Editing Fitch's paradox of knowability (section)

=== Generalisations ===
The proof uses minimal assumptions about the nature of '''K''' and '''L''', so other modalities can be substituted for "known". Joe Salerno gives the example of "caused by God": rule (C) becomes that every true fact ''could have been'' caused by God, and the conclusion is that every true fact ''was'' caused by God. Rule (A) can also be weakened to include modalities that don't imply truth. For instance instead of "known" we could have the [[doxastic logic|doxastic]] modality "believed by a rational person" (represented by '''B'''). Rule (A) is replaced with:

{|
|- 
|(E)
| '''B'''''p'' → '''BB'''''p'' || – rational belief is transparent; if ''p'' is rationally believed, then it is rationally believed that ''p'' is rationally believed.
|-
|(F)|| ¬('''B'''''p'' & '''B'''¬''p'') || – rational beliefs are consistent
|}

This time the proof proceeds:

{|
|- 
|1. Suppose '''B'''(''p'' & ¬'''B'''''p'') 
|- 
|2. '''B'''''p'' & '''B'''¬'''B'''''p'' || from line 1 by rule (B)
|-
|3. '''B'''''p'' || from line 2 by conjunction elimination
|-
|4. '''BB'''''p'' || from line 3 by rule (E)
|-
|5. '''B'''¬'''B'''''p'' || from line 2 by conjunction elimination
|-
|6. '''BB'''''p'' & '''B'''¬'''B'''''p'' || from lines 4 and 5 by [[conjunction introduction]]
|-
|7. ¬('''BB'''''p'' & '''B'''¬'''B'''''p'') || by rule (F)
|-
|8. ¬'''B'''(''p'' & ¬'''B'''''p'') || from lines 6 and 7 by [[reductio ad absurdum]], discharging assumption 1
|}

The last line matches line 6 in the previous proof, and the remainder goes as before. So if any true sentence could possibly be believed by a rational person, then that sentence is believed by one or more rational persons.

Some anti-realists advocate the use of [[intuitionistic logic]]; however, except for the last line, which moves from ''there are no unknown truths'' to ''all truths are known'', the proof is, in fact, intuitionistically valid.