TheInfoList

In
logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...

, more precisely in
deductive reasoning Deductive reasoning, also deductive logic, is the process of reasoning Reason is the capacity of consciously applying logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making ...
, an
argument In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, lab ...
is sound if it is both valid in form and its premises are true. Soundness also has a related meaning in
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal sys ...
, wherein logical systems are sound
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, l ...
every
formula In , a formula is a concise way of expressing information symbolically, as in a mathematical formula or a . The informal use of the term ''formula'' in science refers to the . The plural of ''formula'' can be either ''formulas'' (from the mos ...
that can be proved in the system is logically valid with respect to the
semantics Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference Reference is a relationship between objects in which one object designates, or acts as a means by which to connect to or link to, another ...
of the system.

# Definition

In
deductive reasoning Deductive reasoning, also deductive logic, is the process of reasoning Reason is the capacity of consciously applying logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making ...
, 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 well-known
syllogism A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument In logic and philosophy, an argument is a series of statements (in a natural language), called the premises or premisses (bo ...
: : All men are mortal. : Socrates is a man. : 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 because, assuming the premises are true, the conclusion must be true. However, the first premise is false. Not all birds can fly (penguins, ostriches, kiwis etc.) For an argument to be sound, the argument must be valid ''and'' its premises must be true.

# Use in mathematical logic

## Logical systems

In
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal sys ...
, a
logical system A formal system is used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A formal system is essentiall ...
has the soundness property
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, l ...
every
formula In , a formula is a concise way of expressing information symbolically, as in a mathematical formula or a . The informal use of the term ''formula'' in science refers to the . The plural of ''formula'' can be either ''formulas'' (from the mos ...
that can be proved in the system is logically valid with respect to the
semantics Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference Reference is a relationship between objects in which one object designates, or acts as a means by which to connect to or link to, another ...
of the system. In most cases, this comes down to its rules having the property of ''preserving
truth Truth is the property of being in accord with fact A fact is something that is true True most commonly refers to truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In ...

''. The
converse Converse may refer to: Mathematics and logic * Converse (logic), the result of reversing the two parts of a categorical or implicational statement ** Converse implication, the converse of a material implication ** Converse nonimplication, a logical ...
of soundness is known as completeness. A logical system with syntactic
entailment Logical consequence (also entailment) is a fundamental concept Concepts are defined as abstract ideas A mental representation (or cognitive representation), in philosophy of mind Philosophy of mind is a branch of philosophy that studies th ...
$\vdash$ and semantic entailment $\models$ is sound if for any
sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

$A_1, A_2, ..., A_n$ of
sentences ''The Four Books of Sentences'' (''Libri Quattuor Sententiarum'') is a book of theology Theology is the systematic study of the nature of the Divinity, divine and, more broadly, of religious belief. It is taught as an Discipline (academia), aca ...
in its language, if $A_1, A_2, ..., A_n\vdash C$, then $A_1, A_2, ..., A_n\models C$. In other words, a system is sound when all of its
theorem In mathematics, a theorem is a statement (logic), statement that has been Mathematical proof, proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the the ...
s 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, 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 In propositional logic Propositional calculus is a branch of logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from ...

. (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.

### Soundness

Soundness of a
deductive system A formal system is used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A formal system is essentiall ...
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 Logical consequence (also entailment) is a fundamental concept Concepts are defined as abstract ideas or general notions that occur in the mind, in speech, or in thought. They are understood to be the fundamental building blocks of thoughts ...
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 In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

, 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 In mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (alge ...
.

## 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 Logical consequence (also entailment) is a fundamental concept Concepts are defined as abstract ideas A mental representation (or cognitive representation), in philosophy of mind Philosophy of mind is a branch of philosophy that studies ...
of a set of sentences Γ can be derived in the
deduction system A formal system is used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A formal system is essentiall ...
from that set. In symbols: whenever , then also . Completeness of
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal system A formal system is an used for inferring theorems from axioms according to a set of rules. These rul ...
was first explicitly established by Gödel, though some of the main results were contained in earlier work of
Skolem Thoralf Albert Skolem (; 23 May 1887 – 23 March 1963) was a Norwegian mathematician who worked in mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set ...
. 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

) is restricted to the intended one. The original completeness proof applies to ''all'' classical models, not some special proper subclass of intended ones.

*
Soundness (interactive proof)Soundness is a property of interactive proof systems that requires that no prover can make the verifier accept for a wrong statement y \not\in L except with some small probability. The upper bound of this probability is referred to as the soundness e ...

# Bibliography

* * *Boolos, Burgess, Jeffrey. ''Computability and Logic'', 4th Ed, Cambridge, 2002.