Rule of inference

Definition

Rules of inference are ways of deriving conclusions from premises. They are integral parts of formal logic, serving as 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 then " and "" to the conclusion "", 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.

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Abductive reasoning Absorption (logic)Abstract logic Ackermann set theory Admissible rule Affirming a disjunct Affirming the consequent Aleph number Algebraic logic Algorithms Alphabet (formal languages)Ancient Greece Ancient India Ancient history Argument Argument from analogy Argument from authority Argumentation scheme Aristotelian logic Aristotle Aristotle's logic Arithmetic Arity Associative property Atomic formula Atomic model (mathematical logic)Atomic sentence Automata theory Automated reasoning Automated theorem prover Automated theorem proving Axiom Axiom of choice Axiom schema Axiomatic system Axiomatization of Boolean algebras Banach–Tarski paradox Biconditional elimination Biconditional introduction Bijection Binary operation Boolean algebra Boolean algebras canonically defined Boolean function Cantor's diagonal argument Cantor's paradox Cantor's theorem Cardinality Cartesian product Categorical theory Category (mathematics)Category of sets Category theory Church encoding Church–Turing thesis Class (set theory)Classical logic Codomain Cognitive bias Cognitive psychology Commutative property Compactness theorem Complement (set theory)Complete theory Completeness (logic)Computability theory Computable function Computable set Computably enumerable set Computation Computer science Concrete category Conjunction elimination Conjunction introduction Conservative extension Consistency Constructible universe Construction of the real numbers Constructive dilemma Constructive proof Constructive set theory Continuum hypothesis Contradiction (logic)Copula (linguistics)Countable set Cut rule De Morgan's laws Decidability (logic)Decision-making Decision problem Deductive reasoning Deductive system Denying the antecedent Deontic logic Destructive dilemma Diagram (mathematical logic)Direct proof Disjunction elimination Disjunction introduction Disjunctive syllogism Domain of a function Domain of discourse Double negation elimination Doxastic logic Early modern philosophy Element (mathematics)Elementary diagram Elementary equivalence Elementary function arithmetic Empty set Enumeration Epistemic modal logic Epistemology of logic Equiconsistency Equivalence relation Euclid's Elements Euclidean geometry Evidentiary standard Existential fallacy Existential generalization Existential instantiation Existential quantification Expert system Experts Expression (mathematics)Extension by definitions Extension by new constant and function names Extensionality Fallacy Fallacy of the undistributed middle Finitary relation Finite-valued logic Finite model theory Finite set First-order logic Fitch notation Fixed-point logic Forcing (mathematics)Formal fallacies Formal fallacy Formal grammar Formal language Formal languages Formal logic Formal proof Formal science Formal sciences Formal semantics (logic)Formal system Formal systems Formation rule Foundations of geometry Foundations of mathematics Free logic Free variables and bound variables Function (mathematics)Functional predicate Fuzzy set General set theory George Boole Gottlob Frege Grothendieck universe Ground expression Ground formula Gödel's completeness theorem Gödel's incompleteness theorems Gödel numbering Halting problem Hereditary set Higher-order logic Hilbert's axioms Hilbert system Hilbert systems History of logic History of mathematical logic Hypothetical syllogism Image (mathematics)Immediate inference Implication elimination Implication introduction Inaccessible cardinal Independence (mathematical logic)Inductive reasoning Inference Inference engine Inference objection Infinite-valued logic Infinite set Information theory Inhabited set Injective function Interpretation (logic)Interpretation (model theory)Interpretation function Intersection (set theory)Intuitionistic logic Isomorphism Jaakko Hintikka Kleene logic Knowledge base Knowledge representation Kolmogorov complexity Kripke's theory of truth Kripke–Platek set theory Lambda calculus Large cardinal Law Laws of thought Legal norms Lemma (mathematics)Lindström's theorem List of Hilbert systems List of axioms List of first-order theories List of formal systems List of mathematical theories List of rules of inference List of set identities and relations List of statements independent of ZFC Logic Logic programming Logical biconditional Logical conjunction Logical connective Logical consequence Logical constant Logical disjunction Logical equality Logical equivalence Logical form Logical operator Logical reasoning Logical truth Logicism Löwenheim–Skolem theorem Many-valued logic Many-valued logics Map (mathematics)Material conditional Mathematical formula Mathematical function Mathematical induction Mathematical logic Mathematical object Mathematical proof Mathematical proofs Mathematics Medical diagnosis Medieval philosophy Metalanguage Metavariable Minimal axioms for Boolean algebra Modal logic Model complete theory Model theory Modus ponens Modus tollens Monadic predicate calculus Monadic second-order logic Morse–Kelley set theory NP (complexity)Naive set theory Natural deduction Negation Negation elimination Negation introduction New Foundations Non-Euclidean geometry Non-classical logic Non-logical symbol Non-standard model Non-standard model of arithmetic Nyaya Open formula Operation (mathematics)Ordinal analysis Ordinal number P (complexity)P versus NP problem Paraconsistent logics Paradoxes of set theory Partition of a set Peano axioms Philosophy of logic Philosophy of mathematics Power set Predicate (logic)Predicate (mathematical logic)Predicate logic Predicate variable Premise Prime model Primitive recursive arithmetic Primitive recursive function Principia Mathematica Principle of explosion Probabilistic logic Probability Prolog Proof by contradiction Proof of impossibility Proof system Proof theory Proposition Propositional calculus Propositional formula Propositional logic Propositional variable Quantifier (logic)Quantifier rank Quod erat demonstrandum Recursion Recursive set Reductio ad absurdum Relation (mathematics)Relation (philosophy)Reproducibility Reverse mathematics Robinson arithmetic Rules of replacement Russell's paradox Satisfiability Saturated model Schröder–Bernstein theorem Scientific experiments Second-order arithmetic Second-order logic Self-verifying theories Semantic theory of truth Semantics of logic Sentence (mathematical logic)Sequent Sequent calculi Sequent calculus Set (mathematics)Set theory Signature (logic)Singleton (mathematics)Singular term Skolem arithmetic Soundness Spectrum of a sentence Spectrum of a theory Square of opposition Strength (mathematical logic)String (formal languages)Structural rule Structure (mathematical logic)Substitution (logic)Substructure (mathematics)Supertask Surjective function Susan Haack Syllogism Syllogisms Symbol (formal)Symbolic logic Syntax (logic)T-schema Tarski's axiomatization of the reals Tarski's axioms Tarski's theory of truth Tarski's undefinability theorem 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