<s>
In	O
mathematical	O
logic	O
,	O
monoidal	B-General_Concept
t-norm	I-General_Concept
based	I-General_Concept
logic	I-General_Concept
(	O
or	O
shortly	O
MTL	B-General_Concept
)	O
,	O
the	O
logic	O
of	O
left-continuous	O
t-norms	B-General_Concept
,	O
is	O
one	O
of	O
the	O
t-norm	B-General_Concept
fuzzy	I-General_Concept
logics	I-General_Concept
.	O
</s>
<s>
T-norms	B-General_Concept
are	O
binary	O
functions	O
on	O
the	O
real	O
unit	O
interval	O
[0,1],	O
which	O
in	O
fuzzy	O
logic	O
are	O
often	O
used	O
to	O
represent	O
a	O
conjunction	O
connective	O
;	O
if	O
are	O
the	O
confidences	O
we	O
ascribe	O
to	O
the	O
statements	O
and	O
respectively	O
,	O
then	O
one	O
uses	O
a	O
t-norm	B-General_Concept
to	O
calculate	O
the	O
confidence	O
ascribed	O
to	O
the	O
compound	O
statement	O
‘	O
and	O
’	O
.	O
</s>
<s>
Another	O
way	O
of	O
putting	O
it	O
is	O
that	O
the	O
t-norm	B-General_Concept
can	O
only	O
take	O
into	O
account	O
the	O
confidences	O
as	O
numbers	O
,	O
not	O
the	O
reasons	O
that	O
may	O
be	O
behind	O
ascribing	O
those	O
confidences	O
;	O
thus	O
it	O
cannot	O
treat	O
‘	O
and	O
’	O
differently	O
from	O
‘	O
and	O
,	O
where	O
we	O
are	O
equally	O
confident	O
in	O
both’	O
.	O
</s>
<s>
A	O
confidence	O
counterpart	O
of	O
the	O
implication	O
connective	O
may	O
in	O
fact	O
be	O
defined	O
directly	O
as	O
the	O
residuum	O
of	O
the	O
t-norm	B-General_Concept
.	O
</s>
<s>
For	O
a	O
general	O
t-norm	B-General_Concept
there	O
remains	O
the	O
possibility	O
that	O
has	O
a	O
jump	O
discontinuity	O
at	O
,	O
in	O
which	O
case	O
could	O
come	O
out	O
strictly	O
larger	O
than	O
even	O
though	O
is	O
defined	O
as	O
the	O
least	O
upper	O
bound	O
of	O
s	O
satisfying	O
;	O
to	O
prevent	O
that	O
and	O
have	O
the	O
construction	O
work	O
as	O
expected	O
,	O
we	O
require	O
that	O
the	O
t-norm	B-General_Concept
is	O
left-continuous	O
.	O
</s>
<s>
The	O
residuum	O
of	O
a	O
left-continuous	O
t-norm	B-General_Concept
thus	O
can	O
be	O
characterized	O
as	O
the	O
weakest	O
function	O
that	O
makes	O
the	O
fuzzy	O
modus	O
ponens	O
valid	O
,	O
which	O
makes	O
it	O
a	O
suitable	O
truth	O
function	O
for	O
implication	O
in	O
fuzzy	O
logic	O
.	O
</s>
<s>
Left-continuity	O
of	O
the	O
t-norm	B-General_Concept
is	O
the	O
necessary	O
and	O
sufficient	O
condition	O
for	O
this	O
relationship	O
between	O
a	O
t-norm	B-General_Concept
conjunction	O
and	O
its	O
residual	O
implication	O
to	O
hold	O
.	O
</s>
<s>
Truth	O
functions	O
of	O
further	O
propositional	O
connectives	O
can	O
be	O
defined	O
by	O
means	O
of	O
the	O
t-norm	B-General_Concept
and	O
its	O
residuum	O
,	O
for	O
instance	O
the	O
residual	O
negation	O
In	O
this	O
way	O
,	O
the	O
left-continuous	O
t-norm	B-General_Concept
,	O
its	O
residuum	O
,	O
and	O
the	O
truth	O
functions	O
of	O
additional	O
propositional	O
connectives	O
(	O
see	O
the	O
section	O
Standard	O
semantics	O
below	O
)	O
determine	O
the	O
truth	O
values	O
of	O
complex	O
propositional	O
formulae	O
in	O
 [ 0 , 1 ] 	O
.	O
</s>
<s>
Formulae	O
that	O
always	O
evaluate	O
to	O
1	O
are	O
then	O
called	O
tautologies	O
with	O
respect	O
to	O
the	O
given	O
left-continuous	O
t-norm	B-General_Concept
or	O
tautologies	O
.	O
</s>
<s>
The	O
set	O
of	O
all	O
tautologies	O
is	O
called	O
the	O
logic	O
of	O
the	O
t-norm	B-General_Concept
since	O
these	O
formulae	O
represent	O
the	O
laws	O
of	O
fuzzy	O
logic	O
(	O
determined	O
by	O
the	O
t-norm	B-General_Concept
)	O
that	O
hold	O
(	O
to	O
degree	O
1	O
)	O
regardless	O
of	O
the	O
truth	O
degrees	O
of	O
atomic	O
formulae	O
.	O
</s>
<s>
Some	O
formulae	O
are	O
tautologies	O
with	O
respect	O
to	O
all	O
left-continuous	O
t-norms	B-General_Concept
:	O
they	O
represent	O
general	O
laws	O
of	O
propositional	O
fuzzy	O
logic	O
that	O
are	O
independent	O
of	O
the	O
choice	O
of	O
a	O
particular	O
left-continuous	O
t-norm	B-General_Concept
.	O
</s>
<s>
These	O
formulae	O
form	O
the	O
logic	O
MTL	B-General_Concept
,	O
which	O
can	O
thus	O
be	O
characterized	O
as	O
the	O
logic	O
of	O
left-continuous	O
t-norms	B-General_Concept
.	O
</s>
<s>
The	O
language	O
of	O
the	O
propositional	O
logic	O
MTL	B-General_Concept
consists	O
of	O
countably	O
many	O
propositional	O
variables	O
and	O
the	O
following	O
primitive	O
logical	O
connectives	O
:	O
</s>
<s>
Unlike	O
in	O
BL	B-General_Concept
and	O
stronger	O
fuzzy	O
logics	O
,	O
weak	O
conjunction	O
is	O
not	O
definable	O
in	O
MTL	B-General_Concept
and	O
has	O
to	O
be	O
included	O
among	O
the	O
primitive	O
connectives	O
.	O
</s>
<s>
Bottom	O
(	O
nullary	O
—	O
a	O
propositional	O
constant	O
;	O
or	O
are	O
common	O
alternative	O
tokens	O
and	O
zero	O
a	O
common	O
alternative	O
name	O
for	O
the	O
propositional	O
constant	O
(	O
as	O
the	O
constants	O
bottom	O
and	O
zero	O
of	O
substructural	O
logics	O
coincide	O
in	O
MTL	B-General_Concept
)	O
.	O
</s>
<s>
Well-formed	O
formulae	O
of	O
MTL	B-General_Concept
are	O
defined	O
as	O
usual	O
in	O
propositional	O
logics	O
.	O
</s>
<s>
A	O
Hilbert-style	O
deduction	O
system	O
for	O
MTL	B-General_Concept
has	O
been	O
introduced	O
by	O
Esteva	O
and	O
Godo	O
(	O
2001	O
)	O
.	O
</s>
<s>
The	O
traditional	O
numbering	O
of	O
axioms	O
,	O
given	O
in	O
the	O
left	O
column	O
,	O
is	O
derived	O
from	O
the	O
numbering	O
of	O
axioms	O
of	O
Hájek	O
's	O
basic	B-General_Concept
fuzzy	I-General_Concept
logic	I-General_Concept
BL	B-General_Concept
.	O
</s>
<s>
The	O
axioms	O
(	O
MTL4a	O
)	O
–	O
( MTL4c	O
)	O
replace	O
the	O
axiom	O
of	O
divisibility	O
(	O
BL4	O
)	O
of	O
BL	B-General_Concept
.	O
</s>
<s>
Like	O
in	O
other	O
propositional	O
t-norm	B-General_Concept
fuzzy	I-General_Concept
logics	I-General_Concept
,	O
algebraic	O
semantics	O
is	O
predominantly	O
used	O
for	O
MTL	B-General_Concept
,	O
with	O
three	O
main	O
classes	O
of	O
algebras	O
with	O
respect	O
to	O
which	O
the	O
logic	O
is	O
complete	O
:	O
</s>
<s>
Algebras	O
for	O
which	O
the	O
logic	O
MTL	B-General_Concept
is	O
sound	O
are	O
called	O
MTL-algebras	O
.	O
</s>
<s>
Important	O
examples	O
of	O
MTL	B-General_Concept
algebras	O
are	O
standard	O
MTL-algebras	O
on	O
the	O
real	O
unit	O
interval	O
 [ 0 , 1 ] 	O
.	O
</s>
<s>
Further	O
examples	O
include	O
all	O
Boolean	O
algebras	O
,	O
all	O
linear	O
Heyting	O
algebras	O
(	O
both	O
with	O
)	O
,	O
all	O
MV-algebras	O
,	O
all	O
BL-algebras	O
,	O
etc	O
.	O
</s>
<s>
Since	O
the	O
residuation	O
condition	O
can	O
equivalently	O
be	O
expressed	O
by	O
identities	O
,	O
MTL-algebras	O
form	O
a	O
variety	O
.	O
</s>
<s>
The	O
connectives	O
of	O
MTL	B-General_Concept
are	O
interpreted	O
in	O
MTL-algebras	O
as	O
follows	O
:	O
</s>
<s>
With	O
this	O
interpretation	O
of	O
connectives	O
,	O
any	O
evaluation	O
ev	O
of	O
propositional	O
variables	O
in	O
L	O
uniquely	O
extends	O
to	O
an	O
evaluation	O
e	O
of	O
all	O
well-formed	O
formulae	O
of	O
MTL	B-General_Concept
,	O
by	O
the	O
following	O
inductive	O
definition	O
(	O
which	O
generalizes	O
Tarski	O
's	O
truth	O
conditions	O
)	O
,	O
for	O
any	O
formulae	O
A	O
,	O
B	O
,	O
and	O
any	O
propositional	O
variable	O
p	O
:	O
</s>
<s>
A	O
formula	O
A	O
is	O
said	O
to	O
be	O
valid	O
in	O
an	O
MTL-algebra	O
L	O
if	O
it	O
is	O
fully	O
true	O
under	O
all	O
evaluations	O
in	O
L	O
,	O
that	O
is	O
,	O
if	O
e(A )	O
=	O
1	O
for	O
all	O
evaluations	O
e	O
in	O
L	O
.	O
Some	O
formulae	O
(	O
for	O
instance	O
,	O
p	O
→	O
p	O
)	O
are	O
valid	O
in	O
any	O
MTL-algebra	O
;	O
these	O
are	O
called	O
tautologies	O
of	O
MTL	B-General_Concept
.	O
</s>
<s>
The	O
notion	O
of	O
global	O
entailment	O
(	O
or	O
:	O
global	O
consequence	O
)	O
is	O
defined	O
for	O
MTL	B-General_Concept
as	O
follows	O
:	O
a	O
set	O
of	O
formulae	O
entails	O
a	O
formula	O
A	O
(	O
or	O
:	O
A	O
is	O
a	O
global	O
consequence	O
of	O
)	O
,	O
in	O
symbols	O
if	O
for	O
any	O
evaluation	O
e	O
in	O
any	O
MTL-algebra	O
,	O
whenever	O
e(B )	O
=	O
1	O
for	O
all	O
formulae	O
B	O
in	O
,	O
then	O
also	O
e(A )	O
=	O
1	O
.	O
</s>
<s>
Informally	O
,	O
the	O
global	O
consequence	O
relation	O
represents	O
the	O
transmission	O
of	O
full	O
truth	O
in	O
any	O
MTL-algebra	O
of	O
truth	O
values	O
.	O
</s>
<s>
The	O
logic	O
MTL	B-General_Concept
is	O
sound	O
and	O
complete	O
with	O
respect	O
to	O
the	O
class	O
of	O
all	O
MTL-algebras	O
(	O
Esteva	O
Godo	O
,	O
2001	O
)	O
:	O
</s>
<s>
A	O
formula	O
is	O
provable	O
in	O
MTL	B-General_Concept
if	O
and	O
only	O
if	O
it	O
is	O
valid	O
in	O
all	O
MTL-algebras	O
.	O
</s>
<s>
The	O
notion	O
of	O
MTL-algebra	O
is	O
in	O
fact	O
so	O
defined	O
that	O
MTL-algebras	O
form	O
the	O
class	O
of	O
all	O
algebras	O
for	O
which	O
the	O
logic	O
MTL	B-General_Concept
is	O
sound	O
.	O
</s>
<s>
A	O
formula	O
A	O
is	O
a	O
global	O
consequence	O
in	O
MTL	B-General_Concept
of	O
a	O
set	O
of	O
formulae	O
if	O
and	O
only	O
if	O
A	O
is	O
derivable	O
from	O
in	O
MTL	B-General_Concept
.	O
</s>
<s>
Like	O
algebras	O
for	O
other	O
fuzzy	O
logics	O
,	O
MTL-algebras	O
enjoy	O
the	O
following	O
linear	O
subdirect	O
decomposition	O
property	O
:	O
</s>
<s>
Every	O
MTL-algebra	O
is	O
a	O
subdirect	O
product	O
of	O
linearly	O
ordered	O
MTL-algebras	O
.	O
</s>
<s>
(	O
A	O
subdirect	O
product	O
is	O
a	O
subalgebra	O
of	O
the	O
direct	O
product	O
such	O
that	O
all	O
projection	O
maps	O
are	O
surjective	B-Algorithm
.	O
</s>
<s>
An	O
MTL-algebra	O
is	O
linearly	O
ordered	O
if	O
its	O
lattice	O
order	O
is	O
linear	O
.	O
)	O
</s>
<s>
In	O
consequence	O
of	O
the	O
linear	O
subdirect	O
decomposition	O
property	O
of	O
all	O
MTL-algebras	O
,	O
the	O
completeness	O
theorem	O
with	O
respect	O
to	O
linear	O
MTL-algebras	O
(	O
Esteva	O
Godo	O
,	O
2001	O
)	O
holds	O
:	O
</s>
<s>
A	O
formula	O
is	O
provable	O
in	O
MTL	B-General_Concept
if	O
and	O
only	O
if	O
it	O
is	O
valid	O
in	O
all	O
linear	O
MTL-algebras	O
.	O
</s>
<s>
A	O
formula	O
A	O
is	O
derivable	O
in	O
MTL	B-General_Concept
from	O
a	O
set	O
of	O
formulae	O
if	O
and	O
only	O
if	O
A	O
is	O
a	O
global	O
consequence	O
in	O
all	O
linear	O
MTL-algebras	O
of	O
.	O
</s>
<s>
Standard	O
are	O
called	O
those	O
MTL-algebras	O
whose	O
lattice	O
reduct	O
is	O
the	O
real	O
unit	O
interval	O
 [ 0 , 1 ] 	O
.	O
</s>
<s>
They	O
are	O
uniquely	O
determined	O
by	O
the	O
real-valued	O
function	O
that	O
interprets	O
strong	O
conjunction	O
,	O
which	O
can	O
be	O
any	O
left-continuous	O
t-norm	B-General_Concept
.	O
</s>
<s>
The	O
standard	O
MTL-algebra	O
determined	O
by	O
a	O
left-continuous	O
t-norm	B-General_Concept
is	O
usually	O
denoted	O
by	O
In	O
implication	O
is	O
represented	O
by	O
the	O
residuum	O
of	O
weak	O
conjunction	O
and	O
disjunction	O
respectively	O
by	O
the	O
minimum	O
and	O
maximum	O
,	O
and	O
the	O
truth	O
constants	O
zero	O
and	O
one	O
respectively	O
by	O
the	O
real	O
numbers	O
0	O
and	O
1	O
.	O
</s>
<s>
The	O
logic	O
MTL	B-General_Concept
is	O
complete	O
with	O
respect	O
to	O
standard	O
MTL-algebras	O
;	O
this	O
fact	O
is	O
expressed	O
by	O
the	O
standard	O
completeness	O
theorem	O
(	O
Jenei	O
Montagna	O
,	O
2002	O
)	O
:	O
</s>
<s>
A	O
formula	O
is	O
provable	O
in	O
MTL	B-General_Concept
if	O
and	O
only	O
if	O
it	O
is	O
valid	O
in	O
all	O
standard	O
MTL-algebras	O
.	O
</s>
<s>
Since	O
MTL	B-General_Concept
is	O
complete	O
with	O
respect	O
to	O
standard	O
MTL-algebras	O
,	O
which	O
are	O
determined	O
by	O
left-continuous	O
t-norms	B-General_Concept
,	O
MTL	B-General_Concept
is	O
often	O
referred	O
to	O
as	O
the	O
logic	O
of	O
left-continuous	O
t-norms	B-General_Concept
(	O
similarly	O
as	O
BL	B-General_Concept
is	O
the	O
logic	O
of	O
continuous	O
t-norms	B-General_Concept
)	O
.	O
</s>
