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I understand that $\bot \rightarrow A$ is a tautology, but I dont fully understand what it means that you can conclude anything after proving/assuming $\bot$.

For example:
Premise: $(A\rightarrow B)\rightarrow\bot$
Goal: $A\land(B\rightarrow\bot)$

You can prove this by considering the cases $B$ and $B\rightarrow\bot$.

So consider the case where $B$. Now, all conclusions that we make depend on the fact that $B$, right? So if we can conclude $A\rightarrow B$, we actually have $B\rightarrow (A\rightarrow B)$?

So since $B$, we can in fact conclude $A\rightarrow B$. And since we know that $(A\rightarrow B)\rightarrow\bot$, we can use modus ponens and conclude that $\bot$, so then i guess $(B\rightarrow\bot)$?

Here is the part I dont understand:
We proved $\bot$, which implies $A\land(B\rightarrow\bot)$, but I dont understand why.

Is it because $\bot\rightarrow A\land(B\rightarrow\bot)$ is a tautology? Therefore we can put anything on the righthand side, and since $\bot$, it must be that $A\land(B\rightarrow\bot)$

For example if you had $A$, you couldnt just say $A\rightarrow C$ or $A\rightarrow Y$, because those might not be tautologies.

I guess the concept of $\bot$ (false) being true is also making this confusing.

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  • $\begingroup$ As a statement of the principle of vacuous truth, you may find the tautology $\neg B \to (B \to A)$ easier to work with than $\bot \rightarrow A$. In other words, if $B$ is false, then the implication $B\to A$ must be true. The consequent $A$ will not necessarily be true, however. $\endgroup$
    – Dan Christensen
    Jun 2 at 16:40

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Given that $\bot \to A$ for any $A$ is a tautology, if you have $\bot$, concluding $A$ is simply an application of modus ponens on $\bot \to A$ and $\bot$.

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  • $\begingroup$ What does it mean to "Have $\bot$"? Does it mean that $\bot$ is true? How can false be true? $\endgroup$
    – lightyourassonfire
    Jun 2 at 12:28
  • $\begingroup$ It means assuming/proving $\bot$, as you wrote in your first paragraph. We could for example have gotten that from $A \to B$ and $(A \to B) \to \bot$. $\endgroup$
    – Natalie Clarius
    Jun 2 at 12:30
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    $\begingroup$ We will never get that $\bot$ is true as the final conclusion without further assumptions, as long as our mathematical system is consistent that is. It will always depend on assumptions, which we can then conclude do not hold true. $\endgroup$
    – Natalie Clarius
    Jun 2 at 12:35
  • $\begingroup$ But if it cant be that $\bot$ is true, how could you infer $A$ from $\bot\rightarrow A$, if $\bot$ cannot be true? $\endgroup$
    – lightyourassonfire
    Jun 2 at 12:38
  • $\begingroup$ You can infer it by having it depend on the hypothesis that $\bot$ is true. The situation where "If $\bot$ is true" will simply never happen, but it is still valid to say "If $\bot$ is true then $A$ is true". $\endgroup$
    – Natalie Clarius
    Jun 2 at 12:42

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