A Modal Logic of Quantification and
Substitution
Yde Venema
present address:
Department of Mathematics and
Computer Science
Free University
De Bo elelaan 1081
1081 HV Amsterdam
e-mail: yde@cs.vu.nl
Abstract.
The aim of this pap er is to study the
n
-variable fragment of rst order logic from a
mo dal p erspective. We dene a modal formalism called
cylindric mirror modal logic
,and
showhow it is a modal version of rst order logic with substitution. In this approach,
wecandene a semantics for the language which is closely related to algebraic logic,
as we nd Polyadic Equality Algebras as the mo dal or complex algebras of our system.
The main contribution of the paper is a characterization of the intended `mirror cubic'
frames of the formalisms and, a consequence of the sp ecial form of this characterization,
a completeness theorem for these intended frames. As a consequence, we nd complete
nite yet unortho doxderivation systems for the equational theory of nite-dimensional
representable p olyadic equality algebras.
Keywords: algebraic logic, mo dal logic, logic with nitely manyvariables, completeness,
derivation rules.
1980 Mathematical Sub ject Classication: 03B20, 03B45, 03C90, 03G15.
1
1 Intro duction
This pap er forms part of a program to connect various traditions in logic, viz. rst-order logic,
mo dal logic and algebraic logic. In particular, we will showhowthe
n
-variable fragmentof
the predicate calculus of
n
-ary relations (
n<!
) can b e seen as a mo dal formalism, which
we will call Cylindric Mirror Mo dal Logic (
CMML
)
1
. A distinctive feature of this system is
that simultaneous substitutions of variables for variables in rst-order formulas are treated
as mo dal diamonds. In this way
CMML
is an extension of the formalism
CML
(cf.
Venema
13]) which has diamonds for the existential quantication, and a mo dal constant for the
identity formulas. The connection with algebraic logic lies in the fact that the mo dal algebras
of
CMML
are known in the literature as Polyadic Equality Algebras (of nite dimension)
(cf.
Henkin, Monk & Tarski
6]).
Note that
any
mo dal language can b e interpreted in a relational structure of the appro-
priate signature, i.e. where the structure provides an
n
+ 1-ary
ac
cessibility relation
for each
n
-adic op erator of the language. Therefore, a consequence of the mo dal approachtowards
rst-order logic is that it provides us with a wider class of (Kripke) mo dels for rst-order
logic. Within this more general framework, the standard semantics of rst-order logic forms a
subclass
of frames that will b e called
mirror cubes
here. An interesting asp ect of the `mo dal-
ization' of rst-order logic is that it allows us to play around with this intended semantics
for instance, Andrekaand Nemeti (cf. 2]) have studied an alternative mo del theory for the
predicate calculus where not every
n
-tuple of elements of the domain is
available
for evalua-
tion they show that some negative features of rst-order logic, like its undecidability,do not
apply for this alternative framework.
The main contribution of this note is a nite sound and complete axiomatization of the
CMML
-formulas that are valid in the mirror cub es. As a consequence of results by Andreka
(cf. 1]), a
nite
axiomatization is not p ossible, if we conne ourselves to the ortho doxmodal
derivation rules (Mo dus Ponens, Universal Generalization and Substitution). In fact Andreka
and Tuza show 3] that the variety of Representable Polyadic Equality Algebras is not even
nitely axiomatizable ov
er the variety of Representable Cylindric Algebras. On the mo dal side
of the picture this means that even if one has an oracle providing all
CML
-theorems, one still
has to add innitely many axioms to axiomatize the mirror cub es | under the same restriction
qua derivation rules. The strategy we adopt here to circumvent these negative results is by
considering unortho doxderivation systems. The crucial part of these systems is formed by
a so-called non-
rule such rules originate with
Gabbay
4], and are discussed in detail in
Venema
14]. Our main result concerns the completeness for the class of mirror cub es of a
nite
unorthodox
derivation system. As a corollary of our result, we nd a nite derivation
system for the variety of Representable Polyadic Equality Algebras of nite dimension. It
is interesting to note that this system
is
a nite extension of a complete derivation system
for the Representable Cylindric Algebras. One might conclude that the heart of the non-
nite axiomatizability problem of
RPEA
do es
not
lie in its complexity with resp ect to
RCA
,
but rather in the inadequacy of a
purely equational
approach to axiomatizations in algebraic
logic.
Therefore, the question b ecomes relevant what the exact algebraic counterpart is of non-
rules. We will come back to this matter in section 4, where we will also discuss briey some
1
One can approach the unrestricted predicate calculus from the same mo dal p erspective. As there are many
(mainly technical) problems involved in doing so, wehave conne ourselves to a
fragment
of rst-order logic
here.
2
generalizations to our results that were obtained recently by Sz. Mikulas.
This pap er is organized as follows: in the following section wegointo detail as to how the
n
-variable fragment of the predicate calculus of
n
-ary relations (
n<!
) can be `mo dalized'
into the formalism
CMML
. In section 3 weintro duce a relational (Kripke) semantics for our
language, and we prove our main completeness result. Section 4 contains all the material
on the algebraic connection: in particular, we dene the nite derivation system for the
equational theory of the class
RPEA
n
of Representable Polyadic Equality Algebras.
2 Mo dalizing rst-order logic
In order to explain howthe
n
-variable fragment of the predicate calculus of
n
-ary relations can
b e treated as a mo dal formalism, let us start with an intuitive exp osition, and defer precise
denitions to the end of this section until then, the reader can think of a version of rst-order
logic, where only the rst
n
variables
f
v
0
:::v
n
;
1
g
are available, with the standard semantics
of rst-order logic.
Consider the basic declarative statement in rst-order logic concerning
the truth of a formula in a mo del under an assignment
u
:
M
j
=
u
]
:
(1)
The basic observation underlying our approach is that we can read (1) from an abstract mo dal
p erspective as: \the formula
is true in
M
at the possible world
u
". Note that as wehave only
n
variables at our disp osal, we can identify assignments with maps:
n
(=
f
0
:::n
;
1
g
)
7!
U
,
or equivalently, with
n
-tuples over the domain
U
of the structure | we will denote the set of
such
n
-tuples with
n
U
. Thus we are in a setting of multi-dimensional mo dal logic where the
universe of a mo dal mo del is of the form
n
U
for some base set
U
. Now the truth denition
of the quantiers reads as follows:
M
j
=
9
v
i
u
]
()
there is an assignment
u
0
with
u
i
u
0
and
M
j
=
u
0
]
where
i
is given by
u
i
u
0
()
for all
j
6
=
i
,
u
j
=
u
0
j
.
In other words: the existential quantication b ehaves like a mo dal
diamond
,having
i
as its
accessibility relation
.
As the semantics of the b oolean connectives in the predicate calculus is the same as in
mo dal logic, this shows that the inductive clauses in the truth denition of rst-order logic
t neatly in a mo dal approach. So let us now concentrate on the atomic formulas. Tostart
with, we observe that
identity
formulas do not cause any problem: a formula
v
i
=
v
j
, with
truth denition
M
j
=
v
i
=
v
j
u
]
()
u
i
=
u
j
can b e seen as a mo dal
constant
.
The case of the other atomic formulas is more involved, however as we have conned
ourselves to the calculus of
n
-adic relations, an atomic predicate formula is of the form
P
i
v
(0)
:::v
(
n
;
1)
, where
is a map:
n
7!
n
. In the mo del theory of rst-order logic the
predicate
P
i
will b e interpreted as a subset of
n
U
precisely how the prop ositional variables are
treated in mo dal logic byavaluation. So we will identify the set of prop ositional variables of
the mo dal formalism with the set of predicate symb ols of our rst-order language. However,
this implies that there
cannot be a one-to-one corresp ondence between atomic rst-order
3
formulas and atomic mo dal ones. It follows from our wish to give a modal reading for the
atomic case of (1), that only the formula
P
i
v
0
:::v
n
;
1
will corresp ond to the mo dal atom
p
i
.
For the cases where
is not the identity function, wehave to nd a dierent solution.
Atomic formulas with a
multiple
o ccurrence ofavariable can b e rewritten as formulas
with only `unproblematic' atomic subformulas, for instance
Pv
1
v
0
v
0
() 9
v
2
(
v
2
=
v
0
^
Pv
1
v
2
v
2
)
() 9
v
2
(
v
2
=
v
0
^ 9
v
0
(
v
0
=
v
1
^
Pv
0
v
2
v
2
))
() 9
v
2
(
v
2
=
v
0
^ 9
v
0
(
v
0
=
v
1
^ 9
v
1
(
v
1
=
v
2
^
Pv
0
v
1
v
2
)))
This leaves the case what to do with atoms of the form
P
i
v
(0)
:::v
(
n
;
1)
, where
is a
permutation of
n
, or in other words, atomic formulas where variables have b een substituted
simultaneously
. The previous trickdoes notwork here: for instance, to write a formula like
9
v
3
9
v
4
(
v
3
=
v
0
^
v
4
=
v
1
^9
v
0
9
v
1
(
v
0
=
v
4
^
v
1
=
v
3
^
Pv
0
v
1
v
2
))
whichisequivalentto
Pv
1
v
0
v
2
, one needs
extra
variables as buers.
One might consider a solution where a predicate
P
is translated into
various
mo dal prop o-
sitional variables
p
, one for every p ermutation
of
n
, but this is not very elegant. One might
also forget ab out simultaneous substitutions and conne oneself to a
fragment
of
n
-variable
logic. In
Venema
13] this option is worked out, leading to a mo dal formalism called
Cylindric
Modal Logic
.
Here wewillinvestigate a third p ossibility, whichistotake substitution seriously,soto
sp eak, by adding sp ecial `substitution op erators' to the language. The crucial observation is
that for any p ermutation
,wehavethat
M
j
=
P
i
v
(0)
:::v
(
n
;
1)
u
]
()
M
j
=
P
i
v
0
:::v
n
;
1
u
] (2)
where
u
is the comp osition
2
of
and
u
(recall that
u
is a map:
n
7!
U
). So, if wedene
the relation
1
n
U
n
U
by
u
1
t
()
t
=
u
wehave rephrased (2) in terms of an accessibility relation (in fact, a function):
M
j
=
P
i
v
(0)
:::v
(
n
;
1)
u
]
()
there is a
t
with
u
1
t
and
M
j
=
P
i
v
0
:::v
n
;
1
t
]
So if we add an op erator
to the mo dal language for every p ermutation
,with
1
as
its intended accessibility relation, wehave found the desired mo dal equivalentfor aformula
P
i
v
(0)
:::v
(
n
;
1)
in the form
P
i
.
Our last observation b efore we give the formal denitions of our systems is that wemay
use the fact that we are in a nite-variable fragment of rst-order logic to simplify the lan-
guage a bit. For, recall that every p ermutation of a nite set is a pro duct of
transpositions
,
i.e. p ermutations swapping two elements and leaving every other element in its place. As we
may infer from (2) that
M
j
=
P
i
v
(0)
:::v
(
n
;
1)
u
]
()
M
j
=
P
i
v
0
:::v
n
;
1
(
u
)]
2
In our notation, the
order
of comp osing the two functions should b e read as in the comp osition of two
relations, i.e.,
u
(
i
)=
u
(
(
i
)).
4
we only need mo dal op erators
for transp ositions
: if
=
0
:::
k
,wemay consider
as an
abbreviated
op erator:
:=
0
:::
n
;
1
:
So, to develop some notation concerning transp ositions and their associated accessibility
relations: dene the transp osition
i j
]
n
:
n
7!
n
by
i j
]
n
(
k
)=
8
>
<
>
:
j
if
k
=
i
i
if
k
=
j
k
otherwise
:
(If no confusion can arise, we will drop the sup erscript
n
.) The accessibility relation asso ciated
with transp ositions has a very simple form (assume
i<j
):
u
1
ij
v
()
v
=(
u
0
:::u
i
;
1
u
j
u
i
+1
:::u
j
;
1
u
i
u
j
+1
:::u
n
1
)
:
Nowwe are ready to give formal denitions:
Denition 2.1
Let
n
be an arbitrary but xed natural number. The alphabet of
L
n
and
of
L
r
n
consists of a set of variables
f
v
i
j
i < n
g
, it has got a countable set
Q
of
n
-adic
relation symbols (
P
0
P
1
:::
), identity (=), the Boolean connectives
:
_
and the quantiers
9
v
i
. Formulas of
L
n
and
L
r
n
are dened as usual in rst-order logic, with the restriction that
the atomic formulas of
L
r
n
are of the form
v
i
=
v
j
or
P
l
(
v
0
v
1
:::v
n
)
for
L
n
, we also al low
atomic formulas of the form
P
l
(
v
0
v
1
:::v
n
)
, where
is a permutation on
n
.
A rst-order structure for
L
(
r
)
n
isapair
M
=(
U V
)
such that
U
is a set cal led the domain
of the structure and
V
is
an interpretation function mapping every
P
l
to a subset of
n
U
.
Truth of a formula in a model is dened as usual: let
u
bein
n
U
, then
M
j
=
v
i
=
v
j
u
]
if
u
i
=
u
j
M
j
=
P
l
(
v
0
v
1
:::v
n
)
u
]
if
u
2
V
(
P
l
)
M
j
=
P
l
(
v
0
v
1
:::v
n
)
u
]
if
(
u
(
o
)
:::u
(
n
;
1)
)
2
V
(
P
l
)
M
j
=
9
v
i
u
]
if thereisa
v
with
u
i
v
and
M
j
=
v
]
etc.
An
L
(
r
)
n
-formula
is valid in
M
(notation:
M
j
=
)if
M
j
=
u
]
for al l
u
2
n
U
, rst-order
valid (notation:
j
=
fo
) if it is valid in every rst-order structureof
L
(
r
)
n
.
The
modal
versions of
L
r
n
and
L
n
,onwhichwe will concentrate from now on, are given
in the following denition:
Denition 2.2
Let
n
be an arbitrary but xed natural number.
CMML
n
is the modal simi-
larity type having constants
ij
and unary operators
3
i
,
ij
(for al l
i j < n
). For a set of
propositional variables
Q
, the language of
n
-dimensional cylindric modal formulas in
Q
,or
shortly,
CMML
n
-formulas (in
Q
), is built up as usual: the atomic formulas are the (modal or
boolean) constants and the propositional variables, and a formula is either atomic or of the
form
:
,
_
,
3
i
or
ij
, where
,
are formulas. We abbreviate
2
i
:
3
i
:
.
CML
n
is the fragment of
CMML
n
-formulas in which no mirror operator
ij
occurs.
5