Logical System Soundness
A Logical System Soundness is a Formal System Soundness that is well-formed formula that can be inferred from the axioms and be satisfied by every model of the system.
- AKA: Soundness.
- Counter Example(s):
- See: Truth, Mathematical Logic, Logical System, Formula (Mathematical Logic), Formal Semantics (Logic), Sound Logic Argument, Informal Fallacy.
References
2020a
- (Wikipedia, 2020) ⇒ https://en.wikipedia.org/wiki/Soundness Retrieved:2020-1-9.
- In mathematical logic, a logical system has the soundness property if and only if every formula that can be proved in the system is logically valid with respect to the semantics of the system.
The converse of soundness is known as completeness. In most cases, this comes down to its rules having the property of preserving truth.
...A system with syntactic entailment [math]\displaystyle{ \vdash }[/math] and semantic entailment [math]\displaystyle{ \models }[/math] is sound if for any sequence [math]\displaystyle{ A_1, A_2, ..., A_n }[/math] of sentences in its language, if [math]\displaystyle{ A_1, A_2, ..., A_n\vdash C }[/math], then [math]\displaystyle{ A_1, A_2, ..., A_n\models C }[/math] . 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. 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 Hilbert-style 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.
- In mathematical logic, a logical system has the soundness property if and only if every formula that can be proved in the system is logically valid with respect to the semantics of the system.
2020b
- (Wikipedia, 2020) ⇒ https://en.wikipedia.org/wiki/Formal_system#Logical_system Retrieved:2020-1-9.
- A logical system or, for short, a logic, is a formal system together with its semantics. According to model-theoretic interpretation, the semantics of a logical system describe whether a well-formed formula is satisfied by a given structure. A structure that satisfies all the axioms of the formal system is known as a model of the logical system. A logical system is sound if each well-formed formula that can be inferred from the axioms is satisfied by every model of the logical system. Conversely, a logic system is complete if each well-formed formula that is satisfied by every model of the logical system can be inferred from the axioms.