Appendix to

Alonzo Church’s Referee Reports on Fitch’s ‘A Definition of Value’

by Joe Salerno and Julien Murzi

Fitch’s Operators

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

Lewis and Langford (1932) Definition and Theorems

(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 : ‘-->’ is here used for L&L’s material conditional, and ‘<>’ for possibly]

Fitch’s 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: 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 a’s knowing at t that q empirically necessitates a’s desiring at t that p.[iii]

Def. 3R

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 a’s knowing at t that q empirically necessitates a’s 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]

Fitch’s Axiom and Theorems

Ax. 1 *

(aBtp & (p EN q)) p aBtq

Belief is closed under “empirically necessary” 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]

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[i] An asterisk, ‘*’, 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 Church’s 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.