<s>
In	O
mathematical	O
logic	O
,	O
an	O
atomic	B-Algorithm
formula	I-Algorithm
(	O
also	O
known	O
as	O
an	O
atom	B-Algorithm
or	O
a	O
prime	O
formula	O
)	O
is	O
a	O
formula	O
with	O
no	O
deeper	O
propositional	O
structure	O
,	O
that	O
is	O
,	O
a	O
formula	O
that	O
contains	O
no	O
logical	O
connectives	O
or	O
equivalently	O
a	O
formula	O
that	O
has	O
no	O
strict	O
subformulas	O
.	O
</s>
<s>
Compound	O
formulas	O
are	O
formed	O
by	O
combining	O
the	O
atomic	B-Algorithm
formulas	I-Algorithm
using	O
the	O
logical	O
connectives	O
.	O
</s>
<s>
The	O
precise	O
form	O
of	O
atomic	B-Algorithm
formulas	I-Algorithm
depends	O
on	O
the	O
logic	O
under	O
consideration	O
;	O
for	O
propositional	O
logic	O
,	O
for	O
example	O
,	O
a	O
propositional	O
variable	O
is	O
often	O
more	O
briefly	O
referred	O
to	O
as	O
an	O
"	O
atomic	B-Algorithm
formula	I-Algorithm
"	O
,	O
but	O
,	O
more	O
precisely	O
,	O
a	O
propositional	O
variable	O
is	O
not	O
an	O
atomic	B-Algorithm
formula	I-Algorithm
but	O
a	O
formal	O
expression	O
that	O
denotes	O
an	O
atomic	B-Algorithm
formula	I-Algorithm
.	O
</s>
<s>
For	O
predicate	B-Algorithm
logic	O
,	O
the	O
atoms	O
are	O
predicate	B-Algorithm
symbols	O
together	O
with	O
their	O
arguments	O
,	O
each	O
argument	O
being	O
a	O
term	O
.	O
</s>
<s>
In	O
model	O
theory	O
,	O
atomic	B-Algorithm
formulas	I-Algorithm
are	O
merely	O
strings	O
of	O
symbols	O
with	O
a	O
given	O
signature	O
,	O
which	O
may	O
or	O
may	O
not	O
be	O
satisfiable	O
with	O
respect	O
to	O
a	O
given	O
model	O
.	O
</s>
<s>
The	O
well-formed	O
terms	O
and	O
propositions	O
of	O
ordinary	O
first-order	O
logic	O
have	O
the	O
following	O
syntax	B-Application
:	O
</s>
<s>
that	O
is	O
,	O
a	O
term	O
is	O
recursively	O
defined	O
to	O
be	O
a	O
constant	O
c	O
(	O
a	O
named	O
object	O
from	O
the	O
domain	B-Algorithm
of	I-Algorithm
discourse	I-Algorithm
)	O
,	O
or	O
a	O
variable	O
x	O
(	O
ranging	O
over	O
the	O
objects	O
in	O
the	O
domain	B-Algorithm
of	I-Algorithm
discourse	I-Algorithm
)	O
,	O
or	O
an	O
n-ary	O
function	O
f	O
whose	O
arguments	O
are	O
terms	O
tk	O
.	O
</s>
<s>
Functions	O
map	O
tuples	B-Application
of	O
objects	O
to	O
objects	O
.	O
</s>
<s>
that	O
is	O
,	O
a	O
proposition	O
is	O
recursively	O
defined	O
to	O
be	O
an	O
n-ary	O
predicate	B-Algorithm
P	O
whose	O
arguments	O
are	O
terms	O
tk	O
,	O
or	O
an	O
expression	O
composed	O
of	O
logical	O
connectives	O
(	O
and	O
,	O
or	O
)	O
and	O
quantifiers	B-Language
(	O
for-all	O
,	O
there-exists	O
)	O
used	O
with	O
other	O
propositions	O
.	O
</s>
<s>
An	O
atomic	B-Algorithm
formula	I-Algorithm
or	O
atom	B-Algorithm
is	O
simply	O
a	O
predicate	B-Algorithm
applied	O
to	O
a	O
tuple	B-Application
of	O
terms	O
;	O
that	O
is	O
,	O
an	O
atomic	B-Algorithm
formula	I-Algorithm
is	O
a	O
formula	O
of	O
the	O
form	O
P	O
(	O
t1	O
,…,	O
tn	O
)	O
for	O
P	O
a	O
predicate	B-Algorithm
,	O
and	O
the	O
tn	O
terms	O
.	O
</s>
<s>
All	O
other	O
well-formed	O
formulae	O
are	O
obtained	O
by	O
composing	O
atoms	O
with	O
logical	O
connectives	O
and	O
quantifiers	B-Language
.	O
</s>
<s>
As	O
there	O
are	O
no	O
quantifiers	B-Language
appearing	O
in	O
an	O
atomic	B-Algorithm
formula	I-Algorithm
,	O
all	O
occurrences	O
of	O
variable	O
symbols	O
in	O
an	O
atomic	B-Algorithm
formula	I-Algorithm
are	O
free	O
.	O
</s>
