Propositional Connectives Via Truth Tables

English lets you smuggle meaning in through tone and context. Logic does not. Take If you submit the assignment, then you pass. In English, people hear a promise, a threat, or a causal story. In propositional logic, an implication is a truth function. It is only about the truth values of two statements, not about why they are true.

That difference shows up in two places that matter in proofs. First is ambiguity. The sentence You can take CS201 if you took CS101 can be read as permission, as a prerequisite, or as an exclusive rule. Second is vacuous truth. The claim If $p$ then $q$ is considered true whenever $p$ is false, even if $q$ is false too. That feels wrong if you interpret implication as causation, but it is exactly what makes proof patterns like proof by contradiction and contrapositive work consistently. When an assumption fails, the conditional promise becomes irrelevant instead of breaking the system.

The test of whether you really mean the logical connective is whether you can compute it on all possible input truth values.

The four rows that define the connectives

Let $p$ and $q$ be propositions, each either True or False. A connective is defined by what it outputs on the four possible input rows. Negation has one input, so it just flips True to False and False to True. Conjunction and disjunction behave like you expect from everyday and and or, except that logic or is inclusive.

Implication is the one you have to decide to treat as a definition, not a story. The standard definition is forced by how proofs use it. If $p$ is True and $q$ is False, then $p\to q$ must be False, because this is the only case where you had $p$ but failed to get $q$. In every other row, there is no counterexample to the conditional statement, so $p\to q$ is True.

Fill the columns and watch which row is the unique failure for $p\to q$.

One row matters more than people expect. When $p$ is False and $q$ is False, implication still comes out True. If you treat implication as a promise made only when $p$ happens, that row stops feeling strange.

A compound claim as a filter

Truth tables become useful when you stop reading formulas left to right like English and start treating them as filters on rows. Consider the compound statement $(p\to q)\land\lnot q$. Each row is a possible world, and the compound statement picks out the worlds where both parts are True.

Work row by row. First compute $p\to q$. Then compute $\lnot q$. Only then take the and. A satisfying row is one where the final column is True.

There is a structural question hiding here. If $(p\to q)\land\lnot q$ is True, what must be true about $p$ in that same row. The table answers it without any verbal interpretation of if.