Propositions and Connectives

Propositions #

A proposition is a statement that is either true or false. For example, the statement “The sky is blue” is a proposition. “True” or “false” is considered the truth value of the proposition. In the case of “The sky is blue”, the truth value is “true”. In the case of “The sky is green”, the truth value is “false”.

The truth value “true” is represented by true in Java, True in Python, and \(\top\) in formal logic. The truth value “false” is represented by false in Java, False in Python, and \(\bot\) in formal logic.

Connectives #

A connective is a symbol that connects two propositions. For example, the connective “and” connects two propositions. Examples include:

Connective	Java	Python	Formal logic
and	`&&`	`and`	\(\land\)
or	`\|\|`	`or`	\(\lor\)
xor	`^`	`xor`	\(\oplus\)
not	`!`	`not`	\(\lnot\)
implies	`->`	`->`	\(\rightarrow\)
if and only if	`<->`	`<->`	\(\leftrightarrow\)

Truth Tables #

A truth table is a table that shows the truth values of a proposition and its connectives.

Truth tables for different connectives are shown below.

And #

\(P\)	\(Q\)	\(P\land Q\)
false	false	false
false	true	false
true	false	false
true	true	true

Or #

\(P\)	\(Q\)	\(P\lor Q\)
false	false	false
false	true	true
true	false	true
true	true	true

Xor #

\(P\)	\(Q\)	\(P\oplus Q\)
false	false	true
false	true	true
true	false	true
true	true	true

Not #

\(P\)	\(\lnot P\)
false	true
true	false

Implies #

\(P\)	\(Q\)	\(P\rightarrow Q\)
false	false	true
false	true	true
true	false	false
true	true	true

If and Only If #

\(P\)	\(Q\)	\(P\leftrightarrow Q\)
false	false	true
false	true	false
true	false	false
true	true	true