Predicate logic, also called first-order logic, extends propositional logic with quantifiers and predicates, allowing statements about properties of objects and relations among them. The universal quantifier \(\forall x\, P(x)\) states that every element in a domain satisfies the predicate P, while the existential quantifier \(\exists x\, P(x)\) claims that at least one element does. This added expressiveness enables the analysis of arguments that propositional logic alone cannot capture, such as "all mammals are warm-blooded" or "some philosophers are empiricists." Together with categorical propositions and the square of opposition, predicate logic forms the modern backbone of formal reasoning.
Several advanced tools deepen our ability to evaluate arguments. Abductive reasoning, often called inference to the best explanation, starts from observations and selects the simplest, most likely explanation; it differs from inductive reasoning, which generalizes from specific instances to broader rules. Reductio ad absurdum proves a proposition false by assuming it true and showing the assumption leads to a contradiction or absurd consequence. Analogies argue that because two things are similar in some respects, they are likely similar in a further respect, with strength depending on the relevance of the similarities invoked. A counterexample is a single specific case that disproves a universal claim.
Understanding necessary and sufficient conditions clarifies how requirements relate to outcomes: a necessary condition must be present for an event to occur, while a sufficient condition guarantees it. For instance, being a mammal is necessary for being a dog, and being a dog is sufficient for being a mammal. The principle of charity requires interpreting an argument in its strongest, most reasonable form before critique, rather than attacking a weaker version. An enthymeme is an argument with an implicit premise that the audience is expected to supply, such as "she's a doctor, so she's smart," whose hidden premise is that doctors are smart. Finally, the paradox of the heap (the sorites paradox) shows how vague predicates resist sharp boundaries, illustrating that questions like "when does a collection of grains stop being a heap?" cannot always be answered with classical categories.