I'm currently learning about Logics (Discrete mathematics) and I have a question(Just started learning logics so please don't be harsh 😉 ) that I can't wait to ask the professor about.
Does all hypothises found from the different rules need to be proven?
I use a program called proofweb where I can proof different "things" (forgot what its called). But sometimes it allows me to compete a proof without using all the hypotheses? Is this a valid proof if I don't conclude them all?
Best Answer
There are two issues here.
Hypotheses don't need to be proved or concluded at all. They're hypotheses -- you just assume they are true, and all you prove is that if the hypotheses are true, then the conclusion must be true as well. But at no point do you make any claim about whether the hypotheses are actually true in a particular situation; they might be false under some interpretations or even not satisfiable at all, so the hypotheses themselves are not to be proved. What is proved is the conditional relationship between the hypotheses and the conclusion: Any hypothetical situation where all of the hypotheses are true must also be a situation where the conclusion is true, i.o.w., there can be no situation where all the hypotheses are true but the conclusion is false. What is concluded eventually is the conclusion, and not the hypotheses. What I call "situations" here are in fact interpretations, which, in propositional logic, correspond to assignment functions aka valuation functions.
What you seem to be talking about is not whether hypotheses need to be proved, but whether all of them are needed in a proof. The answer is no: Indeed, it is sometimes possible to prove the conclusion with only part of the hypotheses given, and if this is possible, there is no harm in doing so. Seeing it from the other way round, if you can prove a conclusion from some set of hypotheses, adding additional hypotheses will (at least in classical logic) never render the argument invalid: Once you established a proof that $B$ logically follows from, say, two hypotheses $A_1$ and $A_2$, i.e. you proved that $B$ is true whenever $A_1$ and $A_2$ is true, then it will certainly also be true if $A_1$ and $A_2$ and $A_3$ are true. This property is called monotonicity. So any proof of "If $A_1, ... A_n$, then $B$" with hypotheses $A_1, ..., A_n$ and conclusion $B$ will also be a valid proof of "If $A_1, ..., A_n, A_{n+1}$, then $B$" where $A_{n+1}$ is an additional hypothesis -- if you can show the validity of the argument by using only hypotheses $A_1, ..., A_n$, then this is also a proof of the argument with whatever additional hypotheses that are not needed in the proof.
Lastly, the thing you prove is called an argument, or inference, or entailment, or logical implication.