Conversion from English Language to Logic Symbols

Question

I have a problem in an example of Discrete Mathematics which my teacher worked in his lecture. He gave an argument and proved it that his argument was not valid, but the validity of argument is not what is my concern, What I'm worried about is actual method of conversion from English language to Logic Symbols

Example

An interesting teacher keeps me awake. I stay awake in Discrete Mathematics class. Therefore, my Discrete Mathematics teacher is interesting.

Teacher's Solution

t = My teacher is interesting

a = I stay awake

m = I am in Discrete Mathematics class

Statement of Argument is:

t -> a

a ^ m

therefore m ^ t -------- (and truth table shows that this argument is invalid)

My solution

Now my problem is I tried to solve it on my own but did it wrong because my Argument Statement was:

t -> a

m -> a

therefore m -> t

Now the problem is m -> a is not equivalent to a ^ m in logic but what are the exact rules under which I would know that I should use a ^ m instead of m -> a whereas both are equivalent in simple English. Like

a ^ m ---------I'm awake AND I'm in Discrete Mathematics Class //Teacher's solution

AND

m -> a --------if I'm in Discrete Mathematics Class then I'm awake //my solution

Also

m ^ t ---------I'm in Discrete Mathematics Class AND the teacher who is teaching is interesting //Teacher's solution

AND

m -> t --------If I'm in Discrete Mathematics Class then the teacher who is teaching is interesting //my solution

How are my solution and my teacher's solution different, What are hard and fast rules being applied here which tell us not to use implication but use AND instead ???

Answer 1

The argument :

$t \rightarrow a, a \land m $, therefore : $m \land t$

is not valid, because is it a little bit "more complicated" form of the fallacy known as Affirming the consequent :

(if P, then Q) and Q, therefore P.

---

To understand it, we have to re-phrase a little bit the three statements :

i) : "An interesting teacher keeps me awake" must be rewritten as :

"If My teacher is interesting, then I stay awake"

that is, in symbols : $t \rightarrow a$.

ii) : "I stay awake in Discrete Mathematics class" as :

"I am in Discrete Mathematics class and I stay awake"

that is, in symbols : $m \land a$.

iii) : "My Discrete Mathematics teacher is interesting", i.e. :

"I am in Discrete Mathematics class and My teacher is interesting"

that is : $m \land t$.

Answer 2

Maybe some predicate logic would be usefull

(1) An interesting teacher keeps me awake.

For all all class c, IF ( teacher of class c is interesting) THEN ( I am awake in class c).

(2) I stay awake in discrete mathematics class.

I am akake in class ( Discrete Mathematics).

That is : " Dscrete Mathematics" make the open sentence " I am awake in class c " true.

(3) My Discrete mathematics teacher is interesting

Teacher of class ( Discrete Mathematics) is interesting.

That is : " Discrete mathematics" makes true the open sentence : " Teacher of class c is interesting".