Intuitive answer doesn't match with logical reasoning I used

Question

I'm Trying to see validity of the following argument:
If i get chirstmas bonus, I'll buy stereo
If i sell my motorcycle, I'll buy stereo
therefore if i get a christmas bonus or I sell my motor cycle then I'll buy a stereo
This sounds valid even without any reasoning but when i applied i got a different answer.
CB = Getting chirstmas bonus
BS = Buying Stereo
SM = Selling motor cycle
$CB \rightarrow BS$
$SM \rightarrow BS$
$CB \lor SM \rightarrow BS$
Given
$$CB {\rightarrow} BS$$ $$SM \rightarrow BS$$ Through Generalization $$\therefore (CB \rightarrow BS) \lor (SM \rightarrow BS)$$ $$(\neg CB\lor BS) \lor (\neg SM\lor BS)$$ $$(\neg CB \lor \neg SM) \lor(BS\lor BS)$$ $$(\neg CB \lor \neg SM)\lor(BS)$$ $$\neg (CB \land SM)\lor BS$$ $$\therefore CB \land SM \rightarrow BS$$
This answer doesn't match inuitively, Where did i go wrong?

Answer 1

You have both $CB\rightarrow BS$ and $SM\rightarrow BS$, therefore you should start with $(CB\rightarrow BS)\land (SM\rightarrow BS)$.

What you showed is that if you have either of those, then you'll need both $CB$ and $SM$ to conclude $BS$, which is intuitive to me at least as well.

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Now you have to prove

$$CB→ BS,SM→ BS ⊢ CB∨ SM→ BS$$

To see this follows, you split in cases your hypothesis $CB∨ SM$.

First case. If $CB$ then by $CB→ BS$ you get $BS$ as expected.

Second case. If $SM$ then by $SM→ BS$ you get $BS$ as expected.

Answer 2

As others have pointed: since you have both premises as givens, you should start with $(CB\rightarrow BS)\land (SM\rightarrow BS)$ rather than $(CB\rightarrow BS)\lor (SM\rightarrow BS)$

And with that, you get:

$(CB\rightarrow BS)\land (SM\rightarrow BS) \Leftrightarrow$

$(\neg CB\lor BS)\land (\neg SM\lor BS) \Leftrightarrow$

$(\neg CB\land \neg SM) \lor BS \Leftrightarrow$

$\neg (CB\lor SM) \lor BS \Leftrightarrow$

$(CB\lor SM) \to BS$

I am wondering why you started with $(CB\rightarrow BS)\lor (SM\rightarrow BS)$ ... is it because you saw the 'or' in the desired conclusion "if i get a christmas bonus or I sell my motor cycle then I'll buy a stereo"? If so, then well, yes, you end up with an 'or' o=in the conclusion ($(CB \color{red}\lor SM) \to BS$) ... but this 'or' is really part of the antecedent of the conclusion. But as far as your starting point goes: you have two premises, and so you know that both of $CB\rightarrow BS$ and $SM\rightarrow BS$ are, so the starting point should be $(CB\rightarrow BS)\color{red}\land (SM\rightarrow BS)$

And yes, it may seem surprising that $(CB\rightarrow BS) \color{red}\land (SM\rightarrow BS)$ ends up being equivalent to $(CB \color{red}\lor SM) \to BS$, as it doesn't follow the 'normal' distribution pattern ... but apparently this is what it works out to.