I just don't get exactly how logical implications work.
When using truth tables (albeit impractical for large amounts of variables), it can be quite simple to show truths equate for something such as (p ∧ q) ≡ ¬(¬p ∨ ¬q) where the resulting truths would equate as T F F F on both sides (where p is T T F F and q is T F T F) though a truth table, and i get that semantically p → q ⊨ ¬q → ¬p (eg Sun is not visible(p) if overcast(q), not overcast(q) if sun visible(p)), however when it comes to logical implications such as (p ∨ q) ∧ ¬q ⊨ p, i get confused over how they are realized, does it simply require the same amount of truths in a table? at least 1 truth to match? How do you define if it is actually true based on the table?
Given for example (p ↔ q) ∧ q ⊨ p a truth table (where p and q are as stated above after first example) equate to T F F F ⊨ T T F F, is it logically implied because at least one truth matches an opposing truth, as such if the result was instead F F F T ⊨ T T F F it would not be logically implied?
In addition to a truth table, given it is known that (p ∨ q) ∧ ¬q ⊨ p is a known logical implementation, would (p ∨ q) ∧ ¬q be able to be substituted into (p ↔ q) ∧ q ⊨ p for (p ↔ q) ∧ q ⊨ (p ∨ q) ∧ ¬q as a form of proof? What are relevant alternatives to truth tables (as anything past 3 values becomes quite tedious)?