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Fitch's paradox of knowability is a puzzle ofepistemic logic. It provides a challenge to theknowability thesis, which states that every truth is, in principle, knowable. Theparadox states that this assumption implies theomniscience principle, which asserts that every truth is known. Essentially, Fitch's paradox asserts that the existence of an unknown truth is unknowable. So if all truths were knowable, it would follow that all truths are in fact known.
The paradox is of concern forverificationist oranti-realist accounts of truth, for which theknowability thesis is very plausible,[1] but the omniscience principle is very implausible.
The paradox appeared as a minortheorem in a 1963 paper byFrederic Fitch, "A Logical Analysis of Some Value Concepts". Other than the knowability thesis, his proof makes only modest assumptions on themodal nature ofknowledge and ofpossibility. He also generalised the proof to different modalities. It resurfaced in 1979 whenW. D. Hart wrote that Fitch's proof was an "unjustly neglected logical gem".
Supposep is a sentence that is anunknown truth; that is, the sentencep is true, but it is notknown thatp is true. In such a case, the sentence "the sentencep 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 thatp is true, renderingp no longer anunknown 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 withmodal logic.K andL will stand forknown andpossible, respectively. ThusLK meanspossibly known, in other words,knowable. The modality rules used are:
| (A) | Kp →p | – knowledgeimplies truth. |
| (B) | K(p &q) → (Kp &Kq) | – knowing aconjunction implies knowing each conjunct. |
| (C) | p →LKp | – all truths are knowable. |
| (D) | from ¬p, deduce ¬Lp | – ifp can be proven false without assumptions, thenp is impossible (which is equivalent to therule of necessitation: ifq=¬p can be proven true without assumptions (atautology), thenq isnecessarily true, thereforep is impossible). |
The proof proceeds:
| 1. SupposeK(p & ¬Kp) | |
| 2.Kp &K¬Kp | from line 1 by rule (B) |
| 3.Kp | from line 2 byconjunction elimination |
| 4.K¬Kp | from line 2 by conjunction elimination |
| 5. ¬Kp | from line 4 by rule (A) |
| 6. ¬K(p & ¬Kp) | from lines 3 and 5 byreductio ad absurdum, discharging assumption 1 |
| 7. ¬LK(p & ¬Kp) | from line 6 by rule (D) |
| 8. Supposep & ¬Kp | |
| 9.LK(p & ¬Kp) | from line 8 by rule (C) |
| 10. ¬(p & ¬Kp) | from lines 7 and 9 by reductio ad absurdum, discharging assumption 8. |
| 11.p →Kp | from line 10 by a classicaltautology about thematerial conditional (negated conditionals) |
The last line states that ifp is true then it is known. Since nothing else aboutp was assumed, it means that every truth is known.
Since the above proof uses minimal assumptions about the nature ofL, replacingL withF (seePrior's tense logic (TL)) provides the proof for "If all truth can be known in the future, then they are already known right now".
Some anti-realists advocate the use ofintuitionistic logic; however, except for the last line, which moves fromthere are no unknown truths toall truths are known, the proof is, in fact, intuitionistically valid.
The proof uses minimal assumptions about the nature ofK andL, so other modalities can be substituted for "known". Joe Salerno gives the example of "caused by God": rule (C) becomes that every true factcould have been caused by God, and the conclusion is that every true factwas 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 thedoxastic modality "believed by a rational person" (represented byB). Rule (A) is replaced with:
| (E) | Bp →BBp | – rational belief is transparent; ifp is rationally believed, then it is rationally believed thatp is rationally believed. |
| (F) | ¬(Bp &B¬p) | – rational beliefs are consistent |
This time the proof proceeds:
| 1. SupposeB(p & ¬Bp) | |
| 2.Bp &B¬Bp | from line 1 by rule (B) |
| 3.Bp | from line 2 by conjunction elimination |
| 4.BBp | from line 3 by rule (E) |
| 5.B¬Bp | from line 2 by conjunction elimination |
| 6.BBp &B¬Bp | from lines 4 and 5 byconjunction introduction |
| 7. ¬(BBp &B¬Bp) | by rule (F) |
| 8. ¬B(p & ¬Bp) | from lines 6 and 7 byreductio 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.
Rule (C) is generally held to be at fault rather than any of the other logical principles employed. It may be contended that this rule does not faithfully translate the idea that all truths are knowable, and that rule (C) should not apply unrestrictedly. Kvanvig contends that this represents an illicit substitution into a modal context.
Gödel's Theorem proves that in any recursively axiomatized system sufficient to derive mathematics (e.g. Peano Arithmetic), there are statements which are undecidable. In that context, it is difficult to state that "all truths are knowable" since some potential truths are uncertain.
However, jettisoning the knowability thesis does not necessarily solve the paradox, since one can substitute a weaker version of the knowability thesis called (C').
| (C') | ∃x(((x & ¬Kx) &LKx) &LK((x & ¬Kx) &LKx)) | – There is an unknown, but knowable truth, and it is knowable that it is an unknown, but knowable truth. |
The same argument shows that (C') results in contradiction, indicating that any knowable truth is either known, or it is unknowable that it is an unknown yet knowable truth; conversely, it states that if a truth is unknown, then it is unknowable, or it is unknowable that it is knowable yet unknown.
