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Formal system in mathematical logic
In mathematical logic, an abstract logic is a formal system consisting of a class of sentences and a satisfaction relation with specific properties related
Abstract_logic
Topics referred to by the same term
Abstract logic is a formal system consisting of a class of sentences and a satisfaction relation with specific properties. Abstract logic may also refer
Abstract logic (disambiguation)
Abstract_logic_(disambiguation)
Aspect of mathematical logic
In mathematical logic, abstract algebraic logic is the study of the algebraization of deductive systems arising as an abstraction of the well-known Lindenbaum–Tarski
Abstract_algebraic_logic
1995 live album by Jonas Hellborg, Shawn Lane and Kofi Baker
Abstract Logic is the first collaborative live album by bassist Jonas Hellborg and guitarist Shawn Lane, released in 1995 through Day Eight Music; a remastered
Abstract_Logic_(album)
Reasoning about equations with free variables
the umbrella of classical algebraic logic (Czelakowski 2003). Works in the more recent abstract algebraic logic (AAL) focus on the process of algebraization
Algebraic_logic
Concept in model theory
class in α {\displaystyle \alpha } . Abstract logic Lindström's theorem Heinz-Dieter Ebbinghaus Extended logics: the general framework in K. J. Barwise
Strength_(mathematical_logic)
Smallest cardinal number for which a weak downward Löwenheim–Skolem theorem holds
In mathematical logic the Löwenheim number of an abstract logic is the smallest cardinal number for which a weak downward Löwenheim–Skolem theorem holds
Löwenheim_number
Mathematical model for deduction or proof systems
descriptions of redirect targets Formal science – Study of abstract structures described by formal systems Logic translation – Translation of a text into a logical
Formal_system
Framework for a family of logic languages
in the Standard by their translation to the abstract syntax and semantics of Common Logic. Many other logic-based languages could also be defined as subsets
Common_Logic
mathematical logic, abstract model theory is a generalization of model theory that studies the general properties of extensions of first-order logic and their
Abstract_model_theory
Study of the semantics, or interpretations, of formal and natural languages
by Saul Kripke and others for modal logic and related systems), algebraic semantics (connecting logic to abstract algebra), and game semantics (interpreting
Semantics_(logic)
statements are useful fictions that do not correspond to any actual abstract objects. Logicism asserts that all mathematical truths can be reduced to logical
Mathematical_object
Method of deriving conclusions
of deriving conclusions from premises. They are integral parts of formal logic, serving as the logical structure of valid arguments. If an argument with
Rule_of_inference
Approach to logic
In logic and formal semantics, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to
Term_logic
Subfield of mathematics
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory
Mathematical_logic
Algebraic manipulation of "true" and "false"
developments in abstract algebra and mathematical logic; it is however seen as connected to the origins of both fields. In an abstract setting, Boolean
Boolean_algebra
Symbol representing a property or relation in logic
In logic, a predicate is a non-logical symbol that represents a property or a relation, though, formally, does not need to represent anything at all.
Predicate_(logic)
Class of formal logics
Classical logic (or standard logic) or Frege–Russell logic is the intensively studied and most widely used class of deductive logic. Classical logic has had
Classical_logic
Whether a decision problem has an effective method to derive the answer
effectively determined. Zeroth-order logic (propositional logic) is decidable, whereas first-order and higher-order logic are not. A theory (set of sentences
Decidability_(logic)
Theorem in mathematical logic
known result of what later became known as abstract model theory, the basic notion of which is an abstract logic; the more general notion of an institution
Lindström's_theorem
Study of correct reasoning
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical
Logic
Theorem for proving more complex theorems
lemma Inference objection List of lemmas Objection Porism Such as informal logic, argument mapping, and philosophy. Lemma. Merriam-Webster. Loewen, Nathan
Lemma_(mathematics)
Logical formulation of recursion
In mathematical logic, fixed-point logics are extensions of classical predicate logic that have been introduced to express recursion. Their development
Fixed-point_logic
In logic, a statement which is always true
In mathematical logic, a tautology (from Ancient Greek: ταυτολογία) is a formula that is true regardless of the interpretation of its component terms
Tautology_(logic)
Description of non-logical symbols
In mathematical logic, a signature is a description of the non-logical symbols of a formal language. In universal algebra, a signature lists the operations
Signature_(logic)
Formal system of logic
In mathematics and logic, a higher-order logic (HOL) is a form of logic that is distinguished from first-order logic by additional quantifiers and, sometimes
Higher-order_logic
Process by which desired circuit behavior is turned into a schematic of logic gates
In computer engineering, logic synthesis is a process by which an abstract specification of desired circuit behavior, typically at register transfer level
Logic_synthesis
In model theory, a subfield of mathematical logic, an atomic model is a model such that the complete type of every tuple is axiomatized by a single formula
Atomic model (mathematical logic)
Atomic_model_(mathematical_logic)
Form of logic
A free logic is a logic with fewer existential presuppositions than classical logic. Free logics may allow for terms that do not denote any object. Free
Free_logic
Assignment of meaning to the symbols of a formal language
formal semantics. The most commonly studied formal logics are propositional logic, predicate logic and their modal analogs, and for these there are standard
Interpretation_(logic)
Logical connective AND
In logic, mathematics and linguistics, and ( ∧ {\displaystyle \wedge } ) is the truth-functional operator of conjunction or logical conjunction. The logical
Logical_conjunction
In mathematical logic, a well-formed formula with no free variables
In mathematical logic, a sentence (or closed formula) of a predicate logic is a Boolean-valued well-formed formula with no free variables. A sentence can
Sentence_(mathematical_logic)
Non-contradiction of a theory
In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T {\displaystyle T} is consistent if there is no
Consistency
Set of sentences in a formal language
In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language. In most scenarios a deductive system is first
Theory_(mathematical_logic)
Paradox in set theory
In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox published by the British philosopher and mathematician
Russell's_paradox
System including an indeterminate value
three-valued logic (also trinary logic, trivalent, ternary, or trilean, sometimes abbreviated 3VL) is any of several many-valued logic systems in which
Three-valued_logic
Form of logic that allows quantification over predicates
In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic
Second-order_logic
Mathematical use of "for all" and "there exists"
In mathematical logic, quantifiers are formal counterparts of natural-language adjectives like all, some, most, few, etc. which indicate the number of
Quantifier_(logic)
American record label
the record label it eventually birthed was named after the 1995 CD Abstract Logic (album) by Lane, Hellborg, and drummer Kofi Baker. "I always loved that
Abstract_Logix
Mathematical use of "there exists"
In predicate logic, an existential quantification is a type of quantifier which asserts the existence of an object with a given property. It is usually
Existential_quantification
Mathematical term; concerning axioms used to derive theorems
In mathematics and logic, an axiomatic system or axiom system is a standard type of deductive logical structure, used also in theoretical computer science
Axiomatic_system
Set with algorithmic membership test
computable function, or the empty set. Computably enumerable Decidability (logic) Recursively enumerable language Recursive language Recursion That is, under
Computable_set
Argument whose conclusion must be true if its premises are
In logic, specifically in deductive reasoning, an argument is valid if and only if it takes a form that makes it impossible for the premises to be true
Validity_(logic)
Sequence of words formed by specific rules
In logic, mathematics, computer science, and linguistics, a formal language is a set of strings whose symbols are taken from a set called "alphabet".
Formal_language
Characteristic of some logical systems
In mathematical logic and metalogic, a formal system is called complete with respect to a particular property if every formula having the property can
Completeness_(logic)
of mathematical logic; see also history of logic. 1847 – George Boole proposes symbolic logic in The Mathematical Analysis of Logic, defining what is
Timeline of mathematical logic
Timeline_of_mathematical_logic
Concept in model theory
In model theory, a branch of mathematical logic, the diagram of a structure is the set of sentences with parameters from the structure that are true in
Diagram_(mathematical_logic)
Logic with discrete truth values
In logic, a finite-valued logic (also finitely many-valued logic) is a propositional calculus in which truth values are discrete. Traditionally, in Aristotle's
Finite-valued_logic
Type of logical system
first-order logic (FOL), also called predicate logic, predicate calculus, or quantificational logic, is a type of formal system. First-order logic uses quantified
First-order_logic
Logical operation
In logic, negation, also called the logical not or logical complement, is an operation that takes a proposition P {\displaystyle P} to another proposition
Negation
Logical connective OR
In logic, disjunction (also known as logical disjunction, logical or, logical addition, or inclusive disjunction) is a logical connective typically notated
Logical_disjunction
Metaphysics concept covering the divide between two types of entities
union of the abstract work of the understanding and the concrete input of sensation." Georg Wilhelm Friedrich Hegel: The Science of Logic, Cambridge University
Abstract_and_concrete
Branch of logic
Propositional logic is a branch of classical logic. It is also called statement logic, sentential calculus, propositional calculus, sentential logic, or sometimes
Propositional_logic
Establishment of a theorem using inference from the axioms
In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (known as well-formed formulas when relating to formal language)
Formal_proof
Syntactically correct logical formula
In mathematical logic, propositional logic, and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence
Well-formed_formula
Impossible task in computing
Church and Alan Turing in 1936. By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced using
Entscheidungsproblem
Formal systems of logic that significantly differ from standard logical systems
theory of abstract algebraic logic has also provided means to classify logics, with most results having been obtained for propositional logics. The current
Non-classical_logic
Rules used for constructing, or transforming the symbols and words of a language
In logic, syntax is an arrangement of well-structured entities in the formal languages or formal systems that express something. Syntax is concerned with
Syntax_(logic)
Mathematical logic concept
In mathematical logic, an atomic formula (also known as an atom or a prime formula) is a formula with no deeper propositional structure, that is, a formula
Atomic_formula
Branch of metaphysics regarding abstract objects
to a Theory of Abstract Objects (Thesis). UMass Amherst. doi:10.7275/f32y-fm90. hdl:20.500.14394/12282. Dale Jacquette, Meinongian Logic: The Semantics
Abstract_object_theory
Symbol representing a mathematical concept
In formal systems particularly mathematical logic, a function symbol is a non-logical symbol which represents a function or mapping on the domain of discourse
Function_symbol
Many-valued logic in which truth values comprise a continuous range
In logic, an infinite-valued logic (or real-valued logic or infinitely-many-valued logic) is a many-valued logic in which truth values comprise a continuous
Infinite-valued_logic
Process of generalization
particular ball. In a type–token distinction, a type (e.g., a 'ball') is more abstract than its tokens (e.g., 'that leather soccer ball'). Thinking in abstractions
Abstraction
Mapping of mathematical formulas to a particular meaning
structures are the objects used to define the semantics of first-order logic, cf. also Tarski's theory of truth or Tarskian semantics. For a given theory
Structure (mathematical logic)
Structure_(mathematical_logic)
Set that is not a finite set
Stanford University, retrieved 2019-11-30 Boolos, George (1998). Logic, Logic, and Logic (illustrated ed.). Harvard University Press. p. 262. ISBN 978-0-674-53766-8
Infinite_set
Branch of mathematics that studies sets
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any
Set_theory
Symbol with a fixed meaning in logic
In logic, a logical constant or constant symbol of a language L {\displaystyle {\mathcal {L}}} is a symbol that has the same semantic value under every
Logical_constant
Fragment of first-order logic
In logic, the monadic predicate calculus (also called monadic first-order logic) is the fragment of first-order logic (also called predicate calculus)
Monadic_predicate_calculus
Function, homomorphism, or morphism
Related Abstract logic Algebraic logic Automated theorem proving Category theory Concrete/Abstract category Category of sets History of logic History
Map_(mathematics)
School of thought in philosophy of mathematics
is an extension of logic, some or all of mathematics is reducible to logic, or some or all of mathematics may be modelled in logic. Bertrand Russell and
Logicism
Statement that is taken to be true
well-established, that it is accepted without controversy or question. In modern logic, an axiom is a premise or starting point for reasoning. In mathematics,
Axiom
Concept in logic
Substitution. Logic Journal of the IGPL, 12, 111–124. Curry, H. B. (1952) On the definition of substitution, replacement and allied notions in an abstract formal
Substitution_(logic)
Method of designing specialized integrated circuits
very-large-scale integration (VLSI) layout is encapsulated into an abstract logic representation (such as a NAND gate). Cell-based methodology – the general
Standard_cell
Rule defining the correct structure of expressions in formal grammar
In mathematical logic, formation rules are rules for describing well-formed words over the alphabet of a formal language. These rules only address the
Formation_rule
Variable that can either be true or false
building-blocks of propositional formulas, used in propositional logic and higher-order logics. Formulas in logic are typically built up recursively from some propositional
Propositional_variable
exist” abstractly). No “non-constructive” proofs are allowed (like the classic proof by contradiction without a witness). The main constructive logics are
Constructive_logic
Branch of mathematics
Corry, Leo (January 2000). "The origins of the definition of abstract rings". Modern Logic. 8 (1–2): 5–27. ISSN 1047-5982. Kleiner 2007, pp. 58–59. Kimberling
Abstract_algebra
of inference governing the logic of predicates Propositional calculus, specifies the rules of inference governing the logic of propositions Modal μ-calculus
List_of_formal_systems
Value indicating the relation of a proposition to truth
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical
Truth_value
Token in a mathematical or logical formula
A formal symbol is a fundamental concept in logic, tokens of which may be marks or a configuration of marks which form a particular pattern.[citation
Symbol_(formal)
In abstract algebraic logic, a branch of mathematical logic, the Leibniz operator is a tool used to classify deductive systems, which have a precise technical
Leibniz_operator
Mathematical-logic system
In mathematical logic, the lambda calculus (also written as λ-calculus) is a formal system for expressing computation based on function abstraction and
Lambda_calculus
Limitative results in mathematical logic
Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Axiom of set theory
An Invitation to Abstract Algebra. CRC Press. ISBN 978-1-000-51633-3. Rosenbloom, Paul C. (2005). The Elements of Mathematical Logic. Courier Dover Publications
Axiom_of_choice
Depth of nesting of quantifiers in a formula
In mathematical logic, the quantifier rank of a formula is the depth of nesting of its quantifiers. It plays an essential role in model theory. The quantifier
Quantifier_rank
Type of mathematical variable
In mathematical logic, a predicate variable is a predicate letter which functions as a "placeholder" for a relation (between terms), but which has not
Predicate_variable
Infinite set that is not countable
Injective function Enderton, Herbert (2001). A Mathematical Introduction to Logic (2nd ed.). Hardcourt/Academic Press. p. 9. ISBN 978-0-12-238452-3. Weisstein
Uncountable_set
Infinite cardinal number
MR 0553111. Miller, Jeff. "Earliest uses of symbols of set theory and logic". jeff560.tripod.com. Retrieved 2016-05-05; who quotes Dauben, Joseph Warren
Aleph_number
Approach to static program analysis
Abstract Interpretation" (PDF). In Bruynooghe, Maurice; Wirsing, Martin (eds.). Proc. 4th Int. Symp. on Programming Language Implementation and Logic
Abstract_interpretation
Collection of mathematical objects
Fuzzy set – Sets whose elements have degrees of membership Mathematical logic – Subfield of mathematics Mereology – Study of parts and the wholes they
Set_(mathematics)
Existence and cardinality of models of logical theories
In mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf
Löwenheim–Skolem_theorem
Logical incompatibility between two or more propositions
In traditional logic, a contradiction involves a proposition conflicting either with itself or established fact. It is often used as a tool to detect
Contradiction
Existence of values making formula true
In mathematical logic, a formula is satisfiable if it is true under some assignment of values to its variables. For example, the formula x + 3 = y {\displaystyle
Satisfiability
Form of second-order logic
In mathematical logic, monadic second-order logic (MSO) is the fragment of second-order logic where the second-order quantification is limited to quantification
Monadic_second-order_logic
Relationship where one statement follows from another
penguin}. Abstract algebraic logic Ampheck Boolean algebra (logic) Boolean domain Boolean function Boolean logic Causality Deductive reasoning Logic gate Logical
Logical_consequence
Term in logic and deductive reasoning
In logic, soundness can refer to either a property of arguments or a property of formal deductive systems. An argument is sound if (and only if) it is
Soundness
Term in mathematical logic
In mathematical logic, independence is the unprovability of some specific sentence from some specific set of other sentences. The sentences in this set
Independence (mathematical logic)
Independence_(mathematical_logic)
Concept in mathematical logic
Related Abstract logic Algebraic logic Automated theorem proving Category theory Concrete/Abstract category Category of sets History of logic History
Hereditary_set
Yes-or-no question that cannot ever be solved by a computer
first-order logic statements about natural numbers must be false. Undecidable problems can be related to different topics, such as logic, abstract machines
Undecidable_problem
Components of a mathematical or logical formula
In mathematical logic, a term is an arrangement of dependent/bound symbols that denotes a mathematical object within an expression/formula. In particular
Term_(logic)
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