Editing Fitch's paradox of knowability (section)

== Proof ==
Suppose ''p'' is a sentence that is an ''unknown truth''; that is, the sentence ''p'' is true, but it is not ''known'' that ''p'' is true.  In such a case, the sentence "the sentence ''p'' is an unknown truth" is true; and, if all truths are knowable, it should be possible to know that "''p'' is an unknown truth". But this isn't possible, because as soon as we know "''p'' is an unknown truth", we know that ''p'' is true, rendering ''p'' no longer an ''unknown'' truth, so the statement "''p'' is an unknown truth" becomes a falsity.  Hence, the statement "''p'' is an unknown truth" cannot be both known and true at the same time.  Therefore, if all truths are knowable, the set of "all truths" must not include any of the form "''something'' is an unknown truth"; thus there must be no unknown truths, and thus all truths must be known.

This can be formalised with [[modal logic]]. '''K''' and '''L''' will stand for ''known'' and ''possible'', respectively. Thus '''LK''' means ''possibly known'', in other words, ''knowable''. The modality rules used are:

{|
|- 
|(A)
| '''K'''''p'' → ''p'' || – knowledge [[material conditional|implies]] truth.
|-
|(B)|| '''K'''(''p'' & ''q'') → ('''K'''''p'' & '''K'''''q'') || – knowing a [[logical conjunction|conjunction]] implies knowing each conjunct.
|-
|(C)|| ''p'' → '''LK'''''p'' || – all truths are knowable.
|-
|(D)|| from ¬''p'', deduce ¬'''L'''''p'' || – if ''p'' can be proven false without assumptions, then ''p'' is impossible (which is equivalent to the [[rule of necessitation]]: if ''q=¬p'' can be proven true without assumptions (a [[Tautology (logic)|tautology]]), then ''q'' is [[logical truth|necessarily true]], therefore ''p'' is impossible).
|}

The proof proceeds:

{|
|- 
|1. Suppose '''K'''(''p'' & ¬'''K'''''p'') 
|- 
|2. '''K'''''p'' & '''K'''¬'''K'''''p'' || from line 1 by rule (B)
|-
|3. '''K'''''p'' || from line 2 by [[conjunction elimination]]
|-
|4. '''K'''¬'''K'''''p'' || from line 2 by conjunction elimination
|-
|5. ¬'''K'''''p'' || from line 4 by rule (A)
|- 
|6. ¬'''K'''(''p'' & ¬'''K'''''p'') || from lines 3 and 5 by [[reductio ad absurdum]], discharging assumption 1
|- 
|7. ¬'''LK'''(''p'' & ¬'''K'''''p'') || from line 6 by rule (D)
|- 
|8. Suppose ''p'' & ¬'''K'''''p'' 
|- 
|9. '''LK'''(''p'' & ¬'''K'''''p'') || from line 8 by rule (C)
|- 
|10. ¬(''p'' & ¬'''K'''''p'') || from lines 7 and 9 by reductio ad absurdum, discharging assumption 8.
|-
|11. ''p'' → '''K'''''p'' || from line 10 by a classical [[tautology (logic)|tautology]] about the [[Material conditional#Formal properties|material conditional]] (negated conditionals)
|}

The last line states that if ''p'' is true then it is known. Since nothing else about ''p'' was assumed, it means that every truth is known.

Since the above proof uses minimal assumptions about the nature of '''L''', replacing '''L''' with '''F''' (see [[Temporal logic#Prior's tense logic (TL)|Prior's tense logic (TL)]]) provides the proof for "If all truth can be known in the future, then they are already known right now".

=== 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.