Logique A n a ly s e 143-144 (1993), 245-260
PROOF-THEORETICAL CONSIDERATIONS ABOUT THE LOGIC OF
EPISTEMIC INCONSISTENCY*
Ana Teresa de CASTRO MARTINSt and Tarcfsio H. C. PEQUENO
Abstract
The Logic o f Epistemic Inconsistency (LEI) provides an interesting
example o f a paraconsistent formal system from the proof theoretical
point o f view. Although cut elimination theorem fo r its sequent calculus presentation can only be proved with a restriction, this does not
damage its main desirable outcomes, that are still attained. Therefore,
consistency, subformula property and conservativeness o f logical
constants definitions are we ll preserved. Th e motivation f o r L E I
designing, it s sequent calculus presentation and mo st remarkable
theorems are presented. Instances of the unremovable cut-proof pattern
are analysed.
I. Introduction
The L o g ic o f Epistemic Inconsistency (LEI) is a paraconsistent lo g ic
designed to support reasoning under conditions of incomplete or unaccurate knowledge. In these circumstances, conclusions must be grasped on
the basis o f partial evidences. In the course o f reasoning, partially supported conflicting conclusions may emerge. This kind of situation asks for
a criterious consideration o f the total evidence available in order to solve
these conflicts but, it may eventually happen, the available knowledge not
being able to do so, making some o f these pairs o f conflicting partial
conclusions turn out as unremovable conflicts, being this the best that could
be possibly done under these circumstances.
It is our opinion that, when the term reasoning is used to express a more
general thinking process than mere deduction, as in the situation ju st
described, its role is not just the grasping of conclusions, as for deduction,
but to perform a more comprehensive analysis of the available knowledge.
* research partially supported by PICD/CAPES and CNN.
t PhD student at Federal University of Pernambuco
246
M
A
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& PEQUENO
As a consequence o f this disposition, the whole set o f emerging conclusions, which includes the unresovable conflicting pairs, should be assimilated and reasoned out by a suitable logic, able to do so without
provoking a logical collapse. The kind o f inconsistency so arising should
be called epistemic, by reflecting not an inconsistency in the state of affairs
itself but a deficiency in our knowledge about it [Pequeno & Buchsbaum
91].
Incomplete knowledge reasoning has been the topic o f study o f the so
called nonmonotonic reasoning in artificial intelligence, fo r more than a
decade now. The current attitude in the field has been to avoid the hazadous o f epistemic contradiction by taking a choice between two alternatives. One, which has been called credulous, consists in accepting a ll
conflicting conclusions by splitting them into self consistent subsets [Reiter
80]. The other, sometimes called skeptical, consists in simply rejecting all
those conclusions [McCarthy 80]. In [Pequeno 901, it is suggested to keep
the whole set of conclusions, no matter they possibly being contradictory,
in a single theory, treating them paraconsistently. L EI has been designed
for th is purpose and i t is intended to be used in combination wit h a
nonmonotonic logic called IDL (fo r Inconsistent Default Logic [Pequeno
90]) wh ich takes care o f the dynamics o f this kin d o f reasoning (the
incoming o f knowledge and the dismissing o f no lo n g e r supported
conclusions).
LEI logical stmcture also fits a general pattern o f situations in which a
plurality o f views on a same subject is involved. This is the case, fo r instance, when a same phenomenon is reported by many observers or, in
general, when it is observed in experiments under slightly varying conditions, being these variations undetected and/or unrecorded along the experiment, but anyway enough to affect it. Then, disagreement may also
occur, and again, by a lack of knowledge. This fact make of these situations
instances o f epistemic inconsistencies. It is curious that, in the first case,
epistemic inconsistencies seem to arise fro m the underdetermination o f
conclusions wh ile , in the second one, they seem to come f ro m th e ir
overdetermination. The designing of LEI semantics has been based on the
intuition coming from this last scenario, while its Hilbert style axiomatics
is intended to match the pattern o f reasoning upon incomplete knowledge
described before. LEI soundness and completeness establish an equivalence
between these two perspectives [Pequeno & Buchsbaum 91].
Reasoning upon partial/multiple knowledge is a central issue in many
Artificia l Intelligence applications and its automatization is certainly
profitable. Tableaux systems for L EI have been developed in [Corrêa &
Buchsbaum & Pequeno 93] and [Buchsbaum 91], and a sequent calculus in
[Martins & Pequeno 94]. Ou r aim here, is to discuss proof-theoretical
issues through L E I sequent calculus by stressing L E I metalogical prop-
PROOF-THEORETICAL CONSIDERATIONS A ND EPISTEMIC INCONSISTENCY 247
erties. In section 2, LEI sequent calculus — proved correct and complete
with respect to the given LEI axiomatics — and main L E I theorems are
presented. Proof-theoretical considerations are detailed in section 3 and
conclusions, as well as further developments, are pointed out at the end.
2. LEI sequent calculus and main theorems
LEI has been conceived as a maximal approximation of classical logic able
to support, without trivializing, epistemic contradictions arising from the
application of IDL (default) rules, which introduce defeasible conclusions.
To distinguish these conclusions fro m those irrevocable ones, a ? sign
marking the first ones is adopted in LEI. The intuitive meaning o f a ? is
'there are evidences to a ' o r '
references,
irrefutable, monotonic, formulas w i l l be denoted b y roman
a i s
capital
letters,
?-formulas
p l a u s i b
l e ' . by greek lower letters and sequence of formulas
by
greek
capital
letters.
F
o
r
i s attained i n L E I b y re je ctin g e x fa lso se q u itu r
t Paraconsistency
h
e
quodlibet,
s
a a -•>k (-,ce ---->
e p), to formulas suffixed b y ?. Technically, L E I
differs fro m da Costa paraconsistent calculi Ci [daCosta 74] in some key
o
f
aspects, mainly: first, L EI retains more classical theorems than Ci — a
m
e
t
remarkable example of this is the schema -, (a A -,a ) usually taken as an
a
inner expression of the non-contradiction law. Although rejected in Ci, it is
still a theorem in LEI. Second, a recursive semantics has been provided for
LEI. Whereas d e n o t e s negation in LEI, a paraconsistent negation, - a is
introduced a s a handy abbreviation f o r a ---> p A -ip , b e in g ' p ' a
propositional letter. - can be shown to wo rk as a classical, also called
strong negation in LEI.
LEI sequent rules are:
Structural Rules
Exchange
LXI-, a , P , r I- A
F, f i, a , r I- A
I- A , a , 13, A '
F
A , 0, a, A '
248
MARTINS & PEQUENO
Weakening
LW I - A
I", a , I- A
I' I- A
R
W
F I- a , A
Contradiction
LC I', a , a , I- A
F, a , I- A
I- a , a
IR C
I
a
,
A
I' I- a -4 1 3 , A
A RC
Fa?
A
'
Identity Rules
a I- a [Identity
A I- a , '?'
A F ' , a I- A '
Cut
F, F ' I- A , A '
A
A I- a , A r , a ? F A '
? Cu t
?F' I- A , A '
R
OperationalC
Rules
F - 0 ,
Question mark
a? I- A
Li
•
• F, a ? ? I- A
I- a A
R
,
?
IF I- a ? , A
F, a ? -31.3 ?I- A
I- a - 0 , A
? (a?-0?)? F A
a ? -0 ? , A
Negation
I- A , A
I', - A I- A
F, a , F A
F 1- a
,
A
R
2
?
PROOF-THEORETICAL CONSIDERATIONS AND EPISTEMIC INCONSISTENCY 249
Conjunction
a
A
F,
A
a AP
A
,
A
I', a A [3 A
A
a, A F ' F /3, A R
F, a A P , A , A ' A
Disjunction
LF, a I- A F . P I- A '
F,
R,
a
4
I- A , A '
F 1- a , A
F F 0, A
I- a y 0, A
F a v p, A
R
2
V
Implication
• a , A F
L
R
1
• F ' , a -4 3 A , A '
F
I- a - 0 , A
,
p
1 a ll variables are held constants and, either a is ?-closed or R2 ? and
'
R - areAnot applied
after the first time a occurs justified by being a
F
premise.
,
DoubleaNegation
l
a I- A F F - o t A
D
IT, I ,
a
,
A
Distribution
of ' ---,' over ' y '
A
F, A -43 A
LD,
F, - ( a v P) F A
,'
• - - i o t A -0, A p
LDA r , - ( a A P) F A
• 1-
Distribution of ' -, ' over ' A '
F,
1
A
I' I- - 0 4 ) , A - D 7
y
-0, A
✓ - ( a 4 ) , A RD/7
Distribution o f ' -, ' over
LD - I "'
a A -13
F, (a
0
)
F
A
F I- a A- p, A p
1-
A
-D-7+
250
M
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& PEQUENO
Distribution of ' —1' over '?'
LDT F, (
• 1 '-, --,(a ? ) I- A
a )
?
Distribution
of o v e r
I
LD,? a
-•
7
A ( a v f l ) ? I- A
(-,a)?, A
F ( a
?)
,
A
ROT
v
Classical
Negation
f 3
I
?
IF a )- ? ) ,
a
,
A
First Order A
Rules
A
R
-
Universal Quantification
'
L V
1 ' F, V xo t(x) i- A 1 '
1at is free for x in a (x )
2 x must not occur free in 1
( t
)
,Existential
A
Quantification
I
I- Vxot(x), A
IL3 a ( x )
F
F xce (x), A
IA 3xce(x) 1- A
I1
Ax must not occur free in F, A
2I t is free for x in a (x )
-F
Quantifiers
Equivalence
I
a
3x(--,a) F A
F 3 x ( - 1 a ) , A R —1
(- t )
—,(Vxa)
F
A
---1(Vxa), A
0
2
4
,
Vx (
F V x(--, a ), A R
)
A
1",
A
-F - , ( 3 x a ) , A
0
R
,
a
)
2
3
A
,
A
R
V
PROOF-THEORETICAL CONSIDERATIONS A ND EPISTEMIC INCONSISTENCY 251
Extended Rule for Classical Negation
Classical Negation
a, A
L eF, - a A
I "
F
R
-
,
a I- A
-
a, A e
LEI sequent rules are closely related t o L E I axiomatics presented in
[Pequeno & Buchsbaum 91]. Paraconsistency has been captured by restricting t o classical, ?-free formulas. Double negation rule, distribution rules of o v e r A, y and a n d quantifier equivalences are derivable
in classical logic but not in LEI and must be explicitly stated. Central focus
has been devoted to the new and peculiar ? operator. It has been defined by
operational rules but its structural aspects are also expressed by two new
rules: ?-Cut and ?-RC. The former states that: ' i f a comes from 13 and y
comes from a ? , then certainly y also comes from the stronger a , or from
its proof using T h e latter is closely related to the operational rule
R
premise
a deduction where R
2
2? Question
? w a smark rules reveal operational ?-features. ? states that fro m
aa
p pcan
l iextract
e d its
. plausibility a ? . The opposite is obviously not valid.
. you
Nevertheless,
a
weak
converse of R
?
additional
L2 ? expresses that external ?'s are irrelevant
I—?
—?'s are superfluous.
?
to
an implication
when
both antecedent and succedent are ?-closed (see
—
i
s
R
definition
vC a l 2.2i below).
d . R2 ? provides the propagation of ? along inferences
when
used
in
conjunction
with L —) to simulate a sort of ?-Modus Ponens
?r
(a?,a--->
13
/
?
)
as
in
the
se
t
a
t
e proof
s below:
ts
h
a
t
T
t
T2
o
2a
F
r
h— r
e 1
a>
l
s I
t
?
The
same
is
achieved
in
LEI
, axiomatics using Modus Ponens and rule
t
3
—
a
ha -4 S
r
a
as in the>following:
?
ea ? -4
H
0?
0?
s F
p
?a
a 2
?
a?
m ,
0
?
e a
-?
c ?
4H
o I
1p
n 3?
0
t
?
e ?
,
x
a
t
?
I
H
252
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a —> f3
a? a ? - 0 ?
P?
Distribution o f ? over y is assured by L D . Its symmetrical RD:, is derived. R- states that it is impossible to be sure about a and, at the same
time, t o have an evidence to it s classical negation ( - a )? A theory
including a and ( - a)? is trivia l in LEI. Notice the double line at rules
R
R
2 - d o not hold constant the variables over evidences implicitly quantified
by
? ?. On the other hand, the imp lica tive fo rms o f R
2(a
? >p)—>(a?—>
a n d 0?)R a n -d a, —>-((i a)?)
. wo
e u ld
. ca rry o u t a ? —> a
a —
which is not reasonable.
n Main LEI theorems are:
d
R
Theorem
2.1. all thesis of classical logic hold to ?-free formulas in LEI.
I
Corollary
2.1. the negation symbol b e h a v e s classically for ?-free formulas
in LEI:
t
m
B) —> — > —B)—>
a
A
r
Theorem
k
2.2. The defined symbol - really behaves as classical negation:
s
(a p ) - ) ((a —>- a )
a a
a
r
Ae form of replacement theorem can be recovered with the help of a special
implication
introduced by definition [Pequeno & Buchsbaum 91].
s
t
Definition
2.1. (strong implication) a 0 is a short form for (a —> f3) A
r
i -96 - › —,a a n d (strong double implication) a <=> p is a short form fo r
ca1 0 / 3 ) A - - t— , a )
Theorem
2.3. (replacement) Let a ' be a formula obtained fro m a b y
i
substituting
13 ' fo r some occurrences (not necessarily a ll) o f 13 . Then
o
I
-n
Definition
2.2. (?-closed formula) A formula is ?-closed, i.e., it is under the
Io
vscope
of ?, if it has one of the forms a?,
p, 0 4
are
e
?-closed
formulas
and
/
1
4
,
A
,
V1
,0 y , w h e r e
a
n
d
r
y0
'R
—
e
>
n
ta
p
a
p
i
ll
PROOF-THEORETICAL CONSIDERATIONS A ND EPISTEMIC INCONSISTENCY 253
Theorem 2.4. I f a is a ?-closed formula, then F- a •4 a ? .
A restricted version of the deduction theorem can be kept also.
Theorem 2.5. I f F, a F p and if does not occur any free variable in the
set of premisses, then F F a —>13 unless a is not ?-closed and the L EI
rules R
of
2 being a premise.
?
o r
3. Proof-theoretical issues about LEI
R
—
Cut elimination theorem was established by Gentzen f o r classical and
a
intuitionistic logics and assures that every derivation using sequent calculus
r be transformed into another one with the same endsequent, in which no
can
e occur [Gentzen 35 ]. Prawitz's No rma l Fo rm theorems assert a n
cuts
u
equivalent
result but fo r natural deduction calculus [Prawitz 711. Proofs
theoretical
issues concerned cut-elimination and normalization theorems in
LEI
e will be now discussed.
d Cut elimination in LEI, established in [Martins & Pequeno 94], was
proved
a
with a restriction: some proofs where R
2f
propagation
of ? along ---> to simulate ?-Modus Ponens (a?, a —> p ip ? ).
In
Gentzen transformation o f his Natural
?t fact,
i s appealing
a p p lt oi history,
e d
aDeduction
e i m calculi
i n NgJ and NK , f o r intuitionistic and classical lo g ics
respectively, to his correspondent Logistic Sequent Calculi L I and L K had
tr
h
e
requested the use of cut rule to effectively translate elimination rules to left
tones [Ungar 92]. I t will enforce our argument below in favour o f cut
h
maintenance on such proofs. To illustrate this, consider, as a typical
e
example,
the following proof:
f
i a Fo t
1
r a 3l - a v f i
1
3
1
a v1 3
s 'P
I- a -> v a ? F o t ? ( a v f l ) ? 1 - ( a v ) ? F / 3 - - - ) ( a v i 3 ) f i ? F f l ? ( a v P ) ? i ( a v P ) ?
t
t a? - + ( a v p )? a ? , a ? --)(cx v1 3 ) ? i( a v1 3 ) ? 1 3 ? - ) ( a v1 3 )? 1 3 ? , 1 3 ? --(a v p ) ? F ( a v f i)?
a ? Eja vf i) ?
1
3
?
'
(
a
v
t
r
!
)
i
a ? v( PF (a vp )'? . ( a vt 3 ) ?
m
a ? v p?1- ( c t v 0 ) ?
e
ot?,/ )5? -* v P)?
a
o
c
c
u
r
s
i
n
a
p
254
M
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P E Q UE NO
I f we translate this example in a natural deduction style we get (vide in
[Martins & Pequeno 94] the straightforward derivation o f LEI natural
deduction rules from LEI sequent ones):
1. [ a ? v 13?1
2.
[
a
3.
[
a
4.
a
35.
a
6.
a
7.
(
2 ? ( 5 a)
8. L /
9.
L P
3
lo.
a v
1
?]
]
v)
- ' (a v p)
I
? ( a v p)? 1
v0 )?
E
]
—› (3, 4 )
—› (2, 6 )
p
11. )
12.
a ?
—› (a v 13)?
6
13.
(
a
v
fi)?
— ›
14. ( a v 1
(15. a ? \ / 0 ? ( a v f i ) ?
3 )?
a
In
v this translation, note that we do not have a characterization o f a redundant
inference. In fact, in step 5, fo r instance, we have introduced an
P
implication
that was subsequently eliminated, but only in step 7. We have
)
an intermediate step (the application of 1
permuted
in order to explicit this candidate to redundancy. This proof is
2
indeed
in
normal
? o r
r u lform:
e the
R analytical part corresponds to y elimination —
steps
l
to
14
—
while
the
synthetical part is the -4 introduction — step 15
2
—
and
the
minimum
part
? )
w
h
i
c is the
h conclusion o f the analytical part and the
premise
of
the
synthetical
one,
i.e. step 14. No reduction steps (0 , n ,
c
a
n
n
o
t
can
be
applied
in
order
to
get
rid
o ff redundances. In fact, Normal Form
b
e
theorem was established in LEI [Martins & Pequeno 941 as in classical
logic just adding new )
Nevertheless,
nonadditional Ç reductions were necessary.
3 a n d
Backing
to
the
sequent calculus proof, we can easily see
r e d u c t i correspondent
o
that
ru
le
R
n s
between
2
t
othe elimination of —> through the cut rule and the introduction of
through
Ra r u l e , we have an application o f R
?
d
e
rewriting
process
behind cut elimination, there is no way to be free of such
rl e 2a l l
intermediate
rule
by
permuting
since
this rule cannot be broken
i n g cut upward,
t
o
yw ? . i A p p
t e a ltheorem
through
the
deduction
represented
by
R
---> I n fact, R—> is also
e
d
it s h
h
restricted to cases analogous to the ones related to quantifiers where either
a
l ul o e
q
the discharged
premise a is ?-closed or rules R
w
s2
t
after
the
f iirst toime a appears as a premise. I t is a consequence o f
tn
? a n d
R a
r e
h
m
at
n
o
e
ra p pk l i e d
cr h au
rl a ce
ts e r .
i z a
PROOF-THEORETICAL CONSIDERATIONS A ND EPISTEMIC INCONSISTENCY 255
interpreting ? as a sort of existential quantification of hidden variables over
evidences. Hence, if the discharged premise a is not under the scope of ?
( a is not ?-closed), the application o f R
constant
such ?-quantified hidden variables.
2
we shall show that the main desirable consequences
? Toaend
n this
d section,
R
of
will be kept
even though we have maintained some
- cut-elimination
t
o
a
special
occurrences
of
cuts
in
proofs.
Such cut-proofs can be framed in the
w
i
l
l
following pattern:
n
o
t
h
o
l
d
1
F1
ta
13
I4
The desirable
outcomes
1
Fof cut-elimination are:
3
A
a,
1. Consistency
o f thisl calculus: even i f F
A
?
never prove the empty sequent F-- since
1 above
aI n dproof,
A we would
,
-r
e
x
i
s
t
a
t
l
e
a
st 7
I always
w
e
r
e
A
02HmFI- y, p8 o tr y, y5 1
or
e E
2
2-?a w hnruelers et o id, e n t it y a x io ms u p p e r t o F
o tional
?
F
y, , AI h,
U
t 2
e
A
-23
8AA a a ? ,n
d A3
2
2. Subformula
the following example:
2
amProperty:
n e observe
d
3,c1F o
0
eF3
T
a r?
f a,
r
o1
3
3 2)
a At irp
Im( t
A
hU
i
,- I
r ( a A3
tl 3 ) - ) a ah? 1
F?
A
F
e,a,0 A) ?/- )3a )? ?( a1 A. 0 ) ? - o a ? , ( a A 1 3 ) ? 1 - a ? A , 0 ) ? a (?a A
(S
3
4
- 0 ?t3
aI ( a pA 0 )(pa?.Ap- )4l? 11- a3? ? j (a A c1 3 A) ? I 3 ? ) ? i ft?
I( a- A
1 ,/3- 3 ?) ?
ry c
i
a
t
(a A P)?, ( a A 13)? r a ? " 0 ?
A
1
(
)
F
(
0
(
In
)
ia3 o
Ie
as
5o? pA ?t
p
t f- 3
?
ae
o0 A .
)
c? ?A r1
ae
this
first
? ta
I
n
i
-
it will
( a AP) , ? r a ? A f l ?
( a A/ 3 ) ? - * a ? AP?
p
r3
example
we have two occurrences o f cut rule. Consider, fo r
instance, the one where the cuted formula is ( a AO)? i
such
> p ?-formula
.
N is
o the
t eimplicative
t h form
a otf the formulas which occur in the
v
sequent
A conclusion (i.e., ( a A13)? I
- 1 e23 ? )
o f
t h l, i s
r uyA l e .
.
3
256
M
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-a
F13
a
Af il - a
a A3 1 -
-
A 13) -+ a
N
S
& PEQUENO
a ? l-a ? 1 3 ? 1 -p ?
A1 3 )? 1 -(a At i)? (a A# )? 1 -(a AP) ? a ? , f 3 ? 1 -a ? Af i?
(a A
J3 )
Ap )-4 1 3 A / 3 ) ' - * a ?
? ( a A p ) ? - ) a ? , ( a AP) ? - 3 ? , A f i ) ? F - a ? A p ?
Af3)?-)13?, ( a A p )? f r a ?A/3?
I
* a
(
? .
a
A
(
)
( a A I3)?
a F a 13 ?
3
)
( a AA0 ? - > a ? A p ?
?
f
?
l
)
This second example has the
? same endsequent but the p ro o f tree is
somehow different. Now, the -›same cuted formula ( a A0)?--> 0 ? is not the
implicative f o rm o f ( a A P)?
However,
? h
athe
? consequent
A 1 3 ? of
, , a A p)? f i ? , i. e . 0 ?, can be obtained by
(
application
of
R
t
h
e
a
it
what
in the first Aproof. Hence, whenever the cuted formula is
A
rt
c is o
uv ehappens
p
not
in the cut sequent conclusion, you can
ae implicative
s( the
q
u form
e of formulas
) rewriting steps of cut-elimination [Martins &
rewrite
the
proof
applying
the
A
n
t
?
Pequeno
94]
in order to accomplish
this constraint.
,
p
c
o)
n?
c
l
u
(
The
last
example
is
the
following,
where we consider as hypothesis
Is
- o
a
a <,>i a and
[3 n<,:, /3.?:
A
a
?
1
A a?1- a ?
3
p a?A/3?1- a ?
)
a
?
F
a?
?FGe ?A /3 ?) -4 a ? ( a ?A ? ) ( a -a?A )9?)? a ? ? I- a ?
b
e
?
1 - - ( a ? A 0 7 ) ? - ? ? ( a ? At i ? ) ? - - ) a ? A
? , ( a ? Al 3 ? ) ? 1 - a ?
f
o
(a?,\,(3?)?F af ?
(
a
?A )3?)?1- )3?
r
e
i
(cr?AP?)?, (a ?? A/1 ? )? 1 - a ? Ali?
a
p
(a ? Ap ? )?
p
l
1
(a A fl)?1- a Af i
y
i
o c 7 A
1- (3a ?A 13)? A a A p )
n
g
,
c
u1. u sin g replacement theorem (see theorem 2.3), and the hypothesis
' a •::,> a ?' and 'f3 f i ?'
t
2.
t
h
e same proof of (a ?A p?)?1- a?
.
A
c
In this example, the cuted formula is (a?Ap?)?—> a?? and the cut setquent conclusion
u
is (a ? A P?)?1- a?. Although the former is not the ima
l fo rm o f the latter, you can apply the replacement theorem to
plicative
lsatisfyythis restriction since we have a ? ? <,> a? (see theorem 2.4).
,
PROOF-THEORETICAL CONSIDERATIONS A ND EPISTEMIC INCONSISTENCY 257
To sum up, subformula property is somehow modified to the following:
'all formulas that appear in proofs where the only remaining possible cuts
are those pointed above are either subformulas (or rewriting formulas) o f
those wh ich o ccu r i n th e endsequent o r a re imp lica t ive f o rms o f
subformulas (or rewriting formulas) o f them. Rewriting formulas are those
obtained through distribution rules or by the replacement theorem applied
to strong equivalences'.
3. Inversion Principle: R
2 not have a le ft (operational) ru le to confront with . Indeed, f ro m
it does not make sense to eliminate ? in order to get a —› •
? i s?—> Pa? , n
a nTherefore,
o m a we
l really
o u do not want an elimination rule to deal with such
s case. Thus, the inversion principle, as stated in classical logic, holds to
r allu LEI llogical
e constants but to ?. It does not carry out any dangerous
results. Th e conservative property, the main related result o f the
i
n
inversion principle, still remains by the use of ?-RC (see below) which
t couldh be thought
e as the 'symmetrical' rule of R
s 2 e
n
s4. Conservative
e
Property: prima facie, it would seem that ? is not con?.
t servative
h since we have an ?-rule ( R
a In
t we do not have an operational ?-rule 'to destroy' the effect of
2 fact,
introducing
i ? ) w i t h? over
o u —>,
t i.e.,
t a
n
y
d s y m
o ma-401-.6,
e t r i c a l
1.
e o
sn a? —on
I",
e
-A.
3
?
In truth, this rule will allow to prove (a ? —) )6?) --)(a p ) , and it does
not make sense. Nevertheless, ? is s t ill a conservative operator but
characterized somehow differently fro m Gentzen operators. In Gentzen
(see [Gentzen 35] and [Prawitz 71]), each logical constant is defined
locally by introduction and elimination rules in his natural deduction calculi NJ and NK or by right and left operational rules in his logistic sequent
calculi L i and L K. The introduction o f any lo g ica l symbol does not
presume a premise with a fixed pattern as the implicative formula a —>fl
in the case o f R
logical
symbols, with the exception of ?, obey Gentzen style.
2
? It urges
a n dthe necessity o f characterizing ? properly. We have done it not
only
by operational ?-rules but also by ?-RC and ?-Cut rule. ?-RC and ?a
?
Cut
reflect
a sort of structural property, an ordering of credibility or plau—
›
sibility,
associated
to ?-formulas. Thus, a —)/3 ma y be considered as
[
3
?
stronger than a ? ---)P ? in ?-RC and a stronger than a ? in ?-Cut. Ergo,
i
n
the definition o f the new logical constant ? is characterized not only by
t
h
e
c
a
s
e
o
f
L
258
M
A
R
T
I
N
S
& PEQUENO
(question mark) operational rules as in standard Gentzen logical operators,
but also by the structural rule ?-RC and the identity rule ?-Cut.
To see why it is still a conservative definition, we have only to analyse if
the case where cut is preserved in the proof, i.e., when we use R
keep
property of ?, i.e., all we have introduced by means
2 ? the
, sconservative
t i l l
of application of R
2 Consider
? c athis
n rule:
b e
d i s I' c a h—43,
a A
R
r g 1
e d
2
?
b
- y
a we
n' have
o already said, it has not and could not have an operational
As
t
h1 e left rule 'to cancel' the introduction of ? on the right hand side
symmetrical
r F. This
of
cancellation
?
will be done through the structural rule ?-RC.
Iua
r
l
-?.
e 1"1-a?-3/3?,
A ?—
RIPC - a —q3, A
-?
Thus, a,the operator introduced b y R
2restore
? -Aon
c the
a nthird line
b thee same formula on the first line which was object
0
of
R
e l i m i n a t e d
,
2
b
y
?
AR
?
-4. Conclusions
C and Further
.
Works
a
W
e
p
The
p l Logic o f Epistemic Inconsistency is here presented through sequent
calculus
rules. I t is a remarkable example o f a formal system where a
i c
normal form theorem can be proved but not unrestricted cut elimination. It
a
hast been shown that a special cut-proof pattern cannot be eliminated in
iproofs
o
where R
n
.
2 ?
rewriting
i the
s proof in order to become free o ff cut application since R
cannot
a p? p be
2
l i'sp
e lit' into R-3 (the deduction theorem) due to existential in terpretation
o
f ? over evidences. Fortunately, the preservation o f this cutd .
proof
does
not
damage the desirable outcomes o f cut-elimination: 1 ) the
I
n
calculus
consistency
has still been maintained; 2 ) subformula property,
s
u
although slig h tly modified: th e cuted remaining fo rmu la has a n imc
h
plicational fo rm o f endsequent subformulas. Thus, formulas in a (quasi)
ccut-free
i r proof
c
is still predictable; 3) the inversion principle has not been
u
m
s
attested to ? tsince a symmetrical left operational rule to R
a
c oe e s
2 ?n d
n o t
m a
k
e
s ,
t
h
e
r
e
i
PROOF-THEORETICAL CONSIDERATIONS A ND EPISTEMIC INCONSISTENCY 259
sense, but 4) the conservativeness of all logical constants has been assured,
even for ? where ?-RC plays the role of symmetrical rule to R
2 As
? future work, we aim to extend sequent calculus treatment to those
dynamic aspects of incomplete knowledge reasoning covered by IDL. The
idea is to get an uniform formalization encompassing all the aspects of this
kind of reasoning.
Laborat6rio de inteligencia artificial
Departamento de computa0o
Universidade Federal do Ceará
P.O.Box 12166, Fortaleza, CE 60455-760 Brasil
e-mail: ana@ lia.ufc.br
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1993.
da Costa, N.C.A. 'On the Theory o f Inconsistent Formal Systems.' Notre
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M
A
R
T
I
N
S
& PEQUENO
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