MODAL ITY DE RE A ND VASILIEV'S IMAGINARY LOGICS
V.A. SMIRNOV
The theme o f my report is the reconstruction o f the Vasiliev's
logical systems and the analysis of the modality de re. Unfortunately,
modal syllogistic o f Aristotle has not been adequately reconstructed
till now. Even Charles Peirce in this time remarked that a t the
foundation of all Aristotle's philosophy lies his teaching of modality.
In my opinion, the key problem in the understanding Aristotelean
modal syllogistic is given in —
iVasiliev.
m a g i Vasiliev
n a t i v has
e " published
( n
his original works in years 1910-1914.
oI'll ngive
- the reconstruction of imaginary logic of Vasiliev — the logic
A r i sthe
without
t law
o t of
e contradiction.
l e a n
Then I shall show how to pass on
(from
it) to the modal syllogistic of Aristotle proper. But first of all
)
dwell
on
syllogistic.
l
o assertoric
g
i
c In 1974 1 proposed interpretation of syllogistics in terms o f predicate
calculus. f I proceeded fro m the idea o f Ockam, that positive
o
sentences affirm non-emptiness of the subject. But negative sentences
do not. Lewis Carrot essentialy uses this idea. But he did not analyse
particular negative sentences, that is OSP. Th is operation is a s
follows:
(ASP)
(ESP)'
-1= V x (Sx D T P x )
(JSP)
=
-1(OSP)
3
=x )+ = T (a 1
-1(Ta
3Sa 4 )
+
(=
xx
3
+
Then I gave the following axiomatization of syllogistic C2:
x(=&
1. A S M & AMP D ASP
(S
VaA )S M & EMP D ESP
2.
-x3.
x 1ESP
- D EPS
&
D
(4. ASP D JSP
0
xP
(5.
S JSP T ESP
(xx
p
6. OSP A S P
)S
D JSP D ASS
7.
°xP
&
,x
T
w
)
P
h
x
e
)r
e
0
i
s
&
206
V
.
A . SM I R N O V
I proved the following theorem: a is provable in C2 i f f a i s
provable in one-place predicate culculus.
The proof of this theorem is not trivial.
I think that system C2 is a very natural one. I could not find laws of
identity ASS and ISS in Aristotle's works. These laws are introduced
by Leibnitz.
The system C' — without identity laws may be extended to get a
system definitionally equivalent to Boolean-algebra. It was stated by
Bocharov. I changed his axiomatization as follows
C2D = C2 +
8. ASP D ESP'
9. ASS & ESP' D ASP
10. E (S n P )M D E ( M n S ) P
11. E M (S P ) E M S & EMP
12. E M(S n P)' = EMS' & EMP'
C2D is a definitionally equivalent to Boolean algebra of classes, that
is C2 D + S c P . E S P ' i s equivalent t o B + A S P .
(S
- P) + E S P . S c P ' + J S P —
P
R
+ &
1 (.SS— Pc 'S ) ' )
R
S
Let
us
go
back
to
Vasiliev. I n his first wo rk "About Particular
O
S
S '
Judgements,
about the Triangle o f Opposites, about the Law o f the
)
excluded
Fourth" Vasiliev constructs syllogistics without the " L a w
— the excluded Th ird " . I t is necessary to emphasize that Vasiliev
of
1
distinguishes
between two levels o f logical Laws. He refers to the
( S regulating activities of cognizing subjects as the laws of the first
laws
g
level.
He calls these laws metalogical laws. Vasiliev accepts the
principle
that no statement can be both true and false (The Law o f
P
non-selfcontradiction).
It is also accepted that every statement must
)
be
true
or
false.
The
other
level is the level of ontological character. It
.
is connected with ontological commitments: I t is not true that a is P
and a is not P". This is just the Law of Contradiction. According to
Vassiliev it is possible to think o f the logic without the ontological
laws of contradiction and the tertium non datur.
I shall proceede from the presupposition that the laws of metalogic
are the laws of the logic of propositions. And, according to Vasiliev,
this logic is always a classical logic.
MODALITY DE RE A ND VASILIEWS IMAGINARY LOGIC5.
2 0 7
Non classical elements are expressed by the fact that different types
of predications are possible. Or, if we regard the logical system as a
combined calculus o f propositions and classes then the Logic o f
propositions may be expanded by different theories of classes.
I do not agree with Mal'cev and Kline that Vasiliev had formulated
the idea of manyvalued logics. Of course Vasiliev's ideas prompt to us
the propositional Logics, th a t differ f ro m classical Logics. Th is
approach was very well realized by Aida Aruda. But I think that if we
identify Vasiliev's metalogies with propositional logics as I supposed - then the propositional logic is a unique and it is classical logic.
Vasiliev in his first work "About particular statements..." takes as
basic three type of statements:
ASP - general positive
ESP - general negative
TSP - accidental (particular) - O n l y some S are P".
And statements of each of these types relate to the whole extension of
S. He gives two interpretations to the statement TSP: disjunctive and
accidental.
Some words about the disjunctive interpretation. Sentences " a ll S
are P or Q" Vasiliev interprets as "some S are P and all others are
Q" ; i n symp o lic n o ta tio n o f predicate ca lcu lu s w e h a ve :
dx(Sx &Px) & 3x(Sx & Qx) & Vx(Sx D Px V Qx). That is wh y, the
sentence TSP must be interpreted as 3x(Sx & Px) & 3x(Sx & 1 Px) &
Vx(Sx D Px V —
I It
P is
x )important
,
f o r u s that ASP, ESP and TS P are pairwaise
inconsistent
t h a t in pairs and their disjunction is true (The L a w o f the
excluded
Fourth). No w it is easy to propose the axiomatic o f this
i
s
systems
a
s
3
x I. A S M & AMP A S P
( S 2. A
x S M & EMP D ESP
&
3. ESP D EPS
P
4. T (A S P & ESP)
x
5.) —
&
R A6.
3
7.
V ESP V TSP
Sx ASP
1
P
(
S
8.
(& EESP v ASS
x
T
S
&
S
P
P
&
1
)P
T
x
)
S
.
P
)
208
V
.
A. SMIRNOV
It is easy to show that this Vasiliev's system is definitially equivalent to C2. We add to Vasiliev's system:
JSP --- ASP v TSP
OSP E S P V TSP
and add to C2
TSP
&
OSP
Formally, Vasiliev's syllogistic system is given in his paper "About
particular statements
- i s though
gistic,
a
he himself gives a different interpretation to his system.
d The
e f accidental
i n i t i ointerpretation gives us another system of syllogistic.
the next translation of accidental sentences into S5it :
nI propose
a l
e q u i v
y (A
a l e n t
cD
p (E
t
cD
S
p (T
o
S
v
P
)
In
T
s thist translation
a
P
S
=
)
yn Saxiom
and
P
8
E
d
a
=3lP o)ofwAxiom
fr o SldP
D
replacing
s
8 for E
V
xSA
fs S
V
D
translation.
r =
m
D question
E about completeness have not been discoveyo The
xS
3
red.
Jl
D
Sl S
P
.
x(V
x system,
P
S
C
S
A
given in the work "Imaginative (non Aristotelean)
o Another
S
<
&
i&o g icsOs
L
S
x,o
V n s is non-valid.
P o >
cn
—
Contradiction
i s
Vasiliev supposes that positive atomic
x n u oft the kind " a is P" and only these, are the foundation of
statements
itb s e (P
xD standard
the
Logics. Negative sentences are not atomic sentenfvt usual
o
aS
ol
)
x
(
ces.
They
are
sentences
about the falsity of atomic positive sentences.
frin d
.
&
S
All
negative
statements
are the result o f deduction fro m atomic
o r
tL
g
h
i
O
x
positive
statements and statements about the incompatibility o f prost
e
DFo
perties.
tv t P
a
i r example " a is not re d " is the conclusion fro m " a is
x
O
green"
and
c r
C
e
e "green i s incompatible wit h re d " . I n an imaginary
)
P
situation
is
v
iV
e possible that both positive and negative statements are
&
x
atomic.
In
r
A
n
s this case, it is possible that positive and negative state3
)
ments
may
be compatible. It is possible that sentences "a is and is not
tia
x
P"
i
s
true.
We can regard th e following statements as atomar
s
e
statements:
<
ra
>
r
e
(
se
S
ts
x
tu
&t
l
o
P
o
u
x
sf
)
.
I
MODALITY DE RE A ND VASILIENTS I MAG I NARY LOGICS 2 0 9
a is P (and it is not true, that a is not P)
a is not P (and it is not true, that a is P)
a is and is not P
There are three kinds o f atomic singular sentences in imaginary
logic: positive, negative and indifferent.
I suggest that these singular sentences could been interpreted in
topological terms, correspondently, as a el)°, a e 1
3
where
' is operation interior, + is closure, F is frontier and / is
'" a n d Naturally,
complement.
a
these
E sentences cannot be paarvise true
(
and their disjunction is true (a GP' V a EP" V a c 1 ).
1 constructs "sentences about classes" on the basis o f a
Vasiliev
(
singular sentences. There are three kinds o f general sentences and
a
four kinds o f particular o r, as Vasiliev prefers to say, accidental
E
sentences. This is a basic system of syllogistic in sence, that: every
P
pair of them cannot be simultaneously true and disjuction of all seven
°
sentences is true. Vasiliev considers general positive and general
&
negative sentences to be apodectic. Let us denote them A , SP and
E,SP. aI denote general indefferent sentences " A l l S are P and
E as A . SP. Four kinds o f accidental sentences are the follonon - P"
wings: P(I ) Some S are P and all others are non P , (2) Some S are P
and all /others are P and non P ; (3) Some S are non- P and all others
' non- P; (4) Some S are P, some S are non- P, and all others
are P and
)
are P and
non- P. L e t us denote correspondingly T n
P
T 7SSP,
P, TdSP.
,
TFollowing
1
Vasiliev it is possible to propose the following
translation
:
S
P , of these sentences into predicate calculus:
1
l( p
(a1)(E, SP) = V x (Sx D P"x)
1
A
I)
e (A, SP) = 3 x Sx & Vx (Sx P
S
F
& (T: SP) = 3 x (Sx & P
xP
tp
(TT SP) = 3 x (Sx & P'x) & 3x (Sx & P
a )3
)E1)x(T?
F
1
) SP)
& = 3lx (Sx
x & P'°x) & 3x (Sx & P
(TdSP)
=
3
x
(Sx
& Px)
x=
F
)
(
&
S
x
V
x
( &
S 3xx(Sx & P
P
1
3 )& &
xD
P
xV x of imaginary
( Sv x syllogistic of Vasiliev is the
F
The Proper
axiomatization
°
x
)
&
3
x
( vS x
x
P
P
/
"
°
x
x
)
) Th e se se ve n statements
following:
a re paarwise inconsistent
&
P
S
F
P
,
x( F
F
V )
x
x
)
x& ( )
S
a
V
x
E
xP P
D
(' x
"S v
x& P
/
D
c
)
a
210
V
.
A
.
SMIRNOV
(7 (A , S P & E , S P ), —
1 (A
disjunction
i s t ru e ( A , SP v v E, SP v A„ SP v Tr SP V T V SP v
0T7 SP v TdSP) and
S P & AA, SM & A , MP D A , SP
„
SA , P
SM &) E ,, MP D E , SP
—
A , SM & A MP 0 A, SP
1
are valid.
( A , S P
& In the above mentioned translation all axioms are valid formulas of
predicate
calculus
with operation + and ' (P' P , P Ç P
T
r
,S
P P
" ' )I have
=, not1 checked up whether the operation v is an
Until
now
imbedding
1
.
.
. operation.
)
,
The
system
of syllogistics of Vasiliev possesses specific properties.
)t .
h
General
negative
sentences are not conversible. From E, SP does not
e
i
follow
E , PS (Fro m V x (Sx D P ' ) does not follow V x (Px D S'' x) ).
r
This inconversibility E , is emphasized by Vasiliev himself.
In the terms o f basic statements there ma y be defined other
statements too, for example usual particulars:
.1
0
.1,
S SP A , SP v S P v T7 SP v TdSP
0
P
S
Their translations will be correspondingly :
A
P
vE
, O
vn,S (0 , SP) 3 x (Sx & P'"x)
S
P (1, SP) , -- 3x (Sx & P
S
P
v Fconversible, too.
P
J„ is not
)V
T x)
It is possible to introduce the statements of possibility. They are the
: necessary sentences :
T
duals of
-rS
A
-S
P
O
E
-P
v SP—
JD
S
3V
T
o
0
P
xT
S
0
—
(7
S Pstress that in Vasiliev's system there are no assertoric
We must
P
S
1
S
P proper. It seems to me that Vasiliev's syllogistic system is
S
statements
-0
P
x
V understanding Aristotelean modal syllogistic.
P
a key to
-0
—
&
T
V
-S
1
1T
d
-P
A
1d
S
-'S
0
P
—
S
xP
1
P
)
E
,
S
MODALITY DE RE AND VASILIEV'S IMAGINARY LOGICS 2 1 1
In order E
D b e Assertoric statements are interpreted in the same way as
translations.
in cC2.
o An v
accidental
o e r s i (bilateral possibility).
has moods of the first figure with one assertoric
, This
bE l einterpretation
,
and
D i other necessary premises. This system is the first approximation
of
, tAristotalean
J
syllogistics. There is the following discrepancy : J
D
o n
is
iS
ot P
conversible without additional topological presupposition.
Aristotle
rejects mood B A RO, CO
, s
D
However,
mood BO, CARDO, rejected by Aristotle, is rejected also
0 e
in
,, the
t
h i s proposed by mine.
nb uinterpretation,
ta Irodon't
a n assert
s l athat
t iour
o interpretation
n
o f modal syllogistic is fully
ir u
adequate
s to Aristotaelean texts. A t present Georgian logicians put
forward
pe g r different
o
v interpretations
a
b
l e o f Aristotle's modal syllogistics. In
.n h case,
any
Vasiliev's ideas about the possibilities o f contradictory
e
statements
become transparent and quite acceptable in the above
c t e
mentioned
topologicial
interpretation.
s os
a sr u
V.A. SMIRNOV
y b
s
s t
i
t t
u
a t
e
t S
mf
e o
n r
t S
s *
. i
An
, a
Sl
Pl
a
n
d
J
,
S
P
a
r
e
R e f e r e n ce
I . BomapoE B.A. Bymem aare6pa B Tepmmax omAxormomm. - Aormmeomme momemommm (Tpy al mayqmo-mccxeaomTe)namoro c emmapa no Aorme RacTmTyTa Om000tog AH CCCP). M. , 1983, c . 32-42.
2. BacmBez H.A. 0 qacTaax cymmemmax, o Tpeyroxamme n p o mo n o AomocTel. o 3BROHe IICTUBNeHBOPO geTeepToro. - Ymemae aemmoKM Kaaaamoro ymmepomTeTa. K a s , 1910. c . 3-47.
3. Bacmbea M.A. Boo6pamaemam (meapmoTommema) Aorma. - XypaaA
AmmoTepoTaa Hapommoro Epoommemm. 1913. T. 2-3. c . 207-246.
4. & l e mma H.A. Aormma m meTaAorma. - loroc . 1913. T. 2-3,
c.53-81.
5. Cmmpaoa B.A. Horpymeame cmoTem ommormoTmxm B omomecTmoe
monmAeame upeRmaToa. - Aommeomme mcamenomma (TpyAm Harimo-mccxeloBaTezBoxoro cemmapa MBOTETyTa IIMAOCOctm AH GCCP).
A., 1983, c . 2-16.
6. Cmmpmoa B.A. A4IIHNUMMBH8A examemeaTmoom paommpemmoA climAormoTmmm C2R 6yzeao2 amreOpe. - Rommeomme moozeAotamm
(Tpym maymmo-mcoAmeaTeABomoro oemmaapa no Aormme MHCTMTy—
TB (Ilmococimm AH CCCP). M. , 1983, c . 43-48.