Implication

The implication is important in mathematics. $\phi$ implies $\psi$ is denoted as $\phi\implies\psi$, that means the truth of $\psi$ follows from the truth of $\phi$. And $\phi$ is called antecedent and $\psi$ is consequent. The truth table of $\phi\implies\psi$ as follows:

The T denotes the truth and F denotes the false.

1. A true conclusion from a true assumption, so the first row is true.
2. If that implication is true, that means $\psi$ would have to be T if $\phi$ is T. So we cannot have $\phi$ is T and $\psi$ is F if $\phi\implies\psi$ is T. Hence $\phi\implies\psi$ must be F.
3. We can look at “$\phi$ does not imply $\psi$” ($\phi\nRightarrow\psi$) that is even through $\phi$ is T, $\psi$ is nevertheless F. So $\phi\nRightarrow\psi$ is T if and only if $\phi$ is T and $\psi$ is F. In all other circumstances, $\phi\nRightarrow\psi$ is F, which means $\phi\implies\psi$ is T. So, the third and fourth rows are T.

📝 Note: The implication involves causality. For example, “$\sqrt{2}$ is irrational” does not imply “$1+1=2$”, because this two statements has no relationship, they are independent of each other.

Equivalence

Two statements $\phi$ and $\psi$ are said to be equivalent if each implies the other. It is denoted $\phi\Leftrightarrow\psi$ if $\phi\implies\psi$ and $\psi\implies\phi$. $\phi\Leftrightarrow\psi$ is true if $\phi$ and $\psi$ are both true or both false.

Summary

Assignment 4

4. Proof of modus ponens: The rule of logic stating that if a conditional statement ($\phi\land(\phi\implies\psi)$) is accepted, and the antecedent ($\phi$) holds, then the consequent ($\psi$) may be inferred.

$\phi$	$\psi$	$\phi\implies\psi$	$\phi\land(\phi\implies\psi)$	$[\phi\land(\phi\implies\psi)]\implies\psi$
T	T	T	T	T
T	F	F	F	T
F	T	T	F	T
F	F	T	F	T