Soundness

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 well-formed formula that can be proven in the system is logically valid with respect to the logical semantics of the system.

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 well-known 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, ostriches). For an argument to be sound, the argument must be valid and its premises must be true.^[2]

Some authors, such as Lemmon, have used the term "soundness" as synonymous with what is now meant by "validity",^[3] which left them with no particular word for what is now called "soundness". But this division of the terms is now very widespread.

Use in mathematical logic

Logical systems

In

A logical system with syntactic entailment ${\displaystyle \vdash }$ and semantic entailment ${\displaystyle \models }$ is sound if for any sequence ${\displaystyle A_{1},A_{2},...,A_{n}}$ of sentences in its language, if ${\displaystyle A_{1},A_{2},...,A_{n}\vdash C}$, then ${\displaystyle A_{1},A_{2},...,A_{n}\models C}$. 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.^[