If we construct a system of statements, and the system stems from assuming a false statement is true, do we always reach a false conclusion?

Question

I’m in the process of proving/disproving something and this knowledge would be quite useful. If we start off by assuming a false statement is correct, and then we use logic that would work if the statement was indeed true, would it be possible to arrive at a true conclusion. Is it impossible, I know that if you make a false assumption and it leads to an absurd/false conclusion you can use “proof by contradiction”. However, can contradiction only work if we can guarantee the conclusion is false? Or will the conclusion always be false if all the logic stems from a false assumption?

That $A\implies B$ is always true , if $A$ is false (no matter what $B$ is) does not mean that we can derive $B$ from $A$ in a "reasonable" way. But in fact, a false assumption can lead both to a false (or even absurd) or to a true statement. The conclusion is invalid in both cases. It occurs more often than one would expect that multiple mistakes finally lead to a correct result. — Peter, Aug 09 '22 at 10:10

score 7 · Answer 1 · answered Aug 09 '22 at 09:18

If you start from a false premise you can prove absolutely anything, regardless of whether the conclusion is true or false.

For example: Suppose that $0=1$. Then $0^2 = 1^2 = 1 = 0$, so $0^2=0$. The conclusion is true, but the hypothesis was false.

The point is that if you start with a false premise you cannot learn anything at all about the truth or falsity of your conclusion.

ryang · Answer 2 · 2022-08-09T10:40:34.780

If a system of statements stems from assuming a false statement is true, do we always reach a false conclusion?

No: the perfectly valid argument $$∀x(Cx{→}Hx)∧Cs\implies Hs$$ derives the true conclusion “Socrates is human” from the false premise “All cats are human and Socrates is a cat”.

“proof by contradiction”. However, can contradiction only work if we can guarantee the conclusion is false?

If your argument fails to derive a contradiction, then—even if it is valid—you can't logically conclude anything about its premises; that is, you can logically conclude neither that you'd started with a true premise nor that you'd started with a false premise.

The former is evidenced by the above example, while the latter is evidenced by the fact that a sound argument (i.e., a valid argument with no false premise) derives a true conclusion.

If we construct a system of statements, and the system stems from assuming a false statement is true, do we always reach a false conclusion?

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