1
Sentence Bi-Logic
Arturo Graziano Grappone
Editor-in-Chief of the International Review “Metalogicon”
Key Words: Alassi, Bi-Logic, Psychoanalisis, Sentence Logic , Simassi, Tridim Structure
0. Summary.
Matte Blanco affirmed that our beaviour is based on the interactions of two distinct logics: the
conscious logic which is the standard logic and the unconscious logic which is based on Freud’s laws
of dream. He reduced all Freud’s laws to the sole symmetry principle (whose application is a partial
transformation of non-commutative relations in commutative in any structure). He called bi-logic
structures the achievements of the interaction between unconscious logic and conscious logic and bilogic (or, more recently, super-logic) the logic law system which includes either unconscious or
conscious logic. This paper proves that we can build the predicative calculus of first order (and
therefore all the mathematical structures) from standard sentence logic by addition of Matte Blanco’s
principles on unconscious logic.
1. From Freud’s laws for Unconscious to Matte Blanco’s Laws for Bi-Logics
Freud put the following laws for unconscious: space absence, time absence, condensation of
distinct mind objects and presence of contradiction, displacement of a mind object in another mind
object.
Matte Blanco proved that these laws can be reduced to a partial transformation of non-commutative
relations in commutative relations (symmetry principle). In fact, space disappears when all the
relations among its points become commutative, time disappears when all the relations among its
instants become commutative, two objects condense (i. e. identify) when all their relations become
commutative, contradiction is condensation of opposite object, an object is displaced in another one
when all its relation with a part of this other object become commutative. Matte Blanco calls symmetry
bag a set whose elements have only a commutative relation among them. He proved that displacement
and condensation are the achievement of representation of poly-dimensional objects in the three
dimension of conscious (tridim structures). He introduced other bi-logical concepts: the simassi, i. e.
the union of a symmetry bag and a conscious content, the alassi, i. e. a logical deduction among
simassis with useful achievements, the epistemological swing, i. e. a logical deduction among simassis
without useful achievements, the stratified bi-logical structure only for psychology and the not well
known polarized vector.
2. From Standard Sentence Logic to a Sentence Bi-Logic which Includes it
Let a, b, c, … be generic sentences. Let Na be standard negation of a. Let Aab be standard logical
sum of a and b. Let Vab be AaaNab (i. e., tautology of a and b). Let Bab be AaNb (i. e., converse
implication of b from a). Let Cab be ANab (i. e., implication of b from a). Let Dab be ANaNb (i. e.,
Sheffer’s connective of a and b). Let Eab be NDBabCab (i. e., Equivalence of a and b). Let Fab be Na.
Let Gab be Nb. Let Hab be b. Let Iab be a. Let Jab be NEab (i. e., exclusive or between a and b). Let
2
Kab be NDab (i. e., logical product of a and b). Let Lab be NCab. Let Mab be NBab. Let Xab be NAab.
Let Oab be KaaNab (i. e., contradiction of a and b). Let a, b, c, … be a connective as V, A, B, C, D, E,
F, G, H, I, J, K, L, M, X, O. Let a, b, c, … be a connective as V, A, B, C, D, F, G, H, I, K, L, M, X, O.
Let a, b, c, … be a connective as V, E, F, G, H, I, J, K, L, M, X, O. Let a, b, c, … be natural numbers
and ‘-’ the difference on them. Let a, b, c, … be atomic sentences. Let a, b, c, … be commutative (also
*
empty) sequences of atomic sentences. Let c aa(a)b(b) be a sentence c where all the atomic sentences
(
)
of the sequences a and b are respectively in the former and the later argument of a. Let (a) be
application of symmetry principle among all the sentences of a (Matte Blanco’s symmetry bag). Let [a]
be abbreviatin of N(a)N. Let {a} can be either (a) or [a]. Let a, b, c, … matrices of elements {ma},
{nb}, {pc}, … (tridim structures of symmetry bags) where:
W1: {1a} = {a} ;
( n a) in b means that a contains an atomic sentence c and that there is such n-1 symmetry
bag [md], [pe], [qf], … in b that they contain c and are in the left of ( n a) ;
W3: [ n a] in b means that a contains an atomic sentence c and that there is such n-1 symmetry
bag (md), (pe), (qf), … in b that they contain c and are in the left of [ n a].
W2:
n >1
n >1
n>1
n>1
Let aa, ba, ca, … ab, bb, cb, … ac, bc, cc, … be simassi. Define alassis on them by the
inferences:
a
S1:
aa
Z1:
!
$
M
#
&
S4: ##L {a}{b}{c} L&&
#
{a}{b}{c} &&
#
M
"
%
"
%
"
%
M M
M
M
M
$
'
$
'
$ L ! nab
$ L ! na L'
L mb '
$$
''
$$
''
M
M
M
Z2: #
P1: # M M
&
&
"
%
"
%
M
M
M
M
M
$
'
$
'
$ L na nb
$ L na ! L'
L mb '
$$
''
$$
''
M
M
M
M
#M
&
#
&
"
"M M
%
M
M
M
$M M
$
'
$ L ! ( nab)
$ L ! (nab) L'
$
$
'
U1: $# M M
U2: $# M M
M
M
M '&
"M
"M
M
M
M
M
M %'
$
$
$ L (nb) m a
$ L (na) m b L'
m (n
m( n
$$
$$
''
M
M
M
M
M&
#M
#M
(aabc)a( a, a )
S2:
(aabbc)a( a, b)
"
$
$! L !
$$
#
M
{na}
M
"
M
$
$ ! ! L ! {na}
$$
M
#
{ }
{ }
"M M
M
M %'
$
$ L ! nab L'
'
P2: $$# M M
M
M '&
"M
M
M
M%
$
'
$ L nb
na L'
$$
'
M
M
M '&
#M
"M M
M
M %'
$
$ L ! nab L'
$
'
Q1: $# M M
M
M '& *
c aa a b b
"M
M
M
M%
$
'
$ L na
nb L'
$$
'
M
M
M '&
#M
[
]
[ ] [ ]
(
)
( ( ) ( ))
( ) ( )
! a$
# &
S3:
aa # M & a
## &&
" a%
{ }
[ ]
[ ] [ ]
{ }
[ ]
[ ]
Q2:
!
#
#
#L
#
#
"
M%
'
L'
'
M '&
M%
'
L'
'
M '&
M %'
L'
'
M '&
M %'
L'
'
M '&
"M M
M
M%
$
'
$ L ! (nab) L'
$$
'
M
M '& *
#M M
c aa(a)b(b)
"M
M
M
M%
$
'
$ L (nb) (na) L'
$$
'
M
M
M '&
#M
(
)
M
{ab}{c}
{a}{bc}
M
$
&
&
L&
&
&
%
3
"M
$
$L
$
R1: $# M
"M
$
$L
$$
#M
!L
#
#L
#
N1: #"L
!L
#
#L
##
"L
M
M %'
( nab) L''
M
M '& *
c aa(a)b(b)
M
M
M %'
m a (nb) L'
m( n
''
M
M
M&
! L na N $
na $&
#
&
#L
M &N
M &
&
#
&
N2: #"L mb N &%
mb &%
! L N na $
na N $&
#
&
#L
M &
M &
&
#
&
# L N mb &
mb N &%
"
%
M
!
M
(
)
[ ]
( )
( )
[ ]
( )
[ ]
[ ]
[ ]
( )
abVab
aVabb
abBab
B1:
aBaNbNb
abDab
D1:
aDaNbNb
abFab
F1:
aFabb
abHab
H1:
aHabb
abJab
J1:
aJaNbNb
abLab
L1:
aLaNbNb
abXab
X1:
aXaNbNb
V1:
abVab
aVbab
abBab
B2:
aBbab
abDab
D2:
aDNbNab
abFab
F2:
aFNbNab
abHab
H2:
aHbab
abJab
J2:
aJNbNab
abLab
L2:
aLbab
abXab
X2:
aXNbNab
V2:
"M M
M
M %'
$
$ L ! (nab) L'
$
'
R2: $# M M
M
M '& *
c aa (a)b (b)
"M
M
M
M %'
$
$ L m b (na) L'
m( n
$$
'
M
M
M '&
#M
! N na
! L (na) N pb L$
{ } &
# { }
#
#
#L
M
M
M
L&
#
#
&
#
#
&
N3: "L mc N {qd} L%
N4: " N {mc}
! {na}
! L N na {pb} L$
#
#
&
#L
N# M
M
M
L&
##
##
&&
" {mc}
" L N (mc) {qd} L%
abVab
Vaabb
abBab
B3:
BaaNbNb
abDab
D3:
DNaNaNbNb
abFab
F3:
FNaNabb
abHab
H3:
Haabb
abJab
J3:
JNaNaNbNb
abLab
L3:
LaaNbNb
abXab
X3:
XNaNaNbNb
V3:
(
)
[ ]
[ ]
[ ]
abAab
aAabb
abCab
C1:
aCabb
abEab
E1:
aEabb
abGab
G1:
aGaNbNb
abIab
I1:
aIabb
abKab
K1:
aKabb
abMab
M1:
aMabb
abOab
O1:
aOaNbNb
A1:
abAab
aAbab
abCab
C2:
aCNbNab
abEab
E2:
aEbab
abGab
G2:
aGbab
abIab
I2:
aIbab
abKab
K2:
aKbab
abMab
M2:
aMNbNab
abOab
O2:
aONbNab
A2:
L
{pb}$&
M &
&
{qd}&%
L {pb}$&
O
M &
&
L {qd}&%
O
L
abAab
Aaabb
abCab
C3:
CNaNabb
abEab
E3:
Eaabb
abGab
G3:
GaaNbNb
abIab
I3:
Iaabb
abKab
K3:
Kaabb
abMab
M3:
MNaNabb
abOab
O3:
ONaNaNbNb
A3:
3. Predicative Calculus of First Order as Sentence Bi-Logic
Let A1n , A2n , A3n , … be n-adic predicates. Let a1 , a 2 , a 3 , …, b1 , b2 , b3 , …, c1 , c2 , c3 , … be logical
term variables. Let a1 , a 2 , a 3 , … be term constant. Let F1n , F2n , F3n , … be n-adic functions among terms.
Let t1 , t 2 , t 3 , … be generic terms.
Let
(
A1n t1 Kt n
)
be atomic sentence or atomic sentential function. Let (!a1 ) and (!a1 ) be
(
)
respectively universal and existential quantifier. Let (Qa1 ) be (!a1 ) or (!a1 ) . Let ! "#$K% & 'K
mean that α contains β, γ, δ, … but not ε, ζ, η, ….
Tr1: given any sentence we can always build an equivalent sentence without term constant and
without functions among terms (see also Mendelson 1964).
Tr2: given any sentence, we can always build an equivalent sentence whose atomic sentences
contains the same number of terms and all the possible sequences of the considered terms. In fact we
have, e. g.:
4
( ) ( ) (
)
(
) (
) (
)
KKKA11 t1 A21 t 1 t 2 A13 t1 t 2 t 3 is equivalent to KKKA16 t1 t 1 t1 t 1 t1 t 1 A62 t 1 t1 t 1 t 2 t 2 t 2 A63 t1 t 2 t 3 t 1 t 2 t 3 .
Tr1, Tr2 put the equivalence between the general predicative calculus of first order and a restricted
calculus RPC without constant, functions, and sentences with predicates with distinct argument
number.
Interpretate RPC in sentence bi-logic by the following rules that we give by examples:
&
)
!"
(
+
$
(
+
M
#
n
times
n n n n n n n
+ a ( abcdefg )
TR3: a A1 A2 A1 A3 A1 A2 A4 is equivalent to (
$
(
!$%
+
((
++
' (ace)(bf )( d )( g )*
&
)
!"
(
+
$
(
M #n times+
+ K(,d1 )a a(a1 Kaa d1e1 Ken-a -1)b(b1 Kba + bd1f1 Kfn- a- b-1)c(c1 Kc ad1g1 Kgn -a -1 ) K
TR4: (
$
(
!$%
+
((
++
' K( aK)K *
(
)
(
)
is equivalent to
&
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
'
)
) &( !"
! "
+
+ ( $
$
$
+
+ ( M #a times
M $
+
+ ( $
$
+
%
! $
+ ( !$
+
+
$
ac
! $
+
+ (( ( )
+
+ ( !"
! $
+
+ ( $
$
+
M #n times+ ( M #n , a , 1 times
+ a a(a Ka – e Ke
$
+ K( $
1
a
1
n, a,1 ) b(b1 Kba + b – f1 Kfn,a , b,2 )c(c1 Kc a – g1 Kgn ,a ,1 ) K
+
! $
+ ( !$
%
+
+ (
! $
+
$
+ ( ( b)
+
+ ( "
! $
+
$
+ ( !$
+
M $
+ (M$
#
n
,
a
,
b
,
2
times
+
$
+ ( $
+
! %
+ ( !%
$
+
+ (
K( aK)K
+
* '!
*
&
(
(
TR5: (
(
((
'
(
)
)
!"
+
$
M #n times+
+ K(,d1 )a a( a1 Kaa d1e1 Ken-a -1)b(b1 Kba + bd1f1 Kfn- a -b-1)c(c1 Kc ad1g1 Kgn -a -1) K
$
!$%
+
+
K( aK)K +*
is equivalent to
(
)
5
&
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
'
)
) &( !"
"
+
+ ( $
$
$
+
+ ( M #a times
$
+
+ ( $
$
+
%
+ ( !$
$
+
+ ( ac
$
+
+ (
$
+
+ ( !"
$
+
+ ( $
$
+
#n times+ ( M #n , a , 1 times
+ a a(a Ka – e Ke
$
+K $
1
a
1
n, a,1 ) b(b1 Kba + b – f1 Kfn,a , b,2 )c(c1 Kc a – g1 Kgn ,a ,1 ) K
+
! $
+ ( !$
%
(
+
+
! $
+
$
+ ( b
+
+ (( "
! $
+
$
+ ( !$
+
M $
+ (M$
#n , a , b , 2 times+
$
+
+
! %
+ ( !$
( $
%
+
+
K( aK)K
+
* (' !
*
!
M
!
!
!
M
[ ]
(
)
[]
TR6: a(–K –) is equivalent to a.
4. Example of Set theory Theorem which is Proved by Sentence Bi-Logic
To verify a sentence of first order predicative calculus we turn it to a sentence of standard sentence
logic by converse application of the given inferences.
Theorem 1: For every set x, there is a such set y thatcardinality of y is greater than cardinality of x.
If x is included in y, then the cardinality of x is not greater than the cardinality of y. Every set is
included in V. Hence, V is not a set.
In formalized language:
( ) ( )
CKK (!x1 )( "x 2 )CA11 (x1 ) KA11(x 2 )A21 (x2 x1 )( !x1 )(!x2 )CA22 (x1x2 ) NA12 (x1x2 )(!x1 )A22 x1a1 NA11 a1
by application of TR1:
C( !x 3 )A12 (x 3 )(!x3 ) KA12 (x 3 )CKK ("x1 )(!x2 )CA11 (x1 ) KA11 (x 2 )A21 ( x2 x1 )("x1 )("x 2 )CA22 (x1x2 )NA21 (x1x2 )( "x1 )A22 (x1x 3 ) NA11 (x3 )
by application of TR2:
C( !x 3 )A23 (x 3x 3 )(!x3 ) KA23 (x 3x 3 )CKK ("x1 )(!x2 )CA24 (x1x1 )KA24 (x 2x 2 )A21 ( x2 x1 )("x1 )("x 2 )CA22 (x1x2 )NA21 (x1x2 )( "x1 )A22 (x1x 3 ) NA24 (x 3x 3 )
by application of TR3:
"
%
!
$
'
$
' C((x3 )a (x3 x3 )( (x 3 ) Kb(x3 x3 )CKK ()x1 )((x2 )Cc(x1x1 ) Kd (x 2x 2 )e(x 2x1 )()x1 )( )x 2 )Cf (x1x 2 ) Ng (x1x 2 )()x1 )h(x1x 3 ) Ni (x3 x3 )
!
$
'
$ (ab)(cdi )(eg )( fh)'
#
&
by iterative application of TR4:
" (c) %
"
%
" ( fg ) ! %
" (h )%
!
$
'
$
'
$
'
$ '
$
' C((x )a (x x )( (x ) Kb(x x )CKK $ (ce)' ( (x )Cc( – – )Kd (x x )e (x – )$ ! ( fg )' Cf (– – )Ng ( – –)$ ! ' h( –x ) Ni (x x )
!
3
3 3
3
3 3
2
2
2
2
3
3 3
$
'
$
'
$
'
$ '
$ (ab)(cdi )(eg )( fh)'
$ !'
$ ! ! '
$ !'
#
&
#
&
#
&
# &
by iterative application of TR5:
" bi %
" (c) de %
"
% " a%
" ( fg ) ! %
" ( h)%
!
$
'
$
'
$
' $ '
$
'
$ '
$
' C$ a ' a( – –)$ bhi ' Kb(– – )CKK $ (ce) d ' Cc(– – ) Kd ( – –)e( – –)$ ! ( fg )' Cf ( – –) Ng (– – )$ ! ' h( – – )Ni ( – –)
!
$
'
$
'
$
'
$
'
$
'
$ '
$ (ab)(cdi )(eg )( fh)' $ ! '
$ ! '
$ ! !'
$ ! ! '
$ !'
#
& # &
#
&
#
&
#
&
# &
[]
[]
[ ]
[ ]
[ ]
[]
by iterative application of TR6:
" (c ) de %
"
% " a % " bi %
" ( fg ) ! %
" ( h)%
!
'
$
'
$
' $ ' $
$
'
$ '
$
'
$
'
$
'
$
' C a a bhi KbCKK (ce ) d CcKde $ ! ( fg )' CfNg $ ! ' hNi
!
'
$
'
$
' $ ' $
$
'
$ '
$ (ab)(cdi )(eg )( fh)' $ ! ' $ ! '
$ ! !'
$ ! ! '
$ !'
#
& # & #
&
#
&
#
&
# &
[] [ ]
[] [ ]
by application of W3:
[ ]
[ ]
6
" ( c) 2de %
"
% " a % " bi %
" ( fg ) ! %
" (h )%
!
'
$
'
$
' $ ' $
$
'
$ '
$
'
$
'
$
'
$
'
$
'
!
C a a bhi ' KbCKK $ (ce ) d ' CcKde ! ( fg ) CfNg $ ! ' hNi
$
' $ ' $
$
'
$ '
$ (ab)(cdi )(eg )( fh)' $ ! ' $ ! '
$ ! ! '
$ ! ! '
$ !'
#
& # & #
&
#
&
#
&
# &
[] [ ]
[] [ ]
[
[ ]
]
by application of W2:
" (c ) 2de %
"
% " a % " bi %
" ( fg ) ! %
" (2h)%
!
'
$
'
$
' $ ' $
$
'
$
'
$
' C$ a ' a $ bhi ' KbCKK $ (ce ) d ' CcKde $ ! ( fg )' CfNg $ ! ' hNi
!
'
$
'
$
' $ ' $
$
'
$
'
$ (ab)(cdi )(eg )( fh)' $ ! ' $ ! '
$ ! ! '
$ ! ! '
$ ! '
#
& # & #
&
#
&
#
&
#
&
[ ]
[ ]
[ ]
[ ]
[
[ ]
]
by converse application of U1 and U2:
" ! (cde )%
"
% " a % " bi %
" ( fg ) ! %
" (2h)%
!
'
$
'
$
' $ ' $
$
'
$
'
$
'
$
'
$
'
$
'
$
'
!
C a a bhi ' KbCKK $ ! (cde )' CcKde ! ( fg ) CfNg $ ! ' hNi
$
' $ ' $
$
'
$
'
$ (ab)(cdi )(eg )( fh)' $ ! ' $ ! '
$! ! '
$ ! ! '
$ ! '
#
& # & #
&
#
&
#
&
#
&
[ ]
[ ]
[ ]
[ ]
by converse iterative application of K3:
" ! (cde) fg ! 2h %
"
% " a % " bi %
!
( ) ( )'
'
$
$
' $ ' $
$
'
$
'
$
$
'
!
C a a bhi ' KbC $ ! (cde) ! ( fg ) ! '' KKCcKdeCfNghNi
$
' $ ' $
$ (ab)(cdi )(eg )( fh)' $ ! ' $ ! '
$! ! ! ! ! '
#
& # & #
&
#
&
[] [ ]
[] [ ]
by converse application of C2:
" cde
%
"
% " a % " bi %
!
fg ! 2h '
'
$!
$
' $ ' $
$
'
$
'
$
$
' C a a bhi Kb ! cde ! fg ! ' CKKCcKdeCfNghNi
!
'
$
'
$
' $ ' $
$ (ab)(cdi )(eg )( fh)' $ ! ' $ ! '
$! ! ! ! ! '
#
& # & #
&
#
&
[]
[]
[ ]
[ ]
[ ][ ] [ ]
[ ] [ ]
by converse application of K1:
"
% " a % " bi ! cde fg ! 2h %
!
'
$
' $ ' $
$
' C$ a ' a $ bhi ! cde ! fg ! ' KbCKKCcKdeCfNghNi
!
'
$
' $ ' $
$ (ab)(cdi )(eg )( fh)' $ ! ' $ ! ! ! ! ! ! '
#
& # & #
&
[ ] [ ] [ ][ ] [ ]
[] [ ] [ ] [ ]
by converse application of C3:
"
( a) bi ! cde fg
!
$
$
!
( a) bhi ! cde !
$
$ (ab)(cdi )(eg )( fh) ! ! ! ! !
#
[ ] [ ][ ] ! [2h]%'
[ ] [ ] [ fg ] ! '' CaKbCKKCcKdeCfNghNi
!
! '&
by iterative converse application of Z2:
" !!!(a ) bi cde fg 2h
%
$
'
$
' CaKbCKKCcKdeCfNghNi
!!!
a
bhi
cde
fg
(
)
$
'
$ !!!!!!!(ab)(cdi )(eg )( fh)'
#
&
[ ][ ][ ][ ]
[ ][ ][ ]
by iterative converse application of U1:
" !!!!!!!( abicdefgh ) %
$
'
$
' CaKbCKKCcKdeCfNghNi
!!!!!!( abhicdefg )
$
'
$ !!!!!!!(ab)(cdi )(eg )( fh)'
#
&
by iterative converse application of Z1:
! ( abicdefgh ) $
#
&
# ( abhicdefg ) & CaKbCKKCcKdeCfNghNi
#
&
# (ab)(cdi )(eg )( fh)&
"
%
by iterative converse application of S4:
! (ab)(cdi )(eg )( fh)$
#
&
# (ab)(cdi )(eg )( fh)& CaKbCKKCcKdeCfNghNi
#
&
# (ab)(cdi )(eg )( fh)&
"
%
by iterative converse application of S3:
(ab )(cdi )(eg )( fh )CaKbCKKCcKdeCfNghNi
by iterative converse application of S2:
7
(a )(c)(e )( f )CaKaCKKCcKceCfNefNc
by converse application of S1:
CaKaCKKCcKceCfNefNc
by valid inferences in standard sentence logic:
CaKaCKKCcKceCfNefNc
CaCKKCcKceCfNefNc
CaCKKCceCfNef Nc
CaCKKCceNe fNc
CaCKKNcNefNc
CaCNcNc
CaCcc
Ccc
( ) ( )
So, CKK (!x1 )( "x 2 )CA1 (x1 ) KA1(x 2 )A1 (x2 x1 )( !x1 )(!x2 )CA2 (x1x2 ) NA1 (x1x2 )(!x1 )A2 x1a1 NA1 a1
1
1
2
2
2
2
1
is
deduced from Ccc , but Ccc is a theorem of standard sentence logic (self-impication of a sentences),
1
1
2
2
2
2
1
thus CKK (!x1 )( "x 2 )CA1 (x1 ) KA1(x 2 )A1 (x2 x1 )( !x1 )(!x2 )CA2 (x1x2 ) NA1 (x1x2 )(!x1 )A2 x1a1 NA1 a1
is
( ) ( )
theorem in sentence bi-logic and in predicative calculus of first order too.
5. Conclusions
Sentence bi-logic could give a method for a complete truth calculus of first order predicative
calculus and therefore of any known mathematical structure, but other studies are necessary for this
achievement.
6. References
Malatesta M., The Primary Logic. Instruments for a Dialogue between the two Cultures, Gracewing
Fowler Wright Books, Leominster, Herefordshire, UK, 1997
Matte Blanco I., The Unconscious as Infinite Set. An Essay in Bi-Logic, Duckwort, London, 1975
Matte Blanco I, Thinking, Feeling and Being. Clinical Reflection s on the Fundamental Antinomy of
Human Beings and World, Routlege, London - New York, 1988
Mendelson E., Introduction to Mathematical Logic, Van Nostrand Co., Princeton, New Jersey, 1964