Fitch's paradox of knowability is one of the fundamental puzzles of epistemic logic

Epistemic logic

Epistemic modal logic is a subfield of modal logic that is concerned with reasoning about knowledge. While epistemology has a long philosophical tradition dating back to Ancient Greece, epistemic logic is a much more recent development with applications in many fields, including philosophy,...

. It provides a challenge to the knowability thesis, which states that any truth is, in principle, knowable. The paradox

Paradox

Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...

 is that this assumption implies the omniscience principle, which asserts that any 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 for verificationist

Verificationist

Verificationism is the view that a statement or question is only legitimate if there is some way to determine whether the statement is true or false, or what the answer to the question is...

 or anti-realist accounts of truth, for which the knowability thesis is very plausible, but the omniscience principle is very implausible.

The paradox appeared as a minor theorem

Theorem

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...

 in a 1963 paper by Frederic Fitch

Frederic Brenton Fitch

Frederic Brenton Fitch was an American logician, the inventor of Fitch-style calculus, and a Sterling Professor Emeritus at Yale University...

, "A Logical Analysis of Some Value Concepts". Other than the knowability thesis, his proof makes only modest assumptions on the modal

Modal operator

In modal logic, a modal operator is an operator which forms propositions from propositions. In general, a modal operator has the "formal" property of being non-truth-functional, and is "intuitively" characterised by expressing a modal attitude about the proposition to which the operator is applied...

 nature of knowledge

Knowledge

Knowledge is a familiarity with someone or something unknown, which can include information, facts, descriptions, or skills acquired through experience or education. It can refer to the theoretical or practical understanding of a subject...

 and of possibility

Subjunctive possibility

Subjunctive possibility is the form of modality most frequently studied in modal logic...

. He also generalised the proof to different modalities. It resurfaced in 1979 when W.D. Hart wrote that Fitch's proof was an "unjustly neglected logical gem".

Proof

Suppose p is a sentence which 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

Modal logic

Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...

. 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)	Kp → p	- knowledge implies Implication Implication may refer to:In logic:* Logical implication, entailment, or consequence, a relation between statements* Material implication, or conditional implication, a binary truth functionIn linguistics, specifically in pragmatics:...  truth.
(B)	K(p & q) → (Kp & Kq)	- knowing a conjunction Logical conjunction In logic and mathematics, a two-place logical operator and, also known as logical conjunction, results in true if both of its operands are true, otherwise the value of false....  implies knowing each conjunct.
(C)	p → LKp	- all truths are knowable.
(D)	from ¬p, deduce ¬Lp	- if p can be proven false without assumptions, then p is impossible (which is the converse of the rule of necessitation: if p can be proven true without assumptions, then p is necessarily true).

The proof proceeds:

1. Suppose K(p & ¬Kp)
2. Kp & K¬Kp	from line 1 by rule (B)
3. Kp	from line 2 by conjunction 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 by reductio ad absurdam, discharging assumption 1
7. ¬LK(p & ¬Kp)	from line 6 by rule (D)
8. Suppose p & ¬Kp
9. LK(p & ¬Kp)	from line 8 by rule (C)
10. ¬(p & ¬Kp)	from lines 7 and 9 by reductio ad absurdam, discharging assumption 8.
11. p → Kp	from line 10 by a classical tautology Tautology (logic) In logic, a tautology is a formula which is true in every possible interpretation. Philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921; it had been used earlier to refer to rhetorical tautologies, and continues to be used in that alternate sense...

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.

Generalisations

The proof uses minimal assumptions about the nature of K and L, so other modalities can be substituted for "known". 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 which don't imply truth. For instance instead of "known" we could have the doxastic modality "believed by a rational person" (represented by B). Rule (A) is replaced with:

(E)	Bp → BBp	- rational belief is transparent; if p is rationally believed, then it is rationally believed that p is rationally believed.
(F)	¬(Bp & B¬p)	- rational beliefs are consistent

This time the proof proceeds:

1. Suppose B(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 by conjunction introduction Conjunction introduction Conjunction introduction is the inference that, if p is true, and q is true, then the conjunction p and q is true.For example, if it's true that it's raining, and it's true that I'm inside, then it's true that "it's raining and I'm inside"....
7. ¬(BBp & B¬Bp)	by rule (F)
8. ¬B(p & ¬Bp)	from lines 6 and 7 by reductio ad absurdam, 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 person does believe all true sentences.

Some anti-realists advocate the use of intuitionistic logic

Intuitionistic logic

Intuitionistic logic, or constructive logic, is a symbolic logic system differing from classical logic in its definition of the meaning of a statement being true. In classical logic, all well-formed statements are assumed to be either true or false, even if we do not have a proof of either...

; however, except for the very last line which moves from there are no unknown truths to all truths are known, the proof is, in fact, intuitionistically valid.

The knowability thesis

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.

Berit Brogaard

Berit Brogaard

Berit Oskar Brogaard is a Danish philosopher specializing in the areas of cognitive neuroscience, philosophy of mind, and philosophy of language. Her recent work concerns synesthesia, savant syndrome, blindsight and perceptual reports. She is Professor of Philosophy, Director of the St. Louis...

 and Joseph Salerno offer a criticism of Kvanvig's proposal and then defend a new proposal according to which quantified expressions play a special role in modal contexts. On the account of this special role articulated by Stanley and Szabo, they propose a solution to the knowability paradoxes. Another way to resolve the paradox is to restrict the paradox only to atomic sentences. Brogaard and Salerno have argued against this strategy in several papers that have appeared in journals such as Analysis and American Philosophical Quarterly.

External links

• Fitch's Paradox of Knowability. Article at the Stanford Encyclopedia of Philosophy
  Stanford Encyclopedia of Philosophy
  The Stanford Encyclopedia of Philosophy is a freely-accessible online encyclopedia of philosophy maintained by Stanford University. Each entry is written and maintained by an expert in the field, including professors from over 65 academic institutions worldwide...
  
  , by Berit Brogaard
  Berit Brogaard
  Berit Oskar Brogaard is a Danish philosopher specializing in the areas of cognitive neuroscience, philosophy of mind, and philosophy of language. Her recent work concerns synesthesia, savant syndrome, blindsight and perceptual reports. She is Professor of Philosophy, Director of the St. Louis...
  
   and Joe Salerno.
• Not Every Truth Can Be Known: at least, not all at once. Discussion page on an article of the same name by Greg Restall to appear in Salerno's book
• Joe Salerno