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2. Soundness - Wikipedia

   en.wikipedia.org/wiki/Soundness

   In logic and deductive reasoning, an argument is sound if it is both valid in form and has no false premises. [1] Soundness has a related meaning in mathematical logic, wherein a formal system of logic is sound if and only if every well-formed formula that can be proven in the system is logically valid with respect to the logical semantics of the system.

3. ω-consistent theory - Wikipedia

   en.wikipedia.org/wiki/Ω-consistent_theory

   Every ω-consistent theory is Σ 1-sound, but not vice versa. More generally, we can define an analogous concept for higher levels of the arithmetical hierarchy. If Γ is a set of arithmetical sentences (typically Σ 0 n for some n), a theory T is Γ-sound if every Γ-sentence provable in T is true in the standard model.

4. Consistency - Wikipedia

   en.wikipedia.org/wiki/Consistency

   Such a theory is consistent if and only if it does not prove a particular sentence, called the Gödel sentence of the theory, which is a formalized statement of the claim that the theory is indeed consistent. Thus the consistency of a sufficiently strong, recursively enumerable, consistent theory of arithmetic can never be proven in that system ...

5. Sentence (mathematical logic) - Wikipedia

   en.wikipedia.org/wiki/Sentence_(mathematical_logic)

   A sentence can be viewed as expressing a proposition, something that must be true or false. The restriction of having no free variables is needed to make sure that sentences can have concrete, fixed truth values : as the free variables of a (general) formula can range over several values, the truth value of such a formula may vary.

6. Logic - Wikipedia

   en.wikipedia.org/wiki/Logic

   For example, if the formula () stands for the sentence "Socrates is a banker" then the formula articulates the sentence "It is possible that Socrates is a banker". [127] To include these symbols in the logical formalism, modal logic introduces new rules of inference that govern what role they play in inferences.

7. Validity (logic) - Wikipedia

   en.wikipedia.org/wiki/Validity_(logic)

   If also the premises of a valid argument are proven true, this is said to be sound. [3] The corresponding conditional of a valid argument is a logical truth and the negation of its corresponding conditional is a contradiction. The conclusion is a necessary consequence of its premises. An argument that is not valid is said to be "invalid".

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9. Natural deduction - Wikipedia

   en.wikipedia.org/wiki/Natural_deduction

   The correspondence between the sequent calculus and natural deduction is a pair of soundness and completeness theorems, which are both provable by means of an inductive argument. Soundness of ⇒ wrt. ⊢ If Γ ⇒ A, then Γ ⊢ A. Completeness of ⇒ wrt. ⊢ If Γ ⊢ A, then Γ ⇒ A.