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In logic, an *implication* is a statement formed by two propositions. It takes the form *if p, then q* and is written as \( p \rightarrow q \).
Let's dive into how this works:

• *If p is True* and *q is True* - the statement \(p \rightarrow q\) is True.
• *If p is True* and *q is False* - the statement \(p \rightarrow q\) is False.
• *If p is False* (regardless of q's value) - the statement \(p \rightarrow q\) is always True.

This might look odd, but it's because implication focuses on whether we can find any counterexamples.
If none exist, then the implication is considered true. We'll keep using this rule in each step of building the truth table.

Truth values are simply whether a statement is True (T) or False (F).
In logical expressions, these values help us understand the result of combining different propositions.
For our expression \(((p \rightarrow q) \rightarrow r) \rightarrow s\), we have four variables - \( p, q, r, s\).
Each can be either True or False.
Here's how we approach it:

• We start by listing all possible combinations of these values. There are 16 combinations for 4 variables.
• Then, we compute the truth values step by step using the rules for implications.

This systematic method ensures that we cover every possible scenario, which is the essence of creating a truth table.

Building a truth table is like solving a puzzle. Let's see exactly how the steps come together:
Step 1: List all possible combinations of truth values for \( p, q, r, \) and \ s\.

Step 2: Calculate \( p \rightarrow q \) using the implication rule.
This is done for each combination.

Step 3: Use the results from Step 2 to find \( (p \rightarrow q) \rightarrow r \).
Apply the implication rule again here.

Step 4: Now calculate \(((p \rightarrow q) \rightarrow r) \rightarrow s\).
We take the values from Step 3 and apply the rule one more time.

Step 5: Finally, put all these values into a table.
The columns should include all your propositions and the intermediate results.
This visual layout makes it clear how the final truth values are computed based on each step.
Utilize bullet points and a neat structure to break down each step for easy understanding.