User:Phlsph7/Rule of inference - Lead

Rules of inference are ways of deriving conclusions from premises. They are integral parts of formal logic, serving as norms of the logical structure of valid arguments. If an argument with true premises follows a rule of inference then the conclusion cannot be false. Modus ponens, an influential rule of inference, connects two premises of the form "if $P$ then $Q$" and "$P$" to the conclusion "$Q$", as in the argument "If it rains, then the ground is wet. It rains. Therefore, the ground is wet". There are many other rules of inference for different patterns of valid arguments, such as modus tollens, disjunctive syllogism, constructive dilemma, and existential generalization.

Rules of inference include rules of implication, which operate only in one direction from premises to conclusions, and rules of replacement, which state that two expressions are equivalent and can be freely swapped. Rules of inference contrast with formal fallacies—invalid argument forms involving logical errors.

Rules of inference belong to logical systems and distinct logical systems use different rules of inference. Propositional logic examines the inferential patterns of simple and compound propositions. First-order logic extends propositional logic by articulating the internal structure of propositions. It introduces new rules of inference governing how this internal structure affects valid arguments. Modal logics explore the concepts of possibility and necessity, examining the inferential structure of these concepts. Intuitionistic, paraconsistent, and many-valued logics propose alternative inferential patterns that differ from the traditionally dominant approach associated with classical logic. Various formalisms are used to express logical systems, with some using many intuitive rules of inference to reflect how people naturally reason while others provide minimalistic frameworks to represent foundational principles without redundancy.

Rules of inference are relevant to many areas, such as proofs in mathematics and automatic reasoning in computer science. Their conceptual and psychological underpinnings are studied by philosophers of logic and cognitive psychologists.