# Paradoxes of Material Implication

## Theorems

The conditional operator has the following counter-intuitive properties:

- $\top \dashv \vdash p \implies \top$, or just $\vdash p \implies \top$
- $p \dashv \vdash \top \implies p$

- $\top \dashv \vdash \bot \implies p$, or just $\vdash \bot \implies p$
- $\neg p \dashv \vdash p \implies \bot$

## Proof

### Proof by Truth Table

We apply the Method of Truth Tables to the propositions.

- $\top \dashv \vdash p \implies \top$ and $p \dashv \vdash \top \implies p$:

As can be seen by inspection, the truth values in the appropriate columns match for all boolean interpretations:

$\begin{array}{|c|ccc||c|ccc|} \hline
\top & p & \implies & \top & p & \top & \implies & p \\
\hline
T & F & T & T & F & T & F & F \\
T & T & T & T & T & T & T & T \\
\hline
\end{array}$

$\blacksquare$

- $\top \dashv \vdash \bot \implies p$ and $\neg p \dashv \vdash p \implies \bot$:

As can be seen by inspection, the truth values in the appropriate columns match for all boolean interpretations:

$\begin{array}{|c|ccc||cc|ccc|} \hline
\top & \bot & \implies & p & \neg & p & p & \implies & \bot\\
\hline
T & F & T & F & T & F & F & T & F \\
T & F & T & T & F & T & T & F & F \\
\hline
\end{array}$

$\blacksquare$

## Also presented as

These results are also presented in the following forms:

### True Statement is implied by Every Statement

**If something is true, then anything implies it.**

#### Formulation 1

- $p \vdash q \implies p$

#### Formulation 2

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

### False Statement implies Every Statement

**If something is false, then it implies anything.**

#### Formulation 1

- $\neg p \vdash p \implies q$

#### Formulation 2

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

Also note this counterintuitive result:

### Disjunction of Conditional and Converse

- $\vdash \left({p \implies q}\right) \lor \left({q \implies p}\right)$

## Comment

These counter-intuitive results have caused debate and puzzlement among philosophers for millennia.

In particular, the result $\neg p \vdash p \implies q$ is known as a vacuous truth. It is exemplified by the (rhetorical) argument:

- "If England win the Ashes this year, then I'm a monkey's uncle."

(Alert viewers will note that in 2009 my sister's daughter was indeed a simian. The trend continued into 2011.)

## Sources

- 1946: Alfred Tarski:
*Introduction to Logic and to the Methodology of Deductive Sciences*(2nd ed.) ... (previous) ... (next): $\S \text{II}.13$: Symbolism of sentential calculus - 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): $\S 2.2$: Theorems and Derived Rules - 1973: Irving M. Copi:
*Symbolic Logic*(4th ed.) ... (previous) ... (next): $2.4$: Statement Forms