Annali del Dipartimento di Filosofia (Nuova Serie), XVII (2011), pp. 135-166
Infinitary Modal Logic
and Generalized Kripke Semantics
Pierluigi Minari
This paper deals with the infinitary modal propositional
logic K!1 , featuring countable disjunctions and conjunctions. It is known that the natural infinitary extension
]
LK⇤
!1 (here presented as a Tait-style calculus, TK!1 ) of
⇤
the standard sequent calculus LKp for the propositional
modal logic K is incomplete w.r. to Kripke semantics. It
is also known that in order to axiomatize K!1 one has
to add to LK⇤
!1 new initial sequents corresponding to the
infinitary propositional counterpart BF !1 of the Barcanformula. We introduce a generalization of Kripke semantics, and prove that TK]!1 is sound and complete w.r. to
this generalized semantics. By the same proof strategy, we
show that the stronger system TK!1 , allowing countably
infinite sequents, axiomatizes K!1 , although it provably
doesn’t admit cut-elimination.
Mathematics Subject Classification: 03B45, 03F05.
Keywords: modal logic, infinitary logic, Kripke semantics,
sequent-style calculi, cut-elimination.
1. Introduction
Let LKp be the propositional fragment of Gentzen’s sequent
calculus for classical logic. As is well-known (more or less since
the mid 1950’s, see e.g. [5]), the sequent calculus LK⇤
p obtained
by adding to LKp the inference rule
)'
K (where ⇤ := {⇤ |
⇤ ) ⇤'
2 })
http://www.fupress.com/adf
ISSN 0394-5073 (print) ISSN 1824-3770 (online)
© 2012 Firenze University Press
2
136
Pierluigi Minari
provides an adequate axiomatization of the minimal normal
propositional modal logic K, semantically defined as the set
of all modal formulas which are valid in every Kripke frame.
Furthermore, this axiomatization satisfies the subformula property, since it is not difficult to verify that LK⇤
p admits cutelimination. Of course, the same does hold for the correspondA
ing multimodal system LK⇤
p based on an arbitrarily fixed set
A of agents — below, we will identify without limitations A
with the set of natural numbers !.
Let us now move on to consider the infinitary (multi)modal
version KW!1 of K. In its language, featuring
V the infinitary con(countable disjunction) and
(countable conjuncnectives
tion) in place of the corresponding finitary connectives _ and
^, many interesting infinitary modal operators (typically fixpoints operators) become directly definable — for instance that
of common knowledge, C :
^
C' := {En ' | n ≥ 1}
where E' (everyone knows, that ') is defined as
^
E' := {⇤i ' | i < !}
and
n
z }| {
E ' := E . . . E '
n
It is however a known fact (see e.g. [6], [8], [10]) that the
⇤!
natural infinitary extension LK⇤
!1 (or LK!1 , in the multimodal
⇤!
version) of the sequent calculus LK⇤
p (LKp ), which is simply
obtained by replacing the rules for _ and ^ with their infinitary
counterparts
Γ ) ∆, '
W (' 2 Φ)
Γ ) ∆, Φ
. . . ', Γ ) ∆ . . . (all ' 2 Φ)
W
Φ, Γ ) ∆
and
. . . Γ ) ∆, ' . . . (all ' 2 Φ)
V
Γ ) ∆, Φ
', Γ ) ∆
V
(' 2 Φ)
Φ, Γ ) ∆
3
137
Infinitary Modal Logic and Generalized Kripke Semantics
is not Kripke complete. In particular, the schema
^
^
⇤Φ ! ⇤ Φ
(BF !1 )
that is the infinitary propositional counterpart of the famous
Barcan-formula of quantified modal logic, is trivially valid in
all Kripke frames, but is not derivable in LK⇤
!1 .
Notice that, as a consequence, the basic modal logic of common knowledge KC (see e.g. [1]) cannot be embedded in LK⇤
!1 ,
since BF !1 (together with its converse CBF !1 , which is instead
derivable in LK⇤
!1 ), is essentially needed in order to derive the
fixed point axiom C' $ E' ^ EC' of KC:
C'
$
E1 ' ^ E2 ' ^ E3 ' ^ . . .
l
E' ^ (E(E1 ') ^ E(E2 ') ^ . . .)
#BF !1 " CBF !1
E' ^ E(E1 ' ^ E2 ' ^ . . .)
$
E' ^ EC'
Indeed, BF !1 plays a key role in the axiomatization of K!1 .
It has been proved by Y. Tanaka ([8]; see also [10]) that the
⇤
sequent calculus LK⇤
!1 ⊕BF !1 — that is, LK!1 plus all instances
of ) BF !1 as further initial sequents — axiomatizes K!1 . A
Hilbert-style calculus (KL!1 ) axiomatizing K!1 and featuring
BF !1 as an axiom had been earlier provided by S. Radev in [6].
An alternative route is followed in the present paper. We
work with Tait-style (i.e. one-sided) sequent calculi — but this
is of course not essential — and we hide BF !1 , so to speak,
in the syntax. More precisely, after some preliminaries on the
peculiar NNF -language we adopt (sect. 2), we consider (sect.
3) the two calculi TK]!1 and TK!1 : the essential di↵erence
between them is that sequents are finite (sets of formulas) in
TK]!1 , whereas they can also be countably infinite in TK!1 .
It is a known fact that this di↵erence doesn’t matter as far as
non modal infinitary logic is concerned; it does however matter
when modal operators are added: while TK]!1 is indeed nothing
138
4
Pierluigi Minari
but the one-sided version of LK⇤
!1 , the calculus TK!1 turns out
⇤
to be equivalent to LK!1 BF !1 .
]
As we said, LK⇤
!1 (our TK!1 ) is incomplete w.r. to Kripke
semantics. Yet it is a natural calculus to consider. In sect. 4
we introduce a generalized Kripke semantics (standard Kripke
semantics being but a limit case of the generalized one) and
show that, on the one side, TK]!1 is valid w.r. to the generalized
semantics while, on the other side, BF !1 admits a very simple
generalized countermodel.
In sect. 5 we prove a completeness theorem for TK]!1 w.r. to
the generalized Kripke semantics, thus providing an adequate
relational semantics for this system. Actually, one and the same
proof strategy — a suitable adaptation of the familiar canonical model construction — gives also, as a bonus, a smooth
completeness proof for TK!1 w.r. to standard Kripke semantics, which is alternative to the completeness proofs given in [8],
[10] for LK⇤
BF !1 , and in [6] for KL!1 .
!1
The question concerning cut-free axiomatizability of K!1 is
discussed in the concluding sect. 6. In particular, we show by
suitable counterexamples that TK!1 , as well as some natural
variants of this calculus, do not admit cut-elimination. As far
as we know, only one cut-free axiomatization of K!1 is presently
available, Tanaka’s calculus TLM!1 ([9]), whose sequents are
trees of standard sequents. The cut-elimination theorem for
TLM!1 is however proved only semantically.
2. The infinitary modal language L⇤
!1
The alphabet of our infinitary multimodal propositional language L⇤
!1 comprises the symbols:
• p0 , pe0 , p1 , pe1 , p2 , pe2 . . . : denumerably many propositional
atoms and negated propositional atoms (literals);
V W
e i (i < !) : logical operators.
• , , ⇤i , ⇤
We denote by Lit (Lit+ ) the set of all (positive) literals.
The formulas of the language L⇤
!1 are generated starting from
the literals by applying as usual the modal operators ⇤i and
139
5
Infinitary Modal Logic and Generalized Kripke Semantics
W
e i (i < !), and by forming countable disjunctions ( ) and
⇤
V
conjunctions ( ).
More precisely, the set FM of all L⇤
!1 -formulas is the least
fixed-point of the monotone operator F such that
e i x | x 2 X, i < !}
F (X) = X [ {⇤i x | x 2 X, i < !} [ {⇤
^
_
[ { Z | Z ✓ X, |Z| !} [ { Z | Z ✓ X, |Z| !}
Equivalently,
FM =
[
FM↵
↵<!1
where FM↵ (↵ < !1 ) is defined by transfinite induction as follows:
(i) FM0 = Lit,
(ii) FMβ+1 = F (FMβ ),
S
(iii) FMλ = β<λ FMβ (where λ is a limit ordinal).
The notion of subformula of a formula is defined as usual.
Notational conventions 2.1. Henceforth, the lower-case
Greek letters ', , χ, possibly with indices, are used as metavariables for formulas. Capital Greek letters Γ, ∆, Φ, , . . . will
range over countable subsets of FM, while capital Roman letters
C, D, . . . will range over finite subsets of FM.
Notice that the connective ‘¬’ is not contained in the al⇤
phabet of L⇤
!1 ; officially, L!1 -formulas are always in negation
normal form. It is however convenient to introduce negation in
the metalanguage. The map
' 2 FM 7! ¬' 2 FM
is defined inductively in the natural way:
pk := pk ,
(i) ¬pk := pek , ¬e
e i , ¬⇤
e i := ⇤i ,
(ii) ¬⇤i := ⇤
V
W
W
(iii) ¬
:= {¬ | 2 }, ¬
:=
V
{¬ |
2
}.
Thus ¬¬' and ' are syntactically identical (notation: ¬' ⌘ ').
Observe that ' need not be a subformula of ¬'.
6
140
Pierluigi Minari
Notational conventions 2.2.
• Φ,
:= Φ [ { };
Φ,
:= Φ [ ;
• ¬Φ := {¬' | ' 2 Φ};
V
• ' ^ := {', }; ' _
:= ¬' _ ;
V
W
• > := ;; ? := ;;
:=
• '!
• ⇤i Φ := {⇤i ' | ' 2 Φ};
...;
W
{', };
e i Φ := {⇤
e i ' | ' 2 Φ}.
⇤
Note that ¬> ⌘ ? and ¬? ⌘ >.
We conclude the present section by introducing one further
notion, which however will not be used until sect. 5.
Consider a countable set Γ of formulas: as FM is uncountable,
there exists an ordinal ↵ < !1 such that Γ ✓ FM↵ . This justifies
the following
Definition 2.3 (Environment).
set of FM:
Let Γ be a countable sub-
(i) rk(Γ) := the least limit ordinal λ < !1 s.t. Γ ✓ FMλ ;
(ii) E[Γ] := FMrk( ) .
We call E[Γ] the environment of Γ.
Lemma 2.4. For every countable set Γ of formulas, its environment E[Γ] has the following properties:
(1) E[Γ] is countably infinite;
(2) E[Γ] is closed under subformulas and under negation;
(3) E[Γ] is closed under finite conjunctions and disjunctions,
e i (i < !);
as well as under ⇤i and ⇤
V
(4) for each formula 2 E[Γ]
and
Φ 2 E[Γ], and for every
V
i < !, the formula ⇤i { _ ' | ' 2 Φ} belongs to E[Γ].
Proof. It is easily verified that, for any β < !1 , FMβ satisfies
(1) – (2), and that FMβ also satisfies (3) – (4) whenever β is a
limit ordinal. The conclusion follows by Definition 2.3.
⇤
Infinitary Modal Logic and Generalized Kripke Semantics
1417
3. Tait-style infinitary modal calculi
As anticipated in the introductory section, we work with
Tait-style sequent calculi. Along with the peculiar NNF -syntax
of our language L⇤
!1 , this format allows a considerable economy
and elegance in the presentation of the proof systems under
investigation.
The sequents to be derived are therefore not the usual twosided sequents, but rather one-sided sequents, that is sets of
formulas, having a disjuctive reading in the intended interpretation1.
A choice concerning the cardinality of sequents must however
be taken right from the start: shall we confine to finite sequents
only? Or shall we allow countably infinite sequents too?
As far as infinitary classical truth-functional logic is concerned, it makes no essential di↵erence which of the two alternatives we adopt (see e.g. [2]). Indeed, let us start by considering
three Tait-style non modal propositional calculi T]!1 , T⇤!1 and
T!1 , whose inference rules2 are shown in Figures 1, 2 and 3.
T]!1 derives finite sets of formulas, whereas both T⇤!1 and T!1
derive countable, possibly infinite sets of formulas (recall the
Notational conventions 2.1). Modulo this di↵erence T]!1 and
side, T⇤!1
T⇤!1 have the “same” inference
W rules; on the other
3
and T!1 di↵er only in the -introduction rule . Notice also
1In a classical environment, there is an obvious correspondence between two-
sided and one-sided sequents. ‘ Γ ) ∆’
‘ ¬Γ, ∆’, and ‘ Γ’
‘ ¬Γ1 ) Γ2 ’
for each partition (Γ1 , Γ2 ) of Γ.
2Where formulas are of course intended to belong to the ⇤ - and ⇤ -less
i
i
fragment L!1 of L⇤
!1 . Caution: in all the calculi under consideration in this
paper sequents are sets, not multisets. This means that contraction is hidden in
the logical inference rules: the principal formula of an inference may occur as a
side formula in the premise(s); e.g.
C, ' _ ,
C, ' _
and
Γ, Φ, Φ
Γ, Φ
are instances of OR, respectively OR+ .
3Of course, because of the presence of the rule W, the rule OR is a derived rule
in T!1 .
8
142
Pierluigi Minari
Figure 1. T]!1
p, ¬p
C
C, D
ID (p 2 Lit+ )
C, '
W
C, Φ
OR (' 2
W
C, '
D, ¬'
C, D
CUT
. . . C, ' . . . (all ' 2 Φ)
V
C, Φ
)
AND
Figure 2. T⇤!1
p, ¬p
ID (p 2 Lit+ )
Γ, '
W
Γ, Φ
OR (' 2
Γ
Γ, ∆
W
Γ, '
∆, ¬'
Γ, ∆
CUT
. . . Γ, ' . . . (all ' 2 Φ)
V
Γ, Φ
)
AND
Figure 3. T!1
p, ¬p
ID (p 2 Lit+ )
Γ, Φ
W
Γ, Φ
OR+
Γ
Γ, ∆
W
Γ, '
∆, ¬'
Γ, ∆
CUT
. . . Γ, ' . . . (all ' 2 Φ)
V
Γ, Φ
AND
that the generalized cut-rule with countably many premises
Γ, Φ
. . . ¬', ∆ . . . (all ' 2 Φ)
Γ, ∆
CUT+
Infinitary Modal Logic and Generalized Kripke Semantics
9
143
is admissible (as a derived rule) in T!1 ; actually T!1 is equivalent to the calculus obtained from T⇤!1 by replacing the rules
OR+ and CUT with the rules OR and CUT+ , respectively.
It is immediately verified that
(1) for an arbitrary formula ', the sequents
', ¬'
and
>
are cut-free derivable in any of these calculi, the latter
through a vacuous application of the AND rule;
(2) for every C and ,
T]!1 ` C , T⇤!1 ` C
and T⇤!1 `
) T!1 `
In fact, also the second arrow in (2) above can be reversed and,
in a sense to be specified, the three calculi are equivalent. Let
us write
W
is truth-functionally valid:
• ‘|=!1 ’ to mean that
for every valuation v : Lit+ −! {0, 1} there is some ' 2
such that v(') = 1 (v being extended from positive
literals to arbitrary L!1 -formulas in the natural way);
• ‘`0 ’ to mean that
is cut-free derivable.
Then the relations between the three calculi, their soundness and semantic completeness, as well as the cut-elimination
property for T]!1 , can be summarized as follows on the basis of
known results.
Proposition 3.1. Let C be a finite set of L!1 -formulas, and
{ 1 , . . . , n } be a finite, possibly empty set of countable sets of
L!1 -formulas. Then the following are equivalent:
W
W
(1) T]!1 `0
1, . . . ,
n, C
W
W
(2) T]!1 `
1, . . . ,
n, C
(3) T⇤!1 `
(4) T!1 `
(5) |=!1
1, . . . ,
n, C
1, . . . ,
n, C
1, . . . ,
n, C
10
144
Pierluigi Minari
Proof. (1) ) (2), (3) ) (4), (4) ) (5): trivial.W
(2) ) (3): straightforward, because T⇤!1 ` ¬ Γ, Γ.
(5) ) (1): completeness and cut elimination for T]!1 are proved
in [7].
⇤
Notice that the syntactic proof of cut-elimination for T]!1
given in [7] can be easily adapted to T⇤!1 and T!1 (see e.g. [3];
cp. also [4] and [2]).
Let us now extend the above calculi with modal inference
rules, with the aim of capturing — by means of an adequate
infinitary Tait-style calculus — the infinitary modal propositional logic determined by the class of all Kripke frames, i.e.
the infinitary version K!1 of the (multi)modal system K. Taking into account the need for a preliminary choice concerning
the cardinality of sequents, we shall consider on the basis of the
previous investigation the three candidates TK]!1 , TK⇤!1 and
TK!1 shown in Figure 4.
Figure 4. TK]!1 , TK⇤!1 and TK!1
TK]!1 := T]!1 +
¬C, '
K] (i < !)
¬⇤i C, ⇤i ' i
TK⇤!1 := T⇤!1 +
¬Γ, '
K (i < !)
¬⇤i Γ, ⇤i ' i
TK!1 := T!1 +
¬Γ, '
K (i < !)
¬⇤i Γ, ⇤i ' i
The present scenario turns out to be radically di↵erent from
the previous (non modal) one: now the alternative “finite vs
countable sequents” does matter! Indeed, we have:
(a) TK]!1 admits cut-elimination, but is incomplete (though
clearly sound) w.r. to Kripke semantics;
Infinitary Modal Logic and Generalized Kripke Semantics
11
145
(b) TK⇤!1 too admits cut-elimination and is incomplete (although clearly sound) w.r. to Kripke semantics; but it
is not equivalent to TK]!1 in the sense in which T]!1 and
T⇤!1 are equivalent according to Proposition 3.1;
(c) TK!1 is instead sound and complete w.r. to Kripke semantics; yet it provably doesn’t admit cut-elimination.
As to point (a), a syntactic proof of cut-elimination for TK]!1
can be obtained by adapting Tait’s proof of the cut-elimination
theorem for T]!1 ([7], see also [9]) — the same does hold for
TK⇤!1 . The semantical incompleteness of TK]!1 follows in turn
as a consequence of cut-elimination, because of the following
easily verifiable facts:
Fact 3.2. The schema (“Barcan Formula”)
^
^
,
⇤i ! ⇤i
(BF !1 )
^
^
or, as a finite sequent, ¬ ⇤i , ⇤i
is valid in every Kripke model4.
Fact 3.3. The instance
^
^
¬ {⇤i pn | n < !}, ⇤i {pn | n < !}
of BF !1 has no cut-free derivation in TK]!1 .
Notice that BF !1 is (cut-free) derivable in TK!1 as follows:
�
⇢
ID, W · · · (' 2 )
···
¬ ,'
V
AND
¬ ,
V Ki
¬⇤i , ⇤i
V
V OR+
¬ ⇤i , ⇤i
On the other side, the “converse Barcan Formula”
^
^
⇤i
!
⇤i
4See the next section.
12
146
Pierluigi Minari
is derivable already in TK]!1 :
)
(
¬', '
V
OR
¬ Φ, '
···
V
Ki · · · (' 2 Φ)
¬⇤i Φ, ⇤i '
V V
¬⇤i Φ, ⇤i Φ
AND
As to point (b), it is immediate to verify that TK]!1 and
TK⇤!1 derive the same finite sequents,
(3.1)
TK]!1 ` C , TK⇤!1 ` C
for every C,
like the corresponding non modal calculi. Hence BF!1 is not
derivable in TK⇤!1 as well. On the other side, contrary to Proposition 3.1,
_
Γ
(3.2)
TK⇤!1 ` Γ 6) TK]!1 `
V
For instance, let
:= {¬⇤i p0 , ¬⇤i p1 , ¬⇤i p2 . . . , ⇤i n<! pn }.
Then TK⇤!1 `
�
⇢
ID, W · · · (m < !)
···
{¬pn }n<! , pm
V
AND
{¬pn }n<! , n<! pn
V
Ki
{¬⇤i pn }n<! , ⇤i n<! pn
but, as a semantic
W argument in the next section (Fact 4.4) will
.
show5, TK]!1 0
As to the two claims made in point (c), these will be addressed in section 5, where the semantic completeness of TK!1
is proved, and in section 6, where the question concerning the
non eliminability of the cut rule in this calculus is discussed.
Notice that merely on the basis of what has been established so
far, in particular (3.1) and (3.2) above, we already know that
TK⇤!1 is not closed under the inference rule OR+ characteristic
of TK!1 .
5A simple syntactic argument is also at hand, by using Fact 3.3 and (see
below) point (ii) in the proof of Proposition 3.4.
Infinitary Modal Logic and Generalized Kripke Semantics
13
147
We conclude this section by stating a simple equivalence result, by which the key role played in the present context by the
Barcan Formula BF!1 is made fully evident.
Let TKB]!1 and TKB⇤!1 be the calculi obtained from TK]!1 ,
resp. TK⇤!1 , by adding all the instances of BF!1 as new initial
sequents. Then:
Proposition 3.4. For every countable set Γ of formulas, the
following are equivalent:
(1) TK!1 ` Γ
(2) TKB⇤!1 ` Γ
W
(3) TKB]!1 ` Γ
Proof. (2) ) (1): obvious, since TK!1 ` BF!1 .
(3) ) (2): it is sufficient to observe that the inversion of OR+ is
admissible (as a derived rule) in TK(B)⇤!1 :
⇢
�
ID, W · · · (' 2 Φ)
···
¬', Φ
W
W
AND
Γ, Φ
¬ Φ, Φ
CUT
Γ, Φ
(1) ) (3): first of all, observe that for every Γ, ∆, C :
W W
(i) TK(B)]!1 ` ¬ Γ, (Γ [ ∆)
W
W
(ii) TK(B)]!1 ` ¬ (Γ [ C), Γ, C
W
W W
(iii) TK(B)]!1 ` ¬ (Γ [ ∆), Γ, ∆
The easy verification is left to the reader.
Next, W
given a TK!1 -derivation D ` Γ, we produce a derivation
0
D ` Γ in TKB]!1 arguing by transfinite induction on the
height of h(D) < !1 of D (which is defined in the natural way)
and taking cases according to the final inference R of D. The
case R = ID is trivial; the cases R = W, R = CUT are easily dealt
with using the I.H. together with (i) and (ii) above. Let us spell
out the details only for the cases R = OR+ (R = AND is similar)
and R = Ki .
[ R = OR+ ]: then, taking into account the possibility that the
14
148
Pierluigi Minari
principal formula is a side formula in the premise, D has the
form:
..
W.
∆, Φ, Φ +
W
OR
∆, Φ
W
W
We obtain D0 ` (∆ [ { Φ}) in TKB]!1 as follows:
..
. I.H. W
W
(∆ [ Φ [ { Φ})
W
W
CUT with (ii)
(∆ [ Φ), Φ
W W
CUT with (iii)
∆, Φ
W
W
W CUT with (i)
(∆ [ { Φ}), Φ
W
W
OR
(∆ [ { Φ})
[ R = Ki ]: then D has the form:
..
.
¬∆, '
Ki
¬⇤i ∆, ⇤i '
W
We obtain D0 ` (¬⇤i ∆ [ {⇤i '}) in TKB]!1 as follows:
..
W . I.H.
(¬∆ [ {'})
V
CUT with (ii)
BF!1
¬ ∆, '
W
V
V
K]
¬⇤i ∆, ⇤i ' i
¬⇤i ∆, ⇤i ∆
W
CUT
¬⇤i ∆, ⇤i '
W
CUT with (i)
(¬⇤i ∆ [ {⇤i '}), ⇤i '
W
OR
(¬⇤i ∆ [ {⇤i '})
⇤
4. Generalized (and standard) Kripke frames
As we saw, TK]!1 is incomplete w.r. to Kripke semantics.
In order to provide this calculus with a (possibly natural) adequate semantics, we introduce here a generalization of standard
Kripke models.
15
149
Infinitary Modal Logic and Generalized Kripke Semantics
A generalized (multimodal) Kripke frame is a pair
G = hW, {Ri }i<! i
where W is a non empty set and, for each i < !, Ri is a non
empty family of binary relations over W which is downward
closed w.r. to inclusion, i.e. it satisfies:
(DD)
(8R 2 Ri )(8S 2 Ri )(9T 2 Ri )(T ✓ R \ S)
Standard Kripke frames are of course a special case of generalized Kripke frames, namely the case in which the family Ri
reduces to a singleton {Ri } — singletons trivially satisfy the
condition (DD) — for each i < !.
When dealing with standard frames and models (see below) we
will keep to the familiar way of presentation, by identifying {Ri }
with Ri . In other words, we write S = hW, {Ri }i<! i instead of
S = hW, {{Ri }}i<! i, when S is standard.
A valuation over a generalized frame G is, as usual, a map
v : Lit+ −! 2W
Finally, a generalized Kripke model is a triple
M = hW, {Ri }i<! , vi
where G = hW, {Ri }i<! i is a generalized Kripke frame and v is
a valuation over G.
Given a generalized model M = hW, {Ri }i<! , vi, an element
w of WM = W and a formula ' 2 FM, the relation
M, w ✏g '
is defined inductively as follows:
(i) M, w ✏g pk
(ii) M, w ✏g pek
V
(iii) M, w ✏g
W
(iv) M, w ✏g
i↵ w 2 v(pk )
i↵ w 2
/ v(pk )
i↵ M, w ✏g
i↵ M, w ✏g
for each
for some
2
2
(v) M, w ✏g ⇤i
i↵ there exists a relation R 2 Ri such
(i < !)
that, for every u 2 W, wRu implies M, u ✏g
16
150
Pierluigi Minari
ei
(vi) M, w ✏g ⇤
i↵ for each R 2 Ri there is a state
(i < !)
u 2 W such that wRu and M, u 2g
where ‘M, w 2g '’ is short for ‘not (M, w ✏g ')’.
Notice that, for every formula ', we have
M, w ✏g ¬' i↵ M, w 2g '
as expected.
Truth of a formula ' in a generalized model M, in symbols
M |=g '
as well as generalized universal validity of a formula, in symbols
|=g '
are defined as usual:
• M |=g ' i↵
• |=g ' i↵
M, w ✏g ' for all w 2 WM ;
M |=g ' for all generalized models M.
All these notions are extended to arbitrary countable sets of
formulas Γ according to the following
Notational conventions 4.1.
• M, w ✏g Γ i↵
• M |=g Γ i↵
• |=g Γ i↵
M, w ✏g ' for some ' 2 Γ;
M, w ✏g Γ for all w 2 WM ;
M |=g Γ for all generalized models M.
Caution: note the disjunctive reading of M, w ✏g Γ !
Standard models now coincide with generalized models based
on standard frames S = hW, {Ri }i<! i. In fact, as it is immediately seen, in this case the two non-standard clauses (v) and
(vi) of the above inductive definition become the usual:
i↵ wRi u implies M, u ✏
for every
(v)st M, w ✏ ⇤i
u 2 W (i < !)
ei
i↵ there exists a state u 2 W s.t. wRi u
(vi)st M, w ✏ ⇤
(i < !)
and M, u 2
17
151
Infinitary Modal Logic and Generalized Kripke Semantics
and so the satisfaction relation ‘M, w ✏g '’ boils down to the
familiar one ‘M, w ✏ '’.
Henceforth, when dealing with standard models M, we will
drop the index g from ‘M, w ✏g '’ and from all the related
notions, including those of Convention 4.1.
Trivially, the three calculi TK]!1 , TK⇤!1 and TK!1 are valid
w.r. to the standard Kripke semantics. Yet, after having generalized the notions of Kripke frame, Kripke model and universal
validity in the way described above, we can easily realize that
the soundness of the characteristic schema
⇤(' ! ) ! (⇤' ! ⇤ )
of the minimal normal modal system K is not lost. As a consequence, we have that the infinitary system TK]!1 is in fact
valid w.r. to the generalized semantics.
Theorem 4.2 (g-Validity for TK]!1 ).
For every finite set C of L⇤
!1 -formulas:
`] C
)
|=g C
Proof. It is sufficient to check that the modal rules K]i of TK]!1
preserve truth in any generalized model M = hW, {Ri }i<! , vi;
that is, for every i < !:
M |=g {¬
1 , . . . ¬ n , '}
) M |=g {¬⇤i
1 , . . . ¬⇤i
n , ⇤i '}
So, assume that M |=g {¬ 1 , . . . ¬ n , '} and suppose, towards
a contradiction, that M 6|=g {¬⇤i 1 , . . . ¬⇤i n , ⇤i '}; then
(4.1)
M, w ✏g ⇤i
1,
. . . , M, w ✏g ⇤i
n
and
(4.2)
M, w 2g ⇤i '
for some w 2 W .
By (4.1), there exist relations R1 , . . . , Rn 2 Ri such that
(4.3)
for 1 k n : (8u 2 W )(wRk u ) M, u ✏g
k)
18
152
Pierluigi Minari
Applying the downward directedness condition (DD) to Ri , let
T 2 Ri be such that T ✓ R1 \ . . . \ Rn . Then, by (4.3):
for 1 k n : (8u 2 W )(wT u ) M, u ✏g
(4.4)
k)
On the other side, it follows by (4.2) and T 2 Ri that there
exists a state t 2 W such that
wT t and M, t 2g '
(4.5)
Hence, by (4.4) and (4.5):
M, t 2g {¬
(4.6)
1 , . . . , ¬ n , '}
in contrast with our assumption M |=g {¬
1 , . . . ¬ 1 , '}.
⇤
Now, we show that the Barcan-Formula BF !1 is not universally valid w.r. to the generalized Kripke semantics.
Fact 4.3 (g-countermodel for BF !1 ).
6|=g BF !1 .
Proof. Let us consider the generalized Kripke model
C = hN⇤ , {Ri }i<! , vi
where:
• N⇤ := N [ {⇤} (the natural numbers plus a new state ⇤);
• for every i < !, Ri = R := {RX | X 2 cof (N)}, where:
– cof (N) is the set of all cofinite subsets of N,
– RX := {h⇤, ni | n 2 X};
• v(pn ) := N r {n}, for each pn 2 Lit+ .
The so defined family R of relations satisfies the condition (DD)
since on the one side cofinite subsets of N are closed under finite
intersection, and on the other side, by definition
RX \ RY = RX\Y 2 R (X, Y 2 cof (N)).
Now, for i < ! we have, by construction, that for each n ≥ 0:
(4.7)
whence
(4.8)
(8w 2 N⇤ )(⇤RNr{n} w ) C, w ✏g pn )
C, ⇤ ✏g ⇤i pn
Infinitary Modal Logic and Generalized Kripke Semantics
Thus
(4.9)
C, ⇤ ✏g
19
153
^
{⇤i pn | n < !}
On the other side, for every X 2 cof (N) and every k 2 X:
(4.10)
⇤RX k
and C, k 2g pk
Hence
(4.11)
C, ⇤ 2g ⇤i
^
{pn | n < !}
We conclude from (4.9) and (4.11) that
^
^
C 6|=g {⇤i pn | n < !} ! ⇤i {pn | n < !}
the latter formula being an instance of BF !1 .
⇤
Hence, Theorem 4.2 and Fact 4.3 supplement the underivability proof of BF !1 in TK]!1 , mentioned in the previous section,
with a semantical argument. We can also use the above generalized Kripke
V 4.2, to see that the
W model C, together with Theorem
formula {¬⇤i p0 , ¬⇤i p1 , ¬⇤i p2 . . . , ⇤i n<! pn } is not derivable in TK]!1 , as claimed without proof in sect. 3. Indeed, by
(4.8) and (4.2), we have
W
V
Fact 4.4. C, ⇤ 2g {¬⇤i p0 , ¬⇤i p1 , ¬⇤i p2 . . . , ⇤i n<! pn }.
In conclusion, observe that the results mentioned in point (b)
of section 3 show that TK⇤!1 is not sound w.r. to the generalized Kripke semantics. Of course, both TK⇤!1 and TK!1 are
sound w.r. to the narrower class of all the generalized Kripke
frames G = hW, {Ri }i<! i satisfying the countable downward
directedness condition
(DD!1 ) (8S ✓ Ri )(|S| ! ! (9T 2 Ri )(8S 2 S)(T ✓ S))
However, only TK!1 is also complete w.r. to this special class
of generalized frames. Thus the problem of finding an adequate
Kripke-style semantic characterization of TK⇤!1 remains open.
20
154
Pierluigi Minari
5. Completeness theorems for TK]!1 and TK!1
We are going to prove, via a canonical model technique, a
completeness theorem for TK]!1 w.r. to the generalized Kripke
semantics, and a completeness theorem for TK!1 w.r. to the
standard Kripke semantics. As the reader will see, our proofs
of the two results run closely parallel and, in fact, eventually
diverge only in one very specific key point.
We start with the common part of the two proofs. The notion
of environment E[Γ] of a countable set Γ of formulas (see sect.
2) is employed here for the first time. In the following, we will
often make a tacit use of Lemma 2.4.
By convenience, let us denote by J an arbitrarily fixed element of {TK]!1 , TK!1 }. Recall that the rule OR is available
also in TK!1 as a derived rule.
Definition 5.1. Let Γ ✓ FM:
(i) J B Γ := J ` C for some finite subset C of Γ;
(ii) Γ is J-consistent i↵ J 7 ¬Γ;
(iii) Γ is
(a)
(b)
(c)
J-saturated i↵:
Γ is J-consistent,
Γ [ ¬Γ = E[Γ],
V
V
for all Φ such that Φ 2 E[Γ] : Φ ✓ Γ ) Φ 2 Γ;
V
(iv) Γ is J-uniform i↵ for every Φ such that V
Φ 2 E[Γ]:
if J B ¬Γ, ' for each ' 2 Φ, then J B ¬Γ, Φ.
Lemma 5.2. [Saturated sets] Any J-saturated set Γ satisfies:
(1) For all ' 2 E[Γ], either ' 2 Γ or ¬' 2 Γ, and not both.
(2) For all ' 2 E[Γ], if J B ¬Γ, ' then ' 2 Γ.
V
V
(3) For all Φ such that Φ 2 E[Γ]: Φ ✓ Γ i↵ Φ 2 Γ.
W
W
(4) For all Φ such that Φ 2 E[Γ]: Φ \ Γ 6= ; i↵ Φ 2 Γ.
Proof.
(1): immediate, by (a) and (b) of Definition 5.1.(iii).
/ Γ then, by (1),
(2): suppose J ` ¬C, ' for some C ✓ Γ. If ' 2
¬' 2 Γ and so D := C [ {¬'} is finite subset of Γ such that
155
21
Infinitary Modal Logic and Generalized Kripke Semantics
J ` ¬D, against the J-consistency
of Γ.
V
(3): if Φ is such that Φ 2 E[Γ], then also ΦV ✓ E[Γ] by
Definition 2.3 and Lemma
V 2.4. Suppose now that Φ 2 Γ, and
let ' 2 Φ: as J ` ¬ Φ, ' trivially, we have that J B ¬Γ, '
and so that ' 2 Γ by (2). Hence Φ ✓ Γ. The converse direction
is given by (c) of Definition
5.1.(iii).V
W
(4): by (1) and (3), Φ 2 Γ i↵ ¬Φ 2
/ Γ i↵ ¬Φ * Γ i↵
Φ \ Γ 6= ;.
⇤
V
Fact 5.3 (Distributivity of _ over ).
^
^
Φ $
{ _ ' | ' 2 Φ}
J `(0) _
Proof.
···
···
(
(
)
¬ , ,'
¬', , '
AND
¬( _ '), , '
· · · (' 2 Φ)
V
OR
¬ { _ ' | ' 2 Φ}, , '
V
V
¬ { _ ' | ' 2 Φ}, , Φ
V
V OR
¬ { _ ' | ' 2 Φ}, _ Φ
¬', , '
V
OR )
¬ Φ, , '
¬ , ,'
V
AND
· · · (' 2 Φ)
¬( _ Φ), , '
OR
V
¬( _ Φ), _ '
V
V
¬( _ Φ), { _ ' | ' 2 Φ}
AND
AND
⇤
Notational convention 5.4. For i < !:
p
i
Γ := {' | ⇤i ' 2 Γ}.
Lemma 5.5. For every countable set Γ of formulas:
(1) If Γ is J-consistent then, for all 2 E[Γ], either Γ [ { }
or Γ [ {¬ } is J-consistent.
p
(2) If Γ is J-saturated and ⇤i 2 E[Γ] r Γ then i Γ [ {¬ }
is J-consistent.
(3) If Γ is finite, then Γ is J-uniform.
22
156
Pierluigi Minari
(4) If Γ is J-uniform then, for every
J-uniform.
2 E[Γ], Γ [ { } is
Proof. (1) is immediate by the CUT rule, and
(3) is trivial.
p
i
(2): under the assumptions suppose that Γ [ {¬ } is not Jconsistent. Then, for some finite set C such that ⇤i C ✓ Γ,
we have J ` ¬C, and so, by an application of the rule K] ,
J ` ¬⇤i C, ⇤i . Thus J B ¬Γ, ⇤i , and it now follows by (2)
of Lemma 5.2 that ⇤i 2 Γ, against the assumption.
(4): assume that Γ is J-uniform,Vand suppose J B ¬Γ, ¬ , '
for all ' 2 Φ. Then J B ¬Γ, {¬ _ ' | ' 2 Φ} by the
J-uniformity of Γ, plus the closure properties of environments
E[Γ], in particular (4) of Lemma
2.4. The conclusion
V
V follows by
CUT using the sequent ¬ {¬ _ ' | ' 2 Φ}, ¬ , Φ which is
derivable in J by Fact 5.3.
⇤
Lemma 5.6 (Saturation). Every J-consistent and J-uniform
countable set of formulas Γ can be extended to a countable set
Γ⇤ ◆ Γ which is J-saturated.
Proof. Assume that Γ is both J-consistent and J-uniform. First
of all, let χ0 , χ1 , χ2 . . . be an arbitrarily fixed enumeration of the
environment E[Γ] of Γ (recall that |E[Γ]| = ! by (1) of Lemma
2.4).
Define now inductively, for n ≥ 0, a set Γn ✓ E[Γ] provably
satisfying:
(1) Γn is J-consistent and J-uniform;
(2) Γ ✓ Γn ✓ Γn+1 .
Basis:
• Γ0 := Γ
Γk+1 :
Step Γk
Supposing that Γ0 , . . . , Γk have been defined in a way that
(1) and (2) are satisfied, consider the formula χk . Since Γk is
J-consistent, at least one of the sets Γk [ {χk } and Γk [ {¬χk }
must be J-consistent by (1) of Lemma 5.5.
— If Γk [ {χk } is J-consistent, we set:
Infinitary Modal Logic and Generalized Kripke Semantics
23
157
• Γk+1 := Γk [ {χk }.
— If Γk [ {χk } is not J-consistent, and so Γk [ {¬χk } is
J-consistent, we set:
• Γk+1 := Γk [ {¬χk },
V
in case χk is not of the form Φ for some Φ;
• Γk+1 := Γk [ {¬χk , ¬'},
V
in case χk is of the form V Φ, where ' 2 Φ is chosen in
a way such that Γk [ {¬ Φ, ¬'} is J-consistent.
Observe that in the latter case such a formula ' 2 Φ always
exists. Indeed, otherwise we would have
^
(5.1)
for every ' 2 Φ : J B ¬(Γk [ {¬ Φ}), '
V
and in turn, since Γk [ {¬ Φ} is J-uniform by (4) of Lemma
5.5 and the fact that Γk is J-uniform:
^
^
J B ¬(Γk [ {¬ Φ}), Φ
(5.2)
^
i.e. J B ¬(Γk [ {¬ Φ})
against the J-consistency of Γk [ {¬χk }.
Clearly, (1) and (2) are preserved in the step Γk
Finally, set:
[
Γn
Γ⇤ :=
Γk+1 .
n≥0
⇤
⇤
Now Γ ◆ Γ and Γ is easily seen to be saturated. Indeed, it
⇤
is J-consistent by (1) and (2), and Γ⇤ [ ¬Γ⇤ = E[Γ]
V = E[Γ ] by
/ Γ⇤ .
construction. As to the last condition, suppose
Φ = χn 2
V
Then, by the construction, Γn+1 = Γn [ {¬ Φ, ¬'} for some
/ Γ⇤ . Hence
' 2 Φ; so ¬' 2 Γ⇤ and, by J-consistency of Γ⇤ , ' 2
⇤
⇤
Φ*Γ .
The following Lemma, which makes an essential use of the
Barcan Formula, does hold only for the calculus TK!1 .
Lemma 5.7. Let Γ be a TK!1 -saturatedpset of formulas, and
/ Γ. Then i Γ [ {¬ } is TK!1 let 2 E[Γ] be such that ⇤i 2
uniform.
158
24
Pierluigi Minari
Proof. Under
p the assumptions, suppose that for each ' 2 Φ,
TK!1 B ¬( i Γ [ {¬ }), ' ; that is
p
i
(5.3)
for each ' 2 Φ : TK!1 B ¬ Γ, , '
Then, by applying OR twice, followed by Ki , we have:
p
i
(5.4)
for each ' 2 Φ : TK!1 B ¬⇤i Γ, ⇤i ( _ ')
p
and, by taking into account that ⇤i i Γ ✓ Γ:
(5.5)
for each ' 2 Φ :
TK!1 B ¬Γ, ⇤i ( _ ')
The set Γ is TK!1 -saturated by assumption,
V and by Lemma 2.4
the formulas ⇤i ( _ ') (for ' 2 Φ) and {⇤i ( _ ') | ' 2 Φ}
all belong to E[Γ]. Thus (5.5), together with (2) and (3) of
Lemma 5.2, yields:
^
(5.6)
⇤i { _ ' | ' 2 Φ} 2 Γ
On the other side, we have
^
^
Φ)
(5.7)
TK!1 ` ¬ ⇤i { _ ' | ' 2 Φ}, ⇤i ( _
which is shown as follows by making use of the fact that BF !1
is derivable in TK!1 (see sect. 3):
V
V Fact 5.3
¬ { _ ' | ' 2 Φ}, _ Φ
V
V
K
¬⇤i { _ ' | ' 2 Φ}, ⇤i ( _ Φ)
CUT with BF !1
V
V
¬ ⇤i { _ ' | ' 2 Φ}, ⇤i ( _ Φ)
Now (5.6) and (5.7) yield
(5.8)
TK!1 B ¬Γ, ⇤i ( _
^
Φ)
V
It follows by (2) p
of Lemma 5.2 that ⇤i ( _ p Φ) 2 Γ, and so
V
V
that _ Φ 2 i Γ. Therefore TK!1 B ¬ i Γ, _ Φ and
finally
^
p
i
TK!1 B ¬( Γ [ {¬ }), Φ
(])
using a CUT with TK!1 ` ¬(χ1 _ χ2 ), χ1 , χ2 .
⇤
We are now ready to define the canonical models for TK!1
and TK]!1 . Let
Infinitary Modal Logic and Generalized Kripke Semantics
25
159
• SAT !1 := {Γ | Γ ✓ FM, |Γ| !, Γ is TK!1 -saturated};
• SAT!] 1 := {Γ | Γ ✓ FM, |Γ| !, Γ is TK]!1 -saturated}.
Definition 5.8 (TK!1 - and TK]!1 -universal model).
(1) U!1 is the standard Kripke model
U!1 = hSAT !1 , {Ri }i<! , vi
where
p
• ΓRi ∆ :, i Γ ✓ ∆ (Γ, ∆ 2 SAT !1 , i < !);
• v(p) := {Γ 2 SAT !1 | p 2 Γ} for every p 2 Lit+ .
(2) U!] 1 is the generalized Kripke model
U!] 1 = hSAT!] 1 , {Ri }i<! , vi
where
C
• for i < !, Rp
i := {Ri | C ✓fin FM}, with
ΓRiC ∆ :, i Γ \ C ✓ ∆ (Γ, ∆ 2 SAT ]!1 , i < !)
• v(p) := {Γ 2 SAT ]!1 | p 2 Γ} for every p 2 Lit+ .
Note that, trivially,
RiC[D ✓ RiC \ RiD and RiC[D 2 Ri ,
so that hSAT!] 1 , {Ri }i<! i satisfies the characteristic condition (DD) of a generalized frame.
Theorem 5.9.
(1) For every Γ 2 SAT !1 , for every formula ' 2 E[Γ]:
U!1 , Γ ✏ ' , ' 2 Γ
(2) For every Γ 2 SAT ]!1 , for every formula ' 2 E[Γ]:
U!] 1 , Γ ✏g ' , ' 2 Γ
Proof. For both (1) and (2) we argue by (transfinite)
V induction
W
on (the rank of) '. The cases ' 2 Lit and ' ⌘ Φ, Φ are
immediate by Definition 5.9 and Lemma 5.2, (3) – (4). The case
' ⌘ ⇤i requires instead separate arguments for U!1 and U!] 1 ,
e i easily reduces to the presee below. Finally, the case ' ⌘ ⇤
vious one by (1) of Lemma 5.2.
(U!1 ) : ' ⌘ ⇤i .
26
160
Pierluigi Minari
[(]: immediate by the definition of Ri and the induction hypothesis.
p
/ Γ. Then ⇥ := i Γ [ {¬ } is TK!1 [)]: suppose that ⇤i 2
consistent by (2) of Lemma 5.5 as well as TK!1 -uniform by
Lemma 5.7. Applying Lemma 5.6 to ⇥ we find a ∆ 2 SAT !1
such that ΓRi ∆ and 2
/ ∆. The conclusion U!1 , Γ 2 ⇤i follows by applying the induction hypothesis.
(U!] 1 ) : ' ⌘ ⇤i .
[(]: suppose ⇤i 2 Γ, and let C := { }. Then RiC 2 Ri
and, for every ∆ 2 SAT ]!1 such that ΓRiC ∆ we obviously have
2 ∆, and so also U!] 1 , ∆ ✏g
by the induction hypothesis.
]
Hence U!1 , Γ ✏g ⇤i .
[)]: suppose that ⇤i 2
/ Γ. To conclude U!] 1 , Γ 2g ⇤i it
is sufficient to find, for every finite set C of formulas, a set
∆ 2 SAT ]!1 such that ΓRiC ∆ and U!] 1 , ∆ 2g .
p
This is done as follows: given C, let D := ( i Γ \ C) [ {¬ }. D
is TK]!1 -consistent, for otherwise by the rule K]i we would have
TK]!1 B ¬Γ, ⇤i and so by (2) of Lemma 5.2 ⇤i 2 Γ against
the assumption. On the other side, D is finite and so is also
TK]!1 -uniform by (3) of Lemma 5.5. It now follows by Lemma
5.6 that there exists a set ∆ 2 SAT ]!1 such that D ✓ ∆, and so
ΓRiC ∆, as well as ¬ 2 ∆, 2
/ ∆, and finally U!] 1 , ∆ 2g by
the induction hypothesis.
⇤
Corollary 5.10. For every finite set C ✓ FM:
TK]!1 0 C
)
U!] 1 6|=g C
Hence TK]!1 is sound and complete w.r. to generalized Kripke
semantics.
Proof. If TK]!1 0 C then ¬C is clearly TK]!1 -consistent; being
a finite set, ¬C is also TK]!1 -uniform by (3) of Lemma 5.5.
Then by Lemma 5.6 there exists a set ⇥ 2 SAT!] 1 such that
¬C ✓ ⇥, and so also C \ ⇥ = ;. It follows by (2) of Theorem
5.9 that U!] 1 , ⇥ 2g C.
⇤
27
161
Infinitary Modal Logic and Generalized Kripke Semantics
Corollary 5.11. For every countable set Γ ✓ FM:
TK!1 0 Γ
)
U!1 6|= Γ
Hence TK!1 is sound and complete w.r. to standard Kripke
semantics.
W
Proof. If TK!1 0 Γ then
W also TK!1 0 Γ. So we can argue
as above taking C = { WΓ} and using (1) of Theorem 5.9 to
conclude that U!1 , ⇥ 2W Γ, hence also U!1 , ⇥ 2 Γ, for some
⇤
⇥ 2 SAT!1 containing Γ.
6. TK!1 does not admit cut-elimination
As we anticipated, the CUT rule cannot be eliminated from
TK!1 . This will be now demonstrated by exhibiting a suitable
example of a valid sequent which is not cut-free derivable in the
calculus under investigation.
Below, {qnk | k, n ≥ 0} is a set of pairwise distinct positive
literals. As previously done, we write ‘ `0 ’ to denote cut-free
derivability.
Fact 6.1. For every m ≥ 0 and every Φ ✓ {qnk | n ≥ 0, k m},
if
^
�_
¬⇤i qnk | k ≥ 0 , ⇤i qkk
⇥ ✓ ∆ := ¬⇤i Φ,
n
k
then TK!1 00 ⇥.
Proof. We argue by transfinite induction on the height h(D) of
TK!1 -derivations D.
Let ⌧ < !1 . Assume (I.H.) that for no set ⇥ satisfying the
hypotheses there is a TK!1 -derivation D `0 ⇥ with h(D) < ⌧ .
Suppose, by way of contradiction, that for some m ≥ 0, some
Φ ✓ {qnk | n ≥ 0, k m} and some ⇤ ✓ ∆ there is a cut-free
derivation D of ⇤ with h(⇤) = ⌧ . Let R be the final inference
of D. Clearly R must be one of W, OR+ , Ki .
If R = W we are immediately
in contradiction with the I.H.
W
If R = OR+ , let n ¬⇤i qnj (for some j ≥ 0) be the principal
formula of the inference and D0 be the subderivation of the
premise ⇤0 . Then ⇤0 ✓ ∆{qnk |n≥0, kr} , where r = max(m, j).
28
162
Pierluigi Minari
Since h(D0 ) < ⌧ , we are in contradiction with the I.H. again.
Finally, if R = Ki , it is easily seen that from the subderivation
of the premise we would also get a derivation
^
D0 ` ¬Φ0 , qkk
k
0
for some Φ ✓ Φ. But this is clearly impossible by the bound⇤
edness condition on Φ and the soundness of TK!1 .
Proposition 6.2. The calculus TK!1 does not admit cutelimination. For instance, the sequent
^
�_
¬⇤i qnk | k ≥ 0 , ⇤i qkk
∆ :=
n
k
is derivable, but not cut-free derivable, in TK!1 .
Proof. TK!1 00 ∆ by Fact 6.1, since ∆ ⌘ ∆ with Φ = ;.
On the other side, ∆ can be derived as follows by making use
of a CUT with an appropriate instance of BF !1 (which we know
being derivable in TK!1 ):
¬qkk , qkk
)
Ki
k
k
¬⇤i qk , ⇤i qk
··· W
OR ··· (k≥0)
k
k
..
n ¬⇤i qn , ⇤i qk
.
W , AND V
V
W
V
¬ k ⇤i qkk , ⇤i k qkk
{ n ¬⇤i qnk | k ≥ 0}, k ⇤i qkk
W
V
{ n ¬⇤i qnk | k ≥ 0}, ⇤i k qkk
(
CUT
⇤
We conclude with a further negative result, showing how
a seemingly natural way out of the difficulty emerging from
Proposition 6.2 is in turn doomed to failure.
Let us consider the calculus TK◦!1 obtained from TK!1 by
replacing the rule OR+ with the stronger (and clearly sound)
rule
Γ, {Φm }m2I
W
OR (I countable)
Γ, { Φm }m2I
by means of which countably many disjunctions can be simultaneously introduced.
29
163
Infinitary Modal Logic and Generalized Kripke Semantics
Indeed, the sequent ∆ of Fact 6.2 becomes cut-free derivable
in TK◦!1 :
. . . ¬qkk , qkk . . . (k ≥ 0)
W, AND
V
{¬qkk | k ≥ 0}, k qkk
Ki
V
{¬⇤i qkk | k ≥ 0}, ⇤i k qkk
W
V
{¬⇤i qnk | k ≥ 0, n ≥ 0}, ⇤i k qkk
V
W
OR
{ n ¬⇤i qnk | k ≥ 0}, ⇤i k qkk
But unfortunately, a new counterexample to cut-elimination
comes out! For n ≥ 0, let
⇢
if n = 0;
¬⇤i p0 ,
'n := V
{⇤i p0 , . . . , ⇤i pk , ¬⇤i pk+1 }, if n = k + 1.
Fact 6.3. For every X ✓fin !, if
⇥ ✓ ΓX := {'n }n≥0 , {¬⇤i pm }m2X , ⇤i
then TK◦!1 00 ⇥.
^
pn
n
Proof. As in the proof of Fact 6.2 we argue by transfinite induction on the height of TK◦!1 -derivations.
Let ⌧ < !1 . Assume (I.H.) that for no set ⇥ satisfying the
hypotheses there is a TK◦!1 -derivation D `0 ⇥ with h(D) < ⌧ .
Suppose, by way of contradiction, that for some set X ✓fin !
and some ⇤ ✓ ΓX there is a cut-free derivation D of ⇤ with
h(⇤) = ⌧ . Let R be the final inference of D. Necessarily R is be
one of W, AND, Ki .
If R = W we are immediately
V in contradiction with the I.H.
If R = AND, let 'j+1 = {⇤i p0 , . . . , ⇤i pj , ¬⇤i pj+1 } (for some
j ≥ 0) be the principal formula of the inference, and let D0
be the subderivation of the j + 1-th premise ⇤0 (the one having ¬⇤i pj+1 as secondary formula) of this inference. Then
⇤0 ✓ ΓX[{j+1} , and since h(D0 ) < ⌧ we are in contradiction
with the I.H. again.
If R = Ki , then there would be a finite set Y = X [ {0} ✓ !
30
164
Pierluigi Minari
and a derivation
D0 ` {¬pk }k2Y ,
^
pn
n
which is clearly impossible by the finiteness of Y and the sound⇤
ness of TK◦!1 .
Proposition 6.4. The calculus TK◦!1 does not admit cutelimination. For instance, the sequent
^
:= {'n }n≥0 , ⇤i pn
n
is derivable, but not cut-free derivable, in TK◦!1 .
Proof. TK◦!1 00 by Fact 6.3, since ⌘ X with X = ;.
On the other side, can be derived (in fact, already in TK!1 )
by making use of CUT. First of all, we verify:
(6.1)
TK!1 `0 '0 , . . . , 'm , ⇤i pm
for each m ≥ 0.
This is easily proved by induction on m:
— m = 0:
— m = k + 1:
¬p0 , p0
¬⇤i p0 , ⇤i p0
Ki
¬pk+1 , pk+1
'0 , ⇤i p0
' , . . . , 'k , ⇤i pk ¬⇤i pk+1 , ⇤i pk+1
V 0
'0 , . . . , 'k , {⇤i p0 , . . . , ⇤i pk , ¬⇤i pk+1 }, ⇤i pk+1
I.H.
...
I.H.
Ki
W, AND
Next, using BF !1 , we obtain the following derivation of
TK!1 :
⇢
�
(6.1)
···
··· (m≥0)
..
'0 , . . . , 'm , ⇤i pm
.
W , AND V
V
V
¬ n ⇤i pn , ⇤i n pn
{'n }n≥0 , n ⇤i pn
V
{'n }n≥0 , ⇤i n pn
in
CUT
⇤
Infinitary Modal Logic and Generalized Kripke Semantics
31
165
References
[1] R. Fagin, J. Y. Halpern, Y. Moses and M. Y. Vardi, Reasoning About
Knowledge, The MIT Press, Cambridge 1995.
[2] S. Feferman, Lectures on proof theory, in M. H. Löb (ed.), Proceedings of the Summer School in Logic, Leeds 1967, Lecture Notes in
Mathematics 70, Springer, Berlin 1967, pp. 1 – 107.
[3] E. G. K. Lopez-Escobar, An interpolation theorem for denumerably
long formulas, Fundamenta Mathematicæ , 57, 1965, pp. 253 – 272.
[4] E. G. K. Lopez-Escobar, Remarks on an Infinitary Language with
Constructive Formulas, The Journal of Symbolic Logic , 32, 1967,
pp. 305 – 318.
[5] M. Ohnishi and K. Matsumoto, Gentzen method in modal calculi,
Osaka Mathematical Journal , 9, (1957), pp. 113 - 130.
[6] S. Radev, Infinitary propositional normal modal logic, Studia Logica , 46, 1987, pp. 291 – 309.
[7] W. W. Tait, Normal derivability in classical logic, in J. Barwise (ed.),
The Syntax and Semantics of Infinitary Languages, Lecture Notes in
Mathematics 72, Springer, Berlin 1968, pp. 204 – 236.
[8] Y. Tanaka, Kripke Completeness of Infinitary Predicate Multimodal
Logics, Notre Dame Journal of Formal Logic , 40, 1999, pp. 327 –
339.
[9] Y. Tanaka, Cut-elimination Theorems for Some Infinitary Modal Logics, Mathematical Logic Quarterly , 47, 2001, pp. 326 – 340.
[10] Y. Tanaka and H. Ono, Rasiowa-Sikorski Lemma and Kripke
completeness of predicate and infinitary modal logics, in M. Zakharyaschev, K. Segerberg, M. de Rijke and H. Wansing (eds.), Advances in Modal Logic 98, Vol. 2, CSLI Publications, Stanford 2000,
pp. 419 – 437.
© Copyright 2024 Paperzz
