The propositional sentence $F$ is: $F: \neg P \vee (\text{if} \, (\neg Q) \, \text{then} \, S \, \text{else} \, (P \vee Q))$

To break down the logical structure and find the "sentence" parts, let's focus on the individual propositional components, which represent atomic propositions or sentences within the larger formula.

1. Atomic Sentences:

   • $P$, $Q$, and $S$ are the atomic propositions in the expression. These are the fundamental components or sentences that cannot be broken down further in terms of propositional logic.

2. Negations:

   • $\neg P$ and $\neg Q$ are negated atomic sentences.

3. Disjunction (or):

   • The main disjunction is $\neg P \vee \text{(conditional expression)}$, where the first part is $\neg P$ (a negation of an atomic sentence) and the second part is a conditional expression.

4. Conditional (if-then-else):

   • The condition is $\neg Q$.
   • If $\neg Q$ is true, the result is $S$.
   • If $\neg Q$ is false, the result is $P \vee Q$ (a disjunction between atomic sentences $P$ and $Q$).

Summary of Sentence Parts:

The "sentence parts" of this formula are the atomic propositions $P$, $Q$, and $S$, since these are the basic building blocks that represent propositions or statements in propositional logic.

Would you like further clarification or a more detailed breakdown of this? Here are some related questions for practice:

1. What is the truth table for the expression $F: \neg P \vee (\text{if} \, (\neg Q) \, \text{then} \, S \, \text{else} \, (P \vee Q))$?
2. How does the conditional statement behave in terms of truth values when dealing with $\neg Q$?
3. Can you rewrite $\neg P \vee (\text{if} \, (\neg Q) \, \text{then} \, S \, \text{else} \, (P \vee Q))$ using only basic logical operations?
4. How would the expression change if we replaced the "if-then-else" with just a standard implication?
5. What role do the negations $\neg P$ and $\neg Q$ play in determining the overall truth value of the expression?

Tip: Always break complex propositional formulas down into atomic components and logical operations to simplify their analysis!