I have a silly question.
From the relationship of entailment, we know that,
$(x=0) ⊨ ( xy = 0 )$ holds, (Let's say, $x$ and $y$ are integers. )
And, we can infer that, $x=0$, right?
But, it's not necessary that $x$ is zero, $x$ might have some value while $y=0$ might be actually true, making $xy=0$.
My question is, does inference necessarily reflect truth or just logical consistency?
Additionally, if we make moves based on inferred data from a system we know to be true, isn't it possible that we might make a wrong move cause what we inferred wasn't really actual truth, that is, it just seemed plausible?
A popular inference rule is, modus ponens,
given , a implies b = true and, a= true,
we can infer that, b is true.
Hence, given the validity of the premise, we can infer correct, true conclusions, with respect to the premise.
So, a sound inference procedure indeed produces true propositions, given the premise is true.