Whether an argument is logically valid — whether its conclusion genuinely follows from its premises — is a question about structure, not content. Propositional logic exploits that fact directly: it strips an argument down to simple statements connected by logical operators like "and," "or," "not," and "if... then," and a truth table can check the validity of that stripped-down structure by mechanically working through every possible combination of true and false values, with no need to understand or even know what the original statements were actually about.
Reducing an argument to letters and connectives
Propositional logic represents each simple statement in an argument with a single letter, and represents the logical relationships between those statements with a small set of standard connectives: conjunction ("and"), disjunction ("or"), negation ("not"), and the conditional ("if... then"). An argument like "if it rains, the match is cancelled; it is raining; therefore the match is cancelled" gets reduced to the same structural pattern regardless of what specific statements fill in for "it rains" and "the match is cancelled" — the argument's validity depends entirely on that structural pattern (a well-known valid form called modus ponens), not on the specific real-world content of the statements involved.
Checking every possible world at once
A truth table lists every possible combination of true and false values the component statements in an argument could take, and works out, row by row, what the argument's premises and conclusion would evaluate to under each combination. An argument is logically valid if and only if there's no row in the table where all the premises come out true but the conclusion comes out false — in other words, no possible combination of truth values makes the premises hold while the conclusion fails. This gives propositional logic a genuinely mechanical, purely syntactic test for validity: two people who disagree completely about whether the original premises are actually true in the real world can still agree, by simply working through the table, on whether the argument's logical structure is valid, because validity here is a claim about the relationship between the statements, not about whether they're true.
What we're still unsure about
The mechanics of propositional logic and truth tables, and their use for checking argument validity, are formally rigorous and not in dispute as mathematics. What truth tables genuinely can't settle on their own is whether an argument's premises are actually true in the real world, or whether translating a natural-language argument into propositional logic in the first place has faithfully captured what the original argument actually meant — that translation step often involves real interpretive judgement, and getting it wrong can make a genuinely weak real-world argument look formally valid, or vice versa, which is exactly why logicians treat validity-checking and the translation into logical form as two separate skills, not one.
This sits inside Propositional Logic & Truth Tables, one of seven topics in Logic, one of five domains in Philosophy, one of seventeen subjects the app can quiz you on.