True Statement is implied by Every Statement

Theorem

If something is true, then anything implies it.

Formulation 1

\(\ds p\)

\(\)

\(\ds \)

\(\ds \vdash \ \ \)

\(\ds q \implies p\)

\(\)

\(\ds \)

Formulation 2

$\vdash q \implies \paren {p \implies q}$

Also see

• Paradoxes of Material Implication, in which category this result is grouped

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
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): implication: 1. (material implication)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): implication: 1. (material implication)