User:IssaRice/Computability and logic/Some important distinctions and equivalences in introductory mathematical logic: Difference between revisions
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* decides: decidable set/relation vs deciding a sentence vs a theory being decidable vs deciding every sentence vs a logic being decidable (decidability of logic has several equivalent formulations; this is the topic of the ''Entscheidungsproblem'') | * decides: decidable set/relation vs deciding a sentence vs a theory being decidable vs deciding every sentence vs a logic being decidable (decidability of logic has several equivalent formulations; this is the topic of the ''Entscheidungsproblem'') | ||
* soundness: sound logic (soundness theorem) vs sound theory | * soundness: sound logic (soundness theorem) vs sound theory. soundness of logic is about truth in all interpretations, while soundness of theory is about truth in a specific interpretation. unless the axioms of a theory are just valid sentences (in which case, the axioms are not really adding anything beyond what the logic already has, assuming the logic is complete), the axioms of a theory will in general be false in some interpretations -- this is what makes theories interesting, because they have non-logical content. | ||
* truth in all interpretations (validity) vs truth in the intended interpretation (natural reading, standard interpretation): see [[../Intended interpretation versus all interpretations]] | * truth in all interpretations (validity) vs truth in the intended interpretation (natural reading, standard interpretation): see [[../Intended interpretation versus all interpretations]] | ||
* structure vs interpretation vs model: synecdoche problem | * structure vs interpretation vs model: synecdoche problem | ||