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 the implication "${\displaystyle \phi }$ implies ${\displaystyle \psi }$", ${\displaystyle \phi }$ 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 while "${\displaystyle X}$ is mortal" is the consequent of the 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 and "I am the king of France" is the consequent.

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

• If ${\displaystyle x=1}$ then ${\displaystyle y=2}$,.

"${\displaystyle x=1}$" is the antecedent and "${\displaystyle y=2}$" is the consequent of this hypothetical proposition.

See also

• Consequent
• 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