** **

**Appendix to**

**Alonzo ChurchÕs Referee Reports on FitchÕs ŌA Definition of ValueÕ**

**by Joe Salerno and Julien Murzi**

** **

p SI q = p p q = p strictly implies q

p EN q = p empirically necessitates q

EPp = p is empirically possible

p EC q = p is empirically consistent
with q

*a*KN*t*p = *a* knows at time *t* that p

*a*B*t*p = *a* believes at *t* that p

*a*VL*t*p =
*a* values at *t *that p

*a*DS*t*p =
*a* desires at *t *that p

(p p q) =_{df }~<>(p
& ~q)

11.2 (p & q) p p

11.6 ((p p q) p (q p r)) p (p p r)

12.42 (p p ~q) p (q p ~p)

15.1 ((p --> q) & (q --> r)) p (p --> r)

16.2 ((p p q) & (p p r) & T) p (p p (q & r)): T = ((q & r) p (r & q))

[nb : Ō-->Õ is here used for L&LÕs material conditional, and
Ō<>Õ for possibly]

**FitchÕs Definitions**[i]

*Def. 2 **

Def. 2 is Fitch's definition of knowledge. All we know from Church's use is that it justifies the
principle that *a*'s
knowing at time *t*
that p strictly implies p: *a*KN*t*p p p.[ii]*
*

* *

*Def. 3 **

*a*VL*t* p =_{df } $q(q & (*a*KN*t*q EN *a*DS*t*p)).

Value is what one would desire given sufficient knowledge: it is valuable to *a* at *t *that p if and only if there is a
true proposition q, such that *a*Õs knowing at *t* that q empirically necessitates *a*Õs* *desiring at *t *that p.[iii]

* *

*Def. 3R *

*a*VL*t* p =_{df }$q(q
& EP(*a*KN*t*q) & (*a*KN*t*q EN *a*DS*t*p)).

Value is what one would desire given sufficient knowledge: it is valuable to *a* at *t* that p if and only if there is a truth q that
it is empirically possible to know and *a*Õs knowing at *t* that q empirically necessitates *a*Õs* *desiring at *t *that p.[iv]

*Def. 5 **

(p EN ~p) p ~(p EC p).

Necessarily, if p empirically necessitates ~p, then p is not
(empirically) consistent with itself.[v]

* *

*Def. 6 * *

~(p
EC p) =_{df }~EPp.

p is not (empirically) consistent with itself just in case p is not
empirically possible.[vi]

**FitchÕs Axiom and Theorems **

*Ax. 1 * *

(*a*B*t*p & (p EN q)) p *a*B*t*q

Belief is closed under Ņempirically necessaryÓ implication: necessarily, if *a* believes at *t* that p and p empirically necessitates q, then *a* believes at *t* that q.[vii]

* *

*Th. 1 * *

(p p q) p (p EN q)

Strict implication strictly implies empirical necessitation: necessarily, if p strictly implies q
then p empirically necessitates q.[viii]

*Th. 3 **

*a*KN*t*(p & q) p (*a*KN*t*p & *a*KN*t*q)

Knowing a conjunction strictly implies knowing the conjuncts: necessarily,
if *a* knows at *t* that both p and q, then *a* knows at *t* that p and *a* knows at *t *that q.[ix]

[i] An asterisk, Ō*Õ,
indicates that the principle does not appear explicitly in the reports, and
therefore, that we have hypothesized its content.

[ii] Church's applications appear in Report 1: 2
and Report 2:1.

[iii] Our formulation of Def. 3 is based on ChurchÕs
trivialization argument against it. Compare Report 1: 2 and Report 2: 1--2.

[iv] Report 2: 2.

[v] Report 2: 1.

[vi] Report 2: 1.

[vii] The discussion at Report 1: 2--3 suggests that
Ax. 1 is this closure principle for belief. Alternatively, it is an
unrestricted closure principle for knowledge (viz., knowledge is closed under
necessary empirical implication).

[viii] See for instance, Church's use in Report 2:
1.

[ix] Report 2: 1.