General info on logic

See also: types of valid arguments, direct proof, derivable arguments and inconsistent premises

argument

The notes then provide a formal definition for the structure of a proof:

“Definition: Any implication (i.e. $A\to B$) is called an argument. A general form of an argument is of the form

$(P_1\wedge P_2\wedge \cdots \wedge P_n)\to Q(\ast )$”

IF $P_1$ is true, AND $P_2$ is true, AND… AND $P_n$ is true, THEN $Q$ must be true. We say that the argument $(\ast )$ is valid if $(P_1\wedge P_2\wedge \cdots \wedge P_n)\to Q$ is a tautology. An argument is valid if its logical structure is perfect. It means that it is impossible for all the premises to be true and the conclusion to be false simultaneously.

Denote the argument as:

$\begin{array}{c}{P}_1\\ P_2\\ ⋮\\ P_n\\ Q\end{array}$

For any (compound) statements $A$ and $B$, if the implication $A\to B$ is a tautology, then we say that A logically implies B and we write $A⟹B$.

“Note that '$\to$' and '$⟹$' are different! ($P⟹Q$ means that $P\to Q$ is a tautology)”