Logic/Reasoning

Question

Ok more logic questions..

James would like to determine the relative salaries of three coworkers using two facts. First, he knows that if Fred is not the highest paid of the three, the Janice is. Second, he knows that if Janice is not the lowest paid, then Maggie is the most. Is it possible to determine the relative salaries of Fred, Maggie, and Janice from what James knows? If so, who is paid the most and who is the least? Explain your reasoning.

I denoted let $p = \text{Fred}$, $q = \text{Janice}$, and $r = \text{Maggie}$. From there I broke it down as Not Fred then Janice "exclusive or" Not Janice, then Maggie.

So I had $(\neg p\rightarrow q) \oplus (\neg q \rightarrow r)$ and constructed a truth table. It didn't work. Obviously I am wrong because I found more than one solution. Maybe this problem doesn't even need a truth table. So how do I begin trying to solve this problem? Thanks! :)

Answer 1

We know that if Fred is not the highest paid, then Janice is; this implies that the highest paid of the three is either Fred or Janice. We also know that if Janice is not the lowest paid, then Maggie is the highest paid. But Maggie is not the highest paid — Fred or Janice is — so Janice is the lowest paid. That means that Fred must be the highest paid, Janice the lowest paid, and Maggie the one in the middle.

If you want to express it in terms of propositional logic, let $F_h$ be Fred is the highest paid, $J_h$ be Janice is the highest paid, $M_h$ be Maggie is the highest paid, and $J_\ell$ be Janice is the lowest paid. Then we’re told that $\neg F_h\to J_h$ and $\neg J_\ell\to M_h$.

Since $p\to q$ is logically equivalent to $\neg p\lor q$, $\neg F_h\to J_h$ is logically equivalent to $\neg\neg F_h\lor J_h$, i.e., to $F_h\lor J_h$: Fred is the highest paid, or Janice is the highest paid.

Since $p\to q$ is also logically equivalent to $\neg q\to\neg p$, $\neg J_\ell\to M_h$ is logically equivalent to $\neg M_h\to\neg\neg J_\ell$, i.e., to $\neg M_h\to J_\ell$: if Maggie is not the highest paid, then Janice is the lowest paid.

It’s clear from the meanings that $(F_h\lor J_h)\to\neg M_h$: if Fred or Janice is the highest paid, then Maggie is not the highest paid. And we know that $F_h\lor J_h$ is true, so $\neg M_h$ is true, and therefore (from the last paragraph) $J_\ell$ is true. Clearly $J_\ell\to\neg J_h$, and $\neg J_h\land(F_h\lor J_h)=F_h$, so Fred is the highest paid, Janice is the lowest paid, and Maggie is in the middle.

Answer 2

The most important thing, in my opinion, is to find the most suitable formalization.

In this case the simplest thing seems to have $\;f,j,m\;$ as (whole or real) numbers representing the salaries. We can then use our knowledge of numbers, specifically the fact that they are totally ordered.

It seems the intention of the problem is that all three salaries are different, but the wording is not entirely clear on this point. So let's assume for now that some or all salaries could be equal.

Now we can formalize James's knowledge as \begin{align} (0)\;\;\; & f<j \lor f<m \;\Rightarrow\; f \leq j \land m \leq j \\ (1)\;\;\; & f<j \lor m<j \;\Rightarrow\; f \leq m \land j \leq m \\ \end{align}

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In my experience, logic reasoning is often made simpler by expanding $\;P \Rightarrow Q\;$ to $\;\lnot P \lor Q\;$. Here, using DeMorgan, this gives us the equivalent \begin{align} (0')\;\;\; & (j \leq f \land m \leq f) \;\lor\; (f \leq j \land m \leq j) \\ (1')\;\;\; & (j \leq f \land j \leq m) \;\lor\; (f \leq m \land j \leq m) \\ \end{align}

Now the similarities within $(1')$ suggest it can be simplified: \begin{align} & (j \leq f \land j \leq m) \;\lor\; (f \leq m \land j \leq m) \;\;\;\;\;\text{-- $(1')$} \\ \equiv & \;\;\;\;\;\text{"logic: $\;\land\;$ distributes over $\;\lor\;$"} \\ & (j \leq f \lor f \leq m) \;\land\; j \leq m \\ \equiv & \;\;\;\;\;\text{"ordering: $\;j \le m\;$ implies the left conjunct"} \\ (*)\;\;\;\phantom\equiv & j \leq m \\ \end{align}

We can use $(*)$ to simplify $(0')$: \begin{align} & (j \leq f \land m \leq f) \;\lor\; (f \leq j \land m \leq j) \;\;\;\;\;\text{-- $(0')$} \\ \equiv & \;\;\;\;\;\text{"ordering: $\;j \leq m\;$ makes $\;j \leq f\;$ superfluous;} \\ & \;\;\;\;\;\phantom"\text{ordering: $\;j \leq m\;$ implies equality"} \\ (**)\;\;\;\phantom\equiv & m \leq f \;\lor\; f \leq m = j \\ \end{align}

Combining $(*)$ and $(**)$, we see that $\;(0) \land (1)\;$ is equivalent to $$ j \leq m \leq f \;\lor\; f \leq m = j $$

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So one solution, obviously the intended one, is that the 'salary ladder' is Janice as the lowest, then Maggie, and then Fred as the highest.

But if salaries are allowed to be equal, an alternative solution is that Janice and Maggie have the same highest salary, and Fred the lowest.