a joint project between

Department of Information Studies - University of Copenhagen

and

Department of Communication and Psychology - Aalborg University

This text has been transcribed by Max Cresswell. Folder 7.

Download the PDF of this item Read automatically extracted version of the PDF
Please note that this text is extracted from the PDF, and is as such most likely not styled properly.

UNIVERSITY OF CALIFORNIA, LOS ANGELES

DEPARTMENT OF PHILOSOPHY

LOS ANGELES

,

CALIFORNIA

90024

November 22, I968

Dear Professor Prior

Thank

you for your letter and especially the information about Fine

and Bull. I had earlier thought

about

some of the matters

in my letter to you (in

particular S5 with quantifiers), and reading your article

stimulated

me to prepare

the enclosed abstracts, which I submitted for the

January A.S.L. meetings. But

now

it appears that these results are

not new. I hope that Fine and Bull will send me copies of

their papers

(you

did not send me their addresses or I would write directly to them). [Marginal note reads: Both have

written since this letter sent to the typist (Ed)]

I

wonder if Fine has axiom sets with strong separation p

roperty

I describe in S5 with Multiple

Possibility.

I formulated my axiom in a slightly more complex way than

necessary in

the (vain) hope of

achieving strong separability.

The

theory known to be decidable is what Church calls singulary 2nd

order

logic with identity,

not

necessarily unifor

reading my letter to you, I see that I expressed

theory is decidable, hence the

which

are expressible in the

sublanguage is also decidable. It is clear from the proof

of decidability for the

full

theory that

for quantification

theory

(Tautologies,

Distribution, Vacuous Quantification, Specification) for both individual a

nd m

onadic

predicate

variables, plus the usual identity axioms (

x

x

=

x

,

Leibniz law). Since identity is definable with

predicate

variables (

"

$

for

F

(

F

"

6

F

$

)),

the identity

formulas and axioms are superfluous. Thus since

F

[

Fx

v

G

(

Gx

6

x

(

Fx

6

Gx

))] is valid, it must

be provable just from the usual quantification axioms.

And

so it is. The

F

in question is, intuitively, singleton

x

({

x

}), so take for the formula

F

"

,

H

(

Hx

6

Hx

).

Obviously,

/

H

(

Hx

6

Hx

) and

/

G

(

Gx

6

y

(

H

(

Hx

6

Hy

)

Gy

)), thus

/

F

[

Fx

v

G

(

Gx

6

y

(

Fy

6

Gy

))], hence

/

F

[

Fx

v

G

(

Gx

6

x

(

Fx

6

Gx

responding axiom

in S5

ntifiers was independent.

-2-

It

seems to

validity, so possibly I have still not answered your question about the subtheory with the axiom (A)

(A)

F

[

Fx

v

G

(

Gx

6

y

(

Fx

6

Gx

)].

Are

you thinking of singulary

s

econd order logic sans identity and with only one individual

variable

(say

) ? The formulas of this last theory (call it

V) are completely isomorphic

to those of S5

quant

fact the converse of the 2 of S5 with Quantifiable Propositional Variables will

translate

any formula (in the language of V into

a formula in the language of S5Q.

Thus my completeness

proof

for S5Q tells us that we can axiomatize V with the usual axioms pl

proof for my axiom (8) tells us that (A) is not derivable from the usual axioms so

long as the

derivation

stays within the language of V.

If you still have reprints of Egocentric Logic, I would like one, and other reprints also.

Should

a vi

siting posi

or at a nearby institution for a logician become available I will

certainly mention Fine. (or is a student position e.g. teaching assistant appropriate?)

Sincerely,

David Kaplan

2

DK:bs

P.S. there has been a 3 week delay in typing this letter for which I apologize.

following formulae appear to have been written on the flap

of the envelope. Its not

obviously Kaplans, and could be Priors since it is partly, though not entirely, in Polish notation.]

(

*

)(

(

) (nec.{

*

. {

(

. ~nec.}

(

(

p

)(

q

) (nec.

Np

.

Nq

. ~nec.

Nq

)

(

*

)(

(

) (nec.

F

*

F

(

F

(

A

pN

*

p

N

(

p

N

A

p

(

p

[Transcribed

by M.J.

Cresswell, from a version obtained by OCR from Kaplans typewritten letter.

The

Virtual Lab does not list any letters from Prior to Kaplan.]

DEPARTMENT OF PHILOSOPHY

LOS ANGELES

,

CALIFORNIA

90024

November 22, I968

Dear Professor Prior

Thank

you for your letter and especially the information about Fine

and Bull. I had earlier thought

about

some of the matters

in my letter to you (in

particular S5 with quantifiers), and reading your article

stimulated

me to prepare

the enclosed abstracts, which I submitted for the

January A.S.L. meetings. But

now

it appears that these results are

not new. I hope that Fine and Bull will send me copies of

their papers

(you

did not send me their addresses or I would write directly to them). [Marginal note reads: Both have

written since this letter sent to the typist (Ed)]

I

wonder if Fine has axiom sets with strong separation p

roperty

I describe in S5 with Multiple

Possibility.

I formulated my axiom in a slightly more complex way than

necessary in

the (vain) hope of

achieving strong separability.

The

theory known to be decidable is what Church calls singulary 2nd

order

logic with identity,

not

necessarily unifor

reading my letter to you, I see that I expressed

theory is decidable, hence the

which

are expressible in the

sublanguage is also decidable. It is clear from the proof

of decidability for the

full

theory that

for quantification

theory

(Tautologies,

Distribution, Vacuous Quantification, Specification) for both individual a

nd m

onadic

predicate

variables, plus the usual identity axioms (

x

x

=

x

,

Leibniz law). Since identity is definable with

predicate

variables (

"

$

for

F

(

F

"

6

F

$

)),

the identity

formulas and axioms are superfluous. Thus since

F

[

Fx

v

G

(

Gx

6

x

(

Fx

6

Gx

))] is valid, it must

be provable just from the usual quantification axioms.

And

so it is. The

F

in question is, intuitively, singleton

x

({

x

}), so take for the formula

F

"

,

H

(

Hx

6

Hx

).

Obviously,

/

H

(

Hx

6

Hx

) and

/

G

(

Gx

6

y

(

H

(

Hx

6

Hy

)

Gy

)), thus

/

F

[

Fx

v

G

(

Gx

6

y

(

Fy

6

Gy

))], hence

/

F

[

Fx

v

G

(

Gx

6

x

(

Fx

6

Gx

responding axiom

in S5

ntifiers was independent.

-2-

It

seems to

validity, so possibly I have still not answered your question about the subtheory with the axiom (A)

(A)

F

[

Fx

v

G

(

Gx

6

y

(

Fx

6

Gx

)].

Are

you thinking of singulary

s

econd order logic sans identity and with only one individual

variable

(say

) ? The formulas of this last theory (call it

V) are completely isomorphic

to those of S5

quant

fact the converse of the 2 of S5 with Quantifiable Propositional Variables will

translate

any formula (in the language of V into

a formula in the language of S5Q.

Thus my completeness

proof

for S5Q tells us that we can axiomatize V with the usual axioms pl

proof for my axiom (8) tells us that (A) is not derivable from the usual axioms so

long as the

derivation

stays within the language of V.

If you still have reprints of Egocentric Logic, I would like one, and other reprints also.

Should

a vi

siting posi

or at a nearby institution for a logician become available I will

certainly mention Fine. (or is a student position e.g. teaching assistant appropriate?)

Sincerely,

David Kaplan

2

DK:bs

P.S. there has been a 3 week delay in typing this letter for which I apologize.

following formulae appear to have been written on the flap

of the envelope. Its not

obviously Kaplans, and could be Priors since it is partly, though not entirely, in Polish notation.]

(

*

)(

(

) (nec.{

*

. {

(

. ~nec.}

(

(

p

)(

q

) (nec.

Np

.

Nq

. ~nec.

Nq

)

(

*

)(

(

) (nec.

F

*

F

(

F

(

A

pN

*

p

N

(

p

N

A

p

(

p

[Transcribed

by M.J.

Cresswell, from a version obtained by OCR from Kaplans typewritten letter.

The

Virtual Lab does not list any letters from Prior to Kaplan.]

18-12-2018 13:50:23 (GMT+1) |