We remove all the ones that are false because we assume that all the premises are true.
In logic we are usually interestred in wff’s up to logical equivalence.
Replacing a premise/conclusion in an argument with an equivalent wff has ni impact on the validity of the argument.
A set of wff’s is said to be tautlogically consistent iff there exists a valuation tat makes all the wff’s true together.
Example: {P v Q, -P, Q^R}
This set is tautologically consistent because there is some valuation (P=0, R=1, Q=1) when they are true,
{α1….αn} is tautologically consistent iff a1^…^an is a contradiction.
If I have an argument with n premises and a conclusion γ, the argument a1…an is tautologically valid iff the following set {a1…an : -γ} is tautologically inconsistent.
Truth tables are nice because they always work but sometimes they can become very big.
Can we do something that is better?
Often yes.
But probably not always.
You can do indirect methods, like truth trees (tableaux).
However, sometimes the indirect method requires exponentially much work
but maybe there exists yet another method that is always shorter? We don’t know.