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Frege’s
“Dialog mit Pünjer über Existenz”
Formal Reconstruction
On the background of his interpretation of syllogistic
forms and singular propositions, Frege in “Dialog mit
Pünjer über Existenz” proved Kant’s famous dictum,
that existence is not a real predicate (i.e. it is not the
property of things and it is not the characteristic of
concepts). The paper is a formal reconstruction of
Frege’s proof as – in the frame of alethic modal logic –
twice double reductio ad absurdum.
Borut Cerkovnik
University of Ljubljana
The ways, on which we can something (ontologicaly) says
basis – propositional function
(1) singular proposition
(2) particular proposition
(3) universal proposition
Fx
Fa
(x) Fx
(x) Fx
Frege’s interpretation of
syllogistic forms and form of singular propositions
A*:
E*:
I*:
O*:
(x) (Sx  Px)
(x) (Sx  Px)
(x) (Sx  Px)
(x) (Sx  Px)
(i)*
Aa
L
(x)(a =x  Ax)
Frege’s interpretation of Kant’s dictum
“existence is not a real predicate”
• Existence is not the characteristic of concepts.
• Existence is not the property of things.
First double reductio ad absurdum
(1) w1
(2) w1
(3)
(4)
(5) w1
(6) w1
(7)
(x)(Ax  Ex)
existence is a characteristic of concepts
(assumption for RAA)
◊(x)(Ax  Ex)
existence is contingent (assumption for RAA)
w 2:
(x)(Ax  Ex)
from (2)
w2 :
(x)(Ax  Ex)
from (3)
(a) duality
(b) (P  Q)  (P  Q)
(c) deMorgan law
contradiction in (4): “something, what exist, does not exist”
◊(x)(Ax  Ex)
(1)-(4): reductio ad absurdum
□(x)(Ax  Ex)
from (5): duality
contradiction (1), (2) & (6):
“existence, which should be contingent, is necessary”
existence is not a characteristic of concepts
(1)-(6): reductio ad absurdum
»Ax« is an arbitrary predicate
»Ex« expresses the concept exist or to be existent”
Second double reductio ad absurdum
(1) w0
(2) w0
(3)
(4)
(5) w0
(6) w0
(7)
(x)(a=x  Ex)
existence is a property of objects
(assumption for RAA)
◊(x)(a=x  Ex)
existence is contingent (assumption for RAA)
w 1:
(x)(a=x  Ex)
from (2)
w1 :
(x)(ax  Ex)
from (3)
(a) duality
(c) deMorgan law
contradiction in (5):
“a is an empty sound
or “there’s nothing”
and designate nothing”
 ◊(x)(a=x  Ex)
(1)-(5): reductio ad absurdum
□(x)(a=x  Ex)
from (6): duality
contradiction (1), (2) & (7):
“existence, which should be contingent, is necessary”
existence is not a property of objects
(1)-(7): reductio ad absurdum
»a« is an arbitrary name
»Ex« expresses the concept exist or to be existent”