Modus Ponendo Ponens/Also known as

Proof Rule

Modus Ponendo Ponens is also known as:

• Modus ponens, abbreviated M.P.
• The rule of implies-elimination
• The rule of arrow-elimination
• The rule of (material) detachment
• The process of inference

1910: Alfred North Whitehead and Bertrand Russell: Principia Mathematica: Volume $\text { 1 }$, seemingly uneasy with the language they are using, state:

The process of the inference cannot be reduced to symbols.

Having said that, they then go on to write:

... we shall write instead

"$\vdash p \supset \, \vdash q$,"

which is to be considered as a mere abbreviation of the threefold statement

"$\vdash p$" and "$\vdash \paren {p \supset q}$" and "$\vdash q$."

Remember that 1910: Alfred North Whitehead and Bertrand Russell: Principia Mathematica use $\supset$ to denote the implication function.

Sources

• 1910: Alfred North Whitehead and Bertrand Russell: Principia Mathematica: Volume $\text { 1 }$ ... (previous) ... (next): Chapter $\text{I}$: Preliminary Explanations of Ideas and Notations
• 1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $3$: The Method of Deduction: $3.1$: Formal Proof of Validity: Rules of Inference: $1.$