Rule of Material Implication/Formulation 1

Theorem

$p \implies q \dashv \vdash \neg p \lor q$

This can be expressed as two separate theorems:

Forward Implication

$p \implies q \vdash \neg p \lor q$

Reverse Implication

$\neg p \lor q \vdash p \implies q$

Proof by Truth Table

We apply the Method of Truth Tables.

As can be seen by inspection, the truth values under the main connectives match for all boolean interpretations.

$\begin {array} {|ccc||cccc|} \hline p & \implies & q & \neg & p & \lor & q \\ \hline \F & \T & \F & \T & \F & \T & \F \\ \F & \T & \T & \T & \F & \T & \T \\ \T & \F & \F & \F & \T & \F & \F \\ \T & \T & \T & \F & \T & \T & \T \\ \hline \end {array}$

$\blacksquare$

Sources

• 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $2$: The Propositional Calculus $2$: $2$: Theorems and Derived Rules: Theorem $49$
• 1988: Alan G. Hamilton: Logic for Mathematicians (2nd ed.) ... (previous) ... (next): $\S 1$: Informal statement calculus: $\S 1.2$: Truth functions and truth tables: Exercise $4$
• 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.1$: What is a Set?
• 2000: Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems ... (previous) ... (next): $\S 1.2.4$: Provable equivalence
• 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.3.3$
• 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.10$: Exercise $2.3$