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Soundness vs Completeness - What's the difference?

soundness | completeness |

In lang=en terms the difference between soundness and completeness

is that soundness is the property of a logical theory that whenever a wff is a theorem then it must also be valid. Symbolically, letting T represent a theory within logic L, this can be represented as the property that whenever T \vdash \phi is true, then T \vDash \phi must also be true, for any wff φ of logic L while completeness is the property of a logical theory that whenever a wff is valid then it must also be a theorem. Symbolically, letting T represent a theory within logic L, this can be represented as the property that whenever T \vDash \phi is true, then T \vdash \phi must also be true, for any wff φ of logic L.

As nouns the difference between soundness and completeness

is that soundness is the state or quality of being sound while completeness is the state or condition of being complete.

soundness

English

Noun

  • (uncountable) The state or quality of being sound.
  • (countable) The result or product of being sound.
  • (logic) The property (of an argument) of not only being valid, but also of having true premises.
  • (logic) The property of a logical theory that whenever a wff is a theorem then it must also be valid. Symbolically, letting T'' represent a theory within logic ''L'', this can be represented as the property that whenever T \vdash \phi is true, then T \vDash \phi must also be true, for any wff ''φ'' of logic ''L .

    • completeness

    English

    (wikipedia completeness)

    Noun

    (-)
  • the state or condition of being complete
  • (logic) The property of a logical theory that whenever a wff is valid then it must also be a theorem. Symbolically, letting T'' represent a theory within logic ''L'', this can be represented as the property that whenever T \vDash \phi is true, then T \vdash \phi must also be true, for any wff ''φ'' of logic ''L .
  • *
    • THEOREM 37°. (Gödel's completeness theorem 1930.) In the predicate calculus H'':
    (a) ''If'' \vDash F [''or even if'' \aleph_0-\vDash F], ''then'' \vdash F. ''If'' E_1, . . . , E_k \vDash F [''or even if'' E_1, . . . , E_k \ \aleph_0-\vDash F], ''then     E_1, . . . , E_k \vdash F.
    (b) [...]

    Synonyms

    *(state of being complete ): completion

    Antonyms

    * incompleteness