Paradoxes of Strict Implication

Theorems

The strict implication operator has the following counter-intuitive properties:

Necessary Proposition is Strictly Implied by Every Proposition

If a proposition is necessarily true, then every proposition strictly implies it.

Let $P$ be a proposition of modal logic.

Let $P$ be necessarily true.

Then:

$\forall Q: Q \implies P$

where $Q$ is an arbitrary proposition in the universe of discourse.

Impossible Proposition Strictly Implies Every Proposition

If proposition is not possibly true, then it strictly implies every proposition.

Let $P$ be a proposition of modal logic.

Let $P$ be not possibly true.

Then:

$\forall Q: P \implies Q$

where $Q$ is an arbitrary proposition in the universe of discourse.

Examples

Example: Grass is Blue

The proposition:

Grass is blue strictly implies that $2 + 2 = 4$

is an example of the Paradoxes of Strict Implication:

Necessary Proposition is Strictly Implied by Every Proposition.

Example: $2 + 2 = 5$

The proposition:

$2 + 2 = 5$ strictly implies that grass is blue

is an example of the Paradoxes of Strict Implication:

Impossible Proposition Strictly Implies Every Proposition.

Sources

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