Modis Ponens, also known as Conditional Elimination and Modus ponendo ponens [1][2], is a rule of inference in propositional logic that states that if we have a material condtional that has a true antecedent, then we may infer the consequent of the antedent. In other words, if P implies Q is true and P is true, then we may infer that Q is true. This is often stated symbolically in rule form as:

\frac{P \rightarrow Q, P}{\therefore Q.}

In sequent notation, it could be expressed as:

P \rightarrow Q, P \vdash Q.

As a theorem of propositional logic, it is expressed as: ((P \rightarrow Q)\wedge P) \rightarrow Q).

In all three expressions above, P and Q are metasyntactic variables representing well-formed formulas of propositional logic.


  1. Ikenaga, B. "Rules of Inference and Logic Proofs". Retrieved from
  2. Margaris, A. (1967). First Order Mathematical Logic. Blaisdell Publishing Company.

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