Axiom:Rule of Simplification

Context

The rule of simplification is one of the axioms of natural deduction.

The rule

This is two axioms in one.

1. If we can conclude $p \land q$, then we may infer $p$: $p \land q \vdash p$
2. If we can conclude $p \land q$, then we may infer $q$: $p \land q \vdash q$

This is sometimes known as the rule of and-elimination.

It can be written:

$\displaystyle {p \land q \over p} \land_{e_1} \qquad \qquad {p \land q \over q} \land_{e_2}$

• Abbreviation: $\land \mathcal E_1$ or $\land \mathcal E_2$
• Deduced from: The pooled assumptions of $p \land q$.
• Depends on: The line containing $p \land q$.

Explanation

Note that there are two axioms here in one. The first of the two tells us that, given a conjunction, we may infer the first of the conjuncts, while the second says that, given a conjunction, we may infer the second of the conjuncts.

At this stage, such attention to detail is important.

Demonstration by Truth Table

$\begin{array}{|ccc||c|c|} \hline p & \land & q & p & q \\ \hline F & F & F & F & F \\ F & F & T & F & T \\ T & F & F & T & F \\ T & T & T & T & T \\ \hline \end{array}$

As can be seen, when $p \land q$ is true so are both $p$ and $q$.

$\blacksquare$

Sources

• Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning (1964): $\text{II}: \S 3$
• E.J. Lemmon: Beginning Logic (1965): $\S 1.3$: Theorems $14, \ 15$
• Alan G. Hamilton: Logic for Mathematicians (1978): $\S 1.2$: Example $1.8 \ \text{(a)}$
• Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems (2000): $\S 1.2.1$