# Converting arguments into propositions

I'm trying to make sure i did this right.

Here is the argument:

(a) All cheaters sit in the back row. George sits in the back row $\therefore$ George is therefore a cheater

Here it is expressed using propositional logic:

$C$ = is a cheater

$x$ = students

$B$ = Sits in back row

$y$ = George

$\exists$$x$$\exists$$y(C(x) \implies B(x) \implies ($$B(y)$$\implies$$C(y)$$)$

Which in english means: There exists some students and there exists a George. Some students are cheaters, which implies they sit in the back row, which implies that George, who sits in the back row is therefore a cheater.

I'm fairly new to writing this all mathematically, please correct me, thanks!

You have to use predicate logic, in order to express "all".

All cheaters sit in the back row. George sits in the back row ∴ George is therefore a cheater.

We have to use two predicates and one individual constant :

$C(x)$ = "$x$ is a cheater"

$B(x)$ = "$x$ sits in back row"

$G$ = George.

1) $\forall x(C(x) \to B(x))$ --- 1st premise

2) $B(G)$ --- 2nd premise

3) $C(G) \to B(G)$ --- from 1) by universal instantiation

==== and we cannot conclude with $C(G)$.

• Wouldn't the last one be: $B(G)$ $\rightarrow$ $C(G)$? Since George sits in the back row, it implies george is a cheater. What you're saying is that George is a cheater, that implies he sits in the back row Oct 2, 2015 at 16:41
• I'm trying to figure out how this make sense. "All cheaters sit in the back row".. but that does not mean that just because you sit in the back row you are a cheater. Oct 2, 2015 at 16:50
• @AjeetKljh - from the premises : "All cheaters sit in the back row" and "George sits in the back row" you cannot conclude with "George is a cheater". Counterexample : "All man are mammals", "My dog is a mammal", therefore "My dog is a man". Oct 2, 2015 at 18:46
• I forgot to include the actual question. i am supposed to determine if it is even valid. "All cheaters sit in the back row. George sits in the back row ∴ George is therefore a cheater." doesn't make sense Oct 2, 2015 at 21:52