I have a logical argument in English which says.
All Humans are Mortal.
Zeus is not Mortal.
therefore Zeus is not Human.
And I tried to convert it from English to logic. and did this
h = is Human, z = is Zeus, m = is Mortal
h $ \rightarrow $ m
z $ \rightarrow $ ~m
$ \therefore $ z $ \rightarrow $ ~h
and did it wrong answer as my truth table shows this argument is valid but the original English argument according to Venn Diagram is invalid that meant my conversion was wrong. O.K. then I tried with this
h ^ m
z ^ ~m
$ \therefore $ z ^ ~h
And this time I can't get both premise true in a single row in my truth table, I mean both premise are never true at a same time.
What is the actual way to solve this problem with logical symbols.
(by the way this is not my homework assignment it's just an exercise in Venn Diagram Section of my course and I thought to do it with logical symbols too but could not)
Best Answer
In the language of first-order logic, this could be formalized as follows (there are in fact many ways, I'll just show one of them):
Let $h$ denote the unary predicate "... is human", let $m$ denote the unary predicate "... is mortal", and let $z$ denote "Zeus".
Then your assumptions are $$\forall x.\;h\ x\to m\ x$$ (read: "for all things, their humanness implies their mortality"), and $$\neg\ m\ z$$ (read: "Zeus' mortality is not given").
By modus tollens, these two statements imply the conclusion $$\neg\ h\ z$$ (read: "Zeus' humanness is not given").
In short: $$\frac{\displaystyle\forall x.\;h\ x\to m\ x\quad\quad\neg\ m\ z}{\displaystyle\neg\ h\ z}$$