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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
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
First-order_logic
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
Formal system of logic
In mathematics and logic, a higher-order logic (abbreviated HOL) is a form of logic that is distinguished from first-order logic by additional quantifiers
Higher-order_logic
Type of propositional logic
A second-order propositional logic is a propositional logic extended with quantification over propositions. A special case are the logics that allow second-order
Second-order propositional logic
Second-order_propositional_logic
Method of deriving conclusions
discourse. An important difference between first-order and second-order logic is that second-order logic is incomplete, meaning that it is not possible
Rule_of_inference
Branch of logic
zeroth-order logic. Sometimes, it is called first-order propositional logic to contrast it with System F, but it is distinct from first-order logic. It deals
Propositional_logic
Topics referred to by the same term
derivative is the second Second-order logic, an extension of predicate logic Second-order perturbation, in perturbation theory Second-order cybernetics, the
Second-order
Subfield of mathematics
classical logics such as second-order logic or infinitary logic are also studied, along with Non-classical logics such as intuitionistic logic. First-order logic
Mathematical_logic
Branch of mathematical logic
languages expressible by sentences of existential second-order logic; that is, second-order logic excluding universal quantification over relations,
Descriptive_complexity_theory
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
Mathematical use of "for all" and "there exists"
\exists } . Other quantifiers are only definable within second-order logic or higher-order logics. Quantifiers have been generalized beginning with the
Quantifier_(logic)
Book on the philosophy of mathematics
the definition of conservativeness and Field's use of metalogic and second-order logic. Following the release of the book, other philosophers worked to extend
Science_Without_Numbers
Non-contradiction of a theory
(1924), von Neumann (1927) and Herbrand (1931). Stronger logics, such as second-order logic, are not complete. A consistency proof is a mathematical proof
Consistency
On linear-time algorithms for graph logic
the statement that every graph property definable in the monadic second-order logic of graphs can be decided in linear time on graphs of bounded treewidth
Courcelle's_theorem
Set of sentences in a formal language
first-order logic, the most important case, it follows from the completeness theorem that the two meanings coincide. In other logics, such as second-order logic
Theory_(mathematical_logic)
Mathematical system
In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative
Second-order_arithmetic
Application of logical methods to philosophical problems
in first-order logic. But they can be expressed in second-order logic with only a few axioms. But despite this advantage, first-order logic is still much
Philosophical_logic
Index of articles associated with the same name
(graph theory) In logic, model theory and type theory: Zeroth-order logic First-order logic Second-order logic Higher-order logic Order (journal), an academic
Order_(mathematics)
Book on mathematical logic
extensions into a specific form of logic, many-sorted logic. Beyond many-sorted logic, its topics include second-order logic (including its incompleteness
Extensions of First Order Logic
Extensions_of_First_Order_Logic
Extension of first-order logic with atoms expressing variable dependencies
Dependence logic is a logical formalism, created by Jouko Väänänen, which adds dependence atoms to the language of first-order logic. A dependence atom
Dependence_logic
Study of the scope and nature of logic
of logics in contrast to one universally true logic. These logics can be divided into classical logic, usually identified with first-order logic, extended
Philosophy_of_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)
Programming paradigm based on formal logic
Logic programming is a programming, database, and knowledge representation paradigm based on formal logic. A logic program is a set of sentences in logical
Logic_programming
does not hold for monadic second order logic. As pointed out by Yuri Gurevich, zero-one law was proven for first-order logic by Yu. V. Glebskii, D. I.
Zero–one_law_(logic)
Theorem that every set can be well-ordered
Zorn's lemma.) In second-order logic, however, the well-ordering theorem is strictly stronger than the axiom of choice: from the well-ordering theorem one may
Well-ordering_theorem
Fundamental theorem in mathematical logic
theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. The completeness theorem
Gödel's_completeness_theorem
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
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
Extension of classical first-order logic
Independence-friendly logic (IF logic; proposed by Jaakko Hintikka and Gabriel Sandu [fr] in 1989) is an extension of classical first-order logic (FOL) by means
Independence-friendly_logic
Mapping of mathematical formulas to a particular meaning
view, 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
Structure (mathematical logic)
Structure_(mathematical_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)
Along with first-order guarded logic objects, there are objects of second-order guarded logic. It is known as Guarded Second-Order Logic and denoted GSO
Guarded_logic
Logical formulation of graph properties
first-order logic of graphs concerns sentences in which the variables and predicates concern individual vertices and edges of a graph, while monadic second-order
Logic_of_graphs
Existence of values making formula true
respect to a fixed logic defining the syntax of allowed symbols, such as first-order logic, second-order logic or propositional logic. Rather than being
Satisfiability
Philosophical view explaining systems in terms of smaller parts
Notre Dame Journal of Formal Logic. 34 (4): 539–563. doi:10.1305/ndjfl/1093633905. Väänänen, J. (2001). "Second-Order Logic and Foundations of Mathematics"
Reductionism
Claimed as largest named number
being well-defined, because any axiomatization of the language of second-order logic will have non-isomorphic models, under which Rayo's number could correspond
Rayo's_number
Topics referred to by the same term
fantasy massively multiplayer online role-playing game Existential second-order logic ESO (motorcycles) Eso (town), Orhionmwon, Edo State, Nigeria European
ESO_(disambiguation)
Class of languages studied in formal language theory in computer science
languages are precisely the ones definable in a particular monadic second-order logic called S1S. Wolfgang Thomas, "Automata on infinite objects." In Jan
Omega-regular_language
Existential second order logic captures NP
states that the set of all properties expressible in existential second-order logic is precisely the complexity class NP. It was proven by Ronald Fagin
Fagin's_theorem
Axioms for the natural numbers
respective functions and relations are constructed in set theory or second-order logic, and can be shown to be unique using the Peano axioms. Addition is
Peano_axioms
Symbol representing a property or relation in logic
In the semantics of logic, predicates are interpreted as relations. For instance, in a standard semantics for first-order logic, the formula R ( a ,
Predicate_(logic)
Number denoting a graph's closeness to a tree
logic of graphs using monadic second order logic, then it can be solved in linear time on graphs with bounded treewidth. Monadic second order logic is
Treewidth
Proof technique in model theory
enough to characterise definability in monadic second-order logic. An analogous game for modal logic is the bisimulation game. The main idea behind the
Ehrenfeucht–Fraïssé_game
Assignment of meaning to the symbols of a formal language
explicitly included in first-order theories when equality is treated this way. This second approach is sometimes called first order logic with equality, but many
Interpretation_(logic)
Form of mathematical proof
is a second-order quantifier, which means that this axiom is stated in second-order logic. Axiomatizing arithmetic induction in first-order logic requires
Mathematical_induction
Spanish mathematician (born 1950)
segundo orden [General systems of second-order logic], was supervised by Jesús Mosterín. She is a professor of logic and the philosophy of science at the
María_Manzano
Existence and cardinality of models of logical theories
characterize first-order logic. In general, the Löwenheim–Skolem theorem does not hold in stronger logics such as second-order logic. In its general form
Löwenheim–Skolem_theorem
comprehension schema This formula in second-order logic: (∃x)Φ → (∃Y)(∀x)(Yx ↔ Φ). Aristotelian logic The traditional logic developed by Aristotle, based on
Glossary_of_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
Type of formal logic
Modal logic is a kind of logic used to represent statements about necessity and possibility. In philosophy and related fields it is used as a tool for
Modal_logic
German philosopher, logician, and mathematician (1848–1925)
"Frege's Logic, Theorem, and Foundations for Arithmetic". Frege's logic, now known as second-order logic, can be weakened to so-called predicative second-order
Gottlob_Frege
Inference rule in logic, proof theory, and automated theorem proving
theorem-proving technique for sentences in propositional logic and first-order logic. For propositional logic, systematically applying the resolution rule acts
Resolution_(logic)
Branch of logic
Gödel's completeness theorem, and the method of ultraproducts for first-order logic (FO). These invalidities all follow from Trakhtenbrot's theorem. While
Finite_model_theory
between first-order logic and second-order logic. They are being used as a basis for Hintikka's and Gabriel Sandu's independence-friendly logic. The simplest
Branching_quantifier
Formal language theorem
regular if and only if it can be defined by a formula in monadic second-order logic (MSO). The theorem is due to Julius Richard Büchi, Calvin Elgot, and
Büchi–Elgot–Trakhtenbrot theorem
Büchi–Elgot–Trakhtenbrot_theorem
Generalization of depth-first search trees
planar graph. A characterization of Trémaux trees in the monadic second-order logic of graphs allows graph properties involving orientations to be recognized
Trémaux_tree
Topics referred to by the same term
programming Q&A site Sony's mobile phones in Japan SO (complexity), second-order logic in descriptive complexity Special orthogonal group, a subset of an
SO
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
Complexity class used to classify decision problems
corresponds precisely to the set of languages definable by existential second-order logic (Fagin's theorem). NP can be seen as a very simple type of interactive
NP_(complexity)
Metatheorem
that states that the Peano axioms of arithmetic can be derived in second-order logic from Hume's principle. It was first proven, informally, by Gottlob
Frege's_theorem
Kind of proof calculus
work on natural deduction, and included applications for modal and second-order logic. In natural deduction, a proposition is deduced from a collection
Natural_deduction
Topics referred to by the same term
Leadership, a business degree Microsoft Online Services Monadic second-order logic, a form of logic in which one can quantify over sets Msol or solar mass, also
MSOL
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
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
Theorem in formal logic
is the conjecture of Gaisi Takeuti that a sequent formalisation of second-order logic has cut-elimination (Takeuti 1953). It was settled positively: By
Takeuti's_conjecture
American scientist (1839–1914)
contributions to logic, such as theories of relations and quantification. C. I. Lewis wrote, "The contributions of C. S. Peirce to symbolic logic are more numerous
Charles_Sanders_Peirce
Class of computational complexity
complexity theory is that it is the set of problems expressible in second-order logic with the addition of a transitive closure operator. A full transitive
PSPACE
Philosophical concept
what is traditionally called the logic of second intentions, or what is handled very roughly by second order logic in contemporary parlance, and continuing
Categories_(Peirce)
Paradox in set theory
strong higher-order logic, while Zermelo employed second-order logic, and ZFC can also be given a first-order formulation. The first-order 'description'
Russell's_paradox
frame Predicate logic First-order logic Infinitary logic Many-sorted logic Higher-order logic Lindström quantifier Second-order logic Soundness theorem
List of mathematical logic topics
List_of_mathematical_logic_topics
Theorem in mathematical logic
mathematical logic, Lindström's theorem (named after Swedish logician Per Lindström, who published it in 1969) states that first-order logic is the strongest
Lindström's_theorem
Subfield of automated reasoning and mathematical logic
revised second edition in 1927. Russell and Whitehead thought they could derive all mathematical truth using axioms and inference rules of formal logic, in
Automated_theorem_proving
1999 book by Neil Immerman
chapters, roughly grouped into five chapters on first-order logic, three on second-order logic, and seven independent chapters on advanced topics. The
Descriptive_Complexity
American philosopher and logician (1940–1996)
Boolos argued that if one reads the second-order variables in monadic second-order logic plurally, then second-order logic can be interpreted as having no
George_Boolos
Japanese mathematician (1926–2017)
Takeuti's conjecture speculates that a sequent formalisation of second-order logic has cut-elimination. He is also known for his work on ordinal diagrams
Gaisi_Takeuti
Mathematical logic concept
In mathematical logic and philosophy, Skolem's paradox is the apparent contradiction that a countable model of first-order set theory could contain an
Skolem's_paradox
Basic framework of mathematics
and this means that Peano arithmetic is what is presently called a Second-order logic. This was not well understood at that times, but the fact that infinity
Foundations_of_mathematics
Mathematical model for deduction or proof systems
which, in order to avoid confusion, are usually called metatheorems. A logical system is a deductive system (most commonly first order logic) together
Formal_system
Template that specifies one or more axioms
semantics for second-order logic. Analogously, some first-order set-theoretic schemata can be represented by quantifying over classes or higher-order objects
Axiom_schema
elimination. These logics have fewer inference rules than classical logic. On the other hand, classical logic was a first-order logic, which means roughly
Philosophy_of_mathematics
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
Unsolved problem in computer science
logics over finite structures. By Fagin's theorem, NP is exactly the class of properties of finite structures expressible in existential second-order
P_versus_NP_problem
Theorem in mathematical logic
In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model
Compactness_theorem
Theories in mathematical logic
In first-order logic, a first-order theory is given by a set of axioms in some language. This entry lists some of the more common examples used in model
List_of_first-order_theories
Translation of a text into a logical system
the translation of the English sentence "some men are bald" into first-order logic as ∃ x ( M ( x ) ∧ B ( x ) ) {\displaystyle \exists x(M(x)\land B(x))}
Logic_translation
Computational problem with high complexity
the Weak Monadic Second-Order Logic of One Successor (WS1S) Satisfiability of W. V. O. Quine's fluted fragment of first-order logic β-convertibility of
Nonelementary_problem
Term in mathematical logic
relational symbols, then ψ can be regarded as a sentence in existential second-order logic (ESOL) quantified over the relations, over the empty vocabulary. A
Spectrum_of_a_sentence
Field of computer science
monadic second-order logic and state machines in the form of digital circuits. Program synthesis Model checking Church, Alonzo (1962). "Logic, arithmetic
Reactive_synthesis
American mathematician
of 1939, Henkin took a second course of Logic with Nagel, in which formal systems of propositional logic and first-order logic were addressed. These constituted
Leon_Henkin
Axiom set used in first-order logic
specifically for that portion of Euclidean geometry that is formulable in first-order logic with identity (i.e. is formulable as an elementary theory). As such,
Tarski's_axioms
Statement that is taken to be true
requires the use of second-order logic. The Löwenheim–Skolem theorems tell us that if we restrict ourselves to first-order logic, any axiom system for
Axiom
Computer science field
generally implies the tractability of model checking for monadic second-order logic), bounding the degree of every domain element, and more general conditions
Model_checking
Mathematical notion of infinitesimal difference
the usual real numbers, but the completeness axiom (which involves second-order logic) does not hold. Nevertheless, this suffices to develop an elementary
Differential_(mathematics)
Number of arguments required by a function
In logic, mathematics, and computer science, arity (/ˈærɪti/ ) is the number of arguments or operands taken by a function, operation or relation. In mathematics
Arity
Standard system of axiomatic set theory
constructed in first-order logic. Some formulations of first-order logic include identity; others do not. If the variety of first-order logic in which one is
Zermelo–Fraenkel_set_theory
Reasoning about equations with free variables
logic, algebraic logic is the reasoning obtained by manipulating equations with free variables. What is now usually called classical algebraic logic focuses
Algebraic_logic
Overview of and topical guide to logic
Classical logic Computability logic Deontic logic Dependence logic Description logic Deviant logic Doxastic logic Epistemic logic First-order logic Formal
Outline_of_logic
System for representing and reasoning about time
In logic, a temporal logic is any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time (for example
Temporal_logic
3-volume treatise on mathematics, 1910–1913
stance to a fully extensional stance also restricts predicate logic to the second order, i.e. functions of functions: "We can decide that mathematics
Principia_Mathematica
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