Logic Question. Steve, Halen and Jack in Math class I need to find how the 3 people (Steve, Helen and Jack) did in the Math Class.
I know that:
1. If Steve did not get the highest grade, then Jack did.
2. If Jack did not get the lowest grade, then Helen got the highest grade.

Determine the ranking of the 3 people.
I can write the primitive statement:
s: Steve got the highest grade.
j: Jack got the highest grade.
h: Helen got the highest grade.

The first statement is simple, ~p->j.
How do I translate "did not get the lowest grade"? how do I calculate the ranking? 
 A: It gets easier when you think of them as "or" statements:


*

*Steve or Jack got the highest grade.

*Jack got the lowest grade or Helen got the highest grade.


From 1, we see that Helen didn't get the highest grade (it has to be Steve or Jack). Then, from 2, since Helen didn't get the highest grade, Jack must have gotten the lowest grade. 
Since either Steve or Jack got the highest grade, and it wasn't Jack, that leaves Steve on top. Top and bottom are now accounted for, so Helen must land in the middle.
Final ordering (From lowest grade to highest): Jack, Helen, Steve
A: I used a truth Table to solve this problem 
Statement:
s: Steve has the highest score;
J: Jack has the highest score;
j: Jack has the lowest score;
h: Helen has the highest score;

The truth table: (Table has 2 parts, above: J=1, j=0, below: J=j=0 )
s J j h  |  ~s -> J  |  ~j-> h

0 1 0 1  |     1     |    1   ---- False, more than 1 person has highest score.
0 1 0 0  |     1     |    0   ---- condition on the right fails
0 0 1 1  |     0     |    1   ---- condition on the right fails
0 0 1 0  |     0     |    1   ---- condition on the right fails
1 1 0 1  |     1     |    1   ---- False, more than 1 person has highest score.
1 1 0 0  |     1     |    0   ---- condition on the right fails
1 0 1 1  |     1     |    1   ---- False, more than 1 person has highest score.
1 0 1 0  |     1     |    1   ---- Ture!!!!

0 0 0 1  |     0     |    1   ---- condition on the right fails
0 0 0 0  |     0     |    0   ---- condition on the right fails
1 0 0 1  |     1     |    1   ---- False, more than 1 person has highest score.
1 0 0 0  |     1     |    0   ---- condition on the right fails

(Thank you  @Joffan)

Only possible occasion is s =1, j=1, h=0. Meaning Steve has the highest score, jack has the lowest score, Helen did not receive the highest score. 
A: If you're interested in getting the ranking of the 3 people and we assume that ties cannot occur, it's not too hard to reason out what must happen.  
First, suppose that Steve does not have the highest grade.  Then Jack would have the highest grade by statement 1.  But this means Jack does not have the lowest grade, and therefore Helen has the highest grade by statement 2.  This contradicts the fact that Jack already has the highest grade.  Thus, our assumption is wrong and Steve must have the highest grade.
Second, we know that Jack cannot be 2nd since that would mean Helen is 1st by statement 2, contradicting the fact that we have already proven that Steve has the highest grade.  Thus, Jack is last.
This leaves only one possible ordering (from lowest to highest): Jack, Helen, Steve
A: Assuming that you can't create new symbols, you can't express these propositions into propositional functions. That's because you want to say "Jack didn't get the lowest grade", but that can't be expressed with only these symbols.
However, you can state that $j\implies$ Jack did not get the lowest grade, but those statements aren't equivalent. That's because Jack could get the middle grade, so he didn't get the lowest grade, but also didn't get the highest grade. From there, if you assume that Steve did not get the highest grade, you can use sentences $1$ and $2$ and arrive at a contradiction. Work from there and you can see that there's only one possibility.
A: Let's try the alternative approach of formalizing, and actually calculating the answer, and thereby discover that there are really two different answers, depending how you interpret the problem.
As in many cases, the most important point is how to formalize the problem.  The OP and several earlier answers use variables for propositions like 'Steve has the highest grade', etc.  Instead, it seems to me a lot simpler to use variables for the grades themselves: $\;s,h,j\;$ stand for Steve's, Helen's, and Jack's grades respectively.$
\newcommand{\calc}{\begin{align} \quad &}
\newcommand{\op}[1]{\\ #1 \quad & \quad \unicode{x201c}}
\newcommand{\hints}[1]{\mbox{#1} \\ \quad & \quad \phantom{\unicode{x201c}} }
\newcommand{\hint}[1]{\mbox{#1} \unicode{x201d} \\ \quad & }
\newcommand{\endcalc}{\end{align}}
\newcommand{\Ref}[1]{\text{(#1)}}
\newcommand{\then}{\Rightarrow}
$
Now we can easily formalize 'Steve has the highest grade' as $\;h \le s \land j \le s\;$, 'Jack has the lowest grade' as $\;j \le s \land j \le h\;$, etc.
Note an important point here: I chose to use $\;\le\;$ instead of $\;\lt\;$, since the problem does not clearly say or imply that all the three grades are different: perhaps one of them has the lowest grade, and both others the highest?
With this formalization, we are given two things:
\begin{align}
\tag{1}
\lnot(h \le s \land j \le s) \;\then\; h \le j \land s \le j
\\
\tag{2}
\lnot(j \le s \land j \le h) \;\then\; s \le h \land j \le h
\end{align}
and we are asked to determine the relative order of $\;s,h,j\;$.

As an earlier answer already suggests, it is often easier to write 'then' ($\;\then\;$) in terms of 'or' ($\;\lor\;$), using $\;\phi \then \psi \;\equiv\; \lnot \phi \lor \psi\;$.  Doing that results in the equivalent and more symmetrical
\begin{align}
\tag{1a}
(h \le s \land j \le s) \;\lor\; (h \le j \land s \le j)
\\
\tag{2a}
(j \le s \land j \le h) \;\lor\; (s \le h \land j \le h)
\end{align}
Looking at these, it seems that $\Ref{2a}$ is easiest to simplify, because both sides contain $\;j \le h\;$:
$$\calc
    \tag{2a}
    (j \le s \land j \le h) \;\lor\; (s \le h \land j \le h)
\op\equiv\hint{logic: $\;\land\;$ distributes over $\;\lor\;$}
    (j \le s \lor s \le h) \;\land\; j \le h
\op\equiv\hints{left hand part follows from $\;j \le h\;$, since $\;\le\;$ is a}\hint{total order}
    j \le h
    \tag{2b}
\endcalc$$
Now we can combine all and simplify further:
$$\calc
    \Ref{1} \;\land\; \Ref{2}
\op\equiv\hint{use $\Ref{1a}$ and $\Ref{2b}$}
    ((h \le s \land j \le s) \;\lor\; (h \le j \land s \le j)) \;\land\; j \le h
\op\equiv\hint{logic: $\;\land\;$ distributes over $\;\lor\;$}
    (h \le s \land j \le s \land j \le h) \;\lor\; (h \le j \land s \le j \land j \le h)
\op\equiv\hint{simplify}
    j \le h \le s \;\lor\; s \le j = h
\endcalc$$
And we have found our ordering(s).

Interpreting this result, we see that there are two possibilities: either Jack has the lowest grade and Steve has the highest; or Jack and Helen both have the highest grade.
(And if there are no ties, then that of course prevents the second possibility.)
