Modus Ponendo Ponens/Proof Rule

Proof Rule

Modus ponendo ponens is a valid argument in types of logic dealing with conditionals $\implies$.

This includes propositional logic and predicate logic, and in particular natural deduction.

As a proof rule it is expressed in the form:

If we can conclude $\phi \implies \psi$, and we can also conclude $\phi$, then we may infer $\psi$.

Thus it provides a means of eliminating a conditional from a sequent.

It can be written:

$\ds {\phi \qquad \phi \implies \psi \over \psi} \to_e$

Tableau Form

Let $\phi \implies \psi$ be a well-formed formula in a tableau proof whose main connective is the implication operator.

The Modus Ponendo Ponens is invoked for $\phi \implies \psi$ and $\phi$ as follows:

			Pool:					The pooled assumptions of $\phi \implies \psi$
								The pooled assumptions of $\phi$
			Formula:					$\psi$
			Description:					Modus Ponendo Ponens
			Depends on:					The line containing the instance of $\phi \implies \psi$
								The line containing the instance of $\phi$
			Abbreviation:					$\text{MPP}$ or $\implies \EE$

Also known as

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, or just Rule of Detachment
• The Process of Inference

Also see

• This is a rule of inference of the following proof systems:
  • Definition:Natural Deduction
  • Definition:Hilbert Proof System Instance 1 for Predicate Logic
  • Definition:Hilbert Proof System/Instance 1
  • Definition:Hilbert Proof System/Instance 2

Linguistic Note

Modus Ponendo Ponens is Latin for mode (or method) that by affirming, affirms.

The shorter form Modus Ponens means mode that affirms, of method of affirming.

Technical Note

When invoking Modus Ponendo Ponens in a tableau proof, use the {{ModusPonens}} template:

{{ModusPonens|line|pool|statement|first|second}}

or:

{{ModusPonens|line|pool|statement|first|second|comment}}

where:

line is the number of the line on the tableau proof where Modus Ponendo Ponens is to be invoked

pool is the pool of assumptions (comma-separated list)

statement is the statement of logic that is to be displayed in the Formula column, without the $ ... $ delimiters

first is the first of the two lines of the tableau proof upon which this line directly depends, the one in the form $p \implies q$

second is the second of the two lines of the tableau proof upon which this line directly depends, the one in the form $p$

comment is the (optional) comment that is to be displayed in the Notes column.

Sources

• 1946: Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences (2nd ed.) ... (previous) ... (next): $\S \text{II}.15$: Rules of inference
• 1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): $\S 3.9$: Derivation by Substitution
• 1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): $\S 4.2$: The Construction of an Axiom System: $RST \, 3$
• 1964: Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning ... (previous) ... (next): $\text{I}$: 'NOT' and 'IF': $\S 3$
• 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $1$: The Propositional Calculus $1$: $2$ Conditionals and Negation
• 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 1$: Some mathematical language: Axioms
• 1980: D.J. O'Connor and Betty Powell: Elementary Logic ... (previous) ... (next): $\S \text{II}$: The Logic of Statements $(2): \ 1$: Decision procedures and proofs: $1$
• 2000: Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems ... (previous) ... (next): $\S 1.2.1$: Rules for natural deduction