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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)

Danish ALI
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  • $\to$ does not mean "is", so $z\to\sim m$ does not mean "Zeus is not mortal". – MJD Nov 24 '14 at 00:25
  • @MJD it's functionally equivalent though. "If Zeus then not mortal" or in other words "Either not Zeus or not mortal." – Matt Samuel Nov 24 '14 at 00:30

2 Answers2

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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: $$\frac{\displaystyle\forall x.\;h\ x\to m\ x\quad\quad\neg\ m\ z}{\displaystyle\neg\ h\ z}$$

  • this is a bit complex method for me as I'm just a newbie to Discrete Mathematics yet if I try to do it this way how can I make truth table for your final given form for this argument ? – Danish ALI Nov 24 '14 at 18:20
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This is a simplified approach.

$z$ = zeus, $h$ = human, $m$ = mortal

We have:

Statement 1: $h \Rightarrow m$

Statement 2: $z \Rightarrow \lnot m$

By contraposition of Statement 1: $\lnot m \Rightarrow \lnot h $

So we see that Statement 2 gives: $z \Rightarrow \lnot m \Rightarrow \lnot h$

and thus $z \Rightarrow \lnot h$ as desired.

Zachary
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  • Yes this is simplified approach and I fully understand this method but this time the solution is incorrect as in reality this argument is invalid and here our statement form is making it valid. – Danish ALI Nov 24 '14 at 18:23
  • Can you elaborate? What do you mean by "this time the solution is incorrect" and "in reality this argument is invalid"? In particular, where are these claims coming from? – Zachary Nov 24 '14 at 21:41
  • I mean ----All Humans are Mortal. -- Zeus is not Mortal. -- therefore Zeus is not Human. -- This is an invalid Argument if you do this by Venn Diagram BUT h-->m , z-->~m therefore z-->~h is valid so this means the translation from original English statement into logical symbols was incorrect – Danish ALI Nov 26 '14 at 14:35
  • Actually, this works fine by Venn Diagram. Let $H = {humans}, M = {mortals}$ be sets. Then we have: $H\subset M$ and $z\notin M$ $\Rightarrow z \notin H$ (where $z$ is the single element zeus). – Zachary Nov 26 '14 at 22:27
  • Oh My! where have been I, I'm sorry I was too confused maybe and kept thinking it were an invalid argument based on another example from my book, That was my mistake. And thankyou I get it now, it makes sense by taking countrapositive of first premise. – Danish ALI Nov 27 '14 at 21:11
  • Happy to help :) – Zachary Nov 27 '14 at 22:17