In logic and deductive reasoning, an argument is sound if it is both valid in form and its premises are true.^{[1]} Soundness has a related meaning in mathematical logic, wherein a formal system of logic is sound if and only if every wellformed formula that can be proven in the system is logically valid with respect to the logical semantics of the system.
YouTube Encyclopedic

1/5Views:5 6519 8579 3364581 250

Soundness and Completeness: Introduction to Mathematical Logic #4

[Logic] Soundness #1

5. Soundness and Completeness  Logic for Beginners

What is Logic? #83: The Soundness Theorem for Propositional Logic

Soundness and Completeness for Proof Trees  Attic Philosophy
Transcription
Definition
In deductive reasoning, a sound argument is an argument that is valid and all of its premises are true (and as a consequence its conclusion is true as well). An argument is valid if, assuming its premises are true, the conclusion must be true. An example of a sound argument is the following wellknown syllogism:
 (premises)
 All men are mortal.
 Socrates is a man.
 (conclusion)
 Therefore, Socrates is mortal.
Because of the logical necessity of the conclusion, this argument is valid; and because the argument is valid and its premises are true, the argument is sound.
However, an argument can be valid without being sound. For example:
 All birds can fly.
 Penguins are birds.
 Therefore, penguins can fly.
This argument is valid as the conclusion must be true assuming the premises are true. However, the first premise is false. Not all birds can fly (for example, penguins). For an argument to be sound, the argument must be valid and its premises must be true.^{[2]}
Use in mathematical logic
Logical systems
In mathematical logic, a logical system has the soundness property if every formula that can be proved in the system is logically valid with respect to the semantics of the system. In most cases, this comes down to its rules having the property of preserving truth.^{[3]} The converse of soundness is known as completeness.
A logical system with syntactic entailment and semantic entailment is sound if for any sequence of sentences in its language, if , then . In other words, a system is sound when all of its theorems are tautologies.
Soundness is among the most fundamental properties of mathematical logic. The soundness property provides the initial reason for counting a logical system as desirable. The completeness property means that every validity (truth) is provable. Together they imply that all and only validities are provable.
Most proofs of soundness are trivial.^{[citation needed]} For example, in an axiomatic system, proof of soundness amounts to verifying the validity of the axioms and that the rules of inference preserve validity (or the weaker property, truth). If the system allows Hilbertstyle deduction, it requires only verifying the validity of the axioms and one rule of inference, namely modus ponens. (and sometimes substitution)
Soundness properties come in two main varieties: weak and strong soundness, of which the former is a restricted form of the latter.
Weak soundness
Weak soundness of a deductive system is the property that any sentence that is provable in that deductive system is also true on all interpretations or structures of the semantic theory for the language upon which that theory is based. In symbols, where S is the deductive system, L the language together with its semantic theory, and P a sentence of L: if ⊢_{S} P, then also ⊨_{L} P.
Strong soundness
Strong soundness of a deductive system is the property that any sentence P of the language upon which the deductive system is based that is derivable from a set Γ of sentences of that language is also a logical consequence of that set, in the sense that any model that makes all members of Γ true will also make P true. In symbols where Γ is a set of sentences of L: if Γ ⊢_{S} P, then also Γ ⊨_{L} P. Notice that in the statement of strong soundness, when Γ is empty, we have the statement of weak soundness.
Arithmetic soundness
If T is a theory whose objects of discourse can be interpreted as natural numbers, we say T is arithmetically sound if all theorems of T are actually true about the standard mathematical integers. For further information, see ωconsistent theory.
Relation to completeness
The converse of the soundness property is the semantic completeness property. A deductive system with a semantic theory is strongly complete if every sentence P that is a semantic consequence of a set of sentences Γ can be derived in the deduction system from that set. In symbols: whenever Γ ⊨ P, then also Γ ⊢ P. Completeness of firstorder logic was first explicitly established by Gödel, though some of the main results were contained in earlier work of Skolem.
Informally, a soundness theorem for a deductive system expresses that all provable sentences are true. Completeness states that all true sentences are provable.
Gödel's first incompleteness theorem shows that for languages sufficient for doing a certain amount of arithmetic, there can be no consistent and effective deductive system that is complete with respect to the intended interpretation of the symbolism of that language. Thus, not all sound deductive systems are complete in this special sense of completeness, in which the class of models (up to isomorphism) is restricted to the intended one. The original completeness proof applies to all classical models, not some special proper subclass of intended ones.
See also
References
 ^ Smith, Peter (2010). "Types of proof system" (PDF). p. 5.
 ^ Gensler, Harry J., 1945 (January 6, 2017). Introduction to logic (Third ed.). New York. ISBN 9781138910584. OCLC 957680480.
{{cite book}}
: CS1 maint: location missing publisher (link) CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)  ^ Mindus, Patricia (20090918). A Real Mind: The Life and Work of Axel Hägerström. Springer Science & Business Media. ISBN 9789048128952.
Bibliography
 Hinman, P. (2005). Fundamentals of Mathematical Logic. A K Peters. ISBN 1568812620.
 Copi, Irving (1979), Symbolic Logic (5th ed.), Macmillan Publishing Co., ISBN 0023248807
 Boolos, Burgess, Jeffrey. Computability and Logic, 4th Ed, Cambridge, 2002.