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:

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

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

The 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 $\implies \EE$

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

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.

When invoking 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 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.

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