Rule of inference

Rules of inference are ways of deriving conclusions from <a href="/facts/Premise/HluL8lcd">premises</a>. They are integral parts of <a href="/facts/Formal_logic/SaI7Y3wN">formal logic</a>, serving as norms of the <a href="/facts/Logical_form/foVBtMX4">logical structure</a> of <a href="/facts/Validity_(logic)/oMGkywGC">valid</a> arguments. If an argument with true premises follows a rule of inference then the conclusion cannot be false. <a href="/facts/Modus_ponens/Wq4NcmrG">Modus ponens</a>, an influential rule of inference, connects two premises of the form "if 
 
 
 
 P
 
 
 {\displaystyle P}
 
 then 
 
 
 
 Q
 
 
 {\displaystyle Q}
 
" and "
 
 
 
 P
 
 
 {\displaystyle P}
 
" to the conclusion "
 
 
 
 Q
 
 
 {\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 <a href="/facts/Modus_tollens/sqf3PTlf">modus tollens</a>, <a href="/facts/Disjunctive_syllogism/cfnyp2wY">disjunctive syllogism</a>, <a href="/facts/Constructive_dilemma/86Beulx6">constructive dilemma</a>, and <a href="/facts/Existential_generalization/PT2l8J0f">existential generalization</a>.
Rules of inference include rules of implication, which operate only in one direction from premises to conclusions, and <a href="/facts/Rules_of_replacement/uTyqA71f">rules of replacement</a>, which state that two expressions are equivalent and can be freely swapped. Rules of inference contrast with <a href="/facts/Formal_fallacies/1c3P1BEH">formal fallacies</a>—invalid argument forms involving logical errors.
Rules of inference belong to <a href="/facts/Formal_system/KaeDAjK7">logical systems</a>, and distinct logical systems use different rules of inference. <a href="/facts/Propositional_logic/MNt5v31V">Propositional logic</a> examines the inferential patterns of simple and compound <a href="/facts/Proposition/0877MFMD">propositions</a>. <a href="/facts/First-order_logic/lcISqWIg">First-order logic</a> extends propositional logic by articulating the internal structure of propositions. It introduces new rules of inference governing how this internal structure affects valid arguments. <a href="/facts/Modal_logic/I7uXvYYT">Modal logics</a> explore concepts like possibility and necessity, examining the inferential structure of these concepts. <a href="/facts/Intuitionistic_logic/zI7ulCnM">Intuitionistic</a>, <a href="/facts/Paraconsistent_logic/6HLVEvrr">paraconsistent</a>, and <a href="/facts/Many-valued_logics/qNI26RdF">many-valued logics</a> propose alternative inferential patterns that differ from the traditionally dominant approach associated with <a href="/facts/Classical_logic/ls62MAms">classical logic</a>. Various formalisms are used to express logical systems. Some employ many intuitive rules of inference to reflect how people <a href="/facts/Natural_deduction/PPtgwoaJ">naturally reason</a> while others provide minimalistic frameworks to represent foundational principles without redundancy.
Rules of inference are relevant to many areas, such as <a href="/facts/Mathematical_proof/7ECDrU80">proofs</a> in <a href="/facts/Mathematics/pxTouaz4">mathematics</a> and <a href="/facts/Automated_reasoning/x5EdkKoU">automated reasoning</a> in <a href="/facts/Computer_science/lN3pEyPz">computer science</a>. Their conceptual and psychological underpinnings are studied by <a href="/facts/Philosophy_of_logic/1AGMNRaf">philosophers of logic</a> and <a href="/facts/Cognitive_psychology/GnzIgx2I">cognitive psychologists</a>.

Rule of inference open-in-new

Rule of inference