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Please how do I use a truth table to check the validity of the following argument.

All teenagers are smart.
All smart people study hard.
Jake does not study hard.
Therefore, Jake is not a teenager.

Edit:

While the answer below is right, how to I proceed If I let

teenagers by $t$, smart by $s$, study hard by $h$ and Jake by $j$, then 'all teenagers are smart would be represented by $t→s$; Likewise, all smart people study hard, by $s→h$. How do I represent Jake does not study hard. How do I represent Jake is not a teenager?

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1 Answer

up vote 2 down vote accepted

There's a difference in the types of $t$ and $s$ and the type of $j$. You wanted to represent $t$ and $s$ as truth values, but in fact they are functions from the set $P$ of people to the truth values. The function $t$ assigns to a person the value true if this person is a teenager, other persons get the value false, and likewise the function $s$ gives the value true for smart people and the value false for non-smart people. Now the sentence "all teenagers are smart" can be seen as (the notation $\forall p$ means "for every person $p$") $$ \forall p(t(p)\to s(p)),\tag{1} $$ With a truth table, we can find that $t(p)\to s(p)$ is equivalent to $\neg s(p)\to\neg t(p)$. Note that the truth value of $t(p)\to s(p)$ is only false if $t(p)$ is true but $s(p)$ is false. Likewise, the value of $\neg s(p)\to\neg t(p)$ is only false if $\neg s(p)$ is true and $\neg t(p)$ is false. Therefore, we get the following truth table

t(p) | s(p) | t(p) -> s(p) | ~t(p) | ~s(p) | ~s(p) -> ~t(p)
-----------------------------------------------------------
 0   |  0   |  1           |  1    |  1    |  1
 0   |  1   |  1           |  1    |  0    |  1
 1   |  0   |  0           |  0    |  1    |  0
 1   |  1   |  1           |  0    |  0    |  1

and we see that the statement in (1) can be equivalently written down as $$ \forall p(\neg s(p)\to \neg t(p)). $$ It reads: for all persons it holds that if that person is not smart then it is not a teenager. Similarly, $h$ is a function from $P$ to $\{\mathrm{true},\mathrm{false}\}$ that sends the value true to the people who study hard and the value false to the people who do not study hard. The sentence "all smart people study hard" is then represented as $$ \forall p(s(p)\to h(p)), $$ which is equivalent to $$ \forall p(\neg h(p)\to \neg s(p)) $$ by the truth table

s(p) | h(p) | s(p) -> h(p) | ~s(p) | ~h(p) | ~h(p) -> ~s(p)
-----------------------------------------------------------
 0   |  0   |  1           |  1    |  1    |  1
 0   |  1   |  1           |  1    |  0    |  1
 1   |  0   |  0           |  0    |  1    |  0
 1   |  1   |  1           |  0    |  0    |  1

Now we can look at Jake. Jake is a person $j$ for which $\neg h(j)$ is true. The statement $\neg h(j)\to \neg s(j)$ is also assumed to be true, so it follows by the above truth table that $\neg s(j)$ is true. Now we look in the first truth table and see that, because the statement $\neg s(j)\to \neg t(j)$ is assumed to be true, it follows that $\neg t(j)$ is true.

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The question was "... how do I use a truth table to check ... ". I'm going to go out on a limb here, and suggest that you maybe haven't answered the question. –  user22805 May 24 '12 at 9:07
    
@David Wallace: I'm sorry, my intention was mainly to give a clarification on the difference between $t$, $s$ and $j$, because he was confusing these things. But I had both missed $h$ and didn't draw the truth table to pass from $t(p)\to s(p)$ to $\neg s(p)\to \neg t(p)$. I'll remove my answer if you think it's better. –  Egbert May 24 '12 at 9:21
    
Your answer looks really good now. I've given you +1. –  user22805 May 24 '12 at 11:00
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