Mathematical logic and foundations Definition

Implication

Definition

Let $p$ and $q$ be propositions. The implication from $p$ to $q$ is the proposition $p \to q$ , read "if $p$ , then $q$ ". The proposition $p$ is the hypothesis, and the proposition $q$ is the conclusion. The proposition $p \to q$ is false when $p$ is true and $q$ is false. In all other cases, $p \to q$ is true.

$p$	$q$	$p \to q$
T	T	T
T	F	F
F	T	T
F	F	T

Remarks

Another notation for $p \to q$ is $p \Rightarrow q$ . The proposition $p \to q$ has the same truth value as $\neg p \lor q$ . When $p$ is false, $p \to q$ is true. This is vacuous truth. The converse of $p \to q$ is $q \to p$ . It can have a different truth value.

Reading path

To understand this page from the ground up, read these 2 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Proposition concept
Implication definition

Where this leads

1103 pages build on this one.

Tag	`0ZNX6DQ3EJ`
URI	`m2://0ZNX6DQ3EJ`
Kind	definition
Writing	W1 stated
MSC	03B05
Defines	Implication
Uses	Proposition

Implication

Definition

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

What links here

Used by