<s>
In	O
mathematics	O
and	O
logic	O
,	O
a	O
higher-order	B-Algorithm
logic	I-Algorithm
is	O
a	O
form	O
of	O
predicate	O
logic	O
that	O
is	O
distinguished	O
from	O
first-order	O
logic	O
by	O
additional	O
quantifiers	B-Language
and	O
,	O
sometimes	O
,	O
stronger	O
semantics	O
.	O
</s>
<s>
Higher-order	B-Algorithm
logics	I-Algorithm
with	O
their	O
standard	O
semantics	O
are	O
more	O
expressive	O
,	O
but	O
their	O
model-theoretic	O
properties	O
are	O
less	O
well-behaved	O
than	O
those	O
of	O
first-order	O
logic	O
.	O
</s>
<s>
The	O
term	O
"	O
higher-order	B-Algorithm
logic	I-Algorithm
"	O
,	O
abbreviated	O
as	O
HOL	O
,	O
is	O
commonly	O
used	O
to	O
mean	O
higher-order	O
simple	O
predicate	O
logic	O
.	O
</s>
<s>
Simple	O
types	O
is	O
nowadays	O
sometimes	O
also	O
meant	O
to	O
exclude	O
polymorphic	B-Application
and	O
dependent	O
types	O
.	O
</s>
<s>
Higher-order	B-Algorithm
logic	I-Algorithm
is	O
the	O
union	O
of	O
first-	O
,	O
second-	O
,	O
third-	O
,	O
...	O
,	O
nth-order	O
logic	O
;	O
i.e.	O
,	O
higher-order	B-Algorithm
logic	I-Algorithm
admits	O
quantification	B-Language
over	O
sets	O
that	O
are	O
nested	O
arbitrarily	O
deeply	O
.	O
</s>
<s>
There	O
are	O
two	O
possible	O
semantics	O
for	O
higher-order	B-Algorithm
logic	I-Algorithm
.	O
</s>
<s>
In	O
the	O
standard	O
or	O
full	O
semantics	O
,	O
quantifiers	B-Language
over	O
higher-type	O
objects	O
range	O
over	O
all	O
possible	O
objects	O
of	O
that	O
type	O
.	O
</s>
<s>
For	O
example	O
,	O
a	O
quantifier	B-Language
over	O
sets	O
of	O
individuals	O
ranges	O
over	O
the	O
entire	O
powerset	B-General_Concept
of	O
the	O
set	O
of	O
individuals	O
.	O
</s>
<s>
Thus	O
,	O
in	O
standard	O
semantics	O
,	O
once	O
the	O
set	O
of	O
individuals	O
is	O
specified	O
,	O
this	O
is	O
enough	O
to	O
specify	O
all	O
the	O
quantifiers	B-Language
.	O
</s>
<s>
However	O
,	O
by	O
a	O
result	O
of	O
Kurt	O
Gödel	O
,	O
HOL	O
with	O
standard	O
semantics	O
does	O
not	O
admit	O
an	O
effective	O
,	O
sound	O
,	O
and	O
complete	O
proof	B-Application
calculus	I-Application
.	O
</s>
<s>
Thus	O
,	O
for	O
example	O
,	O
quantifiers	B-Language
over	O
sets	O
of	O
individuals	O
may	O
range	O
over	O
only	O
a	O
subset	O
of	O
the	O
powerset	B-General_Concept
of	O
the	O
set	O
of	O
individuals	O
.	O
</s>
<s>
In	O
particular	O
,	O
HOL	O
with	O
Henkin	O
semantics	O
has	O
all	O
the	O
model-theoretic	O
properties	O
of	O
first-order	O
logic	O
,	O
and	O
has	O
a	O
complete	O
,	O
sound	O
,	O
effective	O
proof	B-Application
system	I-Application
inherited	O
from	O
first-order	O
logic	O
.	O
</s>
<s>
Higher-order	B-Algorithm
logics	I-Algorithm
include	O
the	O
offshoots	O
of	O
Church	O
's	O
simple	O
theory	O
of	O
types	O
and	O
the	O
various	O
forms	O
of	O
intuitionistic	O
type	O
theory	O
.	O
</s>
<s>
Up	O
to	O
a	O
certain	O
notion	O
of	O
isomorphism	O
,	O
the	O
powerset	B-General_Concept
operation	O
is	O
definable	O
in	O
second-order	O
logic	O
.	O
</s>
<s>
Using	O
this	O
observation	O
,	O
Jaakko	O
Hintikka	O
established	O
in	O
1955	O
that	O
second-order	O
logic	O
can	O
simulate	O
higher-order	B-Algorithm
logics	I-Algorithm
in	O
the	O
sense	O
that	O
for	O
every	O
formula	O
of	O
a	O
higher-order	B-Algorithm
logic	I-Algorithm
,	O
one	O
can	O
find	O
an	O
equisatisfiable	O
formula	O
for	O
it	O
in	O
second-order	O
logic	O
.	O
</s>
<s>
The	O
term	O
"	O
higher-order	B-Algorithm
logic	I-Algorithm
"	O
is	O
assumed	O
in	O
some	O
context	O
to	O
refer	O
to	O
classical	O
higher-order	B-Algorithm
logic	I-Algorithm
.	O
</s>
<s>
However	O
,	O
modal	O
higher-order	B-Algorithm
logic	I-Algorithm
has	O
been	O
studied	O
as	O
well	O
.	O
</s>
