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 ${\displaystyle P}$ then ${\displaystyle Q}$" and "${\displaystyle P}$" to the conclusion "${\displaystyle Q}$", 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 concepts like 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. Some employ 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 automated reasoning in computer science. Their conceptual and psychological underpinnings are studied by philosophers of logic and cognitive psychologists.

A rule of inference is a way of drawing a conclusion from a set of premises. Also called inference rule and transformation rule, it is a norm of correct inferences that can be used to guide reasoning, justify conclusions, and criticize arguments. As part of deductive logic, rules of inference are argument forms that preserve the truth of the premises, meaning that the conclusion is always true if the premises are true. An inference is deductively correct or valid if it follows a valid rule of inference. Whether this is the case depends only on the form or syntactical structure of the premises and the conclusion. As a result, the actual content or concrete meaning of the statements does not affect validity. For instance, modus ponens is a rule of inference that connects two premises of the form "if ${\displaystyle P}$ then ${\displaystyle Q}$" and "${\displaystyle P}$" to the conclusion "${\displaystyle Q}$", where ${\displaystyle P}$ and ${\displaystyle Q}$ stand for statements. Any argument with this form is valid, independent of the specific meanings of ${\displaystyle P}$ and ${\displaystyle Q}$, such as the argument "If it rains, then the ground is wet. It rains. Therefore, the ground is wet". In addition to modus ponens, there are many other rules of inference, such as modus tollens, disjunctive syllogism, hypothetical syllogism, constructive dilemma, and destructive dilemma.

There are different formats to represent rules of inference. A common approach is to use a new line for each premise and separate the premises from the conclusion using a horizontal line. With this format, modus ponens is written as:

${\displaystyle {\begin{array}{l}P\to Q\\P\\\hline Q\end{array}}}$

Some logicians employ the therefore sign (${\displaystyle \therefore }$) together or instead of the horizontal line to indicate where the conclusion begins. The sequent notation, a different approach, uses a single line in which the premises are separated by commas and connected to the conclusion with the turnstile symbol (${\displaystyle \vdash }$), as in ${\displaystyle P\to Q,P\vdash Q}$. The letters ${\displaystyle P}$ and ${\displaystyle Q}$ in these formulas are so-called metavariables: they stand for any simple or compound proposition.