w/D˙a].v/by definition.
Introspection 2: ifv2a].w/; thenxva yif and only ifxwa y
Supposev 2a].w/, thenvwa ufor someu, and sov2 a.w/andvu. By the
former observation we note that by condition 6 on ipnms,a.w/Da.v/.
Now from right to left, ifx wa y this means thatx; y 2 a.w/andx 2 a.y/.
So, asx; y 2 a.w/anda.w/ D a.v/we know thatx; y 2 a.v/. Therefore, as
x; y2a.v/andx2a.y/we havexva yby definition ofva.
For the left to right direction the argument proceeds analogously. We begin by showing the ordering defined is a well-preorder. Transitivity
Supposex w
a yandy wa z. This means thatx; y; z 2a.w/andx2 a.y/and
y2a.z/.
Now, asx2a.y/we know˙a.x/˙a.y/, and asy2a.z/we know˙a.y/ ˙a.z/by condition 1 on ipnms. Therefore, by transitivity of the subset relation we know that˙a.x/˙a.z/. Furthermore, by condition 3 we know thatx 2a.x/, whencefxg 2˙a.x/, from which it follows by the previous observation thatfxg 2 ˙a.z/, and sox2a.z/. Therefore,xwa zby definition ofwa.
Reflexivity Follows as a corollary of factivity established above.
For every sets fvj 9uWvwa ugthere existsv2ssuch thatvwa ufor allu2s
Suppose for some sets fvj 9uW vwa ugthat for everyv 2 sthere exists some
u2ssuch thatv—w a u.
Letv2sbe arbitrary, and instantiateusuch thatv—w
a u. Asv—wa uit must be the
case that eitherv; u…a.w/orv…a.u/.
So, asv 2 swe know that9xWv wa x, and so by definition ofwa this means that
v; x 2 a.w/andv 2 a.x/. Analogous reasoning applies tou, and therefore we
know thatv; u2a.w/. From this it follows thatv…a.u/. And, asv; u2a.w/
it follows by condition 6 on ipnms thata.v/ D a.w/ D a.u/, and therefore we
know thatv2a.u/andu2a.v/.
Now, asv 2 —a.u/we know that˙a.v/ ˙a.u/by condition 2. From this it
follows thatu…—a.v/. For, suppose otherwise. Then, asu2—a.v/we knowfug 2
condition 3 on ipnms thatu2
a.u/and by condition 4 thata.u/\—a.u/D ;.
Therefore, we have derived a contradiction.
So, asu…—a.v/whileu2a.v/we know by condition 5 thatu2a.v/. Therefore,
by definition ofwa we have thatu <wa v, and furthermore we knowu¤vasvwa v
by the fact thatwa is reflexive, as established above.
Asv 2 swas arbitrary we have shown that for anyv 2 sthere exists someu ¤ v
such thatu <wa v. Yet, we know thatsis finite asW is finite. Therefore, for some
yit must be the case that there is nozsuch thatz <wa y, whence we have derived a contradiction.
We now show the remaining properties required forM]to be an ipm are satisfied.
˙a].w/is an issue overa].w/, wherea].w/´ fvjvwa ufor some ug
We have defined˙a].w/as˙a.w/, and as˙a.w/is an issue overa.w/, it will suffice
to show thata].w/Da.w/.
So, from left to right supposev 2 a].w/. Then,v wa ufor someu. So, given the
definition ofw
a it must be the case thatv; u 2 a.w/andv 2 a.u/for someu,
whence by the former conjunctv2a.w/. Therefore,a].w/a.w/.
Conversely, ifv 2 a.w/then by condition 3 thatv 2 a.v/, and therefore asv 2
a.w/andv2a.v/it follows by definition ofwa thatvwa v, whencev2 ] a.w/.
Therefore,a.w/a].w/.
Lemma 7.5.4. Given an arbitrary ipnmMand a corresponding ipmM]:
1. ˙a.w/D˙ ];
a .w/
2. ˙—a.w/D˙a];—.w/
Proof. Recall˙a.w/D.}.a.w//\˙a.w//and˙—a.w/D.}.—a.w//\˙a.w//,
while˙a];.w/´}.a];.w//\˙a].w/and˙a];—.w/´}.a];—.w//\˙a].w/. So,
as˙a].w/D˙a.w/, showing botha.w/D ];
a .w/anda.w/D ];
a .w/will be
sufficient to establish the lemma. Furthermore, recalla];.w/´ fv2a].w/jvwa
wganda];—.w/´ fv2a].w/jv—wa wg.
1 From left to right supposev2a.w/. Then by condition 5 on ipnms we know that
v 2 a.w/, and be analogous reasoning using condition 3 we knoww 2 a.w/.
So, asv; w 2 a.w/andv 2 a.w/, thenv wa wby definition ofwa, whence
v2a];.w/.
Conversely, ifv 2 a];.w/then it must be the casev wa w. So,v 2 a.w/and
v2a.w/.
2 We know by lemma 7.5.3 thata.w/ D a].w/, and by the previous case we know
a.w/D ];
a .w/, so it must be the case that.a.w/ a.w//D. ]
a.w/ a];.w//.
Therefore, as.a.w/ a.w// D —a.w/by condition 5 on ipnms, and.a].w/
a];.w//D fvjv—wa wg D ];—
Proof of theorem 7.5.1. LetM be an arbitrary ipnm and takeM], the ipm constructed fromM, given by the mapping defined above. We claimM; siffM]; s, for all
sW and formulas.
Proof is via induction on the complexity of, and here we consider only the cases unique toIPL, with the others established as in the proof of theorem 5.2.1.
i5)´Ea'
By lemma 7.5.4 we know that˙a.w/D˙a];.w/. So,M; sEa'iff8w2s;8t 2
˙a.w/; M; t 'iff (by the induction hypothesis and noted lemma)8w 2s;8t 2 ˙a];.w/; M]; t'iffM]; sEa'.
i6)´Ea—'
Analogous to the previous case. i7)´Ea'
Immediate.