<s>
T-norm	B-General_Concept
fuzzy	I-General_Concept
logics	I-General_Concept
are	O
a	O
family	O
of	O
non-classical	O
logics	O
,	O
informally	O
delimited	O
by	O
having	O
a	O
semantics	B-Application
that	O
takes	O
the	O
real	O
unit	O
interval	O
 [ 0 , 1 ] 	O
for	O
the	O
system	O
of	O
truth	O
values	O
and	O
functions	O
called	O
t-norms	B-General_Concept
for	O
permissible	O
interpretations	O
of	O
conjunction	O
.	O
</s>
<s>
T-norm	B-General_Concept
fuzzy	I-General_Concept
logics	I-General_Concept
belong	O
in	O
broader	O
classes	O
of	O
fuzzy	O
logics	O
and	O
many-valued	O
logics	O
.	O
</s>
<s>
In	O
order	O
to	O
generate	O
a	O
well-behaved	O
implication	O
,	O
the	O
t-norms	B-General_Concept
are	O
usually	O
required	O
to	O
be	O
left-continuous	O
;	O
logics	O
of	O
left-continuous	O
t-norms	B-General_Concept
further	O
belong	O
in	O
the	O
class	O
of	O
substructural	O
logics	O
,	O
among	O
which	O
they	O
are	O
marked	O
with	O
the	O
validity	O
of	O
the	O
law	O
of	O
prelinearity	O
,	O
(	O
AB	O
)	O
(	O
BA	O
)	O
.	O
</s>
<s>
Both	O
propositional	O
and	O
first-order	O
(	O
or	O
higher-order	B-Algorithm
)	O
t-norm	B-General_Concept
fuzzy	I-General_Concept
logics	I-General_Concept
,	O
as	O
well	O
as	O
their	O
expansions	O
by	O
modal	O
and	O
other	O
operators	O
,	O
are	O
studied	O
.	O
</s>
<s>
Logics	O
that	O
restrict	O
the	O
t-norm	B-General_Concept
semantics	B-Application
to	O
a	O
subset	O
of	O
the	O
real	O
unit	O
interval	O
(	O
for	O
example	O
,	O
finitely	O
valued	O
Łukasiewicz	B-General_Concept
logics	I-General_Concept
)	O
are	O
usually	O
included	O
in	O
the	O
class	O
as	O
well	O
.	O
</s>
<s>
Important	O
examples	O
of	O
t-norm	B-General_Concept
fuzzy	I-General_Concept
logics	I-General_Concept
are	O
monoidal	B-General_Concept
t-norm	I-General_Concept
logic	I-General_Concept
(	O
MTL	B-General_Concept
)	O
of	O
all	O
left-continuous	O
t-norms	B-General_Concept
,	O
basic	O
logic	O
(	O
BL	B-General_Concept
)	O
of	O
all	O
continuous	O
t-norms	B-General_Concept
,	O
product	O
fuzzy	O
logic	O
of	O
the	O
product	O
t-norm	B-General_Concept
,	O
or	O
the	O
nilpotent	O
minimum	O
logic	O
of	O
the	O
nilpotent	O
minimum	O
t-norm	B-General_Concept
.	O
</s>
<s>
Some	O
independently	O
motivated	O
logics	O
belong	O
among	O
t-norm	B-General_Concept
fuzzy	I-General_Concept
logics	I-General_Concept
,	O
too	O
,	O
for	O
example	O
Łukasiewicz	B-General_Concept
logic	I-General_Concept
(	O
which	O
is	O
the	O
logic	O
of	O
the	O
Łukasiewicz	O
t-norm	B-General_Concept
)	O
or	O
Gödel	B-Language
–	O
Dummett	O
logic	O
(	O
which	O
is	O
the	O
logic	O
of	O
the	O
minimum	O
t-norm	B-General_Concept
)	O
.	O
</s>
<s>
As	O
members	O
of	O
the	O
family	O
of	O
fuzzy	O
logics	O
,	O
t-norm	B-General_Concept
fuzzy	I-General_Concept
logics	I-General_Concept
primarily	O
aim	O
at	O
generalizing	O
classical	O
two-valued	O
logic	O
by	O
admitting	O
intermediary	O
truth	O
values	O
between	O
1	O
(	O
truth	O
)	O
and	O
0	O
(	O
falsity	O
)	O
representing	O
degrees	O
of	O
truth	O
of	O
propositions	O
.	O
</s>
<s>
In	O
propositional	O
t-norm	B-General_Concept
fuzzy	I-General_Concept
logics	I-General_Concept
,	O
propositional	O
connectives	O
are	O
stipulated	O
to	O
be	O
truth-functional	O
,	O
that	O
is	O
,	O
the	O
truth	O
value	O
of	O
a	O
complex	O
proposition	O
formed	O
by	O
a	O
propositional	O
connective	O
from	O
some	O
constituent	O
propositions	O
is	O
a	O
function	O
(	O
called	O
the	O
truth	O
function	O
of	O
the	O
connective	O
)	O
of	O
the	O
truth	O
values	O
of	O
the	O
constituent	O
propositions	O
.	O
</s>
<s>
The	O
truth	O
functions	O
operate	O
on	O
the	O
set	O
of	O
truth	O
degrees	O
(	O
in	O
the	O
standard	O
semantics	B-Application
,	O
on	O
the	O
 [ 0 , 1 ] 	O
interval	O
)	O
;	O
thus	O
the	O
truth	O
function	O
of	O
an	O
n-ary	O
propositional	O
connective	O
c	O
is	O
a	O
function	O
Fc	O
:	O
[0,1]n	O
 [ 0 , 1 ] 	O
.	O
</s>
<s>
T-norm	B-General_Concept
fuzzy	I-General_Concept
logics	I-General_Concept
impose	O
certain	O
natural	O
constraints	O
on	O
the	O
truth	O
function	O
of	O
conjunction	O
.	O
</s>
<s>
In	O
general	O
t-norm	B-General_Concept
fuzzy	I-General_Concept
logics	I-General_Concept
,	O
therefore	O
,	O
only	O
left-continuity	O
of	O
is	O
required	O
,	O
which	O
expresses	O
the	O
assumption	O
that	O
a	O
microscopic	O
decrease	O
of	O
the	O
truth	O
degree	O
of	O
a	O
conjunct	O
should	O
not	O
macroscopically	O
decrease	O
the	O
truth	O
degree	O
of	O
conjunction	O
.	O
</s>
<s>
These	O
assumptions	O
make	O
the	O
truth	O
function	O
of	O
conjunction	O
a	O
left-continuous	O
t-norm	B-General_Concept
,	O
which	O
explains	O
the	O
name	O
of	O
the	O
family	O
of	O
fuzzy	O
logics	O
(	O
t-norm	B-General_Concept
based	O
)	O
.	O
</s>
<s>
Particular	O
logics	O
of	O
the	O
family	O
can	O
make	O
further	O
assumptions	O
about	O
the	O
behavior	O
of	O
conjunction	O
(	O
for	O
example	O
,	O
Gödel	B-Language
–	O
Dummett	O
logic	O
requires	O
its	O
idempotence	O
)	O
or	O
other	O
connectives	O
(	O
for	O
example	O
,	O
the	O
logic	O
IMTL	O
(	O
involutive	B-Algorithm
monoidal	B-General_Concept
t-norm	I-General_Concept
logic	I-General_Concept
)	O
requires	O
the	O
involutiveness	B-Algorithm
of	O
negation	O
)	O
.	O
</s>
<s>
All	O
left-continuous	O
t-norms	B-General_Concept
have	O
a	O
unique	O
residuum	O
,	O
that	O
is	O
,	O
a	O
binary	O
function	O
such	O
that	O
for	O
all	O
x	O
,	O
y	O
,	O
and	O
z	O
in	O
[0,1],	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
or	O
bi-residual	O
equivalence	O
Truth	O
functions	O
of	O
propositional	O
connectives	O
may	O
also	O
be	O
introduced	O
by	O
additional	O
definitions	O
:	O
the	O
most	O
usual	O
ones	O
are	O
the	O
minimum	O
(	O
which	O
plays	O
a	O
role	O
of	O
another	O
conjunctive	O
connective	O
)	O
,	O
the	O
maximum	O
(	O
which	O
plays	O
a	O
role	O
of	O
a	O
disjunctive	O
connective	O
)	O
,	O
or	O
the	O
Baaz	O
Delta	O
operator	O
,	O
defined	O
in	O
 [ 0 , 1 ] 	O
as	O
if	O
and	O
otherwise	O
.	O
</s>
<s>
In	O
this	O
way	O
,	O
a	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
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
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
as	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
a	O
larger	O
class	O
of	O
left-continuous	O
t-norms	B-General_Concept
;	O
the	O
set	O
of	O
such	O
formulae	O
is	O
called	O
the	O
logic	O
of	O
the	O
class	O
.	O
</s>
<s>
Important	O
t-norm	B-General_Concept
logics	I-General_Concept
are	O
the	O
logics	O
of	O
particular	O
t-norms	B-General_Concept
or	O
classes	O
of	O
t-norms	B-General_Concept
,	O
for	O
example	O
:	O
</s>
<s>
It	O
turns	O
out	O
that	O
many	O
logics	O
of	O
particular	O
t-norms	B-General_Concept
and	O
classes	O
of	O
t-norms	B-General_Concept
are	O
axiomatizable	O
.	O
</s>
<s>
The	O
completeness	O
theorem	O
of	O
the	O
axiomatic	O
system	O
with	O
respect	O
to	O
the	O
corresponding	O
t-norm	B-General_Concept
semantics	B-Application
on	O
 [ 0 , 1 ] 	O
is	O
then	O
called	O
the	O
standard	O
completeness	O
of	O
the	O
logic	O
.	O
</s>
<s>
Besides	O
the	O
standard	O
real-valued	O
semantics	B-Application
on	O
[0,1],	O
the	O
logics	O
are	O
sound	O
and	O
complete	O
with	O
respect	O
to	O
general	O
algebraic	O
semantics	B-Application
,	O
formed	O
by	O
suitable	O
classes	O
of	O
prelinear	O
commutative	O
bounded	O
integral	O
residuated	O
lattices	O
.	O
</s>
<s>
Some	O
particular	O
t-norm	B-General_Concept
fuzzy	I-General_Concept
logics	I-General_Concept
have	O
been	O
introduced	O
and	O
investigated	O
long	O
before	O
the	O
family	O
was	O
recognized	O
(	O
even	O
before	O
the	O
notions	O
of	O
fuzzy	O
logic	O
or	O
t-norm	B-General_Concept
emerged	O
)	O
:	O
</s>
<s>
Łukasiewicz	B-General_Concept
logic	I-General_Concept
(	O
the	O
logic	O
of	O
the	O
Łukasiewicz	O
t-norm	B-General_Concept
)	O
was	O
originally	O
defined	O
by	O
Jan	O
Łukasiewicz	O
(	O
1920	O
)	O
as	O
a	O
three-valued	B-Language
logic	I-Language
;	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
variants	O
,	O
both	O
propositional	O
and	O
first-order	O
.	O
</s>
<s>
Gödel	B-Language
–	O
Dummett	O
logic	O
(	O
the	O
logic	O
of	O
the	O
minimum	O
t-norm	B-General_Concept
)	O
was	O
implicit	O
in	O
Gödel	B-Language
's	O
1932	O
proof	O
of	O
infinite-valuedness	O
of	O
intuitionistic	O
logic	O
.	O
</s>
<s>
A	O
systematic	O
study	O
of	O
particular	O
t-norm	B-General_Concept
fuzzy	I-General_Concept
logics	I-General_Concept
and	O
their	O
classes	O
began	O
with	O
Hájek	O
's	O
(	O
1998	O
)	O
monograph	O
Metamathematics	O
of	O
Fuzzy	O
Logic	O
,	O
which	O
presented	O
the	O
notion	O
of	O
the	O
logic	O
of	O
a	O
continuous	O
t-norm	B-General_Concept
,	O
the	O
logics	O
of	O
the	O
three	O
basic	O
continuous	O
t-norms	B-General_Concept
( Łukasiewicz	O
,	O
Gödel	B-Language
,	O
and	O
product	O
)	O
,	O
and	O
the	O
'	O
basic	O
 '	O
fuzzy	O
logic	O
BL	B-General_Concept
of	O
all	O
continuous	O
t-norms	B-General_Concept
(	O
all	O
of	O
them	O
both	O
propositional	O
and	O
first-order	O
)	O
.	O
</s>
<s>
The	O
book	O
also	O
started	O
the	O
investigation	O
of	O
fuzzy	O
logics	O
as	O
non-classical	O
logics	O
with	O
Hilbert-style	O
calculi	O
,	O
algebraic	O
semantics	B-Application
,	O
and	O
metamathematical	O
properties	O
known	O
from	O
other	O
logics	O
(	O
completeness	O
theorems	O
,	O
deduction	O
theorems	O
,	O
complexity	O
,	O
etc	O
.	O
</s>
<s>
Since	O
then	O
,	O
a	O
plethora	O
of	O
t-norm	B-General_Concept
fuzzy	I-General_Concept
logics	I-General_Concept
have	O
been	O
introduced	O
and	O
their	O
metamathematical	O
properties	O
have	O
been	O
investigated	O
.	O
</s>
<s>
Some	O
of	O
the	O
most	O
important	O
t-norm	B-General_Concept
fuzzy	I-General_Concept
logics	I-General_Concept
were	O
introduced	O
in	O
2001	O
,	O
by	O
Esteva	O
and	O
Godo	O
(	O
MTL	B-General_Concept
,	O
IMTL	O
,	O
SMTL	O
,	O
NM	O
,	O
WNM	O
)	O
,	O
Esteva	O
,	O
Godo	O
,	O
and	O
Montagna	O
(	O
propositional	O
ŁΠ	O
)	O
,	O
and	O
Cintula	O
(	O
first-order	O
ŁΠ	O
)	O
.	O
</s>
<s>
The	O
logical	O
vocabulary	O
of	O
propositional	O
t-norm	B-General_Concept
fuzzy	I-General_Concept
logics	I-General_Concept
standardly	O
comprises	O
the	O
following	O
connectives	O
:	O
</s>
<s>
In	O
the	O
context	O
of	O
other	O
than	O
t-norm-based	O
fuzzy	O
logics	O
,	O
the	O
t-norm-based	O
implication	O
is	O
sometimes	O
called	O
residual	O
implication	O
or	O
R-implication	O
,	O
as	O
its	O
standard	O
semantics	B-Application
is	O
the	O
residuum	O
of	O
the	O
t-norm	B-General_Concept
that	O
realizes	O
strong	O
conjunction	O
.	O
</s>
<s>
Weak	O
conjunction	O
(	O
binary	O
)	O
,	O
also	O
called	O
lattice	O
conjunction	O
(	O
as	O
it	O
is	O
always	O
realized	O
by	O
the	O
lattice	O
operation	O
of	O
meet	O
in	O
algebraic	O
semantics	B-Application
)	O
.	O
</s>
<s>
In	O
the	O
logic	O
BL	B-General_Concept
and	O
its	O
extensions	O
(	O
though	O
not	O
in	O
t-norm	B-General_Concept
logics	I-General_Concept
in	O
general	O
)	O
,	O
weak	O
conjunction	O
is	O
definable	O
in	O
terms	O
of	O
implication	O
and	O
strong	O
conjunction	O
,	O
by	O
The	O
presence	O
of	O
two	O
conjunction	O
connectives	O
is	O
a	O
common	O
feature	O
of	O
contraction-free	O
substructural	O
logics	O
.	O
</s>
<s>
Bottom	O
(	O
nullary	O
)	O
;	O
or	O
are	O
common	O
alternative	O
signs	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
t-norm	B-General_Concept
fuzzy	I-General_Concept
logics	I-General_Concept
)	O
.	O
</s>
<s>
(	O
Weak	O
)	O
disjunction	O
(	O
binary	O
)	O
,	O
also	O
called	O
lattice	O
disjunction	O
(	O
as	O
it	O
is	O
always	O
realized	O
by	O
the	O
lattice	O
operation	O
of	O
join	O
in	O
algebraic	O
semantics	B-Application
)	O
.	O
</s>
<s>
Top	O
(	O
nullary	O
)	O
,	O
also	O
called	O
one	O
and	O
denoted	O
by	O
or	O
(	O
as	O
the	O
constants	O
top	O
and	O
zero	O
of	O
substructural	O
logics	O
coincide	O
in	O
t-norm	B-General_Concept
fuzzy	I-General_Concept
logics	I-General_Concept
)	O
.	O
</s>
<s>
Some	O
propositional	O
t-norm	B-General_Concept
logics	I-General_Concept
add	O
further	O
propositional	O
connectives	O
to	O
the	O
above	O
language	O
,	O
most	O
often	O
the	O
following	O
ones	O
:	O
</s>
<s>
Also	O
called	O
the	O
Baaz	O
Delta	O
,	O
as	O
it	O
was	O
first	O
used	O
by	O
Matthias	O
Baaz	O
for	O
Gödel	B-Language
–	O
Dummett	O
logic	O
.	O
</s>
<s>
Truth	O
constants	O
are	O
nullary	O
connectives	O
representing	O
particular	O
truth	O
values	O
between	O
0	O
and	O
1	O
in	O
the	O
standard	O
real-valued	O
semantics	B-Application
.	O
</s>
<s>
Involutive	B-Algorithm
negation	O
(	O
unary	O
)	O
can	O
be	O
added	O
as	O
an	O
additional	O
negation	O
to	O
t-norm	B-General_Concept
logics	I-General_Concept
whose	O
residual	O
negation	O
is	O
not	O
itself	O
involutive	B-Algorithm
,	O
that	O
is	O
,	O
if	O
it	O
does	O
not	O
obey	O
the	O
law	O
of	O
double	O
negation	O
.	O
</s>
<s>
A	O
t-norm	B-General_Concept
logic	I-General_Concept
expanded	O
with	O
involutive	B-Algorithm
negation	O
is	O
usually	O
denoted	O
by	O
and	O
called	O
with	O
involution	B-Algorithm
.	O
</s>
<s>
Even	O
though	O
standard	O
in	O
contraction-free	O
substructural	O
logics	O
,	O
in	O
t-norm	B-General_Concept
fuzzy	I-General_Concept
logics	I-General_Concept
it	O
is	O
usually	O
used	O
only	O
in	O
the	O
presence	O
of	O
involutive	B-Algorithm
negation	O
,	O
which	O
makes	O
it	O
definable	O
(	O
and	O
so	O
axiomatizable	O
)	O
by	O
de	O
Morgan	O
's	O
law	O
from	O
strong	O
conjunction	O
:	O
</s>
<s>
Additional	O
t-norm	B-General_Concept
conjunctions	O
and	O
residual	O
implications	O
.	O
</s>
<s>
Some	O
expressively	O
strong	O
t-norm	B-General_Concept
logics	I-General_Concept
,	O
for	O
instance	O
the	O
logic	O
ŁΠ	O
,	O
have	O
more	O
than	O
one	O
strong	O
conjunction	O
or	O
residual	O
implication	O
in	O
their	O
language	O
.	O
</s>
<s>
In	O
the	O
standard	O
real-valued	O
semantics	B-Application
,	O
all	O
such	O
strong	O
conjunctions	O
are	O
realized	O
by	O
different	O
t-norms	B-General_Concept
and	O
the	O
residual	O
implications	O
by	O
their	O
residua	O
.	O
</s>
<s>
Well-formed	O
formulae	O
of	O
propositional	O
t-norm	B-General_Concept
logics	I-General_Concept
are	O
defined	O
from	O
propositional	O
variables	O
(	O
usually	O
countably	O
many	O
)	O
by	O
the	O
above	O
logical	O
connectives	O
,	O
as	O
usual	O
in	O
propositional	O
logics	O
.	O
</s>
<s>
First-order	O
variants	O
of	O
t-norm	B-General_Concept
logics	I-General_Concept
employ	O
the	O
usual	O
logical	O
language	O
of	O
first-order	O
logic	O
with	O
the	O
above	O
propositional	O
connectives	O
and	O
the	O
following	O
quantifiers	B-Language
:	O
</s>
<s>
Algebraic	O
semantics	B-Application
is	O
predominantly	O
used	O
for	O
propositional	O
t-norm	B-General_Concept
fuzzy	I-General_Concept
logics	I-General_Concept
,	O
with	O
three	O
main	O
classes	O
of	O
algebras	O
with	O
respect	O
to	O
which	O
a	O
t-norm	B-General_Concept
fuzzy	I-General_Concept
logic	I-General_Concept
is	O
complete	O
:	O
</s>
<s>
General	O
semantics	B-Application
,	O
formed	O
of	O
all	O
-algebras	O
—	O
that	O
is	O
,	O
all	O
algebras	O
for	O
which	O
the	O
logic	O
is	O
sound	O
.	O
</s>
<s>
Linear	O
semantics	B-Application
,	O
formed	O
of	O
all	O
linear	O
-algebras	O
—	O
that	O
is	O
,	O
all	O
-algebras	O
whose	O
lattice	O
order	O
is	O
linear	O
.	O
</s>
<s>
Standard	O
semantics	B-Application
,	O
formed	O
of	O
all	O
standard	O
-algebras	O
—	O
that	O
is	O
,	O
all	O
-algebras	O
whose	O
lattice	O
reduct	O
is	O
the	O
real	O
unit	O
interval	O
 [ 0 , 1 ] 	O
with	O
the	O
usual	O
order	O
.	O
</s>
<s>
In	O
standard	O
-algebras	O
,	O
the	O
interpretation	O
of	O
strong	O
conjunction	O
is	O
a	O
left-continuous	O
t-norm	B-General_Concept
and	O
the	O
interpretation	O
of	O
most	O
propositional	O
connectives	O
is	O
determined	O
by	O
the	O
t-norm	B-General_Concept
(	O
hence	O
the	O
names	O
t-norm-based	O
logics	O
and	O
t-norm	B-General_Concept
-algebras	O
,	O
which	O
is	O
also	O
used	O
for	O
-algebras	O
on	O
the	O
lattice	O
 [ 0 , 1 ] 	O
)	O
.	O
</s>
<s>
In	O
t-norm	B-General_Concept
logics	I-General_Concept
with	O
additional	O
connectives	O
,	O
however	O
,	O
the	O
real-valued	O
interpretation	O
of	O
the	O
additional	O
connectives	O
may	O
be	O
restricted	O
by	O
further	O
conditions	O
for	O
the	O
t-norm	B-General_Concept
algebra	O
to	O
be	O
called	O
standard	O
:	O
for	O
example	O
,	O
in	O
standard	O
-algebras	O
of	O
the	O
logic	O
with	O
involution	B-Algorithm
,	O
the	O
interpretation	O
of	O
the	O
additional	O
involutive	B-Algorithm
negation	O
is	O
required	O
to	O
be	O
the	O
standard	O
involution	B-Algorithm
rather	O
than	O
other	O
involutions	B-Algorithm
that	O
can	O
also	O
interpret	O
over	O
t-norm	B-General_Concept
-algebras	O
.	O
</s>
<s>
In	O
general	O
,	O
therefore	O
,	O
the	O
definition	O
of	O
standard	O
t-norm	B-General_Concept
algebras	O
has	O
to	O
be	O
explicitly	O
given	O
for	O
t-norm	B-General_Concept
logics	I-General_Concept
with	O
additional	O
connectives	O
.	O
</s>
