What does it mean "p is provable from G"

Question

I´ve just been introduced to this subject, but I am bit confused when it comes to answering some of the question I am given. So, say I have a set of premises G and I am asked to show that a preposition p is provable from G. My question is that if I show that p is provable from G am I saying that p is true? If given G and I can only show that p is false does it still count as "G proves p"?

Answer 1

To show that $p$ is provable from $G$ means that we can construct a finite sequence of well formed sentences, each element of which is either a logical axiom, an element of $G$, or follows from two previous statements in the sequence by an inference rule (i.e modus ponens).

Note that if $p$ is provable from $G$ it doesn't necessarily follow that $p$ is a theorem of propositional calculus (which I believe you mean by "truth" here). For example consider a tautology $q$ in propositional calculus and let $G$ be the set containing only the premise $\lnot q$. Then from $G$ we can easily prove $\lnot q$, as the proof sequence literally contains just one element. However, in propositional calculus $\lnot q$ is a contradiction, so most certainly not "true".

In this case we have that propositional calculus with $G$ is an inconsistent theory. It is actually a theorem of first order logic that any well formed sentence is provable within a contradictory theory. In this case then $G$ does indeed prove $p$ for any $p$.

If, however, we know that propositional calculus with $G$ is consistent and you prove $\lnot p$ from $G$ it most certainly does not follow that "$G$ proves $p$". Indeed if you prove any statement $p$ in a consistent theory then its negation is never "true".