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

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 applica ...

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

paradox A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically u ...

is that this assumption implies the ''omniscience 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 for

verificationist Verificationism, also known as the verification principle or the verifiability criterion of meaning, is the philosophical doctrine which maintains that only statements that are empirically verifiable (i.e. verifiable through the senses) are cognit ...

anti-realist In analytic philosophy, anti-realism is a position which encompasses many varieties such as metaphysical, mathematical, semantic, scientific, moral and epistemic. The term was first articulated by British philosopher Michael Dummett in an argume ...

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 In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...

in a 1963 paper by

Frederic Fitch Frederic Brenton Fitch (September 9, 1908, Greenwich, Connecticut – September 18, 1987, New Haven, Connecticut) was an American logician, a Sterling Professor at Yale University. Education and career At Yale, Fitch earned his B.A in 1931 and ...

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

knowledge Knowledge can be defined as Descriptive knowledge, awareness of facts or as Procedural knowledge, practical skills, and may also refer to Knowledge by acquaintance, familiarity with objects or situations. Knowledge of facts, also called pro ...

and of

possibility Possibility is the condition or fact of being possible. Latin origins of the word hint at ability. Possibility may refer to: * Probability, the measure of the likelihood that an event will occur * Epistemic possibility, a topic in philosophy ...

. He also generalised the proof to different modalities. It resurfaced in 1979 when

W. D. Hart Wilbur Dyre Hart (born 1943) is an American philosopher and Professor Emeritus of Philosophy at the University of Illinois at Chicago. He taught at the University of Michigan from 1969 to 1974, the University College London from 1974 to 1991, and ...

wrote that Fitch's proof was an "unjustly neglected logical gem".

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: The proof proceeds: 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 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 Doxastic logic is a type of logic concerned with reasoning about beliefs. The term ' derives from the Ancient Greek (''doxa'', "opinion, belief"), from which the English term ''doxa'' ("popular opinion or belief") is also borrowed. Typically, a ...

modality "believed by a rational person" (represented by B). Rule (A) is replaced with: This time the proof proceeds: 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 Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems ...

; 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.

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. 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'). The same argument shows that (C') results in contradiction, indicating that any knowable truth is either known, or it 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.

Notes

References

* Frederick Fitch,
A Logical Analysis of Some Value Concepts
.

Journal of Symbolic Logic The '' Journal of Symbolic Logic'' is a peer-reviewed mathematics journal published quarterly by Association for Symbolic Logic. It was established in 1936 and covers mathematical logic. The journal is indexed by ''Mathematical Reviews'', Zentralb ...

Vol. 28, No. 2 (Jun., 1963), pp. 135–142 * W. D. Hart. "The Epistemology of Abstract Objects", Proceedings of the Aristotelian Society, suppl. vol. 53, 1979, pp. 153–65. * Johnathan Kvanvig
The Knowability Paradox
Oxford University Press, 2006. * Joe Salerno, ed
New essays on the knowability paradox
. Oxford University Press, 2009.

External links