A NERVE NE T SYSTEM I N MO DA L LOGI C*
KURT W. BACK
Duke University
1. INTRODUCTION
The applic at ion of mathematical models t o social science, especially psychology, social psychology, and sociology, has f ound t wo k inds
of pitfalls. Either the mathematical met hod is so specific that any application has t o be c onf ined t o very restricted or t riv ial situations or
they us e mat hemat ic s p r i ma r i l y f o r illus t rat iv e purposes. Th e ac ceptance of many new mat hemat ic al and logic al techniques i n social
science f ollows frequently a coresponding pattern, of first acceptance
of t he n e w concepts widely wit hout part ic ular concern f or the exact
procedure inv olv ed, t hen applic at ion of t he technique by itself, and
f inally restriction t o problems whic h hav e only marginal relat ion t o
the problem under consideration wi t h possible f urt her expansion by
analogy. This seems to have been the fate of such new techniques as
inf ormat ion theory, game theory. Mark of f chains, o r net work theory.
A f undament al reason f or this dilemma is i n the state of the k nowledge of social phenomena and, possibly, i n t he nat ure o f t he dat a
themselves. The mult iplic it y of factors t o be considered and t he uncertainties s t ill present i n t he meas urement o f human beings f orc e
excessive constraints t o apply rigorous mat hemat ic al methods. I n stead of f orc ing events int o a structure t o whic h these methods fit, i t
may be preferable to loosen up the logical procedure t o make it more
congenial t o the subject matter. I f this is done systematically, the procedures used can s t ill have referents in the more exact rules of logic.
One such modif ic at ion is giv en i n t he t heory of modal logic. A d dition of possibility and necessity to the logical operators makes logic
more applic able t o t he s t udy o f ac t ual h u ma n behav ior. W e s hall
examine t he t heory of nerv e nets whic h — besides it s int rins ic i mportance — is an ex plic it analysis o f f ormal logic i n its relat ion t o
life sciences. B y s howing t he dif f ic ult ies i n ac t ual applic at ion o f
f ormal logic, we can see the needs of a logic adapted to interpersonal
• Paper presented a t Th ird Bionic s Sy mpos ium, Day ton, O h io , Marc h 1963.
Preparation of this paper was supported by a grant f ro m the Office o f Nav al
Research, G roup Psychology Branc h, u n d e r Contrac t No n r 1181(11), Projec t
NR 177-740. Reproduc tion in whole o r in part is permitted f or any use of the
United States Government.
22
behavior and the manner i n whic h modal logic meets some of these
requirements. I t wi l l als o be s hown h o w modal logic can be represented i n net work s s o t hat t he grounding i n strict procedure is not
lost.
1.2 N e r v e N e t Th e o r y a n d L o g ic
Nerve net t heory as dev eloped b y Mc Culloc h a n d Pit t s ('), v o n
Neumann (
2
ferent
input -out put relationships, t hus s imulat ing s ome behav ior o f
liv
) , ing organisms. The nerv e net work s essentially as a binary f ormal
logical
representing t he operations o f Boolean algebra, i n S h a system,
n
cluding
n o n counting, and is thus capable o f perf orming ordinary arit hmetic.
a
n I t can detect o r correct errors, achieving any degree of desired
accuracy. Howev er, this is paid f or at the cost of bulk and c omplex id
ties.
M Th
o i s mode o f representation, f o r a l l it s advantages, has s ome
obvious
o
r deficiences i n t he repres ent ing o f h u ma n behav ior. S o me
of
these
are connected wi t h t he nat ure of deductive logic. This logic
e
ignores
the
content of a proposition; the only measure of a proposition
C
assessed
is
its t rut h value. Hence, any t rue or any false propositions
)
are
equiv
alent
t o each ot her, a n d any c ombinat ion o f propositions
a
is eit her true or false. I n consequence, in this system any proposition
n
is relevant to any other.
d
This indifference to content contributes to the mat hemat ic al elegano
ce o f t he t heory b u t mak es f o r dif f ic ult ies i n applic at ion. I t ev en
t
results i n needless complexities in the establishment of accuracy: erh
rors i n t he transmission of a proposition such as «Two and t wo are
e
four» mus t be checked i n t he s ame wa y as those f o r a propos it ion
r
like «I t is raining today.» I t becomes an even greater obstacle i n the
sunderstanding o f psychological processes, as i n ac t ual prac t ic e t he
ctruth and falsity of any proposition do not have implic at ions f or the
atruth o r f als it y of a l l ot her propositions as required i n Boolean aln
r e
minent
in Nerv ous Ac tiv ity ., Bullet in o f Mathematic al Biophysics, 1943, 115e )
133.
M
p
(
r C
Organisms
f r o m Un re lia b le Components ., i n A ut omat a Studies , Princ et on
2
C
e
Univ
)U . Press, 1956.
s (V
L
e
Relays»,
3
jo u rn a l of t he Frank lin Ins titute, Sept. and Oct., 1956, 191-208, 281O
L
297.
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gebra. There it is a consequence of the def init ion of mat erial implic ation; t hat is, t hat a t rue propos it ion is implied by any propos it ion
and a false proposition implies any proposition. At t ent ion t o content
requires a dif f erent iat ion in t he meaning of truth, whic h is not equivalent t o a s imple probabilit y measure but t o a dis t inc t ion bet ween
analytic and synthetic t rut h. I t also leads t o t he not ion of relevance.
A f unc t ioning nervous system operates wi t h continuous input and
output. I t has been surmised that it typically operates i n an analogue
way a n d t hat digit al analy s is constitutes a lat e a n d ref ined p r o cedure (
digital
manner because the nerve net consists of binary units, neurons
4
and
) ( synapses. The v alue of this t heory wi l l be increased i f features
5of t he analogue operat ion o f t he nerv ous system c an be translated
into
) . essentially binary calculus o f nerv e net analysis. One such approach
N
e i s t he t rans it ion f r o m t he classical logic al algebra t o modal
rlogic.
v
e
n
2. MODAL LOGIC
e
t
logic includes a l l t he princ iples of Boolean algebra and i n
t Modal
h
addition the operations of necessity and possibility. I t is also at least
e
a t hree-v alued logic admit t ing a v a l u e o f «indet erminat e». Thes e
o r
features lead t o t he inc lus ion of strict implic at ion, wh i c h is def ined
y
,
as I t is not possible that a is true and b is not true.» I f this relat iona
ship holds bet ween t w o propositions, b c a n b e deduc ed f r o m a
n
because of some necessary connection bet ween t he t wo propositions.
a
If t hey bot h o n l y happen t o be t rue, s t ric t implic at ion i s indet erlminate. Deduction is thus used in its ordinary usage, and the meaning
yof a proposition is recognized, besides its t rut h value. The addit ions
smade by modal logic permit a closer approx imat ion of net work s t o
eactual processes of organisms.
s Since renewal of interest in modal logic t hrough C. I. Lewis ' paper
i in 1918 (°) (
tapplication
7
t o nerv e net t heory a system is required wh i c h has exs)
i s (e v e r
(4
na l
hav
) ioral Science, Oct., 1962, 430-442.
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plicit transformation rules, a mi n i mu m of primit iv e concepts, and no
differentiation o f impulses by content. Th e best ex ample o f such a
system was found in Prior's system Q (
8 Prior defines t he modal operators as apply ing t o t ime: necessity
«t rue at a l l times», possibility «t rue at some t ime», t rue and
)means
.
false ref er only to the present time. As k nowledge about ot her times
may be imperfect, we have to admit a t hird value — indeterminate —
for all times except the present one. This is a three-valued logic; although Prior assigns numeric al values t o the three states, he does not
use t hem in any ordinal sense but only as a nominal distinction. We
shall ident if y t hem, therefore, by symbols + , —, o r O. Each proposition is t hen characerized by a s t ring of these symbols, theoretically
infinite, each symbol representing the t rut h of the proposition during
a part ic ular time interval. The first symbol must be + o r T h e logic
is based on f our pimit iv e operations: negation, conjunction, necessity,
and possibility. Rules are given f or the transformation of each symbol
under eac h operat ion. This , t hen, i s t he required dev elopment o f
modal logic. I t can be represented in a nerve net or a c omput er if it
is possible t o use t he three-valued strings as input and t o def ine t he
operations i n a way in whic h the nerve net system can handle them.
3. RULES FOR PRIOR'S SYSTEM Q
Let us first define the f our operations. The definitions given here are
equivalent to those used by Prior (
9points useful f or later application.
) b u t
m o d i f i e d
t
o
b3.1 Not
r at ion
i
n
g
o
u
t
c Propositions
e
r
t designated
a
i
are
by s mall letters, a, b, . . . ; operations by
ncapital letters N, K, L, or M. Operation symbols are put before the propositions they ref er to. Propositions consist of a s t ring of symbols + ,
—, or 0, but the first symbol cannot be O. Three of the operations have
only one argument : Na (negation of a), La (a is necessary), and Ma
(a is possible). The f ourt h operat ion has t wo arguments: Kab (c onjunction of a and b). Operations can be c umulat ed and are perf ormed
(
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8
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)
P
P
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25
f rom right to left; e.g. NMNa: it is not possible that not a, or NKaNB:
it is negated t hat a and not b, i.e. mat erial implic at ion.
3.2 Trans f ormat ion Rule of Indeterminacy (0)
Any operat ion whic h includes a 0 wi l l res ult i n a O. I n a s inglevalued operation 0 is invariant. I n a t wo-v alued operation 0 in eit her
of the t wo arguments wi l l result in O. Thus none of the operations wi l l
make indet erminat e k nowledge determinate. I nf ormat ion c annot be
gained t hrough f ormal logic of any kind.
3.3 Trans f ormat ion Rules f or N and K
Both of these operators wo r k on single symbols wit hout regard t o
the rest of the structure. N changes plus to minus and minus to plus.
K operates on a pair of coresponding symbols in the sequences in the
same way as ordinary conjunction does, i.e. t wo plusses mak e a plus,
all ot her combinations are minus . Zeros are dealt wi t h according t o
rule 3.2. These t wo operations and t heir combinations mak e all operations o f Boolean logic possible.
3.4 Trans f ormat ion Rules f o r L and M
The operators change t he symbols according t o t he type of whole
sequence. W e dis t inguis h t hree types ac c ording t o t he presence o f
plusses in the sequence. Ty pe one consists of sequences whic h contain
only plusses (propositions wh i c h are alway s t rue); t y pe t wo o f sequences whic h contain plusses and ot her symbols (propositions whic h
are sometimes t rue); t y pe t hree o f those wh i c h c ont ain n o plusses
(propositions of whic h i t is not k nown i f they are ev er true). Ty pe
one and type three stay inv ariant under both L and M transformation.
In t y pe t wo operat or L changes bot h plus and minus t o minus , and
operator M changes bot h t o plus . Again, z ero stays inv ariant . I n
other words , these rules mean t hat a proposition is necessary at any
time if it was t rue at all times, and it wi l l be possible at all times if
it is true at any time, except, of course, at those times at whic h not hing
is k nown about the t rut h of the proposition.
3.5 Meaning and Degrees of Trut h
These rules a l l o w a l l t he operations of modal logic , i n part ic ular
strict implication, which is defined as NMKaNb, it is not possible that
26
a and not-b occur jointly. For t wo propositions to be connected in that
way, b must be true whenev er a is true. I f the t rut h of b is not k nown
for any t ime period when a is k nown t o be true, t here is n o necessary relationship between the t wo propositions, alt hough they are not
inconsistent wit h each other. Thus t wo propositions can be indif f erent
t oward each other. Meaning is t hus def ined as t he pat t ern of t imes
in whic h a propositions is true.
This system thus permits discussion of several degrees of truth, can
differentiate between different propositions beyond t heir present t rut h
value, admits unconnected propositions and does this in a way whic h
consists of simple transformations of one another. We shall n o w construct a scheme of a nerve net whic h can perf orm according t o t his
system.
4. THE NETWORK SYSTEM
4.1 Th e Ty pe of I nput
In order t o adapt t his modal system t o use i n nerv e net theory,
we have to specify an input in such a way that a three-valued signal
can be transmitted t hrough an of f -on circuit and make the operations
conform t o t he basic operations o f the nerv e net. This means essentially that we can specify an out put pattern f or each c ombinat ion of
inputs. A system specified in this manner can perf orm the basic operations of Boolean algebra but cannot distinguish between t wo input s
to an organ bey ond an on o r of f state. I f these restrictions are observed, i t has been s hown t hat nerv e nets c an be d r a wn by s imple
rules f r o m t he input -out put c ondit ions ("). Cons equent ly i t is s uf f icient to present here only the in-out matrices.
As stated above, P r i o r conceives o f eac h propos it ion as a s t ring
of three-valued symbols. Our nerve net wi l l consist of a set of parallel
organs, one of each symbol. Alt hough the set is theoretically inf init e,
it corresponds i n prac t iv e t o a f init e number, say n. Eac h o f these
organs consists of t wo parts, t he determinac y net (D) and the falsity
net (F). The inputs in the D net are f iring i f the proposition was indeterminate at t he t ime t o wh i c h t he organ refers, and does not f ire
otherwise. The inputs in the F net are f iring if the proposition is false
at t he t ime and do not f ire otherwise. Outputs are read in t he same
(10) CULBERTSON, J . T., »Nerv e Ne t Theory », i n Comput er Applic ations i n
the Behavioral Sciences, Prentice-Hall, 1962.
27
way: I f t he D-effector fires, t he v alue corresponds t o indet erminat e
(zero) n o mat t er what the F-effector does; i f t he D-effector does not
fire, f i r i n g i n t he F-effector means «false», n o f iring means «true».
The whole net work has thus (2n) receptors and (2n) effectors.
The rules or the operations div ide int o t wo kinds: t he t wo Boolean
operations, K and N, and t he modal operations, L and M. Fo r t he
f ormer each organ can f unc t ion separately, and only the three-valued
nature of the logic provides any differences f rom ordinary nerve nets.
For t he lat t er t he whole net work has t o be analyzed and classified
into one of the t hree type before proceeding on each organ.
4.2 I nput -Out put Condit ions f or K and N
For t he operator N t he rules are as f ollows . I n t he D-net work the
output equals t he input . I n t he F-net work t he out put is t he opposite
of the input , dead on a f iring input and vice versa.
K operates wi t h the inputs f rom t wo propositions, a and b; t he inputs f o r net work a r e c ombined s eparat ely ac c ording t o t h e f o l lowing table:
Input
Output
11
1
10
1
01
1
00
0
Apply ing t his s c heme t o t he D-net work mak es t h e ef f ec t or f i r e
whenever one of the propositions is indet erminat e; apply ing it to the
F-network makes t he corresponding effector f ire o n l y wh e n neit her
proposition is false. A s t he F-effector is irrelev ant whenev er t he Dnetwork fires, i t does f ire only when bot h propositions are true.
4.3 I nput -Out put Condit ions f or L and Al
Before apply ing t he operators t he whole input pat t ern has t o be
analyzed and must therefore be stored. The analysis for det erminat ion
of t he type (see above) proceeds as f ollows : The D and F input s are
combined according to t he f ollowing table:
Input
Output
11
0
10
0
01
0
00
1
This results in only the «true» symbols f iring. Af t er this t he number
of f iring f ibers is counted. The det erminat ion of class is as f ollows :
28
X (number o f outputs o f T y p e
previous table f iring)
O<X<n
0
After this each organ is treated separately, but there are t wo kinds
of procedures, one f or types I and I I I and one f or
For types I and I I I the process is the same as f or L and M: t he Dnetwork t rans mit s t h e i n p u t unchanged. Ex c ept f o r indet erminat e
points i n t y pe I I I , t he out put of t he D-net work is alway s zero. Fo r
the F-net work the inputs are reversed to obt ain the meaning of f iring
as «false».
For type l i t h e D-net work transmits the input unchanged. I n the Lorgan a l l outputs i n t he F-net work fire, and i n the M-organ no out put fires. This type corresponds t o a proposition whic h is t rue at least
once, but not at a l l times. I t is t heref ore necessary at no t ime and
possible at all times at whic h any inf ormat ion is available.
Wit h a c ombinat ion of these organs a l l operations of modal logic
are possible. B y giv ing t he activation negative meaning ( i n t he t wo
circuits on means indet erminat e and false respectively), t he required
organs inc lude the NA ND and NOR circuits and thus are convenient
f or transistor circuits (
1
).
5. APPLICATIONS
The use o f modal logic gives f lex ibilit y t o the use of networks and
makes i t possible t o use t hem i n n e w contexts. To s how t wo of its
most interesting features, t wo kinds of applications wi l l be sketched.
5.1 Reliabilit y
The ques tion of possible f ailure of eit her t he input s of the system
has been attacked i n dif f erent ways. Here we c an use t he dif f erentiation bet ween analy t ic a n d synthetic statements. I f we hav e admitted the necessary t rut h of statement (La) at any t ime, i t mus t be
true at all times. Thus, f or any statement of t his k ind an error whic h
(
Computers,
i
Prentic e-Hall, 1963.
l
)
M
A
L
E
Y
,
G
.
A
.
a
29
make t he statement t rue at some t imes and not at others is self-detecting and can then be checked. On the ot her hand, synthetic propositions wh i c h are possible b u t n o t logic ally necessary a l l o w s ome
errors whic h would keep t hem synthetic, and only t heir t rut h at dif ferent t imes wo u l d be stated i n error. I t is lik ely t hat t his div is ion
leads t o economy in actual t hink ing. I n propositions whose universal
truth i s accepted, a n y e r r o r i s aut omat ic ally ex purgat ed. O t h e r
statements depend on correct input ; t hat is, exact empiric al observation I n this connection i t is instructive t o note t hat Cowan (") has
shown he equivalence o f error-correcting codes t o t he Lewis mu l t i valued logic. I t is proposed here that starting wit h a modal logic such
as Lewis ' and Prior's system wi l l lead t o dif f erent and s impler looks
at the problem of reliabilit y .
5.2 Cons is t enc y
Instead of different symbols ref erring to different time units we may
also conceive that they ref er to the relat ion to dif f erent conditions. We
can t hen analy z e dif f erent propositions o n whet her t hey a r e c onsistent, inconsistent, a n d indif f erent under dif f erent c ondit ions a n d
under wh i c h c ondit ions a person ent ert aining these propos it ions is
consistent or inconsistent and what he can do to achieve consistency.
This would t hen be an approach t o oPsycho-Logic» ("). The symbols
can als o be int erpret ed as affective relationships, lik e, dis lik e, a n d
neutral, and consistency of preferences can be analyzed. A number of
current psychological t heories postulate a driv e t oward balanc e o r
consistency (14,15, le, 17). Dav is (") has propos ed a f ormaliz ed system
for these theories. Derivations f rom this system show clearly the limi(
Presence
I
o f Nois e', in Bionic s Symposium, Wrig h t A i r Dev elopment Div is ion,
1960.
2
)(
of C
1At t it udinal Cognition», Behav ioral Science, ja m, 1958, 1-13.
(")
O
3 FESTINGER, L., A Theory o f Cognitiv e Dissonance, Row-Peterson, 1957.
()
W
(")
1
A
A NEWCOMB, T. , " A n Approac h t o t h e St udy o f Communic at iv e A c t s ',
Psychological
Rev iew, Nov . , 1953, 393-404.
5
N
B
) OSGOOD, C. E., Sum, G. J. a n d TANNENBAUM, P. H., Th e Meas urement o f
,(")
E
Meaning,
Univ . o f I llin o is Press, 1957.
JH
L
.(E
S
sonal
1
IO Relations», Americ an J ournal o f Sociology, Jan., 1963, 444-462.
D
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,
ting point o f t he princ iple o f s imilarit y and balance and t he point s
where a theory of dis s imilarit y would have t o enter. Balance theories
can b e t ak en as des c ribing l i mi t i n g c ondit ions (
1
shown
here introduces new concepts whic h wi l l help in investigating
these
erent
D
) . dif
T fh
e sets
p of
r processes.
o c e dThere
u r ise a difference between saying
that indiv iduals tend t oward situations whic h are compatible and that
they t end t o situations i n whic h all points are necessarily connected.
It is not clear whic h of these suppositions are basic to balance theories.
Dealing wit h ideas whic h are possible but not necessary or not even
true at the present t ime is not incompatible wit h organization of one's
cognitions a n d may represent muc h ac t ual human behavior. Understanding of imbalanc e t hrough t he logic of possibility and necessity
wi l l t hen give a more lif elik e model of human behavior.
Duke University
Kurt W. BACK.
(
Mathematical
Met hods i n S m a l l G ro u p Processes, S t anf ord U n i v . Pres s ,
I
1962.
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