Truth Value

Definition

A truth value is the logical status of a proposition.

In basic logic, the two truth values are:

true
false

Example:

2 + 3 = 5

Truth value:

true

Example:

2 + 3 = 9

Truth value:

false

Truth value tells whether a proposition is true or false.

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Truth Value Requires A Proposition

Only propositions have truth values.

Proposition:

It is raining.

This can be true or false.

Not a proposition:

Close the door.

This is a command.

It does not have a truth value.

Truth values apply to statements, not every expression.

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Truth Value Depends On Interpretation

Some propositions depend on context.

Example:

x > 5

If:

x = 10

then:

true

If:

x = 2

then:

false

The symbolic form is the same.

The truth value changes with interpretation.

Truth value depends on what the symbols refer to.

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Truth Values In Compound Statements

Logical operators combine truth values.

Example:

P AND Q

If:

P = true
Q = false

then:

P AND Q = false

The compound truth value is determined by the operator and the truth values of the parts.

Compound statements compute truth values from smaller statements.

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Truth vs Validity

Truth applies to propositions.

Validity applies to arguments.

Example proposition:

2 + 3 = 5

This is true.

Example argument form:

If P, then Q
P
Therefore Q

This is valid.

Truth belongs to statements; validity belongs to reasoning structure.

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Final Summary

• A truth value is true or false.
• Propositions have truth values.
• Commands and questions do not have truth values.
• Truth value can depend on interpretation.
• Logical operators build compound truth values.
• Truth is different from validity.