Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics and to a lesser extent computer science. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct (or valid) and incorrect (or fallacious) inferences. Logicians study the criteria for the evaluation of arguments.
Foundations of logic
Philosophy of logic Analytic-synthetic distinction Antinomy A priori and a posteriori Definition Description Entailment Identity (philosophy) Inference Logical form Logical implication Logical truth Logical consequence Name Necessity Material conditional Meaning (linguistic) Meaning (non-linguistic) Paradox (list) Possible world Presupposition Probability Quantification Reason Reasoning Reference Semantics Strict conditional Syntax (logic) Truth Truth value Validity
Branches of logic
Affine logic Alethic logic Aristotelian logic Boolean logic Buddhist logic Bunched logic Categorical logic Classical logic Computability logic Deontic logic Dependence logic Description logic Deviant logic Doxastic logic Epistemic logic First-order logic Formal logic Free logic Fuzzy logic Higher-order logic Infinitary logic Informal logic Intensional logic Intermediate logic Interpretability logic Intuitionistic logic Linear logic Many-valued logic Mathematical logic Metalogic Minimal logic Modal logic Non-Aristotelian logic Non-classical logic Noncommutative logic Non-monotonic logic Ordered logic Paraconsistent logic Philosophical logic Predicate logic Propositional logic Provability logic Quantum logic Relevance logic Sequential logic Spatial logic Strict logic Substructural logic Syllogistic logic Symbolic logic Temporal logic Term logic Topical logic Traditional logic Zeroth-order logic
Philosophical logic
Informal logic and critical thinking Informal logic Critical thinking Argumentation theory Argument Argument map Accuracy and precision Ad hoc hypothesis Ambiguity Analysis Attacking Faulty Reasoning Belief Belief bias Bias Cognitive bias Confirmation bias Credibility Critical reading Critical thinking Decidophobia Decision making Dispositional and occurrent belief Emotional reasoning Evidence Expert Explanation Explanatory power Fact Fallacy Higher-order thinking Inquiry Interpretive discussion Occam's razor Opinion Practical syllogism Precision questioning Propaganda Propaganda techniques Problem Solving Prudence Pseudophilosophy Reasoning Relevance Rhetoric Rigour Socratic questioning Source credibility Source criticism Theory of justification Topical logic Vagueness
Deductive reasoning
Theories of deduction Anti-psychologism Conceptualism Constructivism Conventionalism Counterpart theory Deflationary theory of truth Dialetheism Fictionalism Formalism (philosophy) Game theory Illuminationist philosophy Logical atomism Logical holism Logicism Modal fictionalism Nominalism Polylogism Pragmatism Preintuitionism Proof theory Psychologism Ramism Semantic theory of truth Sophism Trivialism Ultrafinitism
Fallacies
Fallacy (list) – incorrect argumentation in reasoning resulting in a misconception or presumption. By accident or design, fallacies may exploit emotional triggers in the listener or interlocutor (appeal to emotion), or take advantage of social relationships between people (e.g. argument from authority). Fallacious arguments are often structured using rhetorical patterns that obscure any logical argument. Fallacies can be used to win arguments regardless of the merits. There are dozens of types of fallacies.
Formal logic
Formal logic – Mathematical logic, symbolic logic and formal logic are largely, if not completely synonymous. The essential feature of this field is the use of formal languages to express the ideas whose logical validity is being studied. List of mathematical logic topics
Symbols and strings of symbols
Logical symbols
Logical variables Propositional variable Predicate variable Literal Metavariable Logical constants Logical connective Quantifier Identity Brackets
Logical connectives Logical connective Converse implication Converse nonimplication Exclusive or Logical NOR Logical biconditional Logical conjunction Logical disjunction Material implication Material nonimplication Negation Sheffer stroke
Strings of symbols
Atomic formula Open sentence
Types of propositions
Proposition Analytic proposition Axiom Atomic sentence Clause (logic) Contingent proposition Contradiction Logical truth Propositional formula Rule of inference Sentence (mathematical logic) Sequent Statement (logic) Subalternation Tautology Theorem
Rules of inference
Rule of inference (list) Biconditional elimination Biconditional introduction Case analysis Commutativity of conjunction Conjunction introduction Constructive dilemma Contraposition (traditional logic) Conversion (logic) De Morgan's laws Destructive dilemma Disjunction elimination Disjunction introduction Disjunctive syllogism Double negation elimination Generalization (logic) Hypothetical syllogism Law of excluded middle Law of identity Modus ponendo tollens Modus ponens Modus tollens Obversion Principle of contradiction Resolution (logic) Simplification Transposition (logic)
Formal theories
Formal proof List of first-order theories
Expressions in a metalanguage
Metalanguage Metalinguistic variable Deductive system Metatheorem Metatheory Interpretation
Propositional and Boolean logic
Propositional logic
Absorption law Clause (logic) Deductive closure Distributive property Entailment Formation rule Functional completeness Intermediate logic Literal (mathematical logic) Logical connective Logical consequence Negation normal form Open sentence Propositional calculus Propositional formula Propositional variable Rule of inference Strict conditional Substitution instance Truth table Zeroth-order logic
Boolean logic Boolean algebra (list) Boolean logic Boolean algebra (structure) Boolean algebras canonically defined Introduction to Boolean algebra Complete Boolean algebra Free Boolean algebra Monadic Boolean algebra Residuated Boolean algebra Two-element Boolean algebra Modal algebra Derivative algebra (abstract algebra) Relation algebra Absorption law Laws of Form De Morgan's laws Algebraic normal form Canonical form (Boolean algebra) Boolean conjunctive query Boolean-valued model Boolean domain Boolean expression Boolean ring Boolean function Boolean-valued function Parity function Symmetric Boolean function Conditioned disjunction Field of sets Functional completeness Implicant Logic alphabet Logic redundancy Logical connective Logical matrix Product term True quantified Boolean formula Truth table
Predicate logic and relations
Predicate logic
Atomic formula Atomic sentence Domain of discourse Empty domain Extension (predicate logic) First-order logic First-order predicate Formation rule Free variables and bound variables Generalization (logic) Monadic predicate calculus Predicate (mathematical logic) Predicate logic Predicate variable Quantification Second-order predicate Sentence (mathematical logic) Universal instantiation
Relations
Mathematical relation Finitary relation Antisymmetric relation Asymmetric relation Bijection Bijection, injection and surjection Binary relation Composition of relations Congruence relation Connected relation Converse relation Coreflexive relation Covering relation Cyclic order Dense relation Dependence relation Dependency relation Directed set Equivalence relation Euclidean relation Homogeneous relation Idempotence Intransitivity Involutive relation Partial equivalence relation Partial function Partially ordered set Preorder Prewellordering Propositional function Quasitransitive relation Reflexive relation Serial relation Surjective function Symmetric relation Ternary relation Transitive relation Trichotomy (mathematics) Well-founded relation
Mathematical logic Mathematical logic
Set theory Set theory list Aleph null Bijection, injection and surjection Binary set Cantor's diagonal argument Cantor's first uncountability proof Cantor's theorem Cardinality of the continuum Cardinal number Codomain Complement (set theory) Constructible universe Continuum hypothesis Countable set Decidable set Denumerable set Disjoint sets Disjoint union Domain of a function Effective enumeration Element (mathematics) Empty function Empty set Enumeration Extensionality Finite set Forcing (mathematics) Function (set theory) Function composition Generalized continuum hypothesis Index set Infinite set Intension Intersection (set theory) Inverse function Large cardinal Löwenheim–Skolem theorem Map (mathematics) Multiset Morse–Kelley set theory Naïve set theory One-to-one correspondence Ordered pair Partition of a set Pointed set Power set Projection (set theory) Proper subset Proper superset Range of a function Russell's paradox Sequence (mathematics) Set (mathematics) Set of all sets Simple theorems in the algebra of sets Singleton (mathematics) Skolem paradox Subset Superset Tuple Uncountable set Union (set theory) Von Neumann–Bernays–Gödel set theory Zermelo set theory Zermelo–Fraenkel set theory
Metalogic Metalogic – The study of the metatheory of logic. Completeness (logic) Syntax (logic) Consistency Decidability (logic) Deductive system Interpretation (logic) Cantor's theorem Church's theorem Church's thesis Effective method Formal system Gödel's completeness theorem Gödel's first incompleteness theorem Gödel's second incompleteness theorem Independence (mathematical logic) Logical consequence Löwenheim–Skolem theorem Metalanguage Metasyntactic variable Metatheorem Object language – see metalanguage Symbol (formal) Type–token distinction Use–mention distinction Well-formed formula
Proof theory Proof theory – The study of deductive systems. Axiom Deductive system Formal proof Formal system Formal theorem Syntactic consequence Syntax (logic) Transformation rules
Model theory Model theory – The study of interpretation of formal systems. Interpretation (logic) Logical validity Non-standard model Normal model Model Semantic consequence Truth value
Computability theory Computability theory – branch of mathematical logic that originated in the 1930s with the study of computable functions and Turing degrees. The field has grown to include the study of generalized computability and definability. The basic questions addressed by recursion theory are "What does it mean for a function from the natural numbers to themselves to be computable?" and "How can noncomputable functions be classified into a hierarchy based on their level of noncomputability?". The answers to these questions have led to a rich theory that is still being actively researched. Alpha recursion theory Arithmetical set Church–Turing thesis Computability logic Computable function Computation Decision problem Effective method Entscheidungsproblem Enumeration Forcing (computability) Halting problem History of the Church–Turing thesis Lambda calculus List of undecidable problems Post correspondence problem Post's theorem Primitive recursive function Recursion (computer science) Recursive language Recursive set