ROCH OUELLET
Inclusive First-Order Logic
Abstract. Some authors have studied in an ad hoc fashion the inclusive logics, that is the logics which admit
or include objects or sets without element. These logics have been recently brought into the limelight
because of the use of arbitrary topoi for interpreting languages. (In topoi there are usually many objects
without element.) The aim of this paper is to present, for some inclusive logics, an axiomatization as
natural and as simple as possible. Because of the intended applications to category theory, the logics
studied are many-sorted and intuitionistic.
§ 1. Introduction
1.1
Usually the empty set  is not accepted as the underlying set of interpretations of firstorder logic. However, some authors, namely Mostowski [7], Hailperin [3], and Quine [10], have
studied logics admitting any set (empty or not) as underlying set. Such a problem was considered
before the seventies as an academic one and was studied only casually. Recent intercourse
between logic and category theory has brought this problem into the limelight. Nowadays,
languages are interpreted in arbitrary topoi. And, in some topoi, there are many objects without
element, that is objects which are not the codomain of a morphism from 1. For example, in the
category ouv(X)~ of sheaves over a topological space X, every proper open subset of X gives rise
to a sheaf which ( in ouv(X)~ ) has no element. In such a context, objects without element should
be included as underlying objects; following a terminology by Quine [10], the resulting logics are
called inclusive. The logics which exclude these objects without element are said to be exclusive.
1.2
The aim of this paper is to present, for some inclusive logics, an axiomatization as natural
and as simple as possible. Because of the intended applications to category theory, the logics
studied here are many-sorted and intuitionistic. Moreover, this work will be restricted to finitary
logics. (The infinitary ones were studied by Makkai & Reyes [13].) Parallel developments have
been presented by Coste [2].
1.3
This paper is a summary of our Ph. D. thesis which was supervised by Aubert Daigneault.
André Joyal suggested the subject and guided us during the research. We are grateful to them.
§ 2. Special features of inclusive logics
2.1
As mentioned in 1.2, many-sorted and finitary languages will be considered; they will
have an equality E, applied solely to two terms of the same sort. Their grammars contain also the
symbols 0 (false), 1 (true), the connectives , , , and the quantifiers , . The connectives ¬
and  are defined: for example, ¬ φ is a metaabreviation for φ  0. In order to simplify
substitution rules, disjoint sets of free ( V(s) ) and bound ( (s) ) variables are introduced for each
sort s. The universal closure of φ(v) will be any of the formulas  φ( ) where is a bound
variable, of the same sort as v, which does not occur in φ; however, this closure will be written
"v φ(v) ".
For any well-formed expression a (term or formula), σ(a) will denote the minimal support
of a, i.e. the set of the free variables occurring in a; and λ(a) will denote the minimal level of a,
i.e. the set of all sorts s such that at least one free variable of sort s occurs in a. Finally, theories
will be written down as sets of sequents whose intended meaning will be described in 2.5.
2.2
So far, the inclusive aspect of our logics did not affect their grammars; it will influence
their axiomatizations. Sequents which are valid in traditional exclusive logics are not necessarily
satisfied in domains without element; for example:
1  v (v = v)
v R(v)  v R(v).
(1*)
On the other hand, the following sequents:
v R(v) 
(2*)
R(v)
R(v)  v R(v)
(3*)
are satisfied in any domain, empty or not. In view of (1*), it seems that the transitivity of  is no
longer valid.
These observations had been made by Mostowski [7], Hailperin [3], Joyal [4], and others.
Mostowski gave an axiomatization in which  is not transitive. Hailperin preserved the
transitivity of  by restricting his formal system to sentences. We propose an inclusive logic
which deals with arbitrary formulas and in which  is transitive.
2.3
To simplify matters, let us assume now that the language is one-sorted. The semantical
content of (2*) and (3*) is: for every interpretation || || with X as underlying set,
(4*)
||v R(v) ||  || R(v) ||  || v R(v) ||.
In these inequalities, ||v R(v) || and || v R(v) || must be subobjects of X 1 and not of X 0. When
the sequent " φ   " is interpreted as " ||φ ||  ||  || ", the symbol "" denotes the canonical
preordering on the subobjects of a same power of X. In order to compare ||φ || and ||  ||, the
formulas φ and  must depend either explicitly of implicitly on the same variables. Therefore
||φ|| and || || are defined as subobjects of XCard(σ(φ)σ()): in practice, a subobject ||φ ||v of XCard v is
defined for every finite sequence v of distinct variables which includes σ(φ)σ(); then the
sequent " φ   " is interpreted as " ||φ ||v  ||  ||v ", where v is any enumeration without repetition
of σ(φ)σ(). So, (4*) should be written
||v R(v) ||<v>  || R(v) ||<v>  || v R(v) ||<v>.
From the transitivity of , it follows that:
p* ||v R(v) || = ||v R(v) ||<v>  || v R(v) ||<v> = p* ||v R(v) ||
(5*)
p* ||v R(v) ||  p* ||v R(v) ||,
where p* is the inverse-image functor associated to the morphism p: X 1  X 0. But the semantical
content of (1*) is:
||v R(v) ||  ||v R(v) ||.
(6*)
In order to infer (6*) from (5*), p* needs to reflect the preordering relation, i.e. p* should
be injective. Which is the case when X is a not empty set; hence  is transitive within traditional
exclusive logic, even if the middle term is the interpretation of a formula. However, when X has
no element, the morphism p: X 1  X 0 induces a non injective p*. Therefore, (6*) is not derivable
from (5*) in general.
The paradox of the example (1*)−(2*)−(3*) is due to the lack of a syntactical translation
for the distinction between (5*) and (6*), to the absolute character of (1*). Therefore, the solution
is quite simple: it will suffice to relativize sequents with respect to "levels". What these levels
should be? "Finite sequences of distinct variables" would be a good answer, but it would give
many "equivalent" levels.
2.4
Redundancy from the levels will be eliminated in the following way. Let v and w be finite
sequences of distinct variables. They give rise to the same level when:
||φ ||v  ||  ||v
(7*)
iff
||φ ||w  ||  ||w
for every formulas φ and  whose supports are included both in v and in w. Suppose first that one
of the sequences − say w − contains every coordinate of the other. Let X v be the product
 •••

, where n is the length of v, si is the sort of vi , and
is the underlying object of sort si for
the interpretation || ||; X is defined similarly; and the canonical projection X v  X w which
forgets the factors associated to coordinates of w not in v is denoted by p. Then:
w
||φ ||w = p* || φ ||v
and
|| ||w = p* || ||v .
The "only if" part of (7*) follows from the functorial property of p* (a functor between
preordered sets preserves the preordering relation) without any further hypothesis on v and w.
Moreover, the converse implication is equivalent to:
if p* || φ ||v  p* ||  ||v
then
|| φ ||v  ||  ||v.
As observed by A. Joyal, this follows from the existence of a section q of p, i.e. a morphism
q: X w  X v such that pq = id. Indeed:
|| φ ||v = id* || φ ||v = q* p* || φ ||v  q* p* || ||v = || ||v .
But such a morphism q exists when:
(8*)
for every variable in one sequence there is a variable of the same sort in the other.
Now let v and w be any finite sequences of distinct variables. We may infer (7*) from the
condition (8*). Indeed, let x be any sequence without repetition whose coordinates are those of v
and w; apply the previous proof first to v and x, and then to w and x.
2.5
A level is defined as a finite subset of the set S of all sorts. And a sequent is any
expression of the form
φ T 
where φ and  are formulas, and T is a level which includes λ(φ)λ(). When T is equal
λ(φ)λ(), it is usually omitted.
§ 3. Axiomatization of the inclusive logics
DEFINITION. An interpretation || || of a language L in a topos E is the following data:
3.1
−
for every sort s, an object Xs of E;
−
for every operation symbol  of arity s1, …, sn and of sort s, a morphism ||||:
 Xs ;
−
for every relation symbol R of arity s1, …, sn, a subobject ||R|| of
 ••• 
 •••  .
Such an interpretation is canonically extended: for every formula φ and every finite sequence v of
distinct variables including σ(φ), a subobject ||φ ||v of X v is defined.
Let || || be an interpretation in the topos E. The metaformula
3.2
|| || = φ v 
will mean that, with respect to the preordering relation between subobjects of X v, ||φ ||v is inferior
to || ||v .
LEMMA. Define the minimal level of a sequence v = v1, …, vn of variables as the set λ(v) =
{s1, …, sn} where si (1  i  n) is the sort of vi . Let v and w be finite sequences of distinct
variables including σ(φ)σ(). If v and w have the same minimal level, then:
|| || = φ v 
iff
|| || = φ w .
PROOF. See 2.4
3.3
This lemma allows to define a relation of semantical derivability indexed by the finite
subsets T of S (instead of being indexed by the finite sequences of distinct variables):
|| || = φ T 
iff
there is a finite sequence v such that λ(v) = T and || || = φ v .
The inclusive first-order internal logic of a topos E is expressed by the relation
E = φ T 
iff
for any interpretation || || in E, || || = φ T .
The main purpose of this paper is to give an axiomatization of the inclusive first-order internal
logic of topoi, i.e. to characterize the relation
= φ T 
iff
for any topos E, E = φ T .
3.4
Here are the axioms and rules of inference for this axiomatization. In the following,
[v  t] φ denotes the formula obtained from φ by substituting the term t to the free variable v.
(r)
φ  φ.
(t)
If φ T  and  T , then φ T .
(n)
If φ T  and T  U, then φ U .
(s)
Let t be a term and v be a variable, both of sort s. Let T be a finite subset of S; and
φ,  be formulas. Define: U = λ(φ)  λ() and U' = λ( [v  t] φ )  λ( [v  t] ).
If φ T  U  , then [v  t] φ T  U' [v  t] .
(0)
0  φ.
(1)
φ  1.
()
φ T 1 and φ T 2
iff
φ T 1  2.
()
φ1 T  and φ2 T 
iff
φ1  φ2 T .
()
φ   T 
iff
φ T   .
(Er)
1  vEv.
(Es)
φ  vEv'  [v  v' ] φ.
()
When T  λ(vs φ)  λ() and vs  σ(φ), vs φ T  iff φ T{s} .
()
When T  λ(φ)  λ(vs ) and vs  σ(φ), φ T vs  iff φ T{s} .
3.5
The adequation of these axioms and rules of inference can be verified directly, using the
works by Lawvere and Tierney on topoi. However, it is much simpler to derive them, as in Joyal
[5], from some axiomatization of the higher-order internal logic of topoi.
Because of the levels, the completeness theorem cannot be proven by the "algebra-offormulas" approach. However, as we will show in section 5, the methods of Aczel [1] may be
easily generalized to our context, yielding a first proof of the completeness theorem. Then, one
may use this result to reduce the inclusive many-sorted logic to the traditional exclusive onesorted logic; the methods of Rasiowa & Sikorski [11] may then be applied to construct adequate
models (universal models in the categorical terminology).
3.6
The syntactical relationship between inclusive and exclusive logics is all expressed by the
PROPOSITION. Let  be a set of sequents; φ,  be formulas; T be a finite subset of S including
λ(φ)  λ(); s1, …, sn be an enumeration of T \ (λ(φ)  λ()). For every i  {1, ... , n}, let
be
a free variable of sort si . Then:
 − φ T  iff  −   .
PROOF. Use (); then () n times; then () again.
§ 4. Origin of the axiomatization
4.1
The axioms and rules stated in 3.4 were not found by trials and errors but arose from
general principles regarding inclusive logic. These principles, which are mainly due to A. Joyal,
allow the selection of “the” good axiomatization in a systematic way.
A new approach to the traditional exclusive one-sorted logic will be presented here. Its
natural extension to inclusive many-sorted logic will lead to these principles. Indeed, in this way,
we will construct syntactically the levels and we will obtain every axiom and every rule of the list
in 3.4.
4.2
From a strictly syntactical point of view, logic is the study of sets of formulas. Such
formulas are acted upon by transformations, i. e. by substitutions of variables. This approach will
be formalized here through a category Tr. Introduce first an infinite set V whose elements are to be called variables. Now define a
V-transformation set as a A,  where A is a set and  a V-action on A, i..e. a function :
VV  A A such that idV  a = a and ζ  (η  a) = (ζ ○ η)  a. (Usually, ζ  a will be written
as ζa.) A support of a in A,  is a subset W of V such that
ζVV ηVV if ( ζ | W = η | W ) then ζa = ηa.
A transformation set is said to be locally finite iff every element has a finite support. TrV* − or
simply Tr* − will be the category of locally finite V-transformation sets. (A morphism between
A,  and A', '  is a function h: AA' such that h(ζ  a) = ζ ' h(a).)
The traditional exclusive one-sorted logic studies the objects of Tr* internally endowed
with operations , , , with constants 0, 1, with an equality E, and with operators  and .
Recall that " is an internal operation" means that  is a morphism of Tr*, i.e. that  commutes
with the substitutions ζ.
4.3
From the semantical point of view, every formula induces a predicate which does not
contain variables and hence cannot be directly endowed with a transformation structure.
However, transformations act indirectly on these predicates, by identifying or permuting the
variables, and hence the places of the predicates: for example, the predicate φ(−1, −2, −3)
associated with φ(v1, v2, v3) is ternary while φ(−1, −2, −2) which is associated with φ(v1, v2, v2) is
binary. Therefore, using the language of category theory, the information on transformations can
be completely expressed by one functor F: N S, where N is the full subcategory of S (sets)
whose objects are the = {1, ... , n} where n is a natural number: for example,
F( ) = the set of the n-any predicates
φ(−1, −2, −2) = F1, 2, 2 ( φ(−1, −2, −3) )
where 1,2,2 denotes the function 3  2 which applies 1 on 1, 2 on 2, and 3 on 2.
This predicate approach uses the category S N for the one-sorted logic. But how is the
word "exclusive" translated in this approach? First, observe that N op is (isomorphic to) the initial
algebraic theory N. (Recall, see v.g. Lawvere [6], that the objects of N are the finite powers 1, X,
X  X, X  X  X, …, of a formal object X, and that the morphisms of N are the canonical
projections.) So, a functor F: N S can be thought of as a presheaf on N ; F is in some way the
subobject functor for some formal object X. When the object X has an element 1  X, the
morphism X1 is the coequalizer of the two projections p1: X 2  X and p2: X 2  X. If,
furthermore, X is within a topos E, then the subobject functor  turns colimits into limits: so
(1)  (X) is then the equalizer of (p1) and (p2). Since the morphisms X1, p1 and p2 of
N correspond respectively to the functions 0  1, 1: 1  2, and 2: 1  2, the fact that the
one-sorted logic is exclusive is reflected in taking the full subcategory, A of S N whosc objects are
the' F 's such that F(0  1) is the equalizer of F1 and F2.
Now, let N* be the full subcategory of N obtained by eliminating the object 0. The
correspondance F  F|N* gives rise to an isomorphism between A and S N*. For convenience,
S N* will be used here instead of A.
4.4
The equivalence between the two approaches described in 4.2 and 4.3 is formally
translated by defining an equivalence #: Tr* S N*. Let A be an object of Tr*. Its image A# will
be the functor homTr*(V –, A): N*  S, V denoting the set of variables endowed with the action
"evaluation". Now let h: A  B be a morphism of Tr*. Its image h# is defined as the composition
with h: so, h#n(V n  A) = V n  A  B. One verifies quite easily that the functor # is faithful and
full. Using the hypothesis "V is infinite", one also proves that # is representative: any object F of
S N* is isomorphic to Fb#, where Fb is the set i* (F)(V) endowed with the action
ζ  a = i* (F)(ζ)(a),
i* being here the Kan extension of F: N*  S along the canonical inclusion i*: N*  S.
4.5
Since equivalence of categories is an equivalence relation, we have as a bonus the
COROLLARY. TrV* and TrW*are equivalent categories provided that V and W be both
infinite.
This is the syntactical formalization of the well-known fact that the set of variables −
which is always assumed to be infinite − takes a not intrinsic part in mathematical logic, and that
this set of variables can be replaced by any other infinite set.
4.6
Our study of the inclusive logic will be based on the natural extension of the equivalence
# defined in 4.4. In order to make the transition smoother, let us first study briefly the one-sorted
inclusive logic. This logic admits any functor from N to S since there is then no condition on the
underlying set X. So, semantically, this logic is expressed by the category S N. But which category
will express the syntactical approach to this logic? The problem is to find a category that will
make the following diagram commutative.
Tr* ?

S N * ~ A 
SN
Note that, as far as the predicates are concerned, the transition from exclusive to inclusive logic is
made by not eliminating the 0-th coordinate; that the 0-ary predicates correspond to sentences, i.e.
to formulas for which  is a support. Consequently, the transformation category for the onesorted inclusive logic will be the category TrV defined in this way:
−
The objects of TrV are ordered pairs A1,  : A0  A1 where A1 and A0 are objects of TrV*, where A0 is trivial (in A0,  is a support for every element), and where  : A0  A1 is a
morphism of TrV*.
−
The morphisms of TrV are commutative squares of TrV* of the following form:
A0

A1
h0
A0′
h1
′
A1′
One defines a functor : Tr  S N in the following way. Let A = A1,  : A0  A1 be an
object of Tr. Its image A is the functor homTr(V –, A) where V now is the ordered pair V,   V.
As before, the image h# of a morphism h of Tr is the composition with h. One verifies quite easily
that # is a faithful and full functor. Using the hypothesis "V is infinite", one also proves that # is
representative: any object F of S N is isomorphic to Fb#, where
Fb =  i(F)(V), i(F0  F)(V) .
Here, i(F) is the Kan extension of F along the inclusion functor i: N  S and F0 is the constant
functor from N to S whose value is F(0).
4.7
Let us now consider many-sorted inclusive logic. Let us fix a not-empty set S (whose
elements are to be thought of as sorts) and a family V = Vs | s  S  of mutually disjoint infinite
sets (the elements of Vs are to be thought of as variables of sort s).
How to define here the predicates? Each formula φ and each finite sequence v = v1, ..., vn
of distinct variables which includes σ(φ) give rise to a subobject ||φ ||v  X v. (Here X v =
 •••

, where
is the underlying object of sort si, the sort of the variable vi.) Therefore the
predicate approach could be expressed by the set-valued presheaves defined on the finite products
Xv =
 ••• 
. Such a product is canonically identified with the family ms | s  S , where
ms is the cardinality of {i  | si = s}. Let N / f S be the full subcategory of S / f S whose objects
are the "almost everywhere zero" families of natural numbers. For the predicate approach to
many-sorted inclusive logic, the category of S-valued functors defined on N / f S will be used.
Now let us turn to the transformation side of this logic. First, notice that the category Tr
defined in 4.6 is isomorphic to the category of "locally finite" functors from M(V) to S, M(V)
being the category whose morphisms are id0, 01, and ζ:11 (with ζ  VV) and whose
composition is induced by the functional composition of VV:
M(V) = •
0
•
1
ζ  VV For example, to a functor A: M(V )  S is associated the object A1,  : A0  A1 of Tr where
Ai = A(i)
i = 1, 2
 = A(01)
ζ  VV et a1  A1.
ζ  a1 = A(ζ : 11) (a1)
The transformation object for many-sorted logic is also a "locally finite" functor from M(V ) to S.
However, the domain M(V ) is modified. The objects of the many-sorted M(V ) are the finite
subsets of S. (So, in the two-sorted case, an A: M(Vs, Vt)  S gives rise to a set A() of
sentences, to two sets A({Vs}) and A({Vt}) of formulas in which occur free variables of only one
sort, and to a set A({Vs, Vt}) of formulas in which occur free variables of both sorts.) Moreover,
the set of morphisms homM(V )(T, T') is  if T is not included in T', and is
{ ζ  homS / S (V, V) | s T ζs =
}
if T  T'.
η
ζ if s  T. Finally, the s-th coordinate of T  T '  T " is ηs ○ ζs if sT, and is
M(V )
whose objects are the "locally finite"
Let us write Tr for the full subcategory of S
functors. One defines again the equivalence # by the formula homTr (V −, A). This time, however,
V does not denote a well-defined object of Tr ; here, the letter V is only a convenient notation:
for any object m of N / f S, V m will denote the product ΠsSVs m(s). (Here, Vs : M(V )  S is the
"object of variables of sort s": its value for an object T is either Vs or  according as s belongs or
not to T; its value for a morphism ζ: T  T' is either the function Vs  Vs : vs  ζs(vs) or the
inclusion  Vs (T' ) according as s belongs or not to T.)
Thus, for any object A of Tr, A# is the locally finite functor homTr (V −, A): N / f S  S.
For any morphism h of Tr, h# is the composition with h. One verifies quite easily that the functor
# is faithful and full. Using the hypothesis "each Vs is infinite", one also proves that # is
representative: for, any locally finite F: N / f  S is isomorphic to Fb#, where
Fb = i (F) (  Vs () | s  S  ) : M(V ) S.
(Here, i (F) is the Kan extension of f along the inclusion i : N / f S  S / S.)
4.8
Every first-order many-sorted language gives rise to an object Fm of Tr:
Fm(T) = { φ | (φ)  T}.
This object Fm is endowed with an internal preordering relation  which is the syntactical
translation of the semantical consequence relation. The axiomatization given in 3.4 characterized
formally this preorder .
Since Tr is a subcategory of S M(V ), the internal preorder  on Fm is a family, indexed by
the objects T of M(V ), of preorders T defined on Fm(T). And since the objects of M(V ) are the
finite subsets of S, one obtains in this way an abstract justification for the levels used as indices in
the sequents.
The axiom (r) and the rule (t) express formally that each T is reflexive and transitive,
i.e. that each T is a preordering relation. Since  is internal, these preordering relations T are
natural with respect to T; this gives rise to the rules (n) and (s). (As a matter of fact, these two
rules can be combined into one rule (s'): it is obtained from (s) by writing "and U'  ••• " instead
of "and U' = ••• "; however (n) and (s) were used to draw attention to the enlargements of levels.)
The ordering relation obtained from  endows Fm with a Heyting lattice structure; the
axioms (0) and (1), the rules (), (), and () express this fact. Moreover, Fm has an equality; in
accordance with Halmos' theory of polyadic algebras, such an equality is characterized as a
reflexive and substitutive relation; and hence axioms (Er) and (Es) are introduced into the
axiomatization. Finally, Fm has existential and universal quantifiers; but how will one translate
this?
4.9
Only () is considered; dual considerations apply to (). To begin with (see Ouellet [8]
for details), in the exclusive one-sorted logic, Fm can be endowed with an existential quantifier 
iff the diagonal Δ: Fm  homTr*(V, Fm) has an internal left adjoint. For, homTr*(V, Fm) is
(isomorphic to) the set V × Fm / ~, where
(v, φ) ~ (v', φ') iff v'  σ(φ) \ {v} and φ' = [v  v' ] φ;
moreover, the diagonal Δ: Fm  V × Fm / ~ is defined by
(1*)
Δ(φ) = [v, φ]
where v  σ(φ).
The internal preordering relation  on Fm induces an internal preordering relation ≤ on V × Fm / ~ :
(2*)
[v, φ] ≤ [v', φ'] iff there is w  V \(σ(φ)  σ(φ')) such that [v  w] φ  [v'  w ] φ'.
Therefore, Fm and hom(V, Fm) are endowed with internal category structures, The diagonal Δ,
since it preserves the preordering relation, is an internal functor between these internal categories.
Existential quantifier  and internal left adjoint : hom(V, Fm)  Fm are obtained from each
other through the formula
v(a) =  [v, a].
(3*)
For the inclusive many-sorted logic the rule () is now inferred from the existence of
internal left adjoints  s for the diagonals Δs: Fm  homTr(Vs, Fm). First, let us describe these
diagonals Δs (here us will be a fixed variable of sort s) :
hom(Vs, Fm)(T) =
(4*)
ΔsT (φ) =
if | 
if 
, where  if if  .
The preordering relation ≤T on hom(Vs, Fm)(T) is given by (2*) if s T; when s  T, ≤T is the
restriction of  to the set hom(Vs, Fm)(T).
Now let us assume that every Δs has an internal left adjoint  s. Two cases must be
considered :
 Suppose first that s T. The T-coordinate  sT of  s is left adjoint to the T-coordinate ΔsT
of the diagonal Δs:
(5*)
 sT [v, φ] T  iff [vs, φ] ≤T Δ sT ().
But, Δ sT () = [vs, ] according to (4*) and the hypothesis "vs  σ()" ; moreover  sT [vs, φ]
= vs φ from (3*). Consequently, (5*) means that
vs φ T 
iff
iff
[vs, φ] ≤T Δ sT ()
φ T .
(This last "iff" uses (2*).)

Suppose now that s  T. Then the hypothesis "T  (vs φ)" implies that σ(φ)  Vs  {vs}.
If the variable us used in the construction of hom(Vs, Fm)(T) was vs, then since  sT is a left
adjoint of Δ sT ,
 sT (φ) ≤T  iff φ ≤T  {s} Δ sT (),
i.e.
 sT (φ) ≤T  iff φ ≤T  {s} .
The rule () is obtained by defining: vs φ =  sT (φ).
4.10 REMARK. Let φ be a formula such that σ(φ)  Vs  {vs}. And let T denote (vs φ).
Then φ belongs to hom(Vs, Fm)(T ) and to hom(Vs, Fm)(T  {s}). Therefore, there are two
possible values for vs φ : the first,  sT (φ), belongs to Fm(T ); the other,  sT  {s} (φ), belongs to
Fm(T  {s}).
The intuitive content of this remark is more transparent in the one-sorted (inclusive) case.
Then S = {s} and T = . The element  s (φ) of Fm() is the value of vs (φ) treated as a
sentence. While the element  s{s} (φ) of Fm ({s}) is the value of vs (φ) treated as a formula. In
some sense, the left adjoint  s : { φ | σ(φ)  {vs}}  Fm() transmutes formulas with support
{vs} into sentences.
§ 5. The completeness theorem
5.1
It is well known that an operation symbol  of arity s1, ..., sn and of sort s can be
replaced by a relation symbol R of arity s1, ..., sn, s : one has only to think of the atomic
formula "R(tl, ... , tn, t)" as meaning "R(tl, ..., tn) E t ".
We will therefore prove the completeness theorem only for languages with no operation
nor constant symbols. (The extension of the theorem to languages with such symbols is
straightforward but tedious.)
Our proof will parallel that of Aczel [1].
5.2
Let L be a language containing only relation symbols and in which S is the set of sorts.
And let Cs (s  S) be any set. We will use the following notation:
LC : the language obtained from L by adding the elements of Cs (s  S) as (new)
constants of sort s
StC : the set of sentences of LC
for φ  StC, Ind(φ) = set of constants occurring in φ
for Γ  StC, Ind(Γ ) = γΓ Ind(φ).
for Γ  StC, Cn(Γ ) = set of sentences δ in StInd(Γ ) such that {1  γ | γΓ } − 1  δ.
Note that, since 1 and δ are sentences, the implied level of the sequent 1  δ is the empty set.
5.3
We will also need the following definitions (δ is any sentence of LC and Γ any set of
sentences such that Ind(δ)  Ind(Γ )) :

Γ is a theory iff Cn(Γ ) = Γ.

Γ is δ-consistent iff δ  Cn(Γ ).

Γ is δ-complete iff it is δ-consistent and φ  Cn(Γ ) or φ  δ  Cn(Γ ) for all φ such that
Ind(φ)  Ind(Γ ).

Γ 'is prime iff φ    Cn(Γ ) implies that φ  Cn(Γ ) or   Cn(Γ ).

Γ is existential iff vs φ(vs)  Cn(Γ ) implies φ(cs)  Cn(Γ ) for some constant symbol cs of
sort s.

Γ is δ-saturated iff it is a δ-consistent, prime, existential theory.
When δ is 0, it is usually omitted: by "consistent", we mean "0-consistent ", etc.
5.4
LEMMAS.
A.
Every δ-complete theory is prime.
B.
The union of a chain of δ-consistent (δ-complete) theories is a δ-consistent (δ-complete)
theory.
C.
Every δ-consistent Γ can be extended to a δ-complete theory Γ '.
D. Every δ-consistent Γ can be extended to a δ-complete rich extension Γ ', that is to a δcomplete extension Γ ' of Γ such that vs φ(vs)  Cn(Γ ) implies φ(cs)  Γ ' for some constant
symbol cs of sort s.
PROOFS. The proof of B is straightforward. The proofs of A, C, and D are similar to Aczel [I],
lemmas 1, 2, and 3. The last two use the following version of the Deduction Theorem, which is
easily proved by induction on the length of the derivation:
DEDUCTION THEOREM. Let Σ be a set of sequents; and let γ be a sentence. If Σ {1  γ} −
φ T , then Σ − γ  φ T .
5.5
THEOREM (Aczel [1], theorem 1) Every δ-consistent Γ can be extended to a δ-saturated Γ '.
5.6
COROLLARIES
A. Every (φ  )-consistent theory Γ can be extended to a -saturated theory Γ ' such that
φ  Γ '.
B. Every (vs φ(vs))-consistent theory Γ can be extended to a φ(cs)-saturated theory Γ ', for
some constant cs of sort s.
PROOFS. We need only adapt the proof of theorem 3 in Aczel [1]. For example, for A: use the
Deduction Theorem and () to show that Γ  {φ} is -consistent; then use the theorem 5.5 with
Γ  {φ} in lieu of Γ, and with  in lieu of δ.
5.7
We now introduce extended Kripke structures: in our context, these are interpretations K
of L in a topos SY, where Y = Y, ≤  is a preordered set.
Note that the Kripke structures of Aczel [1] are special instances of ours (except for the
possible use of proper classes): Aczel [1] requires that the images by K of the morphisms yy'
(where y ≤ y') be set inclusions. Also, the "intuitionistic relational system" of Aczel [1] is the
restriction of the functor K to the segment {y'  Y | y ≤ y'} of Y.
5.8
Any formula φ of the language L and any finite sequence v of distinct variables including
σ(φ) give rise to a subobject (or subfunctor) ||φ ||v of K v, that is to a family ||φ ||v (y) K v(y) of set
inclusions, natural with respect to y.
Therefore, we can define a 4-ary relation
c  ||φ ||v (y).
Its components are: first, a formula φ of L; second, a finite sequence v of distinct variables
including σ(φ); third, an element y of Y; fourth, an element c of K v(y). We shall omit the sequence
v and denote this relation by K =yy φ(c). It can be easily shown that the following induction
properties hold:
  RK(y)

K =y R(v1, ... , vn)(c) iff 

K =y (v E w) (c) iff cv = cw

K =y 1 (c)

not K =y 0 (c)

K =y (φ  ) (c) iff K =y φ(c) and K =y  (c)

K =y (φ  ) (c) iff K =y φ(c) or K =y  (c)

K =y (φ  ) (c) iff for any z ≥ y, K =y φ(c|z) implies K =y φ(c|z) where c|z is the image
of c by the function K v(y  z)

K =y vs φ(vs) (c) iff K =y φ(c, cs) for some cs  Ks(y)
, ... ,

5.9
K =y vs φ(vs) (c) iff for any z ≥ y and any cs  Ks(z), K =z φ(c|z , cs).
LEMMA. (Aczel [1], lemma 4) ValK, y is a saturated theory where
ValK, y = { φ(c) | K =y φ(c)}.
5.10 Let M be a set of the same cardinality as St. We define Y as the set of consistent theories
Γ such that Ind(Γ )  M. (We use the set M in order to restrict Y to a set; without the condition
" Ind(Γ )  M ", Y would be a proper class.) Now letY = Y,   be the preordered set of these
consistent theories ordered by the set inclusion .
We define a Kripke structure H. Let s be a sort; Γ,  be consistent theories belonging
to Y; and R be a relation symbol of arity s1, …, sn :
Inds(Γ ) = set of constants of sort s occurring in some γ  Γ
c ~Γ c' iff c  Inds(Γ ) and c'  Inds(Γ ) and cEc'  Cn(Γ )
Hs(Γ ) = Inds(Γ ) / ~Γ
Hs(Γ  ) : [c]Γ  [c]
RH(Γ ) = {[c]Γ  Hs(Γ ) | R(c) Cn(Γ )}.
5.11
THEOREM. (Aczel [l], theorem 5) A theory   Y is saturated iff ValH,  = .
5.12
We now prove the completeness theorem. Let Σ {σ0} be a set of sequents in L such that
not Σ − σ0 .
Using the rules (),(), (), and (), we transform each sequent in Σ {σ0} into a sentence: for
example, if T = λ(φ)  λ() and T  T' = ,
φ T  T' 
equivalent to 1 T  T' φ  
equivalent to 1 T' v1 … vm (φ  )
equivalent to
equivalent to 1   v1 … vm (φ  )
 v1 … vm (φ  )
where σ(φ)  σ() = {v1, ..., vm} and T' = {s1, ..., sn}.
Let γ0 be the sentence so obtained from the sequent σ0 . And let Γ be the set of sentences
obtained from the elements of Σ. The hypothesis "not Σ − σ0" means that γ0  Cn(Γ ). Hence Γ
is γ0-consistent and, by theorem 5.5, can be extended to a γ0-saturated Δ. It follows from theorem
5.11 that Val H, Δ = Δ. Hence, the restriction of H to the segment { Γ'  Y | Δ Γ'} is a model of
Δ (hence of Γ ) that does not satisfy γ0 .
This restriction of H is also a model of  that does not satisfy σ0 . Hence
not  = σ0 .
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ÉCOLE DES HAUTES ÉTUDES COMMERCIALES
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Studia Logica XL (1981), 13-28 (revised version)