Axiom:Modus Ponendo Ponens
Contents |
Context
The modus ponendo ponens is one of the axioms of natural deduction.
The rule
If we can conclude $p \implies q$, and we can also conclude $p$, then we may infer $q$:
- $p \implies q, p \vdash q$
This is also known as:
- Modus ponens;
- The rule of implies-elimination;
- The rule of material detachment.
It can be written:
- $\displaystyle {p \quad p \implies q \over q} \to_e$
- Abbreviation: $\implies \mathcal E$
- Deduced from: The pooled assumptions of each of $p \implies q$ and $p$.
- Depends on: Both of the lines containing $p \implies q$ and $p$.
Explanation
This means: if we know that $p \implies q$, and we also know $p$, then we also know $q$.
Thus it provides a means of eliminating a conditional from a sequent.
Also see
The following are related argument forms:
Alternative Forms
By considering this rule in conjunction with the Rule of Implication and Extended Rule of Implication, this axiom can also be expressed:
- $p \vdash \left({p \implies q}\right) \implies q$
- $\vdash p \implies \left({\left({p \implies q}\right) \implies q}\right)$
Linguistic Note
Modus ponendo ponens is Latin for mode that by affirming, affirms.
Modus ponens means mode that affirms.
Demonstration by Truth Table
$\begin{array}{|c|ccc||c|} \hline p & p & \implies & q & q\\ \hline F & F & T & F & F \\ F & F & T & T & T \\ T & T & F & F & F \\ T & T & T & T & T \\ \hline \end{array}$
As can be seen, when $p$ is true, and so is $p \implies q$, then $q$ is also true.
Sources
- Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning (1964): $\text{I}: \S 3, \ \text{I}: \S 5$: Theorem $\text{T3}$
- E.J. Lemmon: Beginning Logic (1965): $\S 1.2$: Theorem $1$
- Alan G. Hamilton: Logic for Mathematicians (1978): $\S 1.3$: Proposition $1.9$
- H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996): $\S 1.12$
- Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems (2000): $\S 1.2.1$
