✏️ Lots of notes for very little progress

chapter_1/1_2/investigate.md

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+# Investigate!
+
+Q: Little Timmy's Mom tells him, "If you don't eat all your broccoli, then you
+will not get any ice cream." Of course, Timmy loves his ice cream, so he quickly
+eats all his broccoli (which actually tastes pretty good).
+
+After dinner, when Timmy asks for his ice cream, he is told no! Does Timmy have
+a right to be upset? Why or why not?
+
+A: Well, probably, but in the context of this class, I'm guessning no?
+
+Let's think about this logically:
+
+Let $P$ be the predicate "If you don't eat all your broccoli", and let $Q$ be
+the conclusion "you will not get any ice cream."
+
+This is expressed as:
+
+$$ P \to Q $$
+
+If we consult our truth tables from the last section, we have:
+
+| $P$ | $Q$ | $P \to Q$ |
+| --- | --- | --------- |
+| T   | T   | T         |
+| T   | F   | F         |
+| F   | T   | T         |
+| F   | F   | T         |
+
+But remember that Timmy did eat his broccoli, so that is actually $\neg P$.
+
+| $P$ | $\neg P$ |
+| --- | -------- |
+| T   | F        |
+| F   | T        |
+
+This changes our if/then truth table:
+
+| $\neg P$ | $Q$ | $P \to Q$ |
+| -------- | --- | --------- |
+| F        | T   | T         |
+| F        | F   | F         |
+| T        | T   | T         |
+| T        | F   | T         |
+
+Of note is the last three columns. The first of which shows that definitively,
+Timmy's Mom is correct, if Timmy doesn't eat his broccoli, he will definitely
+not get any ice cream:
+
+| $\neg P$ | $Q$ | $P \to Q$ |
+| -------- | --- | --------- |
+| F        | F   | F         |
+
+But notice the last two columns, where "If you eat all your broccoli" is true:
+
+| $\neg P$ | $Q$ | $P \to Q$ |
+| -------- | --- | --------- |
+| T        | T   | T         |
+| T        | F   | T         |
+
+This equates to The mom explicitly saying "If you do eat your broccoli, you may
+or may not get ice cream." Which is true, she never explicitly said that, lol.
+But hey, Timmy's Mom, come on man!!