Affirming the antecedent
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Affirming the antecedent is a valid argument form which proceeds by affirming the truth of the first part (the "if" part, commonly called the antecedent) of a conditional, and concluding that the second part (the "then" part, commonly called the consequent) is true. It is commonly referred to as Modus ponens, or "method of affirming."
- If P, then Q.
- P.
- Therefore, Q.
In logical operator notation, this is symbolized
- <math> p \rightarrow q <math>
- <math> \vdash p, <math>
- <math> \vdash q <math>
Many people assume that this works the other way as well, so that one could say:
- If P then Q.
- Q.
- Therefore P.
In logical operator notation, this is symbolized
- <math> p \rightarrow q <math>
- <math> \vdash q, <math>
- <math> \vdash p <math>
where <math>\vdash<math> represents the logical assertion.
But this is a Logical fallacy called Affirming the consequent. Since P implies Q, but Q does not necessarily imply P.
You can see this if we simply substitute in actuall statements for P. and Q.
- If there is fire here, then there is oxygen here.
- There is oxygen here.
- Therefore, there is fire here.
Sometimes P and Q entail each other, in that case we can say P if and only if Q. (Sometimes the shorthand P iff Q is used rather than writing out if and only if).