# Dictionary Definition

soundness

### Noun

1 a state or condition free from damage or decay
[ant: unsoundness]

3 the firmness and tone of healthy tissue; "his
muscle firmness" [syn: firmness]

# User Contributed Dictionary

## English

### Noun

soundness# Extensive Definition

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.

## Sound arguments

An argument is sound if and only
if

For instance,

- All men are mortal.
- Socrates 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 animals can fly.
- Pigs are animals.
- Therefore, pigs can fly.

Since the first premise is actually false, the
argument, though valid, is not sound.

## Soundness of logical systems

Soundness is among the most fundamental
properties in mathematical logic. A 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). Most axiomatic systems have only the
rule of modus 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 special case 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 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: ?S P,
then also ?L P. In other words, a system is weakly 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 ? of
sentences of that language is also a logical
consequence of that set ?, in the sense that any model that
makes all members of ? true will also make P true. In symbols where
? is a set of sentences of L: if ? ?S P, then also ? ?L P. Notice
that in the statement of strong soundness, when ? is empty, we have
the statement of weak soundness.

## 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
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 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.

## References

- Fundamentals of Mathematical Logic
- Irving Copi. Symbolic Logic, Vol. 5, Macmillian Publishing Co., 1979.
- Boolos, Burgess, Jeffrey. Computability and Logic, Vol. 4, Cambridge, 2002.

portalpar Logic

soundness in German: Korrektheit (Logik)

soundness in Icelandic: Rétt
röksemdafærsla

soundness in Italian: Correttezza (logica
matematica)

soundness in Macedonian: Правилност

soundness in Japanese: 健全性

soundness in Ukrainian: Правильність

soundness in Chinese: 可靠性定理

# Synonyms, Antonyms and Related Words

admissibility, advantageousness,
agreeableness,
aplomb, auspiciousness, authenticity, authoritativeness,
authority, balance, balanced personality,
beneficialness,
benevolence,
benignity, body, calculability, canonicalness, canonicity, class, cogency, common sense, completeness, concreteness, constancy, contact with
reality, cool, cool head,
coolheadedness,
coolness, credit
standing, density,
dependability,
desert, due sense of,
durability, entireness, equilibrium, eupepsia, excellence, expedience, fairness, faithworthiness,
fastness, favorableness, fine
fettle, fine shape, fine whack, fineness, firmness, first-rateness,
force, fullness, glowing health, good
condition, good constitution, good digestion, good health, good
sense, good shape, good trim, goodliness, goodness, gospel truth,
grace, haleness, hardness, healthfulness, healthiness, healthy body,
healthy constitution, healthy mind, helpfulness, high feather,
homeostasis, horse
sense, imperturbability,
intactness, integrity, invariability, invincibility, justifiability, justness, kindness, level head,
level-headedness, levelheadedness,
logic, logicality, logicalness, lucid interval,
lucidity, marbles, mass, materiality, mental balance,
mental equilibrium, mental health, mental hygiene, mental poise,
merit, mind, mint condition, nerve, niceness, normalcy, normality, normalness, orthodoxicalness,
orthodoxism,
orthodoxness,
orthodoxy, palpability, picture of
health, plain sense, plausibility, pleasantness, plenitude, ponderability, practical
mind, practical wisdom, practicality, predictability, profitableness, quality, rationality, reason, reasonability, reasonableness, reliability, religious
truth, rewardingness, right
belief, right mind, rightness, robust health,
rootedness, rude
health, rugged health, sanemindedness, saneness, sang-froid, sanity, secureness, security, sense, senses, sensibleness, skillfulness, sober senses,
sober-mindedness, soberness, sobriety, solidity, solvency, sound mind, sound
sense, soundness of mind, stability, stable equilibrium,
stable state, stableness, staunchness, steadfastness, steadiness, steady nerves,
steady state, stoutness, strength, sturdiness, substance, substantiality, substantialness,
superiority, sweet
reason, tangibility,
temper, the truth, top
shape, toughness,
traditionalism,
trustworthiness,
undeflectability,
unerringness,
uniformity, unindebtedness,
unshakable nerves, unshakableness, usefulness, validity, value, virtue, virtuousness, weight, wholeness, wholesomeness, wit, worth