[Logic] Arguments, Validity, Soundness and Completeness

Arguments:

In logic and philosophy, an argument is a series of statements typically used to persuade someone of something or to present reasons for accepting a conclusion [Wikipedia]

An argument consists of one or more premises and only one conclusion.

Another reference is here.

 Deductive arguments

The conclusion is a logic consequence of the premises. It is based on the premises and then the conclusion follows necessarily.

 Inductive arguments

An inductive argument is an argument that is intended by the arguer merely to establish or increase the probability of its conclusion. [Encyclopedia]

 Validity

 A valid deductive argument guarantees the truth of the conclusion provided the premises are true, regardless of the reality of the premises, by following logic form. Being a valid deductive argument, if all premises are true, the conclusion is impossibly false (must be true).

A --> B
A
--------
B

This is a valid argument no matter whether A and (A-->B) are possibly true or not in real.

Soundness

A sound argument is a valid argument and the premises are true in real.

Validity vs. Soundness

In the example above,

• if A and B stand for "All trains travel faster than cars" and "No one will travel by car", then the argument is valid but not sound because two premises are not true in real. 
•  The argument such as  "All even numbers can be divided by 2; 4 is a even number; therefore 4 can be divided by 2", is valid and sound.

Completeness

A system is said to be complete if something is really true, the system is capable of proving it.

Soundness vs. Completeness

A good way to understand these definitions is that soundness prevents false negatives and completeness prevents false positives. [from note]

False negatives mean "tested to be true actually it is false". Soundness prevents "something is said to be true but actually is not true". It guarantees that "if something is said to be true, it really is true."

False positives mean "tested to be false actually it is true". Completeness prevents "something is said to be false but actually is not false". It guarantees that "all true things are provable."

A sound logic proves only true things. A complete logic proves all true things. [from note]