Antecedent (logic)

An antecedent is the first half of a hypothetical proposition, whenever the if-clause precedes the then-clause. In some contexts the antecedent is called the protasis.[1]

Examples:

• If ${\displaystyle P}$, then ${\displaystyle Q}$.

This is a nonlogical formulation of a hypothetical proposition. In this case, the antecedent is P, and the consequent is Q. In an implication, if ${\displaystyle \phi }$ implies ${\displaystyle \psi }$ then ${\displaystyle \phi }$ is called the antecedent and ${\displaystyle \psi }$ is called the consequent.[2] Antecedent and consequent are connected via logical connective to form a proposition.

• If ${\displaystyle X}$ is a man, then ${\displaystyle X}$ is mortal.

"${\displaystyle X}$ is a man" is the antecedent for this proposition.

• If men have walked on the moon, then I am the king of France.

Here, "men have walked on the moon" is the antecedent.

Let ${\displaystyle y=x+1}$. If ${\displaystyle x=1}$ then ${\displaystyle y=2}$

See also

• Affirming the consequent (fallacy)
• Denying the antecedent (fallacy)
• Necessity and sufficiency

References

1. See Conditional sentence.
2. Sets, Functions and Logic - An Introduction to Abstract Mathematics, Keith Devlin, Chapman & Hall/CRC Mathematics, 3rd ed., 2004