# See if "7<4 implies 7 is ..." Is the following conclusion valid?

For my homework I need to see if the following conclusion is correct. $$7<4 \implies 7\ \text{is not a prime number}\\ \lnot(7<4)\\ -----------------\\ \text{7 is prime number}\\$$

To tell you the truth, I have no idea how to start this, letalone how to finish it, so any help is welcome.

Thanks!!!!

• I don't see your problem. If there's such a homework, you must have learned something, haven't you? Oct 26, 2013 at 11:20
• This deduction is not valid. Oct 26, 2013 at 11:20
• @ftfish I am studying on german language, and that is not my language, so sometimes it is hard for me to concetrate on lectures so I need to ask for help this way Oct 26, 2013 at 11:22

Let , P be " $7<4$ " and Q be " $7\ \text{is not a prime number}$ " So actually you want to know whether the following identity holds

$P \implies Q \\ \lnot P \\ -----------------\\ \lnot Q \\$

Well . Actually $P \implies Q$ can be written as $(\lnot P \lor Q) \\$ . If this is true and $\lnot P$ is true you can not certainly tell that $\lnot Q$ is true . Here a fault remains .

But according to Modus Tollens the following identity is correct . $\lnot Q \\ P \implies Q \\ -----------------\\ \lnot P \\$

Hope this helps .

• Can you direct me to some book I can read to get some more information about logical conclusion. Thanks Oct 26, 2013 at 11:38
• The book named "Discrete Mathematics And Its Applications" by "Kenneth H.Rosen" is very good . I have learnt a lot from this book . In fact according to me this book is the father of logic . Oct 26, 2013 at 11:40

The given argument is not valid. That is, the conclusion does NOT FOLLOW from the premises:

$$\quad (7\lt 4) \implies (7 \;\;\text{ is not a prime number})$$ $$\quad \lnot (7\lt 4)$$ $$\therefore\; 7\;\text{is not a prime number}$$

This employs a fallacy of denying the antecedent. It is an argument of the form $$p \rightarrow q$$ $$\lnot p$$ $$\therefore \lnot q$$

And as such, the argument is not valid, and the conclusion does not follow from the premises. Note that the argument is invalid, even though the conclusion happens to be true. The validity of the argument here has nothing to do with the truth of the conclusion. A valid argument form is such that if the premises were true, then they necessarily imply that conclusion must be true. In this case we have only that the conclusion happens to be true, though its truth is not a function of the truth (or lack thereof) of the premises.

If we have an implication $p \rightarrow q$, then from the added premise $p$, we can logically conclude $\therefore q$, by modus ponens. Or, if we have the added premise $\lnot q$ we can infer $\lnot p$, by modus tollens.

But from the premises $p\rightarrow q$, and $\lnot p$, we cannot infer $\therefore \lnot q$.

• Very nice write - up too, such nice details +1 Oct 27, 2013 at 0:06