Soundness
Encyclopedia
In mathematical logic
, a logical system has the soundness property if and only if
its inference rules prove only formulas that are valid with respect to its semantics. In most cases, this comes down to its rules having the property of preserving truth
, but this is not the case in general. The word derives from the Germanic 'Sund' as in Gesundheit, meaning health. Thus to say that an argument is sound means, following the etymology, to say that the argument is healthy.
is sound if and only if
For instance,
The argument is valid (because the conclusion is true based on the premises, that is, that the conclusion follows the premises) and since the premises are in fact true, the argument is sound.
The following argument is valid but not sound:
Since the first premise is actually false, the argument, though valid, is not sound.
property means that every validity (truth) is provable. Together they imply that all and only validates 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). Most axiomatic systems have only the rule of mods ponens (and sometimes substitution), so it requires only verifying the validity of the axioms and one rule of inference.
Soundness properties come in two main varieties: weak and strong soundness, of which the former is a restricted form of the latter.
is the property that any sentence that is provable in that deductive system is also true on all interpretations or models 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. In other words, a system is sound if each of its theorems (i.e. formulas provable from the empty set) is valid in every structure of the language.
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 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.
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
, a logical system has the soundness property if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
its inference rules prove only formulas that are valid with respect to its semantics. In most cases, this comes down to its rules having the property of preserving truth
Truth
Truth has a variety of meanings, such as the state of being in accord with fact or reality. It can also mean having fidelity to an original or to a standard or ideal. In a common usage, it also means constancy or sincerity in action or character...
, but this is not the case in general. The word derives from the Germanic 'Sund' as in Gesundheit, meaning health. Thus to say that an argument is sound means, following the etymology, to say that the argument is healthy.
Of arguments
An argumentArgument
In philosophy and logic, an argument is an attempt to persuade someone of something, or give evidence or reasons for accepting a particular conclusion.Argument may also refer to:-Mathematics and computer science:...
is sound if and only if
- The argument is validValidityIn logic, argument is valid if and only if its conclusion is entailed by its premises, a formula is valid if and only if it is true under every interpretation, and an argument form is valid if and only if every argument of that logical form is valid....
. - All of its premises are trueTruthTruth has a variety of meanings, such as the state of being in accord with fact or reality. It can also mean having fidelity to an original or to a standard or ideal. In a common usage, it also means constancy or sincerity in action or character...
.
For instance,
- All men are mortal.
- SocratesSocratesSocrates was a classical Greek Athenian philosopher. Credited as one of the founders of Western philosophy, he is an enigmatic figure known chiefly through the accounts of later classical writers, especially the writings of his students Plato and Xenophon, and the plays of his contemporary ...
is a man. - Therefore, Socrates is mortal.
The argument is valid (because the conclusion is true based on the premises, that is, that the conclusion follows the premises) and since the premises are in fact true, the argument is sound.
The following argument is valid but not sound:
- All organisms with wings can fly.
- Penguins have wings.
- Therefore, penguins can fly.
Since the first premise is actually false, the argument, though valid, is not sound.
Logical systems
Soundness is among the most fundamental properties mathematical logic. A soundness property provides the initial reason for counting a logical system as desirable. The completenessCompleteness
In general, an object is complete if nothing needs to be added to it. This notion is made more specific in various fields.-Logical completeness:In logic, semantic completeness is the converse of soundness for formal systems...
property means that every validity (truth) is provable. Together they imply that all and only validates are provable.
Most proofs of soundness are trivial. For example, in an axiomatic system
Axiomatic system
In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A mathematical theory consists of an axiomatic system and all its derived theorems...
, proof of soundness amounts to verifying the validity of the axioms and that the rules of inference preserve validity (or the weaker property, truth). Most axiomatic systems have only the rule of mods ponens (and sometimes substitution), so it requires only verifying the validity of the axioms and one rule of inference.
Soundness properties come in two main varieties: weak and strong soundness, of which the former is a restricted form of the latter.
Soundness
Soundness of a deductive systemDeductive system
A deductive system consists of the axioms and rules of inference that can be used to derive the theorems of the system....
is the property that any sentence that is provable in that deductive system is also true on all interpretations or models 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. In other words, a system is sound if each of its theorems (i.e. formulas provable from the empty set) is valid in every structure of the language.
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 ω-consistent 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 , then also . Completeness of first-order logicFirst-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
was first explicitly established
Gödel's completeness theorem
Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. It was first proved by Kurt Gödel in 1929....
by Gödel
Godel
Godel or similar can mean:*Kurt Gödel , an Austrian logician, mathematician and philosopher*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 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.