Why are these $2$ statements not logically equivalent?

Question

This is my first post on this site. I am stuck on a logic question. The question states: which statements is/are logically equivalent to:

Kimo will pass algebra I only if he studies.

a) If kimo studies, then he will pass Algebra I

b) either kimo studies or he will fail Algebra I

c) if kimo does not study, he will not pass algebra I

d) if kimo is to fail algebra I, then he must not study.

The answer is only b) and c), but to me it seems like all four are true. could someone please help me see how a) and d) are wrong? also, it seems like truth tables are helpful, but i don't really know how to make one. i would appreciate it if someone could teach me how to make one for this problem. could someone also let me know how i can add line breaks? im sorry for the cluttered formatting! Thank you!

Answer 1

but to me it seems like all four are true.

They aren't asking which ones are true. You have no way of knowing which are true as you have no knowledge as to whether Kimo will study or pass or if there is any relation between them. For all we know Kimo could be an octopus.

They are asking you which sentences mean the exact same thing as ""Kimo will pass algebra I only if he studies"

"Kimo will pass algebra I only if he studies" Means that if Kimo doesn't study, he will fail. But if Kimo does study he might pass or he might fail. But he will only pass if studies. If he *doesn't study he is sure to fail.

So which mean the exact same thing.

a) If kimo studies, then he will pass Algebra I

Then says if he studies he is guaranteed to pass. But maybe he will pass anyway if he doesn't study. And if he does study he can't fail. That's not the same thing.

b)either kimo studies or he will fail Algebra I

this means there are two possibilities. He studies. Or he fails. (Or both). If he doesn't study, then it will be inevitable that he fails. If he doesn't fail, then it must be that he studied. It is possible that studies and did fail anyway. So the is the same as ""Kimo will pass algebra I only if he studies" "

c) " if kimo does not study, he will not pass algebra I"

This means if kimo doesn't study, he will fail. If he does study.... we don't know. That's also the same thing.

d) "if kimo is to fail algebra I, then he must not study"

That means the only way that Kimo can fail is if he doesn't study. If he does study then he will pass. If he doesn't study, he might fail, or he might pass. That is not the same thing.

Truth tables are a way of considering cases if "Kimo studies" and "Kimo passes" are compatible and make the sentence true.

Consider the statement

"Kimo will pass algebra I only if he studies"

Now consider "Kimo studies" and "kimo passes" are both true. That is compatible with "Kimo will pass algebra I only if he studies" so we declare that "Kimo will pass algebra I only if he studies" will be true in that case.

$\begin{array}. \text{Kimo Studies}&\text{Kimo Passes}&||&\text{Kimo will pass algebra I only if he studies}\\ \text{TRUE}&\text{TRUE}&&\text{TRUE}\end{array}$

Now consider "Kim studies" is false and "kimo passes" is true. That's incompatible with "Kimo will pass algebra I only if he studies" because Kimo can only pass if he studies. So that makes ""Kimo will pass algebra I only if he studies"" false.

$\begin{array}. \text{Kimo Studies}&\text{Kimo Passes}&||&\text{Kimo will pass algebra I only if he studies}\\ \text{FALSE}&\text{TRUE}&&\text{FALSE}\end{array}$

Now consider "Kimo studies" is true and "kimo passes" is false. Kimo studied, but failed. That is compatible with "Kimo will pass algebra I only if he studies" because that says he won't pass if he doesn't study. It doesn't say that he will pass if he does study. So:

$\begin{array}. \text{Kimo Studies}&\text{Kimo Passes}&||&\text{Kimo will pass algebra I only if he studies}\\ \text{TRUE}&\text{FALSE}&&\text{TRUE}\end{array}$

And finally consider if "Kimos studies" is false and "Kimo passes" is false. Then Kimo didn't study and didn't pass. That's compatible with "Kimo will pass algebra I only if he studies" so

$\begin{array}. \text{Kimo Studies}&\text{Kimo Passes}&||&\text{Kimo will pass algebra I only if he studies}\\ \text{FALSE}&\text{FALSE}&&\text{TRUE}\end{array}$

And that is the truth table for "Kimo will pass algebra I only if he studies"

Now do the same thing with " If kimo studies, then he will pass Algebra I"

$\begin{array}. \text{Kimo Studies}&\text{Kimo Passes}&||&\text{Kimo will pass algebra I only if he studies}\\ \text{TRUE}&\text{TRUE}&&\text{TRUE}\\ \text{FALSE}&\text{TRUE}&& \color{red}{\text{TRUE}}\\ \text{TRUE}&\text{FALSE}&&\color{red}{\text{FALSE}}\\ \text{FALSE}&\text{FALSE}&&\text{TRUE}\end{array}$

Notice this table differs from the table for "Kimo will pass algebra I only if he studies" in two cases so they are not the same.

b) either kimo studies or he will fail Algebra I"

$\begin{array}. \text{Kimo Studies}&\text{Kimo Passes}&||&\text{either kimo studies or he will fail Algebra I}\\ \text{TRUE}&\text{TRUE}&&\text{TRUE}\\ \text{FALSE}&\text{TRUE}&& \text{FALSE}\\ \text{TRUE}&\text{FALSE}&&\text{TRUE}\\ \text{FALSE}&\text{FALSE}&&\text{TRUE}\end{array}$

Note that is exactly the same as "Kimo will pass algebra I only if he studies"

c)"if kimo does not study, he will not pass algebra I"

$\begin{array}. \text{Kimo Studies}&\text{Kimo Passes}&||&\text{if kimo does not study, he will not pass algebra I}\\ \text{TRUE}&\text{TRUE}&&\text{TRUE}\\ \text{FALSE}&\text{TRUE}&& \text{FALSE}\\ \text{TRUE}&\text{FALSE}&&\text{TRUE}\\ \text{FALSE}&\text{FALSE}&&\text{TRUE}\end{array}$

d) "if kimo is to fail algebra I, then he must not study

$\begin{array}. \text{Kimo Studies}&\text{Kimo Passes}&||&\text{if kimo is to fail algebra I, then he must not study}\\ \text{TRUE}&\text{TRUE}&&\text{TRUE}\\ \text{FALSE}&\text{TRUE}&& \color{red}{\text{TRUE}}\\ \text{TRUE}&\text{FALSE}&&\color{red}{\text{FALSE}}\\ \text{FALSE}&\text{FALSE}&&\text{TRUE}\end{array}$

Note this is not equivalent to "Kimo will pass algebra I only if he studies" but it is equivalent to a) " If kimo studies, then he will pass Algebra I"

Answer 2

a) This is not logically equivalent because it is the converse of the original statement. More specifically, a) claims studying will guarantee that Kimo will pass. However, the original statement says that studying is only a requirement, not necessarily a guarantee.
b) and c) These are equivalent since these are all wordings of the contrapositive of the original statement (i.e. "if Kimo does not study, he can not pass").
d) This is not logically equivalent because it is the inverse of the original statement. More specifically, d) states that in order for Kimo to fail, then the only possible way to do so is to not study. However, as mentioned in a), there is nothing ruling out the possibility that Kimo could fail despite having studied.

For truth tables proving why converses and inverses are not logically equivalent while contrapositives are, see here.
Also, keep in mind that all because statements are not logically equivalent does not mean that they can't both be true!

p.s. Line breaks without using blank lines can be added if you add two spaces at the end of your sentences.
Like this.

Answer 3

I find it most useful, when dealing with an implication, $P\implies Q$, to look at its negation, $P\land\lnot Q$. Also, remember that "if $P$, then $Q$" is equivalent to "$P$ only if $Q$"; both are the negation of "$P$ and not $Q$".

To see the LaTeX for the Truth Tables below, right-click on the table and choose "Show Math As > TeX Commands".

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Scheme of Abbreviation
$\text{S}$ - Studies
$\text{D}$ - Doesn't study
$\text{P}$ - Passes
$\text{F}$ - Fails

Truth Tables $$ \text{Kimo will pass algebra I only if he studies}\\ \text{$\lnot$(Kimo will pass algebra I and he does not study)}\\ \begin{array}{l|c|c|} &\text{S}&\text{D}\\\hline \text{P}&T&F\\\hline \text{F}&T&T\\\hline \end{array} $$

$$ \text{a) If Kimo studies, then he will pass Algebra I}\\ \text{$\lnot$(Kimo studies and he does not pass Algebra I)}\\ \begin{array}{l|c|c|} &\text{S}&\text{D}\\\hline \text{P}&T&T\\\hline \text{F}&F&T\\\hline \end{array} $$

$$ \text{b) Either Kimo studies or he will fail Algebra I}\\ \text{$\lnot$(Kimo does not study and he passes Algebra I)}\\ \begin{array}{l|c|c|} &\text{S}&\text{D}\\\hline \text{P}&T&F\\\hline \text{F}&T&T\\\hline \end{array} $$

$$ \text{c) If Kimo does not study, he will not pass algebra I}\\ \text{$\lnot$(Kimo does not study and he passes algebra I)}\\ \begin{array}{l|c|c|} &\text{S}&\text{D}\\\hline \text{P}&T&F\\\hline \text{F}&T&T\\\hline \end{array} $$

$$ \text{d) If Kimo is to fail algebra I, then he must not study}\\ \text{$\lnot$(Kimo fails algebra I and he studies)}\\ \begin{array}{l|c|c|} &\text{S}&\text{D}\\\hline \text{P}&T&T\\\hline \text{F}&F&T\\\hline \end{array} $$