The expression you've provided, ~p → (p → q), is a logical implication involving propositions $p$, $q$, and a negation of $p$, written as $\sim p$ (read as "not p"). Let’s break down and simplify this logical expression step by step:

Step-by-Step Analysis:

1. Expression: $\sim p \rightarrow (p \rightarrow q)$

   • $\sim p$: This means "not $p$", so the truth value of $p$ is false.
   • $p \rightarrow q$: This is a conditional statement that means "if $p$, then $q$".

2. Truth Table Construction: To fully understand this expression, we need to construct a truth table for all possible truth values of $p$ and $q$.

$p$	$q$	$\sim p$	$p \rightarrow q$	$\sim p \rightarrow (p \rightarrow q)$
T	T	F	T	T
T	F	F	F	T
F	T	T	T	T
F	F	T	T	T

Interpretation:

• Case 1: $p = T$, $q = T$:

  • $\sim p = F$, and $p \rightarrow q = T$.
  • So, $\sim p \rightarrow (p \rightarrow q) = T$, because $F \rightarrow T = T$.

• Case 2: $p = T$, $q = F$:

  • $\sim p = F$, and $p \rightarrow q = F$.
  • So, $\sim p \rightarrow (p \rightarrow q) = T$, because $F \rightarrow F = T$.

• Case 3: $p = F$, $q = T$:

  • $\sim p = T$, and $p \rightarrow q = T$.
  • So, $\sim p \rightarrow (p \rightarrow q) = T$, because $T \rightarrow T = T$.

• Case 4: $p = F$, $q = F$:

  • $\sim p = T$, and $p \rightarrow q = T$.
  • So, $\sim p \rightarrow (p \rightarrow q) = T$, because $T \rightarrow T = T$.

Conclusion:

The statement $\sim p \rightarrow (p \rightarrow q)$ is always true, which means this is a tautology. No matter the truth values of $p$ and $q$, the whole expression will evaluate to true.

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Would you like more details on how the tautology is derived or how conditionals work? Here are some related questions:

1. What is the definition of a tautology in propositional logic?
2. How do truth tables help verify logical statements?
3. Can you explain the properties of the conditional (→) operator in more depth?
4. How does the negation ($\sim$) of a proposition affect logical expressions?
5. What are other examples of tautologies in propositional logic?

Tip: A conditional statement is false only when the antecedent is true and the consequent is false.