Where does consistency collide?

A property of a formal theory: the set of its axioms is consistent when no statement and its negation can both be derived from it, so it never proves a contradiction, which is equivalent to the theory having a model. Gödel's 1931 Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I proved that a sufficiently strong consistent theory cannot prove its own consistency, tying the notion to the limits of formal proof. The referent is a syntactic property of a system of axioms, either it can derive a contradiction or it cannot, and it has nothing to do with an estimator converging on a number, a database preserving its rules, or a person keeping their beliefs in step.[1]

Unit

A theory, judged as either consistent or contradictory

Recovery means

Not applicable

Measured by

Established by exhibiting a model, or by a relative consistency proof reducing one theory's consistency to another's

A property of a statistical estimator: it is consistent when, as the sample grows without bound, the estimate converges in probability to the true value of the quantity being estimated, so more data drives the error toward zero. Fisher's 1922 On the mathematical foundations of theoretical statistics introduced consistency as one of the criteria a good estimator should meet. The referent is an asymptotic guarantee about a procedure applied to ever larger samples, a claim about limiting behaviour under repeated sampling, and it is unrelated to a logical system avoiding contradiction, a database honouring its rules, or a mind seeking internal agreement.[2]

Unit

An estimator, judged by its limiting behaviour as sample size grows

Recovery means

Not applicable

Measured by

Shown by proving convergence in probability to the parameter, often via the law of large numbers or bounds on bias and variance that vanish with sample size

The C in the ACID properties of a transaction: each transaction moves the database from one valid state to another, preserving every declared rule and constraint, so an interrupted or partial change never leaves the data violating its own invariants. Haerder and Reuter's 1983 Principles of transaction-oriented database recovery fixed this ACID meaning; in distributed systems the word is reused for consistency models, where Gilbert and Lynch's 2002 proof of Brewer's CAP conjecture shows that strong consistency, availability, and partition tolerance cannot all be guaranteed at once. The referent is a correctness guarantee about stored state under change, a matter of engineered invariants and replica agreement, not a logical theory's freedom from contradiction, an estimator's limit, or a person's inner harmony.[3][4]

Unit

A transaction or a replicated system, judged against declared invariants

Recovery means

A failed transaction is rolled back so the database returns to its last valid state

Measured by

Checked by verifying that constraints hold before and after each transaction, and by testing a system against a formal consistency model such as linearizability

The human tendency to seek internal agreement among attitudes, beliefs, and relationships, so that a person feels pressure to change something when their cognitions clash. Heider's 1946 Attitudes and cognitive organization set out balance theory, in which people prefer a balanced pattern of liking and belief and are pushed toward consistency when the pattern is unbalanced, a founding statement of the consistency theories that also include cognitive dissonance. The referent is a motivational property of a mind, a felt drive toward harmony among thoughts, and it is a different thing entirely from a logical system without contradiction, an estimator converging on a value, or a database preserving its constraints.[5]

Unit

A person's set of related attitudes, judged as balanced or in tension

Recovery means

Not applicable

Measured by

Inferred from experiments in which people resolve conflicting cognitions by changing an attitude, a belief, or a relationship to restore agreement