Rule of Material Implication/Formulation 1/Proof by Truth Table

Theorem

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

Proof

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

• 1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): $\S 2.3$: Basic Truth-Tables of the Propositional Calculus
• 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $2$: The Propositional Calculus $2$: $3$ Truth-Tables
• 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 1$: Some mathematical language: Connectives
• 1988: Alan G. Hamilton: Logic for Mathematicians (2nd ed.) ... (previous) ... (next): $\S 1$: Informal statement calculus: $\S 1.2$: Truth functions and truth tables: Example $1.4 \ \text{(a)}$
• 2000: Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems ... (previous) ... (next): $\S 1.4.1$: The meaning of logical connectives: Exercise $1.8: \ 1$