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
aKNtp = a knows at time t that p
aBtp = a believes at t that p
aVLtp = a values at t that p
aDStp = 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]
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: aKNtp p p.[ii]
Def. 3 *
aVLt p =df $q(q & (aKNtq EN aDStp)).
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 aÕs knowing at t that q empirically necessitates aÕs desiring at t that p.[iii]
aVLt p =df $q(q & EP(aKNtq) & (aKNtq EN aDStp)).
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 aÕs knowing at t that q empirically necessitates aÕs 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 *
(aBtp & (p EN q)) p aBtq
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 *
aKNt(p & q) p (aKNtp & aKNtq)
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.