Answers provided at this link do not satify my question.
How can this English sentence be translated into a logical expression?
In Kenneth Rosan, the answer to this following sentence
“You cannot ride the roller coaster if you are under 4 feet tall unless you are older than 16 years old.”
is given as,
$(r \wedge \neg s) \rightarrow \neg q$
Where,
q: “You can ride the roller coaster.”
r: “You are under 4 feet tall.”
s: “ You are older than 16 years old.”
My solution:
So, I broke down this compound sentence as follows:
“[You cannot ride the roller coaster] if [you are under 4 feet tall] unless [you are older than 16 years old.]”
Now, substituting variables in given compound sentence.
($\neg q$) if (r unless s).
Applying equivalence formula for Q if P $\Leftrightarrow$ P $\to$ Q
(r unless s) $\to$ ($\neg q$)
Now, solving for unless. So, (r unless s) $\Leftrightarrow$ ($\neg s \to r$) ref.
($\neg s \to r$) $\to$ ($\neg q$)
Again solving for $\to$ (implication), we get:
(s $\lor$ r) $\to$ ($\neg q$)
So, my derivation is obviously wrong and does not match with Kennet Rosen.
My Question: What mistake I did? and How to derive the given answer systematically?
Best Answer
As noted by Jay, Kenneth Rosen interprets (r unless s) according to:
$$ \begin{array}{cc|c} r & s & (r \text{ unless } s) \Leftrightarrow (\neg s \to r) \\\hline 0 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 1 \\ \end{array} $$
The issue turns out to be the order of operations for "unless".
You started by breaking it up like this
And substituted variables to get:
If instead we use a different order of operations to group these, it works out with your definition of unless. That is, we have:
Now using $(P \mathbf{\text{ if }} Q) \Leftrightarrow (Q \to P)$
Now using your $(P \mathbf{\text{ unless }} Q) \Leftrightarrow (\neg P \to Q)$
expanding
expanding
This is the same as the other result.
To see this, use associativity of logical or
Then turn it into an implication
Use demorgan's law
(EDIT: Previously I arrived at the answer with the same order of operations, but a different interpretation of unless: $(r \text{ unless } s) = (r \text{ and } \neg s)$. Because "(anything) unless True = True" seriously sounds wrong to me. My interpretation of unless worked in this case, but apparently is not the correct english interpretation. Apologies.)