### Logic: Propositional logic

### Implication

Earlier, we became acquainted with the *logical operators not*, *and*, and *or*. Here we treat two more important operators.

Implication

The **implication** of two propositions is a proposition that is precisely true if the first proposition is false or the second proposition is true. In all other cases, the implication is false.

Implication is denoted by the *if...then*-operator #\rightarrow#. So, if #\blue p# and #\blue q# stand for propositions, then #\blue{ p}\rightarrow \blue{q}# indicates that #\blue p# **implies** #\blue q#.

**Example**

The proposition \(\blue{\textit{"If}}\) #\blue{\textrm{it rains}}# \(\blue{\textit{then}}\) #\blue{\textrm{the ground is wet"}}# is true if #\blue{\textrm{"It rains"}}# and #\blue{\textrm{"The ground is wet"}}# are both true statements.

The implication is also true whenever the proposition #\blue{\textrm{"It rains"}}# is false.

Operator notation: #\blue{\textrm{"It rains"}} \rightarrow \blue{\textrm{"The ground is wet"}}#

Use of implication makes it possible to express equivalence of propositions.

Bi-implication

The **bi-implication **of two propositions is a proposition that is true exactly when both propositions have the same value. This means that the bi-implication is true if the two propositions are either both true or both false. If not, the bi-implication is false.

Bi-implication of two statements #\blue p# and #\blue q# expresses **equivalence** in the sense that #\blue p# and #\blue q# always have the same value (*true*/*false*).

Bi-implication is denoted by the *if and only if*-operator #\leftrightarrow#.

**Example**

The proposition #\blue{\textrm{"I am Janine's child"}}# #\blue{\textit{if and only if} }# #\blue{\textrm{"Janine is my parent"}}# is true since #\blue{\textrm{"I am Janine's child"}}# and #\blue{\textrm{"Janine is my parent"}}# both are either true or false statements at the same time.

Operator notation:

#\blue{\textrm{"I am Janine's child"} }# #\leftrightarrow# #\blue{\textrm{"Janine is my parent"}}#.

Is the following a correct logical consequence?

"I did not study hard, so I will not pass my exam"

Given the proposition ''if I study hard, then I will pass my exam'', we know that studying hard implies passing. However, we do not know what happens when I do not study hard. Therefore, we cannot conclude that I will not pass my exam if I do not study hard.

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