<s>
Laws	B-General_Concept
of	I-General_Concept
Form	I-General_Concept
(	O
hereinafter	O
LoF	O
)	O
is	O
a	O
book	O
by	O
G	O
.	O
Spencer-Brown	O
,	O
published	O
in	O
1969	O
,	O
that	O
straddles	O
the	O
boundary	O
between	O
mathematics	O
and	O
philosophy	O
.	O
</s>
<s>
"	O
Equations	O
of	O
the	O
second	O
degree	O
"	O
(	O
Chapter	O
11	O
)	O
,	O
whose	O
interpretations	O
include	O
finite	B-Architecture
automata	I-Architecture
and	O
Alonzo	O
Church	O
's	O
Restricted	O
Recursive	O
Arithmetic	O
(	O
RRA	O
)	O
.	O
</s>
<s>
"	O
Boundary	B-General_Concept
algebra	I-General_Concept
"	O
is	O
Meguire	O
's	O
(	O
2011	O
)	O
term	O
for	O
the	O
union	O
of	O
the	O
primary	O
algebra	O
and	O
the	O
primary	O
arithmetic	O
.	O
</s>
<s>
Laws	B-General_Concept
of	I-General_Concept
Form	I-General_Concept
sometimes	O
loosely	O
refers	O
to	O
the	O
"	O
primary	O
algebra	O
"	O
as	O
well	O
as	O
to	O
LoF	O
.	O
</s>
<s>
The	O
second	O
series	O
of	O
editions	O
appeared	O
in	O
1972	O
with	O
the	O
"	O
Preface	O
to	O
the	O
First	O
American	O
Edition	O
"	O
which	O
emphasised	O
the	O
use	O
of	O
self-referential	O
paradoxes	B-Application
.	O
</s>
<s>
LoFs	O
mystical	O
and	O
declamatory	O
prose	O
and	O
its	O
love	O
of	O
paradox	B-Application
make	O
it	O
a	O
challenging	O
read	O
for	O
all	O
.	O
</s>
<s>
In	O
accordance	O
with	O
G	O
.	O
Spencer-Brown	O
'	O
s	O
interest	O
in	O
paradoxes	B-Application
,	O
the	O
only	O
sentence	O
that	O
makes	O
a	O
statement	O
that	O
something	O
is	O
,	O
is	O
the	O
statement	O
which	O
says	O
no	O
such	O
statements	O
are	O
used	O
in	O
this	O
book.Felix	O
Lau	O
:	O
Die	O
Form	O
der	O
Paradoxie	O
,	O
2005	O
Carl-Auer	O
Verlag	O
,	O
Except	O
for	O
this	O
one	O
sentence	O
the	O
book	O
can	O
be	O
seen	O
as	O
an	O
example	O
of	O
E-Prime	O
.	O
</s>
<s>
Highlights	O
how	O
syntactically	B-Application
distinct	O
statements	O
in	O
logic	O
and	O
2	O
can	O
have	O
identical	O
semantics	O
;	O
</s>
<s>
Moreover	O
,	O
the	O
syntax	B-Application
of	O
the	O
primary	O
algebra	O
can	O
be	O
extended	O
to	O
formal	O
systems	O
other	O
than	O
2	O
and	O
sentential	O
logic	O
,	O
resulting	O
in	O
boundary	O
mathematics	O
(	O
see	O
below	O
)	O
.	O
</s>
<s>
Also	O
called	O
the	O
"	O
mark	O
"	O
or	O
"	O
cross	O
"	O
,	O
is	O
the	O
essential	O
feature	O
of	O
the	O
Laws	B-General_Concept
of	I-General_Concept
Form	I-General_Concept
.	O
</s>
<s>
The	O
counterpoint	O
to	O
the	O
Marked	O
state	O
is	O
the	O
Unmarked	B-General_Concept
state	I-General_Concept
,	O
which	O
is	O
simply	O
nothing	O
,	O
the	O
void	O
,	O
or	O
the	O
un-expressable	O
infinite	O
represented	O
by	O
a	O
blank	O
space	O
.	O
</s>
<s>
The	O
Marked	O
state	O
and	O
the	O
void	O
are	O
the	O
two	O
primitive	O
values	O
of	O
the	O
Laws	B-General_Concept
of	I-General_Concept
Form	I-General_Concept
.	O
</s>
<s>
The	O
syntax	B-Application
of	O
the	O
primary	O
arithmetic	O
goes	O
as	O
follows	O
.	O
</s>
<s>
There	O
are	O
just	O
two	O
atomic	B-Algorithm
expressions	I-Algorithm
:	O
</s>
<s>
Let	O
the	O
"	O
unmarked	B-General_Concept
state	I-General_Concept
"	O
be	O
a	O
synonym	O
for	O
the	O
void	O
.	O
</s>
<s>
We	O
can	O
now	O
state	O
the	O
"	O
arithmetical	O
"	O
axioms	B-Algorithm
A1	O
and	O
A2	O
,	O
which	O
ground	O
the	O
primary	O
arithmetic	O
(	O
and	O
hence	O
all	O
of	O
the	O
Laws	B-General_Concept
of	I-General_Concept
Form	I-General_Concept
)	O
:	O
</s>
<s>
After	O
crossing	O
from	O
the	O
unmarked	O
to	O
the	O
marked	O
state	O
,	O
crossing	O
again	O
(	O
"	O
recrossing	O
"	O
)	O
starting	O
from	O
the	O
marked	O
state	O
returns	O
one	O
to	O
the	O
unmarked	B-General_Concept
state	I-General_Concept
.	O
</s>
<s>
This	O
suggests	O
that	O
every	O
primary	O
arithmetic	O
expression	O
can	O
,	O
by	O
repeated	O
application	O
of	O
A1	O
and	O
A2	O
,	O
be	O
simplified	O
to	O
one	O
of	O
two	O
states	O
:	O
the	O
marked	O
or	O
the	O
unmarked	B-General_Concept
state	I-General_Concept
.	O
</s>
<s>
Starting	O
from	O
an	O
initial	O
marked	O
or	O
unmarked	B-General_Concept
state	I-General_Concept
,	O
"	O
complicating	O
"	O
an	O
expression	O
by	O
a	O
finite	O
number	O
of	O
repeated	O
application	O
of	O
A1	O
and	O
A2	O
cannot	O
yield	O
an	O
expression	O
whose	O
simplification	O
differs	O
from	O
the	O
initial	O
state	O
.	O
</s>
<s>
Thus	O
the	O
relation	B-Algorithm
of	O
logical	O
equivalence	O
partitions	O
all	O
primary	O
arithmetic	O
expressions	O
into	O
two	O
equivalence	O
classes	O
:	O
those	O
that	O
simplify	O
to	O
the	O
Cross	O
,	O
and	O
those	O
that	O
simplify	O
to	O
the	O
void	O
.	O
</s>
<s>
The	O
primary	O
arithmetic	O
is	O
analogous	O
to	O
the	O
following	O
formal	O
languages	O
from	O
mathematics	O
and	O
computer	B-General_Concept
science	I-General_Concept
:	O
</s>
<s>
Logical	O
equivalence	O
is	O
an	O
equivalence	O
relation	B-Algorithm
over	O
the	O
set	O
of	O
primary	O
algebra	O
formulas	O
,	O
governed	O
by	O
the	O
rules	O
R1	O
and	O
R2	O
.	O
</s>
<s>
An	O
initial	O
is	O
a	O
primary	O
algebra	O
equation	O
verifiable	O
by	O
a	O
decision	O
procedure	O
and	O
as	O
such	O
is	O
not	O
an	O
axiom	B-Algorithm
.	O
</s>
<s>
While	O
a	O
proof	O
of	O
T13	O
would	O
require	O
induction	B-Algorithm
,	O
the	O
intuition	O
underlying	O
it	O
should	O
be	O
clear.C2	O
or	O
its	O
equivalent	O
is	O
named	O
:	O
</s>
<s>
Perhaps	O
the	O
first	O
instance	O
of	O
an	O
axiom	B-Algorithm
or	O
rule	O
with	O
the	O
power	O
of	O
C2	O
was	O
the	O
"	O
Rule	O
of	O
(	O
De	O
)	O
Iteration	O
"	O
,	O
combining	O
T13	O
and	O
AA	O
=	O
A	O
,	O
of	O
C	O
.	O
S	O
.	O
Peirce	O
's	O
existential	O
graphs	O
.	O
</s>
<s>
By	O
design	O
,	O
these	O
initials	O
are	O
very	O
similar	O
to	O
the	O
axioms	B-Algorithm
for	O
an	O
abelian	O
group	O
,	O
G1-G3	O
below	O
.	O
</s>
<s>
Demonstrations	O
and	O
proofs	O
invoke	O
an	O
initial	O
as	O
if	O
it	O
were	O
an	O
axiom	B-Algorithm
.	O
</s>
<s>
The	O
result	O
is	O
a	O
normal	O
form	O
,	O
the	O
primary	O
algebra	O
analog	O
of	O
the	O
conjunctive	B-Application
normal	I-Application
form	I-Application
.	O
</s>
<s>
Independence	B-Algorithm
:	O
J1	O
cannot	O
be	O
demonstrated	O
from	O
J2	O
and	O
vice	O
versa	O
(	O
T18	O
)	O
.	O
</s>
<s>
If	O
the	O
Marked	O
and	O
Unmarked	B-General_Concept
states	I-General_Concept
are	O
read	O
as	O
the	O
Boolean	O
values	O
1	O
and	O
0	O
(	O
or	O
True	O
and	O
False	O
)	O
,	O
the	O
primary	O
algebra	O
interprets	O
2	O
(	O
or	O
sentential	O
logic	O
)	O
.	O
</s>
<s>
Given	O
an	O
assignment	O
of	O
every	O
variable	O
to	O
the	O
Marked	O
or	O
Unmarked	B-General_Concept
states	I-General_Concept
,	O
this	O
primary	O
algebra	O
translation	O
reduces	O
to	O
a	O
primary	O
arithmetic	O
expression	O
,	O
which	O
can	O
be	O
simplified	O
.	O
</s>
<s>
Schwartz	O
(	O
1981	O
)	O
proved	O
that	O
the	O
primary	O
algebra	O
is	O
equivalent	O
—	O
syntactically	B-Application
,	O
semantically	O
,	O
and	O
proof	O
theoretically	O
—	O
with	O
the	O
classical	O
propositional	O
calculus	O
.	O
</s>
<s>
Likewise	O
,	O
it	O
can	O
be	O
shown	O
that	O
the	O
primary	O
algebra	O
is	O
syntactically	B-Application
equivalent	O
with	O
expressions	O
built	O
up	O
in	O
the	O
usual	O
way	O
from	O
the	O
classical	O
truth	O
values	O
true	O
and	O
false	O
,	O
the	O
logical	O
connectives	O
NOT	O
,	O
OR	O
,	O
and	O
AND	O
,	O
and	O
parentheses	O
.	O
</s>
<s>
Interpreting	O
the	O
Unmarked	B-General_Concept
State	I-General_Concept
as	O
False	O
is	O
wholly	O
arbitrary	O
;	O
that	O
state	O
can	O
equally	O
well	O
be	O
read	O
as	O
True	O
.	O
</s>
<s>
Another	O
consequence	O
of	O
self-duality	O
is	O
the	O
irrelevance	O
of	O
De	O
Morgan	O
's	O
laws	O
;	O
those	O
laws	O
are	O
built	O
into	O
the	O
syntax	B-Application
of	O
the	O
primary	O
algebra	O
from	O
the	O
outset	O
.	O
</s>
<s>
This	O
primary	O
algebra	O
translation	O
of	O
syllogistic	O
logic	O
also	O
suggests	O
that	O
the	O
primary	O
algebra	O
can	O
interpret	O
monadic	B-Application
and	O
term	O
logic	O
,	O
and	O
that	O
the	O
primary	O
algebra	O
has	O
affinities	O
to	O
the	O
Boolean	O
term	O
schemata	O
of	O
Quine	O
(	O
1982	O
:	O
Part	O
II	O
)	O
.	O
</s>
<s>
Hence	O
the	O
primary	O
algebra	O
is	O
a	O
boundary	B-General_Concept
algebra	I-General_Concept
.	O
</s>
<s>
The	O
axioms	B-Algorithm
for	O
an	O
abelian	O
group	O
,	O
in	O
boundary	O
notation	O
,	O
are	O
:	O
</s>
<s>
Some	O
recursive	O
formulae	O
simplify	O
to	O
the	O
marked	O
or	O
unmarked	B-General_Concept
state	I-General_Concept
.	O
</s>
<s>
Church	O
introduced	O
RRA	O
in	O
1955	O
as	O
an	O
axiomatic	O
formalization	O
of	O
finite	B-Architecture
automata	I-Architecture
.	O
</s>
<s>
The	O
syntax	B-Application
of	O
his	O
alpha	O
existential	O
graphs	O
is	O
merely	O
concatenation	O
,	O
read	O
as	O
conjunction	O
,	O
and	O
enclosure	O
by	O
ovals	O
,	O
read	O
as	O
negation.The	O
existential	O
graphs	O
are	O
described	O
at	O
length	O
in	O
Peirce	O
,	O
C.S.	O
</s>
<s>
The	O
above	O
formalisms	O
are	O
,	O
like	O
the	O
primary	O
algebra	O
,	O
all	O
instances	O
of	O
boundary	O
mathematics	O
,	O
i.e.	O
,	O
mathematics	O
whose	O
syntax	B-Application
is	O
limited	O
to	O
letters	O
and	O
brackets	O
(	O
enclosing	O
devices	O
)	O
.	O
</s>
<s>
A	O
minimalist	O
syntax	B-Application
of	O
this	O
nature	O
is	O
a	O
"	O
boundary	O
notation	O
"	O
.	O
</s>
<s>
The	O
work	O
of	O
Leibniz	O
,	O
Peirce	O
,	O
and	O
Nicod	O
is	O
innocent	O
of	O
metatheory	O
,	O
as	O
they	O
wrote	O
before	O
Emil	O
Post	O
's	O
landmark	O
1920	O
paper	O
(	O
which	O
LoF	O
cites	O
)	O
,	O
proving	O
that	O
sentential	O
logic	O
is	O
complete	O
,	O
and	O
before	O
Hilbert	O
and	O
Łukasiewicz	O
showed	O
how	O
to	O
prove	O
axiom	B-Algorithm
independence	B-Algorithm
using	O
models	O
.	O
</s>
<s>
The	O
lambda	B-Language
calculus	I-Language
;	O
</s>
<s>
Combinatory	B-Application
logic	I-Application
with	O
two	O
(	O
S	O
and	O
K	O
)	O
or	O
even	O
one	O
(	O
X	O
 '	O
)	O
primitive	O
combinators	B-Application
;	O
</s>
<s>
Mathematical	O
logic	O
done	O
with	O
merely	O
three	O
primitive	O
notions	O
:	O
one	O
connective	O
,	O
NAND	O
(	O
whose	O
primary	O
algebra	O
translation	O
is	O
or	O
,	O
dually	O
,	O
)	O
,	O
universal	O
quantification	B-Language
,	O
and	O
one	O
binary	O
atomic	B-Algorithm
formula	I-Algorithm
,	O
denoting	O
set	O
membership	O
.	O
</s>
<s>
"	O
Laws	B-General_Concept
of	I-General_Concept
Form	I-General_Concept
–	O
An	O
Exploration	O
in	O
Mathematics	O
and	O
Foundations.	O
"	O
</s>
<s>
Boundary	B-General_Concept
Algebra	I-General_Concept
:	O
A	O
Simpler	O
Approach	O
to	O
Basic	O
Logic	O
and	O
Boolean	O
Algebra	O
.	O
</s>
<s>
Special	O
Issue	O
,	O
Laws	B-General_Concept
of	I-General_Concept
Form	I-General_Concept
:	O
Spencer-Brown	O
at	O
Esalen	O
,	O
1973	O
.	O
</s>
<s>
Laws	B-General_Concept
of	I-General_Concept
Form	I-General_Concept
,	O
archive	O
of	O
website	O
by	O
Richard	O
Shoup	O
.	O
</s>
<s>
The	O
Laws	B-General_Concept
of	I-General_Concept
Form	I-General_Concept
Forum	O
,	O
where	O
the	O
primary	O
algebra	O
and	O
related	O
formalisms	O
have	O
been	O
discussed	O
since	O
2002	O
.	O
</s>
<s>
The	O
Markable	O
Mark	O
,	O
an	O
introduction	O
by	O
easy	O
stages	O
to	O
the	O
ideas	O
of	O
Laws	B-General_Concept
of	I-General_Concept
Form	I-General_Concept
The	O
BF	O
Calculus	O
and	O
the	O
Square	O
Root	O
of	O
Negation	O
by	O
Louis	O
Kauffman	O
and	O
Arthur	O
Collings	O
;	O
it	O
extends	O
the	O
Laws	B-General_Concept
of	I-General_Concept
Form	I-General_Concept
by	O
adding	O
an	O
imaginary	O
logical	O
value	O
.	O
</s>
<s>
Laws	B-General_Concept
of	I-General_Concept
Form	I-General_Concept
Course	O
-	O
taking	O
people	O
through	O
the	O
main	O
body	O
of	O
the	O
text	O
of	O
Laws	B-General_Concept
of	I-General_Concept
Form	I-General_Concept
by	O
Leon	O
Conrad	O
,	O
Spencer-Brown	O
'	O
s	O
last	O
student	O
,	O
who	O
studied	O
the	O
work	O
with	O
the	O
author	O
.	O
</s>
