<s>
In	O
mathematics	O
and	O
philosophy	O
,	O
Łukasiewicz	B-General_Concept
logic	I-General_Concept
(	O
,	O
)	O
is	O
a	O
non-classical	O
,	O
many-valued	O
logic	O
.	O
</s>
<s>
It	O
was	O
originally	O
defined	O
in	O
the	O
early	O
20th	O
century	O
by	O
Jan	O
Łukasiewicz	O
as	O
a	O
three-valued	B-Language
modal	O
logic	O
;	O
it	O
was	O
later	O
generalized	O
to	O
n-valued	O
(	O
for	O
all	O
finite	O
n	O
)	O
as	O
well	O
as	O
infinitely-many-valued	O
(	O
ℵ0-valued	O
)	O
variants	O
,	O
both	O
propositional	O
and	O
first	O
order	O
.	O
</s>
<s>
It	O
belongs	O
to	O
the	O
classes	O
of	O
t-norm	B-General_Concept
fuzzy	I-General_Concept
logics	I-General_Concept
and	O
substructural	O
logics	O
.	O
</s>
<s>
Łukasiewicz	B-General_Concept
logic	I-General_Concept
was	O
motivated	O
by	O
Aristotle	O
's	O
suggestion	O
that	O
bivalent	O
logic	O
was	O
not	O
applicable	O
to	O
future	O
contingents	O
,	O
e.g.	O
</s>
<s>
For	O
an	O
elementary	O
introduction	O
to	O
the	O
three-valued	B-Language
instantiation	O
Ł3	O
,	O
see	O
three-valued	B-Language
logic	I-Language
.	O
</s>
<s>
The	O
propositional	O
connectives	O
of	O
Łukasiewicz	B-General_Concept
logic	I-General_Concept
are	O
(	O
"	O
implication	O
"	O
)	O
,	O
and	O
the	O
constant	O
(	O
"	O
false	O
"	O
)	O
.	O
</s>
<s>
The	O
original	O
system	O
of	O
axioms	O
for	O
propositional	O
infinite-valued	O
Łukasiewicz	B-General_Concept
logic	I-General_Concept
used	O
implication	O
and	O
negation	O
as	O
the	O
primitive	O
connectives	O
,	O
along	O
with	O
modus	O
ponens	O
:	O
</s>
<s>
Propositional	O
infinite-valued	O
Łukasiewicz	B-General_Concept
logic	I-General_Concept
can	O
also	O
be	O
axiomatized	O
by	O
adding	O
the	O
following	O
axioms	O
to	O
the	O
axiomatic	O
system	O
of	O
monoidal	B-General_Concept
t-norm	I-General_Concept
logic	I-General_Concept
:	O
</s>
<s>
That	O
is	O
,	O
infinite-valued	O
Łukasiewicz	B-General_Concept
logic	I-General_Concept
arises	O
by	O
adding	O
the	O
axiom	O
of	O
double	O
negation	O
to	O
basic	B-General_Concept
fuzzy	I-General_Concept
logic	I-General_Concept
(	O
BL	B-General_Concept
)	O
,	O
or	O
by	O
adding	O
the	O
axiom	O
of	O
divisibility	O
to	O
the	O
logic	O
IMTL	O
.	O
</s>
<s>
Finite-valued	O
Łukasiewicz	B-General_Concept
logics	I-General_Concept
require	O
additional	O
axioms	O
.	O
</s>
<s>
A	O
hypersequent	O
calculus	O
for	O
three-valued	B-Language
Łukasiewicz	B-General_Concept
logic	I-General_Concept
was	O
introduced	O
by	O
Arnon	O
Avron	O
in	O
1991	O
.	O
</s>
<s>
Sequent	O
calculi	O
for	O
finite	O
and	O
infinite-valued	O
Łukasiewicz	B-General_Concept
logics	I-General_Concept
as	O
an	O
extension	O
of	O
linear	O
logic	O
were	O
introduced	O
by	O
A	O
.	O
Prijatelj	O
in	O
1994	O
.	O
</s>
<s>
Hypersequent	O
calculi	O
for	O
Łukasiewicz	B-General_Concept
logics	I-General_Concept
were	O
introduced	O
by	O
A	O
.	O
Ciabattoni	O
et	O
al	O
in	O
1999	O
.	O
</s>
<s>
Infinite-valued	O
Łukasiewicz	B-General_Concept
logic	I-General_Concept
is	O
a	O
real-valued	O
logic	O
in	O
which	O
sentences	O
from	O
sentential	O
calculus	O
may	O
be	O
assigned	O
a	O
truth	O
value	O
of	O
not	O
only	O
0	O
or	O
1	O
but	O
also	O
any	O
real	O
number	O
in	O
between	O
(	O
e.g.	O
</s>
<s>
The	O
truth	O
function	O
of	O
strong	O
conjunction	O
is	O
the	O
Łukasiewicz	O
t-norm	B-General_Concept
and	O
the	O
truth	O
function	O
of	O
strong	O
disjunction	O
is	O
its	O
dual	O
t-conorm	B-General_Concept
.	O
</s>
<s>
The	O
truth	O
function	O
is	O
the	O
residuum	O
of	O
the	O
Łukasiewicz	O
t-norm	B-General_Concept
.	O
</s>
<s>
By	O
definition	O
,	O
a	O
formula	O
is	O
a	O
tautology	O
of	O
infinite-valued	O
Łukasiewicz	B-General_Concept
logic	I-General_Concept
if	O
it	O
evaluates	O
to	O
1	O
under	O
each	O
valuation	O
of	O
propositional	O
variables	O
by	O
real	O
numbers	O
in	O
the	O
interval	O
 [ 0 , 1 ] 	O
.	O
</s>
<s>
The	O
standard	O
real-valued	O
semantics	O
determined	O
by	O
the	O
Łukasiewicz	O
t-norm	B-General_Concept
is	O
not	O
the	O
only	O
possible	O
semantics	O
of	O
Łukasiewicz	B-General_Concept
logic	I-General_Concept
.	O
</s>
<s>
General	O
algebraic	O
semantics	O
of	O
propositional	O
infinite-valued	O
Łukasiewicz	B-General_Concept
logic	I-General_Concept
is	O
formed	O
by	O
the	O
class	O
of	O
all	O
MV-algebras	O
.	O
</s>
<s>
Like	O
other	O
t-norm	B-General_Concept
fuzzy	I-General_Concept
logics	I-General_Concept
,	O
propositional	O
infinite-valued	O
Łukasiewicz	B-General_Concept
logic	I-General_Concept
enjoys	O
completeness	O
with	O
respect	O
to	O
the	O
class	O
of	O
all	O
algebras	O
for	O
which	O
the	O
logic	O
is	O
sound	O
(	O
that	O
is	O
,	O
MV-algebras	O
)	O
as	O
well	O
as	O
with	O
respect	O
to	O
only	O
linear	O
ones	O
.	O
</s>
<s>
Font	O
,	O
Rodriguez	O
and	O
Torrens	O
introduced	O
in	O
1984	O
the	O
Wajsberg	O
algebra	O
as	O
an	O
alternative	O
model	O
for	O
the	O
infinite-valued	O
Łukasiewicz	B-General_Concept
logic	I-General_Concept
.	O
</s>
<s>
A	O
1940s	O
attempt	O
by	O
Grigore	O
Moisil	O
to	O
provide	O
algebraic	O
semantics	O
for	O
the	O
n-valued	O
Łukasiewicz	B-General_Concept
logic	I-General_Concept
by	O
means	O
of	O
his	O
Łukasiewicz	O
–	O
Moisil	O
(	O
LM	O
)	O
algebra	O
(	O
which	O
Moisil	O
called	O
Łukasiewicz	O
algebras	O
)	O
turned	O
out	O
to	O
be	O
an	O
incorrect	O
model	O
for	O
n	O
≥	O
5	O
.	O
</s>
<s>
C	O
.	O
C	O
.	O
Chang	O
's	O
MV-algebra	O
,	O
which	O
is	O
a	O
model	O
for	O
the	O
ℵ0-valued	O
(	O
infinitely-many-valued	O
)	O
Łukasiewicz	B-General_Concept
–	I-General_Concept
Tarski	I-General_Concept
logic	I-General_Concept
,	O
was	O
published	O
in	O
1958	O
.	O
</s>
<s>
For	O
the	O
axiomatically	O
more	O
complicated	O
(	O
finite	O
)	O
n-valued	O
Łukasiewicz	B-General_Concept
logics	I-General_Concept
,	O
suitable	O
algebras	O
were	O
published	O
in	O
1977	O
by	O
Revaz	O
Grigolia	O
and	O
called	O
MVn-algebras	O
.	O
</s>
<s>
In	O
1982	O
Roberto	O
Cignoli	O
published	O
some	O
additional	O
constraints	O
that	O
added	O
to	O
LMn-algebras	O
produce	O
proper	O
models	O
for	O
n-valued	O
Łukasiewicz	B-General_Concept
logic	I-General_Concept
;	O
Cignoli	O
called	O
his	O
discovery	O
proper	O
Łukasiewicz	O
algebras	O
.	O
</s>
<s>
Łukasiewicz	B-General_Concept
logics	I-General_Concept
are	O
co-NP	O
complete	O
.	O
</s>
<s>
Łukasiewicz	B-General_Concept
logics	I-General_Concept
can	O
be	O
seen	O
as	O
modal	O
logics	O
using	O
the	O
defined	O
operators	O
,	O
</s>
