What does decidability mean?
Decidability is the property of being decidable — of a question, problem, or formal system for which a definite answer can be reached. Its most important home is mathematical logic and theoretical computer science, where a problem is called decidable if there exists an effective procedure, or algorithm, that correctly returns yes or no for every instance in a finite amount of time. Propositional logic, for example, is decidable because truth tables mechanically settle any statement, while first-order logic in general is not, as Church and Turing independently demonstrated in the 1930s. Beyond this technical core, the word can be used more loosely for any matter that admits a clear-cut resolution, though such general use is uncommon and somewhat formal. Decidability carries a precise, scholarly connotation and is almost always discussed alongside its opposite, undecidability, which marks problems — like Turing's halting problem — that no algorithm can ever fully resolve. For anyone studying computation, proof, or the limits of mechanical reasoning, it is an indispensable concept.
nounThe quality or property of being decidable; in logic and mathematics, the property of a problem, question, or formal system for which an effective procedure exists that determines the truth or falsehood of any given statement (or yields a yes/no answer) in a finite number of steps.
- (Logic & mathematics) The property of a decision problem or formal system for which there exists an effective (algorithmic) procedure that correctly answers every instance in finite time — e.g. the decidability of propositional logic versus the undecidability of first-order logic established by Church and Turing.
- (General) The quality of being capable of being decided, settled, or determined one way or the other.
"Gödel and Turing showed that decidability fails for certain well-formed questions: no algorithm can settle every case."
"A key result in model theory is the decidability of the theory of real closed fields, proved by Tarski."
"Before committing to the project, the committee wanted assurance of the decidability of the disputed boundary claims."
Turing proved in 1936 that some perfectly well-asked questions can never be answered by any computer — ever — and 'decidability' is the word mathematicians use to draw that line.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Etymology of decidability
Decidability derives from 'decidable', which combines the Latin verb decidere ('to cut off, determine, decide') — from de- ('off, away') plus caedere ('to cut') — with the suffix -able and the noun-forming -ity. The Latin root caedere also gives English words such as 'incise', 'concise', and 'precise', all built on the idea of cutting. The abstract noun took on its specialized logical sense in the late nineteenth and early twentieth centuries, as mathematicians formalized what it means for a question to be settled by a mechanical procedure.
Related word forms
How decidability is actually used
Decidability is primarily a term of art in mathematical logic, computability theory, and computer science, appearing mostly in academic and technical registers. In everyday English it is rare; outside logic contexts it tends to sound stilted compared with plain alternatives like 'settleability' or simply rephrasing with 'can be decided'. The negative counterpart 'undecidability' is arguably even more famous, owing to Turing's halting problem.
Easily confused with decidability
Decisiveness is a personality trait of making choices quickly and firmly, whereas decidability is a technical property of problems or systems having an algorithmic procedure to settle them.
Deciduous describes trees that shed their leaves seasonally; despite sharing the Latin root caedere ('to cut'), it has nothing to do with whether a question can be algorithmically answered.