MARTIN
WEAK
NECESSITY
K.
AND
DAVIES
TRUTH
THEORIES*
Consider the sentence
(1)
Socrates is necessarily a man.
(a) Intuitively we say that the semantic constituents, “Socrates” and
“is a man” of the non-modal sentence
(2)
Socrates is a man
are semantic, and not merely syntactic, constituents of (1); someone who
does not understand that name or that predicate does not understand (1).
In this respect (1) differs from the metalinguistic
“Socrates is a man” is a necessary truth in English.
(b) Intuitively, also, (1) results from the application of a sentential
operator “necessarily” to (2) at least somewhat as
Socrates is not a man
results from the application of sentential negation to (2).
(c) What is more - and this has sometimes seemed to conflict with the
second intuition - the position of “Socrates” in (1) is referentially transparent. If
Cicero is necessarily a man
is true, and “Cicero” denotes the very same object as does “Tully”,
then
Tully is necessarily a man
is true. There is a sharp contrast here with the position of “Socrates” in,
for example,
Fred believes that Socrates is a man.
(d) Finally, when a speaker uses (1) to make an essentialist claim whether it is true or false - he does not thereby deny that the people who
Journal of’i’hiiosophical Logic 7 (1918) 415439.
AN Rights Resewed.
Copyright Q 1978 by D. Reidel Publishing Company,
Dordrecht. Holland.
416
MARTlN
K.
DAVIES
were in fact Socrates’ parents might have remained childless, and he does
not assert that Socrates shares with God and numbers the property of noncontingent existence. “Necessarily” in (1) expresses ‘weak’ necessity.’
Can we provide, for a language containing such sentences as (1) a
semantic theory adequate to these four intuitions? 1 shall argue that such a
theory - in particular, a theory of (absolute) truth - can be given.
It would not be satisfactory to represent any of (a)-(d) as the intuition
that “necessarily” as it occurs in (1) expressesnecessity de re rather than
de ditto. It would be unsatisfactory just because it would invite confusion.
The claim that a modal operator “a ” expressesnecessity de ditto could
mean either that “0” is primarily an operator upon whole sentences, or
that the result of prefixing “~” to a sentence is ultimately to be regarded as
a predication of that sentence. Upon the first reading, a de re modal
operator would be one which is primarily a predicate forming operator upon
predicates (more like “large” than “not”), while upon the second reading
any operator, whether upon sentences or predicates, would be de re
provided that the contained expression is a semantic constituent of the
resultant expression. Confusion is only increased by those who use the
terms to describe various sentences containing a single modal operator, as
when
(3)
0 (Vx)Man(x)
is said to be a de ditto modal sentence, but
(4)
(Vx)oMan(x)
a de re modal sentence. All that this can mean - if (4) is to be well formed is that in (4) there is quantification into the o-context while in (3) there is
not. If it does mean that then surely
qMan(Socrates)
is a de ditto sentence, but since it is just an instance of (4) one can imagine
that there would also be considerable pressure to describe it as de t-e.’
In the first three sections of this paper 1 shall set out my preferred
schema of truth theories for non-modal languages, provide the model theory
and proof theory for a weak necessity operator, and give a truth theory for
a language containing such an operator. In a final section I shall consider
certain difficulties which the theory confronts.
WEAKNECESSITYANDTRUTHTHEORIES
417
I
If a theory of truth is to serve as the core of an adequate total theory of
meaning for the language of a certain population then it is not sufficient
that the truth condition specifying biconditionals be ‘interpretational’. The
truth theory must also meet the following structural constraint (SC): If, but
only if, members of the population who have been taught to understand
sentences sr , s2, . . are able, without further training, to go on to understand a sentence s, then the truth theory should employ in the canonical
derivations of biconditionals for sr ) s2 , . . . resources already sufficient for
the canonical derivation of a biconditional for s.~ A theory will not meet
the SC if it employs in the canonical derivations of biconditionals for
resources already sufficient for the derivation of a biconditional
SI,S2,-..
for an expression which is not a sentence of the language at all.
Suppose that the object language (OL) for which we are to provide a
theory of meaning is a first-order language I, with one one-place predicate
“F”, one two-place predicate “H”, two names “a” and “b”, sentential
connectives “m” and “&“, and the existential quantifier? and suppose that
the metalanguage (ML) extends the OL. A theory of truth in Tarski’s style
employs as a semantic primitive the relation of satisfaction between
sequences of objects and formulae of L, and it yields for every formula of
L - and so, in particular, for every sentence of 1, - a statement of satisfaction conditions. Upon the definition of truth:
u is true iff o is a sentence and u is satisfied by every
sequence
the theory yields a homophonic truth condition specifying biconditional
for each sentence ofL. There is, however. a serious objection to such a
theory for it employs resources sufficient for the derivation of biconditionals for expressions which are not sentences of L. Only a syntactic
restriciion upon the application of the truth prcdicatc prevents the theory
from serving up truth conditions for every formula of I,. Along with
Tr(“Fa”,
we should have
L) ++ Fa
418
MARTIN
K.
Tr(“Fr~e”, L) -
DAVIES
( Vvo)Fvo
It would trivialize the SC if we allowed that it is met, in this case, by the
restriction of the truth predicate to a syntactically distinguished subclassof
the formulae of L, while sentences of L and expressions which are not sentences of L are assigned semantic properties uniformly by the theory. What
is needed, evidently, is to provide an alternative theory - and, in particular,
an axiom for the quantifier - which speaks only of truth and sentences.
Consider first the theory he whose axioms are as follows, where “y”, “6”
range over names, “o”, “T” range over sentences, “a” ranges over one-place
complex predicates,4 and sentential connectives have wider or narrower
scope according to the ordering: “-“,
“-+“, “&“, “w”.
(Ala)
(VY E L)(Wr&‘,
L> -F(d(y,
@lb)
WY, 6 E L)V’d~h9
(A24
d(“a”, L) = a
Wb)
d(“,“,
(A34
(VoEL)(Tr(C-u’,L)--Tr(o,L))
Wb)
(Vu,
(A4)
(V@ E L)(Trf(3
NJ)
(Vx)(37
LB)
W1, L) -Hd(r,
L), $6, L)))
L) = b
T E L)(Tr[u
E LM,
& rl, L) ++ Tr(u, L) & Tr(r, L))
z+)@vr’, L) -
(37 E L)(Tr(r?$,
L)))
L) =x
This theory is semantically quite coherent, and it has the desired feature
of assigning truth conditions only to sentences of L. But because (AU) has
the effect of restricting the range of the ML quantifier which is used in
stating the truth conditions of quantified OL sentences to the domain of
objects named in L, these truth conditions for quantified sentences are, in
general, incorrect.
To preserve the desired feature while getting the truth conditions right,
we could introduce an extension L’ of L, such that L+ contains at least one
name of each object in the range of the quantifier in L. The axioms of X0
would be revised to speak of L’, and one further axiom
(A-0
(Vu E L)(Tr(u,
L) -
Tr(u, L’))
would be added. It would be a mistake to think that the resulting theory
WEAKNECESSITYAXDTRUTHTHEORIES
419
(A,) would be open to an objection similar to that brought against the
satisfaction theory, for since only the names “a” and “b” have denotation
axioms in X1 only the sentences of L are awarded truth conditions by hr.
Nevertheless, A, does have one unattractive feature, for the language L’ is,
in general, not learnable. If the domain of quantification is infinite then L’
contains infinitely many names each of which would, in a theory for L’, be
treated as a semantic primitive. Let this be clear: the use of L’ does not
render the language L unlearnable according to X1 but it is possible, and
perhaps preferable, to replace the appeal to L+ by quantification over a
family of languages each of which is a learnable extension of L.
In the theory A2 ’ which follows, “M” ranges over all syntactic extensions
of L by finitely many names of objects in the domain of quantification in L.
@la)
@lb)
W4
Wb)
(B3a)
(tlM)(tlaE~~(Tr(C-ol,M)--Tr(a,,~)
Wb)
(Vhf)(Va,
rEM)(Tr(-CJ
034)
WM)(v+
EM)(Trfl
& r’,M)
3vJmL’l
-
Tr(o,M) & Tr(r,M))
M)
+-+(3M”‘2M)(37EM’)Tr(%$,M’))
NJ)
(VM)(Vx)(3M’
(BP)
WM)WM’
X)(37EM’)d(7,M’)
>-W(WEM)(d(7,M)
= x
= d(y,M’))
To see these axioms at work, we set out the proof of a very simple
biconditional.
Tr(“(3 ve)(If@e, a) & - Fv,)“,
L)
-(3M>_L)(37EM)Tr~~((y,a)&-FFyl,M)
*(3M2
by 034)
L)(37EM)(TrcH(7,a)l,M)
&-Tr$-F$,M))
by (B3a), (B3b)
-
(3 &f 2 L)(3
7 E MW’(d(7,
W,
d(‘a”,
W)
8~ -
F(d(7,
Ml))
by (Bla), (Blb)
MARTINK.DAVIE.5
420
-
“+”
(3 zh~)(Zf(v~. d(“d’, L)) & - f+,)
-(3~0)(~(~0,~)~-~~0)
by (BU)
by @2a)
That X2 delivers a homophonic biconditional for each sentence of L is a
trivial corollary of the following result:
IfA is any context for n names then
X2\ {@2a), Wb)) t- (VWWrr
, . . , 7n EM)(Tr(rA(71
-A(471
>M), . ’ . t 47,zWN.
f . . . , rn>7J?l
It might be thought that X2 is open to a charge of vicious circularity.
Axiom (B4). which is used in derivations of truth conditions for quantified
sentences of I,, involves quantification over semantic, and not merely
syntactic, extensions of Z,; yet if a language M is to be a semantic extension
of L it must surely agree with L in its assignments of truth conditions to all
sentences, including quantified sentences, of L. This thought would be a
mistake, for although it is indeed the case that
(5)
(VM)(VM'
ZM)(voE
M)(Tr(a,M)
-Tr(a,
M')),
we can regard the extension relation (2) as defined by (BP) and then prove
(5) by induction upon the length of sentences.
of connectives as they operate upon
X2 provides a unified treatment
sentences, as in
and upon compiex predicates, as in
(3 po)(- i*‘vo & NV,, Q))
by taking the former use as primary. This is in contrast to the satisfaction
theory, and accords well with the thought that negation has its home in its
role of forming a sentence by using which someone can deny what another
has said -- a sentence which is true just in case the other man’s sentence is
not true. But despite this contrast with the satisfaction theory it must not
be thought that Al, which avoids sequences, also avoids objects; the
interpretation of the quantifier is not to be confused with the substitution
WEAKNECESSITYAKDTRUTHTHEORIES
421
interpretation. On the contrary, objects are precisely the values of the
denotation function and the satisfaction relation between finite sequences
of objects and open formulae can be defined by the schema
Sat(s, rA(Vi, , . . . , vi,>‘,) ++ df
WW(37l> . . .,r,EM)(sti,)=dtyl,M)&...&ts(i,)
=d(~,,WkT6&71,.
. . ,r,)l,M)).
This direct recursion on truth, which will form the basis of our theory
for a modal language, is faithful to two insights attributable to Frege,
namely, that the sentence is the primary semantic unit and that truth is the
primary semantic concept.6 But the extension to the modal case may seem
to be doomed by an argument - also sometimes attributed to Frege’ which is supposed to show that any context C for sentences which admits
substitution of broadly logical equivalents (as ‘w’ does), and which is
referentially transparent, is in fact truth functional. A version of the argument is as follows. Suppose that s1 and s2 are any two sentences alike in
truth value, and that for any sentence s, %s = 1-lis true iff s is true, and
‘6s = $ is true iff s is false. Then consider the sequence of sentences
(6)
x (sd
(7)
z (Ss, = 1)
(8)
c (Ss2 = 1)
(9)
2
62)
The steps from (6) to (7) and from (8) to (9) preserve truth value since they
rest upon broadly logical equivalences, and the step from (7) to (8)
preserves truth value in view of the referential transparency of C. So C is
truth functional.
There is something right here, and something wrong. To see this, only
consider that %s,’ and r&s,1 are definite descriptions and that definite
descriptions have two kinds of uses which must be represented differently
at the level of input to a semantic theory. 9 In their genuinely referential
(entity invoking) uses they are semantically akin to complex demonstratives, while in their other (Russellan) uses they are akin to two-place
quantifiers. Clearly, someone who offers the truth of
422
MARTIN
a(s1 -6s1
K.
DAVIES
= 1)
as the ground for the step from (6) to (7) is giving r&r1 the non-referential
reading, and the sequence would be better represented thus (where
“(IxAx)Bx” represents “the unique x which is A is B”):
(6)
z @l)
(7’)
C((lx((s,
(8’)
c ((rx((s2 &Lx = 1) v (- s2 & x = 0))) (x = 1))
&x=
1)v (-sr
&x=O)))(x=
1))
The step from (7’) to (8’) is nothing other than the substitution of contingently coextensive predicates.
So what is right in the original argument is that a context Z for sentences
which admits substitution of broadly logically equivalent sentences, and
substitution of contingently co-‘denoting’ descriptions, is in fact truth
functional. What is wrong is to think that this threatens the notion of a
modal sentential operator admitting substitution of co-referential genuine
singular terms, for descriptions - in the relevant kind of use - are no?
genuine singular terms, and this can be shown without appeal to their use
in modal sentences.
The argument does impose a certain obligation, for we cannot proceed
in the derivations of truth conditions by substituting contingently co‘denoting’ descriptions within modal contexts. In particular, we cannot
make substitutions merely on the basis of a denotation axiom
d(“C, L) = a
if that axiom would be equally well represented as
(lx D(‘i.z”, x, L))(x = a).
But this obligation is not different in kind from one which we already have
to meet in providing a homophonic truth theory for a modal language,
namely, to carry through the derivations of truth conditions without
making substitutions on the basis of mere material equivalence. That the
denotation of “a” in L is a is a non-contingent property of L.”
With these preliminaries let us turn to the task of providing a truth
theory for a language containing a modal sentential operator, “I? expresses
WEAKNECESSITYANDTRUTHTHEORIES
423
weak necessity so that, for example, both
oMan(Socrates)
and
0 - Flanet(Socrates)
express arguably true essentialist claims concerning the contingently existing
man Socrates. In the terminology of possible worlds, these say that with
respect to every world in which Socrates exists he is a man and is not a
planet, but the truth theory provided will not itself involve quantification
over worlds.
II
Consider the language resulting from the addition of “W to L (and call the
new language “15” as well). The broad outline of a truth theory for L is
clear ‘r but a certain amount of care is needed at two points. First, the
proof theory used in delivering biconditionals must be restricted so that the
theoremhood of
Fa+ (3x)Fx
and the truth of “OFa” do not jointly suffice for the truth of “0(3x)Fx”.
Thus, either the rule of necessitation
(10)
If I-B then k q B
or else the axiom schema
(11)
w+4)+PP+“4)
must be subject to some restriction. In order to forestall any suspicion that
the restrictions to be imposed are in some way ad hoc, or insufficiently
motivated, I shall prove a completeness theorem for a version of S5 with
weak necessity, and with members of the domain of quantification existing
perhaps only contingently. Second, since the objects denoted by the names
in L exist (we may suppose) contingently and languages have their semantic
properties non-contingently, it is natural to allow that L itself exists
contingently.”
Because of this, the sentence
Tr(“gFa”,
L) +-+ q Tr(“Fa”,
L),
424
MARTIN
K.
DAVIES
which we might have expected to be a theorem of an adequate truth theory,
is in general not true. For suppose that “F” has the sense of “is a parent”,
that ‘5~” denotes George VI, and that “b” denotes Elizabeth II. Then
“oFa” is not true, and yet it is the case that with respect to every world in
which L exists “Fa” is true. For if L exists then Elizabeth II exists and she
cannot exist unless George VI is a parent. I take up these two points in this
section and the next.
Since we accept as a valid sequent
(12)
Fa I- (3x)Fx
and yet reject
(13)
OFa f o(3x)Fx
a first thought would be that the restriction upon the proof theory ought to
speak of the rule of existential generalization and possibly of the dual rule
of universal instantiation also:But this would be an error, for the sequent
(14)
Wa, a) I- n(3 x)H(a, x)
is acceptable: if with respect to every possible situation in which a exists it
is H to itself, then with respect to every such situation there is something
to which a is H. The sequent
(15)
n(Yx)Fx
I- nFa
is also acceptable: if with respect to every possible situation everything is
F, then surely with respect to every possible situation in which a exists it is
F. The feature which renders (13) unacceptable, but is not shared by (14)
and (1 S), is that the name “a” of a contingently existing object occurs in
the premise but does not occur in the conclusion.
Thus I offer as axioms and rules for the system which I shall call “SSC”
(,‘C” for “contingent existence”):
Any basis for first-order predicate calculus, together with all
instances of the T axiom schema
q P+P>
all instances of the S5 axiom schema
OP-+qOP,
425
WEAKNECESSITYANDTRUTHTHEORIES
and the rule of inference (Keen):
Ifill..
. . ,A,
I-B thennAr,.
. . ,nAn FOB
provided that ail the constants and free variables in
A, occur in B.
n,,...,
I assume that “D”, “a”, “&,‘. and “-” are primitive and that other modal
operators, quantifiers and connectives arc definitional abbreviations. Note
that the rule (Necn) could be replaced by (10) without restriction, together
with all instances of (1 I) in which all the constants and free variables in p
occur also in 4.
The associated model theory is as follows.13 An SSC-model structure is a
triple (It/, w*) II) where we E Wj and D is a function from members of
W (worlds) to sets. D(w) is the domain of w. To obtain an SSC-model we
add to a model structure an interpretation fsuch that for each constant c
f(c)ED(w*).
and for each n-place predicate P,j’(P) is a function from worlds such that
f(P)(w) C (D(w))“.
For each such fwe define a sequence-relative partial valuation
V: ( ,gw D(w))~ x (u: u is a wff) x W + {I, 0)
as follows:
(9
!f u is an atomic wff pit,, . . . , tn) where the ti are variables
or constants then
V(s, u, w) = I if (s*(tr), . . . : s*(t,))Ef(P)(w)
V(s, u, w) = 0 if (s*(r,), . . . 2s*(tn))E (D(JvN” W’)(w)
otherwise V(s, u, w) is undefined,
where S*(Z’i) = s(i) and S*(C) =f(c).
(ii)
If u is - r
V(s, u, w)
V(s, (7,w)
otherwise
(iii)
If u is (T &u) then
V(s, u, w) = 1 if V(s, 7, w) = 1 and V(s, V, w) = 1
then
= 1 if V(s, 7. w) = 0
= 0 if V(s, 7, w) = 1
V(s, u, w) is undefined.
426
MARTINK.DAVIES
V(s, (I, w) = 0 if V(s, T, w) and V(s, v, w) are both defined
but not both equal to 1
otherwise V(s, u, w) is undefined.
(iv)
If u is (3 Oi)7 then
I$, u, w) = 1 if for some s’ & s such that
s’(i) ED(w), V(s’, 7, w) = 1
V(s, (I, w) = 0 if for every s’ A s such that
s’(i) ED(w), V(s), 7, w) = 0
otherwise V(s, u, w) is undefined.
(9
If u is 0~ then
V(s, u, w) = 1 if V(s, 7, w) is defined and for no w’ E IV is
V(s, 7, w’) = 0
V(s, u, w) = 0 if I$, 7, w) is defined and for some w’ E IV,
V(s, 7, w’) = 0
if I+, T, w) is undefined then V(s, (I, w) is undefined.
Note that for any wff u, V(s, u, w) is defined just in cases*(f) ED(w) for
all t in u. We can define a partial valuation upon sentences and worlds as
follows:
VaI(u, w) = 1 if (Vs E (D(w))w)V(s, u, w) = 1
Val(u, w) = 0 if (Vs E (D(w))~)I+,
u, w) = 0
otherwise Val(u, w) is undefined.
A sentence u is true in a model just in case Val(u, w*) = I. Since for every
constant c, f(c) E NW*), the function Val(u, w*) upon sentences is total.
A sentence is valid if it is true in every model, and it is easy to see that
every theorem of SSC is valid. The converse result, that every valid sentence
is a theorem, is established in the Appendix by a proof in the style of
Henkin.
The Barcan formula
(Vx) o@x+ qvxpx
is not provable in SK, but its converse is. A falsifying model is easy to
construct. Let IV= (w*,w,},D(w*)=
(e,,e,),D(wi)=
{ei,e,}. L&P
be a one-place atomic predicate such that
WEAK
NECESSITY
AND
TRUTH
THEORIES
427
Then the reader can check that
Val((Vve)“Pve,
w”) = 1
val(qvvo)Pu~,
w*> = 0.
while
On the other hand, the converse formula is easily proved, for we have
(Vx)*x
i- @Y
and all the constants and free variables in the premise occur in the conclusion so by (Necn)
qvxpx
t-may,
whence by generalization since y is not free in =(V x)@x
qvxpx
t-- (VX)O@X.
Since we are aiming at a truth theory for a transparent modal operator
we shah want to add identity to our system. The.additional logical axioms
are
(Idl)
W~o)(~o= vo)
and all instances of
W)
WX)WY)(X =Y + (@@>X)+ W-TYN)
where +(x, y) is the result of replacing some or all occurrences of x in
@(x,x) byy. From (Idl) we have by (Necn)
q(V~o)(~o= vo>
and by the converse of the Barcan formula
(V~o)O(~o= vo).
In any model it is required that
f(=)(w)
= ((x,x>: x ED(w)}.
Since we have as a theorem
428
MARTIN
(~~o)(V~,
K.
DAVIES
)(%I = 01 + q%l = VlN
the resulting system is SS with weak necessity, contingent existence, and
necessary identity.
111
The sentence
(16)
Tr(“oFu”,
L) ++ oTr(“f?‘,
L)
is not in general true, for the right side of the biconditional is weaker than
the left. What is needed is to have a language mentioned on the right whose
contingency of existence exactly matches that of the objects denoted by
names occurring in the sentence mentioned on the left, in short a sublanguage of L containing just the name “a”. This motivates the following
closer approximation to an acceptable theorem:
(VM C_L)((“u"
EM & (VM' C_L)(“Q" EM' + M' > M)) -+
(T~(‘QFQ”, L) -
oTr(“Fa”,
M))) .
Here, and in the theory which follows, “M" ranges over all syntactic
extensions of the language LO = L\(“a”, “b”) by finitely many names of
objects in the domain of quantification of L. Thus, for every language and
every set of names in that language there is a restriction of that language to
that set of names. With respect to every possible situation LO exists and
every sentence of Le has truth in LO conditions.
The way in which the existence of a language depends precisely upon the
existence of the objects denoted by names in that language also motivates
a relaxation of the restriction upon the rule (Necn). Suppose that
A,,...
,A, t--B
where the premises speak of objects not mentioned in the conclusion, but
a language is mentioned in the conclusion whose existence requires the
existence of those objects. Then it is acceptable to pass to
4411...,
d,t-oB.
Similarly, if the premises speak of a language and the conclusion speaks of
objects whose existence suffices for the existence of that language, then it
is also acceptable.13
WEAK
TECESSITY
ANDTRUTHTHEORIES
429
We are now able to state the axioms of a truth theory, X, for our modal
language L. The three-place denotation relation replaces the denotation
function. For the atomic predicates and the names we have
(Cl4
o(viw)tvrEM)(vX)(o~7,X,M)-t
(Tr~.~.M)+-+Rc))
Wb)
c(V1+f)(V7,6
EM)(Vx,~)(oD(y,x,M)&nD(6,y,M)
n(Trt?‘fCy,
@,M)
-Hx,Y)))
04
oD(%“, a, L)
(C2b)
uD(“b”,
q
+
b, L)
and for the connectives and quantifier
(C3a)
o(‘dM)(vaEJ~~(Tr~~~o’,M)--Tr(u,M))
Wb)
~(VM)(Va,7EM)~(Trf,G&?,M)
+-+ Tr(u, M) & Tr(r, M))
(C4)
3(VM)(V@
o(3.M
EM)n(Trc(C(3Vi)@O?,
Ji’f)
?M)(37Ekf’)Tr(%$,h.f’)).
Using “R(u, M, M’)” to abbreviate
uEM’&(VM”
wj(uE‘~“+M”
2M’)
we have for the modal operator
W
rJ(VM)( vu f M)( VM c_Mj(R(0, M. M’)+
~(Tr@~,Al) -
ITr(u,M’)))
Finally, we need
(CUa)
qvIq(vx)(3M
1 M)( 37E
(Cub)
q(VM)(V~EM)(~!~)~D(~,X,M)
(cpj
L~(tlM)(tlJtt
Mymo,
x, M’)
M)(tl7E,~~(tlX)(oD(7,X.,W)
+UD(Y,X?M'))
?.
As will be expected, the adequacy of X follows easily from a lemma
which is in turn proved by induction on the length of sentences. I omit this
relatively straightforward proof but it may be helpful, at this point, to
430
MARTIN
K.
DAVIES
provide a proof of a very simple biconditional:
(17)
n(Tr(“nFu”,
L) -
0Fu)
Note first that from (Cub), (CP), and (C2a) we have
(18)
o(WM c_L)(‘a”EM+
OD(“U”,U,M))
Then (Cl a), (CS), and (18) respectively yield
(19)
(VM)(“a”EM-+(oD(“u”,u,M)
+ O(Tr(“Fu”,M)
(20)
(VM 5 L)(R(“Fu”, L, M) +
o(Tr(“nFu”, L) ++0 Tr (“Fu”, M)))
(21)
( VMC_
L)(‘a” EM + qD(“U”,
-Fu)))
II, M))
From (19) (2 l), and the S4 axiom we have
(22)
qo(Tr(L‘Fu”,Me)
-
Fu)
and from (20)
(23)
q(Tr(“DFu”,
qTr(“Fu”,Me))
L) -
where MO is the restriction of L to the set of names (‘a”). Consider the two
sequents
Fu, Tr(“Fu”, Me) Tr(“Fu”,
Fu k Tr(“Fu”,
Me), Tr(“Fu”, Me) -
Me)
Fu f- Fu.
The reader can check that from these by the relaxation of the rule (Necn)
we can proceed to
oFa, D(Tr(“Fu”,
oTr(“Fu”,
Me) -
MO), o(Tr(“Fu”,
Fu) t- oTr(“Fu”,
Me)
MO) ++ Fu) t- OFU
and thence to
o(Tr(“Fu”,Me)
-
Fu) l-- OTr(“Fu”,Me)
Thus from
(24)
q(Tr(“Fu”,
Me) -
Fiz)
-
OFa.
WEAKNECESSITYANDTRUTHTHEORIES
(25)
Tr(“nFa”,L) - qTr(“Fa”,
431
Me)
we have
(26)
Tr(“Wu”,
L) -
JFU.
Noting that (22) and (23) are the necessitations of (24) and (25) respectively, and that although “Me” does not occur in (26), “u” does occur there,
we have by one further use of the relaxation of (Necn) the desired
biconditional(l7).
We can now check, briefly, that h is adequate to the four intuitions noted
at the beginning.
(a) According to X, the name “a” and the predicate “F” are semantic
constituents of “OFa” for the axioms (C2a) and (C 1a) governing those
lexical items figured crucially in the derivation of (17). Thus if h is to meet
the SC, speakers who understand “oFu” must also understand “Fa”, and
speakers who understand “oFa” and “H(b, b)” must also understand
“H(a, a)” and “Fb”; in short they must understand the name “a” and the
predicate “F”.
(b) The derivation of the canonical biconditional for “oFa” proceeds
via the canonical biconditional for “Fa” (which follows from (19) and (21)
and with the S4 axiom yields (22)), at least somewhat as does the derivation of the canonical biconditional
o(Tr(“-
Fa”,f.)
++-
Fa)
for “- Fa’9.t5
(c) That the positions of genuine singular terms in L are referentially
transparent follows from the fact that their positions in the denotation
axioms are transparent. The provision of X does not, of course, show that
some idiom of some natural language is transparent, but it does show that
our intuitions about our actual language are thus far consistent.
(d) Finally, “0” in L expressesweak necessity, for the truth conditions
of OL sentences containing “0” are stated using the weak necessity operator
of the ML.
IV
I turn now to some difficulties which confront the theory h. Notice first
that the problem discussed in Section III of Peacocke’s paper (call it ‘the
432
MARTIN
K.
DAVIES
Fine Problem’) can arise even when there is no existence predicate in the
.OL. For consider the sentence
(27)
C( 3V~)(FV(J & - O(FTAJ& Fu))
where “a” is a name of Socrates and “F” has the sense of “is a man”. Upon
Peacocke’s reading of the modal operators this sentence is true and the most
obvious kind of satisfaction theory is bound to deliver a false theorem
when it is applied to the result of existentially generalizing upon (27). This
fact suggeststhat, although the language I!, does not contain an existence
predicate, the theory X will similarly deliver false theorems. For if (27) is
true then it is possible that there is a semantic extension of L which contains
a name of a man who cannot possibly coexist with Socrates; whereas there
can be no such language. But this does not threaten X since (27), upon my
reading of the modal operators, is not true. The sentence
(28)
- (Fu & Fb),
for example, is not true with respect to any possible situation in which the
denotations of “a” and “b” do not both exist, and neither is the sentence
(29)
n - (Fa & Fb).
It might be replied that we should allow that (29) is true provided that,
with respect to all situations in which the objects do exist, (28) is true. But
I reject this suggestion, on the ground that to admit that there might be a
situation with respect to which it is true that 0~ and yet not true that p is
to risk losing such grasp as we have upon the notion of necessity.
Second, it might be thought ari objection to h that the proof theory
provided allows one to proceed from
(30)
UFa
(31)
n(3x)x
10
= a
whereas such a sentence as
(1)
Socrates is necessarily a man
may surely be true while
W E A K N 1.; C 1: S S I ‘I‘ Y A N D T R U T I-I T H E 0 R I 1: S
(32)
433
Socrates necessarily exists
is false. This is, in fact, no objection since just as (30) expresses the weak
modal claim thata cannot exist without being F. so (3 1) expresses the
indubitable claim that a cannot exist without existing. But now the
objector can quite correctly reply that I am left with no way to express
what is ordinarily meant by (32), or by
(33)
The number two necessarily exists.”
There are several ways in which one might try to meet this important
objection. One way would be to introduce a new atomic predicate:
“necessarily exists”. This proposal obviously allows us to represent (32) and
(33) but it is likely to be judged unsatisfactory for two reasons. First, it
fails to discern “exists” as a semantic constituent in “necessarily exists”.
Second, it leaves us still without any way of representing sentences which
are intuitively closely related to (32) and (33), such as
(34)
Necessarily, if the number two exists then Socrates exists
(35)
Eiizabeth 11could not have existed without George VI
existing.
In the face of (32) and (33) there might also be a temptation to introduce a
further modal operator “x” to be treated analogously with “c” but
expressing strong necessity so that
WT*,...,Yn)
is true just in case with respect to every possible situation the objects
denoted by yr ) . . . , 7n exist and are ,4 _ Certainly there would be no
difficulty in providing a suitable proof theory for “I? by a restriction upon
the extended rule of necessitation dual to that for ‘Q”. A truth theory for
a language containing only “W could not be constructed exactly parallel to
h since the results of replacing “0” by “W in the axioms of X are not true.
But hopes arc raised by the fact that, if “a” is a name in M, the sequent
U(Tr(“Fa”,M)
*
Fu). ~Tr(“Fa”,M)
k ~Fu
is intuitively valid. For if with respect to every possible situation in which
M, and so a, exists it is the case that
434
MARTIN
Tr(“Fa”,
K.
DAVIES
M) ++ Fa,
and with respect to every possible situation M does exist and Tr(“Fa”, M),
then it surely follows that with respect to every possible situation a exists
and Fa. This a matter of some interest, but it will not be pursued further
here just because, like the other proposal, it leaves (34) and (35) untouched
A more satisfactory way to meet the objection - a way which uncovers
“exists” in “necessarily exists”, finds a single modal operator in (1) and
(32) and provides for (34) and (35) - is to give a favoured treatment to
atomic sentences so that
(30)
OFa
is true just in case with respect to every possible situation a exists and is F,
while
a--Fb
is true just in case with respect to every possible situation in which b exists,
b is not F. Then (1) can be represented as
q((3x)x
= a -+ Fa)
q((3x)x
= b +(3x)x
and (34) as
= a).
The technical difficulties - other than those associated with the Fine
problem - which are confronted in providing a truth theory for such a
modal language are rather different from those we overcame for weak
necessity. in the proof theory, the rule of substitution for theorems has to
be restricted. In the truth theory the necessitations of sentences containing
semantic vocabulary have to be regarded differently from the necessitations
of OL sentences. Otherwise for example - even supposing that “a” is the
only name in the language M - the biconditionals
qTr(“-FI;a”,M)+--+o--t;a
Tr(“o m Fa”, M) *
aTr(“w
q(Tr(“-
- Fa)
Fa”,M) *
Fa”, M)
would all, in general, be false. As Peacocke notes,” the way around this
WEAKNECESSITYANDTRLTHTHEORIES
435
difficulty is to allow that the truth values of ML sentences involving
semantic vocabulary are determined by the truth values of non-semantic
OL sentences, together with the requirement that such biconditionals as
these be true.
Without suggesting that I have any good complaint against this approach
I think it profitable to ask how far we can go towards meeting the original
objection while retaining the main features of a weak necessity operator.
For simplicity I shall assume that existence is to be expressed by a new
atomic predicate “E”, rather than by “(3 x)x = 5”. The idea will be to give
a favoured treatment to just this atomic predicate.
First. we need to make a simple addition to the model theory for SC:
(i>
V(s,Et,,w)=
1 if s*(tr)ED(w)
V(s,Etl, w)=O
if s”(t,)@D(w).
Obviously, V(S, Et,, w) is everywhere defined and
V(s, qEt, , w) = 1 if (Ww)s*(t,)
V(s, CEt, ) w) = 0
ED(w)
otherwise.
Then we shall require some modifications to the proof theory so that, for
example, the sequents
are provable. while the sequents
q(Ea & Fa) b OEU
@(Ea & Fb) t OEa
o(Fa + Eu), 0 Fa f of&
V-b + Ea), q Fb f- OEU
MARTIN
436
q(Fa
K.DAVIES
+ Fb), oFa I- @Fb
are not provable.
The obvious axiom to add to the truth theory is
(Clc)
Then in particular, since []D(“a”,
q(Tr(“k’a”,
a, L) we have
L) +-+ Ea).
Given the earlier doctrine that if a does not exist then L does not exist, this
biconditional must be allowed to say more than that wherever L does exist,
“Ea” is true in I, just in casea exists. For surely from this biconditional
it must follow that if a does not exist then “La” is not true in L, and so
(with (C3a)) “- Ea” is true in L. We must allow then that certain sentences
are true or false in L even with respect to situations in which L does not
exist.
Notice that so far we do not have to allow that “a” denotes a in L with
respect to situations in which a does -not exist. It is in virtue of the fact that
“a” denotes a in L with respect to those situations in which a - along with
the denotations of other names in L - does exist (situations with respect to
which all atomic sentences containing “a” have truth in L conditions) that
it is a’s existence upon which the truth of “Ed’ depends.
But now we must turn to one last difficulty, namely that upon the
modification envisaged the Fine problem does arise. Consider the sentence
0( 3vo)(Evo
& - O(Evo & Ea)).
This is true, and yet in no possible situation is there a language extending
L, and so containing “a”, which contains a name of an object which exists
in that situation and cannot coexist with a. For in such a situation a would
not exist and so no semantic extension of L would exist. This is clearly a
serious problem, and a number of possible solutions deserve attention. One
which holds prospects of being relatively smooth involves forgoing the
earlier doctrine that in a situation in which the denotation of a name does
not exist any language M containing that name does not exist. We can
however remain faithful to the spirit, if not the letter, of that doctrine by
maintaining that with respect to such situations no atomic sentences ofM
WEAK
NECESSITY
AND
TRUTH
THEORIES
437
except those of the form: Ey, have truth in M conditions. One might say
that M exists, but is a mere shadow of its former self. Then in a situation
in which a does not exist, but an object which cannot coexist with a does
exist, the language L does exist and so by (CUa) an extension M of L exists,
containing a name of that object.
The truth in L of “EC depends precisely upon the existence of a, and so
“Ea” is false in L, with respect to this situation. This, however, is not quite
enough for we need, inter ah, that “EC is false in the extension M. We
cannot rely upon the fact that with respect to those situations in which a
and the denotations of all other names in M exist (situations with respect to
which all atomic sentences containing “8 have truth in M conditions) “u”
denotes u in M, because there are no such situations. So we are obliged to
allow that with respect to this situation, in which L exists but a does not,
‘P’ denotes a in L and in M. ” This is not as major a concession as it may
at first appear, because atomic sentences of L - other than “Ea” and “Eb” will still not be provided with truth in L conditions with respect to this
situation. Notice too that (Cub) must be dropped, and a corresponding
alteration made in (C4).
It would be absurd to suggest that we have here more than the sketch
of a suggestion, and it would be rash to deny that with sufficient ingenuity
one might remain faithful to the letter as well as to the spirit of that earlier
doctrine perhaps, for example, by inventing ‘shadow languages’ which exist
in the absence of their less shadowy counterparts. Nevertheless! it is difficult
not to suspect that these further efforts would have less philosophical than
formal interest, and that we have before us enough to decide whether weak
necessity is a viable prospect.
Magdalen College, Oxford
APPENDIX:
COMPLETENESS
IIENKIN COMPLETENESS THEOREM FOR SSC: If u is a sentence which
is consistent with respect to S5C, then there is a model ( W, w*, D, f >such
that Val(o, w*) = 1.
Proof (sketch): l9 For each natural number i, introduce a countable set
Ci of new constants, and let Lo contain the constants in Ce.
438
MARTIN
K. DAVIES
Extend {u) to a maximal consistent set Fe in Lo with the property that
Co)
if (3 x)*x E r0 then for some c E Co, @cE ro.
For each ri and each sentence fl Of Li such that O/.3E ri construct a
subordinate rj of ri as follows. Let T’j” = {flj and let Ly be the maximal
sub-language of Li with the property that
if f is a constant in Li and t does not occur in 6, and there is
or E ri such that
(i) r is inconsistent with 0,
(ii) t occurs in 7, and
(iii) it is not the case that r is p 8~v where 0~ E ri, p is
inconsistent with p, and t does not occur in 1-1,
then t 4 LF.
Let r/ = {p f u (T: T E Lp & or E T’i}. Then I’/ is consistent. Let
Lj = LT u Cj and let T’j 2 I’; be a maximal consistent set in Lj with the
property (Lj).
Define a model in the obvious way with W = {ri : i E o}, w* = ro, and
NT’,) = the set of constants in Li.
Then it is easy to show that Val(a, w*) = 1, as required.
NOTES
* This paper, apart from the final section, is based upon parts of my D.Phil. thesis.
The thesis owed much to the prior and continuing work of Christopher Peacocke, and
I am grateful for his encouragement in the preparation of the material in its present
form.
i See Kripke, ‘Identity and Necessity’ in Munitz (ed.), Identity and fndividuation,
New York University Press, 1971, p. 137. If “0” expresses weak necessity then
f-oA(7,,...
, m)‘is true just in case with respect to every possible situation in which
the denotations of 7,. . . . , 7n exist, they are A.
’ The distinction might be refined to avoid this difficulty, and in particular to rule
that “oMan(Socrates)”
is a de re modal sentence in virtue of its equivalence to
(3x)(x
= Socrates & q Man(x)),
but none of (a)-(d) can be represented as the intuition that (1) is, in this sense, a de re,
rather than a de ditto, modal sentence.
3 This constraint is discussed at length in Section 2 of my Trurh, Quantification, and
Modality, Oxford D.Phil thesis, 1976.
4 In Dummett’s sense; see Frege: Philosophy o~Langrurge, Duckworth, 1973, p. 11
and elsewhere.
WEAK
NECESSITY
AND
TRUTH
THEORIES
439
’ Such a theory
was suggested by Gareth Evans in a lecture in 1974. Because it
stays close to the ideas of Dummett
op cif. (esp. Chapter
2), Evans called it a
‘Fregean’
theory as against the usual ‘Tarskian’
kind.
6 See Foundations
of Arithmetic
(trans. Austin),
Blackwell,
1953, p. (x), and The
Busic Laws of Arithmetic
(trans. Furth),
Cambridge
University
Press, 1964, pp. 89-
90.
’ See ‘On Sense and Reference’
in Geach and Black (eds.). Translations
from the
Philosophical
Writings of Gottlob
Frege, Blackwell,
1956, pp. 62-65.
8 See Plantinga,
TheNature
of,Vecessity,
Oxford
University
Press, 1974, p. 2.
’ For the divided treatment
of descriptions
see Peacocke,
‘Proper Names, Reference
and Rigid Designation
in Blackburn
(cd.), Meaning,
Reference
and Necessify,
Cambridge
University
Press, 1975.
lo See Peacocke,
‘Necessity
and Truth Theories’
(Ik’TT),
Section II.
‘t SeeNTT,
Section
IL
l2 cf.?VTT.
footnote
12, and the present paper Section IV.
I3 This is one variant of what Kripke
called ‘the Frege--Strawson
view’; see
‘Semantical
Considerations
on Modal Logic’ reprinted
in Linsky (ed.),Reference
and
Modality,
Oxford
University
Press. 197 I, p. 66.
l4 This relaxation
constitutes
a proper,
and not a logical, part of the theory.
l5 There is the significant
difference
that the S4 axiom is not needed in the latter case.
l6 See Wiggins, ‘The De Re “Must”:
A Note on the Logical Form of Essentialist
Claims’ in Evans and McDowell
(eds.), Truth ond Meaning,
Oxford
University
Press,
1976, p. 301.
I7 See the second objection
discussed
in NTT, Section IV.
ta One could also arrive at this point by considering
just what it is for M to be an
extension
of L with respect to such a situation.
l9 There is only one crucial difference
between
this proof and that for quantified
S5
without
the Barcan formula;
see Hughes and Cresswell,
An lnrroduction
to Modal
Logic, Methuen,
1972, pp. 174-176.