# Paradoxes of Material Implication

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## Theorems

The conditional operator has the following counter-intuitive properties:

### Tautological Consequent

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

### Tautological Antecedent

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

### Contradictory Antecedent

- $\bot \implies p \dashv \vdash \top$

### Contradictory Consequent

- $p \implies \bot \dashv \vdash \neg p$

## 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

\(\ds p\) | \(\) | \(\ds \) | ||||||||||||

\(\ds \vdash \ \ \) | \(\ds q \implies p\) | \(\) | \(\ds \) |

#### Formulation 2

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

### False Statement implies Every Statement

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

#### Formulation 1

\(\ds \neg p\) | \(\) | \(\ds \) | ||||||||||||

\(\ds \vdash \ \ \) | \(\ds p \implies q\) | \(\) | \(\ds \) |

#### Formulation 2

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

Also note this counterintuitive result:

### Disjunction of Conditional and Converse

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

## Examples

### Red Grass and Green Moon

The compound statement:

**If grass is red then the moon is made of green cheese**

is true, despite being semantically meaningless.

## Historical Note

The Paradoxes of Material Implication 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.*

## 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): Chapter $2$: The Propositional Calculus $2$: $2$: Theorems and Derived Rules - 1973: Irving M. Copi:
*Symbolic Logic*(4th ed.) ... (previous) ... (next): $2$ Arguments Containing Compound Statements: $2.4$: Statement Forms - 1988: Alan G. Hamilton:
*Logic for Mathematicians*(2nd ed.) ... (previous) ... (next): $\S 1$: Informal statement calculus: $\S 1.2$: Truth functions and truth tables: Conditional