SIX PROBLEMS I N " TRA NS L A TI O NA L E QUIV A L E NCE "
by
Francis Jeffry PELLETIER
I wish to first formulate a criterion according to which two systems
of logic might be said to be "really the same system
having
where the difference is in the
- i n different
s p i tvocabulary
e
o — especially
f
logical
operators
each
has.
One
way
o
f
doing this might be to show
t h e i r
that the theorems o f the two systems are validated by precisely the
same structures ; but I am here interested in a more "syntactic" test
for this. On an intuitive level, my desires could be expressed by saying
that I want t o be able t o translate one system in to the other,
preserving theoremhood. F o r this reason I ca ll i t "translational
equivalence" between the two systems.
It is often said that a standard formulation P, of propositional logic
using ; 8 0 as connectives (plus some rule o f inference) and a
formulation P2 using {
—
"merely
, ; —
notational
; }
variants of one another", or "are the same logic",
or
are "translationally equivalent." Why is this?
( (in
p my
l uterminology)
s
pIsn'teit because
r h athe ptwo translations
sI. (p & q ) - - 1 q )
M
o = —,(p&
d
2. (p—>q)
u
s
preserve
P
o all the
n theorems of the two logics, and furthermore, 3 and 4
are
e true?
n
s
)
a3 The rule ofr inference chosen for '8z: is, when the '8L's are replaced
e by i n accordance with I, a derivable rule in P2
4. The rule of inference chosen for i s , when the a r e replaced
by '8L's in accordance with 2, a derivable rule in P
l
At least this is what is said in many textbook discussions. In theory,
.
though, one should say something further; namely that the translations do not introduce any disturbance into the systems. For, so far,
all that has been said is that the translations have to map theorems
onto theorems. But we would also like to see that non-theorems get
424
F
.
J. PELLETI ER
mapped onto statements that " sa y the same thing in th e other
language" preserve equivalance ; t h a t is , ta ke a non-theorem,
translate it into the other language and then translate the result back
into the first language. The result o f this should be equivalent to the
original formula.
Let's make this a bit more precise for our logics P, and P2. We say
that P, and P, are translationally equivalent just because there are two
functions f, and f , which will accomplish the required translations
from P, to P2 and from P2 to P
i
language
P
"Given
P2
i, tr oe s p Ie can
c t express
i v e l ythe
, language P
iP
language
f 2 , P, I can express P2
"a
Tnahsne dy
a
f ,r e
iti, (A)
= A , if A is atomic
f
,
(A) = A , if A is atomic
sf e
d
a f yn i scn te d
u
i
o
f, (—,A) = —if, (A)
t
2
=
-"fn G si v e
a
n
1 t((A
f,
2 &( B))
A =
) ( A ) - - > J O ) t 2 (A B ) = - ft2 o rh l
l oe o
w
1 (f
sfm r to guarantee that P, and P2 are in fact translationally equivalent,
Now
2
( A )
&
( meet
B
)
) criteria, to wit
the
functions
must
certain
:o
t translation
m h
P
e
A. I f A is an axiom o f P, the f, (A) is a theorem of P2
2
B. I f A is an axiom o f P2 the f, (A) is a theorem of PI
tC. I f R is a rule of inference in P, then f, applied to each premise of R
o and to the conclusion of R must be a derivable rule o f P,
P
D. I f R is a rule of inference in P
i2 tand
h etonthe conclusion
f ,
of R must be a derivable rule o f P,
.aE. [pf , p
(f, (A))
l i —
e A]
d is a theorem of P2
S
F.
t [ f , (f,
o (A)) — A] is a theorem of P,
o
e
a
c
h
to say, after translation from one system to the other, the
fpThis is
r
e
m
i
s
original
axioms are still provable, the rule of inference are derivable,
,e
and the translation functions themselves introduce nothing new.
h
o More generally,
f
two systems o f logic S, and S2 are said to be
a
R
translationally
equivalent
if there are translation functions f, and f,
s
which
obey
A
through
F.
t
h
PROBLEM
1: Is this notion of translational equivalence reasonable?
e
That
is,
does it capture the intended force o f "really the same
e
system"
? Is it in any sense trival ?
f
f
e
c
t
o
f
s
a
y
SIX PROBLEMS I N "TRA NS L A TI O NA L EQ UI VALENCE"
4 2 5
In fact my interests concerning translational equivalence have to do
with modal logics. I am interested in determining whether certain
well-known modal logics are translationally equivalent to one another.
For this purpose I shall consider only translation functions that map
non-modal vocabulary into themselves. And I shall assume that each
modal logic has the same set o f atomic propositions, th e same
propositional connectives I & , v,—), •
rules
of inference, and the same class of modal-free wffs. In fact, the
,
only
is that system i has the modal
--)1 ,difference
t h e to be
s encountered
a m e
operators
p r o pD o s i t i o n a l
i a n d are the identity function on everything except the modal
functions
O
operators.
So the question amounts to: can we find systems i and k for
i
which
there are f, and f, obeying A through F?
w It his often
e easy to find one of the required two functions which will
obey
r e the restrictions A and C. But finding a pair of functions so as to
salso ysatisfy B, D, E, and F is much more difficult, and in fact I can
give
s tonly an " a rtificia l" example of it (below).
e I give
m first three examples of the case where the "translation" can
kbe effected in one direction. Logic K is generated from the propositiohnal logic together with these rules and axioms (the notation follows
aChetlas)( t)
s[RE] i f H (A B ) then ( E
E
k A i f —
[RN]
H A then B1
)
l[Def
— E01l kH A k
k
A
k[M] O k (A & B) —) (E
a[C]
k A H & Elk (AE& l B k) O k (A & B)
nB
)
dSystem T is generated in the same way (with E
with
t r ethe
p axiom
l a c i n g
E
l k
O
kt[T] o g eA t h e r
Consider now f, (from T to K ; that is, which says "given K, I can
S
oexpress T" )
t
(O, A) = (E l
h
k f,
to see that f, obeys A and C — that the translation of all T
e It is easy
( A )
t
r &
a (
f
,
n 1
(
A
)
)
s B l )
a r t
i i o
n a
n
C
H
E
426
F
.
J. P E L L E T I E R
axioms are theorems of K, and the translations of the rules o f T are
derivable in K. Fo r example,
H El, (A & B) ( E
H
T 0, A A
A
&
are axioms of T whose translations are, respectively
L I ,
B (Elk )(A & B) & (A & B)) ( ( E k A & A) & (El
1—
k B
&
B ) )
(E l
One can easily see that, since [M] is already present in K , the first
k
must be a theorem of K. And the second is propositionally a theorem
A
of K. The translation of the rules, e.g., [RN], yields
&
ifA HA then H(E l
k) A
&
which is
obviously in K, since [RN] is in K.
A
— )
Thus) f, satisfies A and C, and so in some sense T is definable or
A
modelable within K. The next task would be to find an f
(El
2 s u c h
t h a t
f
k2
function
which will obey B and D, and such that E and F are true.
A There are various other examples, perhaps more interesting, of this
)
kind
of —
=KD45,
o
n e - is defined as K above (with Ed45 replacing E l
axioms
X
w
k ay
,)[D]p
m
o ld ue s
t h e s e
I
lw
l
i
n
g
—
[4]
H
0
0
-h[
c. 10
2 E
.e5
4 .1
1
The
system
S
r]
O
4
4
5
5
a
is
5 T above (with 1E4
e the system
n
A
A
11
.,e
,14.2oreplacing
n —e A— Lic145)
X
—
o
2
i,[G]pr fleI ups l›<) a)L c i n g
tfst- -hh, e e, > I
0
c0 0
G
now
st"E Consider
p
y l se u1 a
sc the translation function f, (from KD45 to S4.2, i.e.
c
1
4
"given
S4
1
[th
oh
1 e. 4m 51 4 ]
.f o b r A m
se
a
o
v
e
m
fi (0d45
2
1 5A A) — 0 1 7
2
I
u
l
a
"
(o
f
w
i
t
eA
d
4 . 2d f i
A
will yield the result that the translation o f all KD45
a en check
te
h
sc- quick
- ( A4 )
d
e
f
n
e- )
o
5
cni• n ei
Die A
tK
0
5
d
ce4
4
—
fm
n.
o
c)2
rd
eA
u
o
a
slf
SIX PROBLEMS I N "TRA NS L A TI O NA L EQUIVALENCE'
4 2 7
axioms and rules of inference will be theorems and derivable in S4.2•
For example, [Def 0] becomes
EK>
4.2
[D] becomes
A
f 0
—0
[4] becomes
0 4
E0
. 1
1 4
2
[5] becomes
4 .A
. 2'
2 A0
It is easy 4
t o check that a ll o f these are theorems o f S 4
A—
.
translation
of the rules also is straightforward. [RN] of KD45 becomes
- ) .
2 . T ifh Ae
3 b 2 then F- 0 0
4
E .02A
which is obviously derivable in S4
l KD45
E - becomes
[RE]
ofA
.
0 1)
2 , s iif nHcA e B ) ithen
t F--(01-1
, -4 A
4 0O
h a s
i
t
.s
- ED
o
which
w
is alsonderivable. (In
2 the presence o f propositional logic and
2 the
1 Erule
[Def
01,
[
R
N
]
0
A40
a
n
d
if H
(A
—
B)
then
H(O
0 A OB)
4
.
[
T
]
. 4
2 . to [RE]. So given H (A —B), the rule [RE] of S4
is equivalent
.
2
.
yield
I— AA — E4.2B), and then the modified [RE] would yield the
2
A
desired
A —
g0 a
i n
I have not been able to find an
2 w o H(071
uN-l d A
B A in to some formula o f KD45 wh ile
t2 which will translate
E 04.2
satisfying B, D, E and
1 F. )
4 .just given (of K and T, and of KD45 and S
The two examples
2
illustrate
System K is included in system T (yet here
4 . 2 different points.
B
we) find that all theorems
of T can be interpreted as certain theorems
- system S
of K). System KD45) and
)
4 ye t a ll the theorems
and
o f KD45 can be interpreted as certain
.theorems o f S 4
systems
X and Y, where X is included in Y, and yet all theorems of X
.2 a r e
i .n F
d oe rp e n d e n t
2
fo u l fl
o
e
c o vn e
t
h
e
ra a ng eo
Ir
s
h
o
u
l
d
f
i
428
F
i
.
PELLETI ER
are theorems of Y. Well – the "change only the subscript of the box"
translation will do here. Thus between K and T for example we would
have
f2(Ek A) = n t f'
2 (A ).
Since system K is included in system T, this translation obeys the
restrictions of mapping axioms into theorems and primitive rules into
derivable rules. Of course, this f
2 t oKgand
make
e tThtranslationally
e r
equivalent in my sense because conditions
w i E and
t hF are not satisfied. Of the two,
t
h
e
Hf
e a r l i e r
2
and
f
(f
i
i f
d
o
e
s
( i
only
the first
is correct.
To see this, let A be D
n
o
t
A(
i P & p) and f2(f1 (A)) = (EiP & p). Sure enough, we have
(Elk
) f
p. T h e n
f
,
(
A
)
) 2(DiP & E d ) .
=
—(
But now let A be Elk IL Then f2 (A) = El
AA
But
t p, a n d
f
)
i
((Elkf P &2 P) 1 ( A ) )
)
=
(
E
l
k
=A
is
not
a
theorem
of
K.
The
f
P
1
i& a n conditions
satisfy
d
f
A through D, however. So, if X is a formula of T
1(
P
and
2
v e • nof K, then we have
P Y gis a)i formula
h
e H Xr iff eHf
f
o1 Y( iff
Xr H) f
s
y 2s
t
e
What
by conditions E and F, and what is missing from
m
sis required
(Y )
these
f, and f
K
2
translating
a
nit into some formula of the other system, we can get back
,d an
to
i sequivalent formula of the original system by applying the other
tT
translation
h a t : function. It is not just that theorems can be translated into
sdtheorems,
t a r but that the entire language can be translated back and forth
introducing anything new. I guess the moral o f this last
towithout
i n g
example
w
i is th a t mutual modeling is insufficient f o r translational
translation functions themselves have to be "innotequivalence.The
h
a
f
o
r
m
u
l
a
o
f
t
h
SIX PROBLEMS IN TRA NS L A TI O NA L EQ UI VALENCE"
4 2 9
cuous" - or perhaps one should say that if one o f them introduces
some non-innocuity, then the other must be able to conteract it.
PROBLEM 2: Does this o n at
e -allw? aDoes
y it, for instance, show that deontic logic is a part of
m otense
d logic
e l (by
i n modeling
g "
KD45 in S
ife K is intuitively clear, is the given interpretation of
h For
4 a instance,
v
a T
. in nK clarifying
y ? Is the interpretation of K in T clarifying ?
s i2 g n i f
l"i example
of two systems which I can
i I now
c) ?
aturnnto
D my
oc "eaertificia
s
t
showc tol bea translationally
equivalent. I say that it is an artificial
r i f y
example
has n o t independently been
a because
n y toneh o f ithensystems
g
discussed.
I
developed
it
in
the
context
of trying to write a logic for
?
vagueness(
2
definite
that" and 0 interpreted as " it is indefinite (vague) whether".
I) then
considered
a
certain intuitive principles about vagueness and
l o to
tried
g express
i
them as theorems of the logic. For example, I wanted
c usual interdefinability of E and 0 . (I use a subscript 'v' for these
the
operators).
w h e
r e i0 , A
n
And
E I thought that a statement was vague if and only if it was not
definite.
w
a
<>, A 4-• —El, A
s
t
o
b(
2 as part of a commentary on John Heintz's "Might There be Vague Objects ?" A
e1981
) n espanded version was read a t a conference "Foundations o f Logic " i n
igreatly
waterloo.
Ontario in 1982. what is more or less the present version was presented at a
I
tSociety
e for Exact Philosophy meeting in Athens, Georgia in 1984. In the course of this, I
f
rhave
p a chance to discuss it with many people, of whom I should especially mention
i had
rAlistair
r e Urquhart, Johan van Bentham, Michael Dunn, and Richard Routley. Routley
me to a series of papers by himself and Montgomery in the late 1960's in
thass pointed
e
Logique
et
Analyse
wherein they are concerned to give a foundation for the usual modal
dt
systems
in
terms
of
a contingency operator V and a non-contingency operator A, rather
p
athan
r necessity (E) and possibility (0) operators. I f one changes theirV to 0 and their A
sto eE. and treats their axioms and rules as defining a new system (rather than a new
of an old system), one gets the results described here.
"definition
s
i e
t n
i t
s e
d
i
t
a
t
a
430
F
.
J. PELLETI ER
This forces upon one the principle that i f a statement is vague
(definite) then its negation must be also
0, A 0 ,
A
D A 0 , --IA
Certain other principles also seemed plausible to me, for example that
if A and B were both definite, so must their conjunction be
H E , A & 0 , B) 0 , (A & B)
I f some formula was provable, then it was definite
if HA then [ - D
A
If two formulae were provably equivalent, then it should be provable
that one was definite just in case the other was
if 1-(A B ) then H(01, A . 0 „ B)
Certain formulae are not true in this logic, fo r example
0,A
DA A
0 , A .1 7 1 0 ,A
0, A - D
0 , (A V& B) ( 0 , A & El, B)
Dv (A AB ) -> (El, A [11, B)
These invalid formulae will be recognized respectively as the analogues of [D], [T], [4], [5], [NIL and an alternate foundation of logic K.
At least I took them to be invalid under the interpretation o f Eh, as
"definite". A false sentence might be definite, so we don't have [T]; I
thought that borderline examples o f definiteness (" it's definite, but
not definitely so") showed against [4] and [5]; a conjuction might be
definite but neither conjuct be (as when they contradict each other);
and an implication might be definite, and its antecedent definite,
without its consequent being definite (as for example ((p & -1 3 ). p )).
Although the analogue of [M
does
] i s seem
n to
o be:
t (
3
v a l i d ,
)a (
s
Not-So-Strange
3
Modal Logic of Indetermacy", Logique et Analyse 108, pp. 415422.
c) l o s e l y
rI e
l
a
t
es d
ph r
i
n
c
i a p
l
e
l
l
n
o
t
a
SIX PROBLEMS I N "TRA NS L A TI O NA L EQ UI VALENCE"
4 3 1
H 0, (A & B) & (A & B)) -) (0 , A & 0 , B),
a principle I call [Mx ] (" if a conjunction is definitely true, then each
conjunct must be definite").
As it turns out, this logic can be axiomatized b y the following,
added on top of propositional logic :
[RE]
[Def
[RN]
[C]
[Mx- ]
[V]
i f
H
i f
H D
H
H
H (A B ) then 1 -(0 , A — 0 , B)
0 , A 0 , --IA
A then H 0, A
, A & E, B) -) n , (A & B)
El, (A & B) & (A & B)) -) (El, A & B )
0, A — EJ„
(The last axiom stands for "vagueness"). This is an "anti-normal"
logic, since it neither includes nor is included in K - it is not included
in K because it has principle [V] which K does not have, and it does
not include K because it lacks principle [M] which K has. Using the
Montague-Scott method, it is straightforward to give a "neighborhood
semantics" fo r this logic. I n the paper mentioned in footnote 3, I
showed that it is complete fo r the class o f "contrary", "partially
supplemented" models that are "closed under intersections" and
"contain the u n it" (I ignore the details here except to remark that
[RE] and [Def 01 are valid in any class of such models, that "contrary" models validate [V], that "partially supplemented
- m name
their
o d efrom
l s the gfactethatt "supplemented models" validate [M]
and "partially supplemented" ones validate that subset fo r which
[Mx] holds, that "closed under intersections" models validate [C],
and "models which contain the u n it" validate [RN]. F o r further
details see Chellas.)
Now I cla im that this logic is translationally equivalent t o T .
Consider the translation functions
fi( D A) = (Elv fi (A) 84 fi (A))
f2( L I J
A)
First, let us look at the axioms and rules o f T, and replace L i
=
We get (note that by f
t (DIA)
A
bthroughout.
y
D
( 0l , A( v0( A)
, A )
i
t r a n s f o r m s
[RE'] f i f H (A — B) then H(A v A & A) ( D v B 84- B))
i
n
t
o
2
(
A
)
v
L
i
-
432
F
_
J
_
PELLETI ER
[Def<>1 1--- -AK>, v
—
(0 , A & A)
[ C] ( ( 0 , , A & A) & (D, B & B) -> (0 „ (A & B) & (A & B))
H (0, (A & B) & (A & B)) ( ( D
[T'] H ( vD , A A & A)
& -) A A )
&
( E
And finally, let us look at fl 2 ( A ) ) —A. In the only interesting case,
v
B
where A = 0 , p: here f2(DvP) = (DtP v Et - &
i((Dv
p ) ,p &ap) ny (D
d
f ,
o
f
B
)
)
tv[ V1 h &( ( i C sv P & P) y i(EN, &s —
-1 1-3 can easily be verified that these are all theorems of system V.
It
0' )- ,let's
)1) Now
'
E
y other P
do lit the
way: take the axioms and rules of V and
hreplace
e 1n2 A in accordance wit h f
c(20, (A
en& o—
t e
t h a t
w
f ) ) .
A
if H A B ) then I - 0 0
e2[RE"]
[Def
0'1
g( 0 , 1 A 1--(El)
A(F-A
y then
O
e[RN"]
t (if
b
e
c
o
m
e
[C"]
0
1
t t EAI-, A y
H(
&
(E
s
A
& yB)) O
tD
,
- B
&
[Mx- "I
tH
1 Wilt(A
A - ) & B)
( y0 0
-7
0
1 1 3 &) )
1
(1-(El
i
[B"]
1
B ) D)
0
(1
A
&
y=
And now we
into the translation functions, using A = E t P.
&
(0 replace
A
A
4
So, f
B
)
y0
&
1
l[T"] H ( V
1
0
B
-E t - P y) , L I I
E
(1 A
B
)B
) see
) that these are all theorems of T.
t1
It
to
)—isp )again& easy
v
-—
=
O
t
P
,O
7
>
( So V and T are translationally equivalent; and, I would claim, this
t
-A
( the same
P logic, just as our earlier formulations o f the
makes
them
O
1
)
t
propositional
logic are "really notational variants of each other". And
v
A
(A
this
is
so
despite
the fact that looking at them in the abstract would
P
)0
v one to determine this - after all, V was even an never
allow
&
)1
aP n t i
normarlogic
and its semantics could only be given via a neighborE
hood
method while T can be given a relational semantics on possible
)
t
worlds.
,
—
a The fact that V and T appear to be so different on the surface and
A
n
)
d
f
2
o
f
t
h
i
SIX PROBLEMS I N "TRA NS L A TI O NA L E Q UI V A LE NCE
—
4 3 3
yet are translationally equivalent makes me wonder whether other
logics might have this feature. Fo r example, perhaps S4 and T are
translationally equivalent? Perhaps they are really the same logic?
Maybe all modal logics (except S O
(Wouldn't that make modal logic easy?
4!
PROBLEM
d translation
functions f
) a r e 3 r: Fin
e a
l l y
lt a restrictions
hn d e f A through F for pairs of the well-known modal logics.
2
s
aw hm i ec h
PROBLEM
o
l
ob
ge: Formulate
i y c
?a criterion which will te ll whether two
arbitrary logics have such translation functions or not.
PROBLEM 5 : Would modal logic really become easier if all systems
were translationally equivalent to one another?
PROBLEM 6 : How does all of this relate to Quinean indeterminacy of
translation and the intertranslatibility o f alternate conceptual
schemes? For example, can a radical translator ever tell whether
he is talking with a native speaker of V as opposed to a native
speaker of T? (The native speaker of V has no single word for
"necessity" b u t can assert as a theorem everything that the
native speaker of T can assert as a theorem. The native speaker
of T has no single word fo r "vagueness" but can assert as a
(
system
4
in which there are an infinite number of distinct modalities (e.g., T) can be
translationally
)
equivalent to it, because there are only a finite number of "things that
can
S be said" in S5 using only the variable p whereas there are an infinite number of
"things
5
to be said" in T using only the variable p. All the other usual modal systems
have
h an infinite number of modal functions of one variable. In passing it should also be
remarked
that the number of distinct modalities is no sure clue to whether two systems
a
ares translationally equivalent, For example, if one adds the [4] axiom to T. yielding S4v
theoresulting system has 14 irreducible modalities. Yet it is translationally equivalent to
thensystem gotten by adding the [4] axiom (with appropriate subscript) to V, and this
system has only 6 irreducible modalties.
l
y
a
f
i
n
i
t
e
n
u
434
F
.
.1. PELLETIER
theorem everything that the native speaker of V can assert as a
theorem.)(
5
University
o f Alberta
Francis Jeffry PELLETIER
)
Dpt. of Philosophy
Edmonton
Canada T6G2E5
(
paper.
5 The work reported in this paper was supported in part by a Canadian NSERC
grant
) (A5525), and in part by grant RO 245/12 from Deutsche Forschungs Gemeinschaft
under
T the auspices of Christian Rohrer of Stuttgart. My thanks to Professors Rohrer and
Franz
h Guenthner (of Tilhingen) for their hospitality during the summer of 1983.
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