I have spent all day reading over SE and other sites trying to understand this but I am having trouble.
The conditional statement: If you are a guitar player, then you are a musician. a → b
\begin{array}{|c|c|c|}
\hline
a& b & a→b \\ \hline
T & T & T \\ \hline
T & F & F \\ \hline
F & T & T \\ \hline
F & F & T \\ \hline
\end{array}
If-then form: If you are a guitar player, then you are a musician.
True, guitars players are musicians.
Converse: If you are a musician, then you are a guitar player.
False, not all musicians play the guitar.
Inverse: If you are not a guitar player, then you are not a musician.
False, even if you don’t play a guitar, you can still be a musician.
Contrapositive: If you are not a musician, then you are not a guitar player
True, a person who is not a musician cannot be a guitar player.
Looking at the truth table above, the last row shows that F,F=T. The inverse statement says this too but there it's false whereas in the truth table it's True. The converse statement doesn't seem to agree with the truth table either.
I understand that the converse is b → a and the inverse is ~a → ~b and the contrapositive is ~b → ~a
What I don't understand is this (apologies for showing a different example)
If it is raining, there are clouds in the sky
a = Raining, b = Clouds
Contrapositive: If there are no clouds in the sky, then it is not raining. (I understand that this is logically equivalent to the conditional statement)
I don’t understand what the use of the truth table is. It’s useful in showing that if it rains, it’s cloudy and that you can’t have rain and then no clouds. But in these two examples you are given if "a" is true or false and then if "b" is true of false. What happens when you are told like in the contrapositive that "b" is false and "a" is false (this is in the opposite order, given "b" first then "a")?
Can you still look at the truth table, look at the last row and say that the conditional statement is True overall?
What has really confused me is also that logically if I know that it is raining then there should be clouds but I also know that just because there are clouds doesn’t necessarily mean it will be raining. This is the same as saying all squares are rectangles but not all rectangles are squares. I don’t see where this is in the truth table.
Sorry again for all of my confusion, I am probably making this more confusing than it is, but I need a step by step explanation
Thank you for your time and answers
Best Answer
Here's the complete table.$$\def\T{\mathsf T}\def\F{\color{blue}{\mathsf F}} \begin{array}{|c:c|c:c|c:c|}\hline a& b & a\to b & \neg b\to\neg a& b\to a&\neg a\to\neg b \\\hline\T & \T & \T & \T & \T & \T \\ \hdashline\T & \F & \F & \F & \T & \T \\ \hdashline\F & \T & \T & \T & \F & \F \\ \hdashline\F & \F & \T & \T & \T & \T \\ \hline \raise{0.5ex}\tiny\text{guitar player}&\small\text{musician}&\text{position}&\tiny\raise{1ex}\text{contraposition}&\text{converse}&\text{inverse}\\ \hline\end{array}$$
This shows that in all interpretations where $a\to b$ is valued $\T$, then $\neg b\to\neg a$ is also valued $\T$. Thus we say $a\to b$ entails $\neg b\to\neg a$. Likewise $a\to b$ is entailed by $\neg b\to\neg a$.
No, all four statements are true in the interpretation of $a=\F$ and $b=\F$, because $\F\to\F$ and $\neg\F\to\neg\F$ are both valued as true.
Now, it holds that $b\to a$ is not entailed by $a\to b$.
That does not mean that $b\to a$ is valued false in all interpretations where $a\to b$ is valued true. It only means it may be false in some interpretation where that happens (and this is so).
Additionally, you appear to be reading these statements as universally quantified predicates rather than propositional statements.