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Study of the properties of logical systems
Metalogic is the metatheory of logic. Whereas logic studies how logical systems can be used to construct valid and sound arguments, metalogic studies
Metalogic
Theory whose subject matter is itself a theory
arguments, metalogic studies the properties of logical systems. Logic concerns the truths that may be derived using a logical system; metalogic concerns
Metatheory
Programming paradigm based on formal logic
more general use of a metalogic or metalanguage to describe and reason about another language, called the object language. Metalogic programming allows object-level
Logic_programming
Common metalanguage and metalogic symbols Symbol Unicode value (hexadecimal) HTML codes LaTeX symbol Logic Name Read as Category Explanation Examples
List_of_logic_symbols
Concrete manifestation of an element (type) in computer science
In computer science, an instance or token (from metalogic and metamathematics) is a specific occurrence of a software element that is based on a type
Instance_(computer_science)
Overview of and topical guide to logic
Intuitionistic logic Linear logic Many-valued logic Mathematical logic Metalogic Minimal logic Modal logic Non-Aristotelian logic Non-classical logic Noncommutative
Outline_of_logic
Term in philosophy
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Antinomy
Statement that attaches a meaning to a term
definitions are usually introduced using extension by definition (so using a metalogic). On the other hand, lambda-calculi are a kind of logic where the definitions
Definition
Term in logic and deductive reasoning
for introductory deductive reasoning contexts and the latter arises in metalogic and mathematical logic. In deductive reasoning, a sound argument is an
Soundness
Distinguishing objects and classes of objects
The distinction is important in disciplines such as logic, linguistics, metalogic, typography, and computer programming. The type–token distinction separates
Type–token_distinction
Study of correct reasoning
application usually happens in the form of extended or deviant logical systems. Metalogic is the field of inquiry studying the properties of formal logical systems
Logic
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)
Subfield of computer science and logic
science, in particular in knowledge representation and reasoning and metalogic, the area of automated reasoning is dedicated to understanding different
Automated_reasoning
Term used to model separate circumstances that cannot exist together
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Impossible_world
One or more words used to refer to something
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Name
Terms to describe a conditional relationship between two statements
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Necessity_and_sufficiency
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Schrödinger_logic
Relationship between objects
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Reference
Type of logic regarding reasoning about beliefs
counterpart of Gödel's incompleteness theorem of metalogic, as well as Löb's theorem, and other metalogical results in terms of belief. To demonstrate the
Doxastic_logic
Type of logical system
proving in first-order logic. First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the
First-order_logic
Mathematical model for deduction or proof systems
Formal Semantics and Logic (PDF). Nousoul Digital Publishers. p. 12. Metalogic can in turn be roughly divided into two parts: proof theory and formal
Formal_system
Axiomatic set theories based on the principles of mathematical constructivism
constructive arithmetic theories. These are features of a fixed theory which metalogically relate judgements of propositions provable in the theory. Particularly
Constructive_set_theory
Problem-solving procedures with certain characteristics
In metalogic, mathematical logic, and computability theory, an effective method or effective procedure is a finite-time, deterministic procedure for solving
Effective_method
British professor, philosopher, and logician (1925-2000)
work titled Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, published in 1971. Hunter, Geoffrey (1971). "Metalogic: An Introduction
Geoffrey_Hunter_(logician)
Form of reasoning
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Deductive_reasoning
Study of mathematics itself
Putnam, Gregory Chaitin, Alfred Tarski, Paul Cohen and Kurt Gödel. Today, metalogic and metamathematics broadly overlap, and both have been substantially
Metamathematics
Number measuring the chance an event occurs
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Probability
Relationship in which one statement follows from another
(1952), Van Nostrand Publishing. p.88. Hunter, Geoffrey (1996) [1971]. Metalogic: An Introduction to the Metatheory of Standard First-Order Logic. University
Logical_consequence
Formal statement in logic
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Strict_conditional
Formal systems of logic that significantly differ from standard logical systems
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Non-classical_logic
Logically self-contradictory statement
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Paradox
Various systems of symbolic logic
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical
Intuitionistic_logic
Reasoning for mathematical statements
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Mathematical_proof
Problem of determining if a Boolean formula could be made true
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Boolean satisfiability problem
Boolean_satisfiability_problem
Book on the philosophy of mathematics
raised concerning the definition of conservativeness and Field's use of metalogic and second-order logic. Following the release of the book, other philosophers
Science_Without_Numbers
Analysis of facts to form a judgment
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Critical_thinking
Two types of knowledge, justification, or argument
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
A_priori_and_a_posteriori
Concept of focusing on form over concept
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Formalism_(philosophy)
Assumed context surrounding an utterance
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Presupposition
Metatheorem
In metalogic and metamathematics, Frege's theorem is a metatheorem that states that the Peano axioms of arithmetic can be derived in second-order logic
Frege's_theorem
Bearer of truth values
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Proposition
List of statements that appear to contradict themselves
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
List_of_paradoxes
System of mathematical set theory
Kripke–Platek set theory (KP) is an axiomatic set theory developed by Saul Kripke and Richard Platek. It is substantially weaker than Zermelo–Fraenkel
Kripke–Platek_set_theory
Translation of a text into a logical system
logic translation is specifically relevant for logic programming and metalogic. A major challenge in logic translation is determining the accuracy of
Logic_translation
Knowledge about knowledge
Metahistory, a book by Hayden White Meta-philosophy Meta-epistemology Metalogic Metamathematics Metaphysics Meta-ethics Meta-ontology Metatheory Metadata
Metaknowledge
Academic discipline
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Logic_in_computer_science
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
List_of_set_theory_topics
Logical connective
{\displaystyle \equiv } is used in reasoning about those logic formulas (e.g., in metalogic). In Łukasiewicz's Polish notation, it is the prefix symbol E {\displaystyle
If_and_only_if
Axiom
realisable in just H A {\displaystyle {\mathsf {HA}}} . For the next metalogical theorem, recall that P A {\displaystyle {\mathsf {PA}}} is non-constructive
Church's thesis (constructive mathematics)
Church's_thesis_(constructive_mathematics)
Rules used for constructing, or transforming the symbols and words of a language
Well-formed formula Dictionary Definition Hunter, Geoffrey (1996) [1971]. Metalogic: An Introduction to the Metatheory of Standard First-Order Logic. University
Syntax_(logic)
Conformity to reality
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Truth
Text for clarification; one of four rhetorical modes
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Description
Type of nonargument
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Simple non-inferential passage
Simple_non-inferential_passage
Type of diagrammatic notation for logic
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Randolph_diagram
Idea that knowledge comes only/mainly from sensory experience
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Empiricism
Variable that stores data about other variables or program structure
doi:10.2178/bsl/1146620060. S2CID 6909703. Hunter, Geoffrey (1996) [1971]. Metalogic: An Introduction to the Metatheory of Standard First-Order Logic. University
Metavariable
Concept of philosophy and logic used to express modal claims
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Possible_world
Class of formal logics
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Classical_logic
Inference seeking the simplest and most likely explanation
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Abductive_reasoning
Study of the scope and nature of logic
philosophy of logic and philosophical logic differently or not at all. Metalogic is closely related to the philosophy of logic as the discipline investigating
Philosophy_of_logic
American philosopher and psychologist
relating to the metalogic of reference 1989–2011, studies relating to clinical psychology 2012- , further work relating to the metalogic of reference Beginning
Steven_James_Bartlett
Rule of logical inference
{\displaystyle P\to Q,\neg Q\vdash \neg P} where ⊢ {\displaystyle \vdash } is a metalogical symbol meaning that ¬ P {\displaystyle \neg P} is a syntactic consequence
Modus_tollens
In mathematical logic (a subtopic within the field of formal logic), two formulae are equisatisfiable if the first formula is satisfiable whenever the
Equisatisfiability
German polymath (1646–1716)
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Gottfried_Wilhelm_Leibniz
Symbolic logic system
argument that tails occurred. If the intuitionistic logic system is metalogically assumed consistent, the syllogism may be read as saying that a constructive
Minimal_logic
Branch of mathematics that studies sets
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Set_theory
Logical operator in modal logic
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Modal_operator
Symbol in mathematical logic
Q {\displaystyle P\vdash Q} can be read as: From P, I know that Q In metalogic, the study of formal languages; the turnstile represents syntactic consequence
Turnstile_(symbol)
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
List_of_fallacies
Type of argument
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Argumentation_scheme
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
Logic founded on unproven premises
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Begging_the_question
Branch of logic
axiomatic systems for propositional logic. For more examples, as well as metalogical theorems that are specific to such axiomatic systems (such as their completeness
Propositional_logic
Branch of logic
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Informal_logic
Statement supporting a conclusion
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Premise
Sufficient evidence/argument for truth
University Press, ISBN 0-521-77911-1. Hunter, Geoffrey (1996) [1971]. Metalogic: An Introduction to the Metatheory of Standard First-Order Logic. University
Proof_(truth)
Logical rule of inference
{\displaystyle P\lor Q,\lnot P\vdash Q} where ⊢ {\displaystyle \vdash } is a metalogical symbol meaning that Q {\displaystyle Q} is a syntactic consequence of
Disjunctive_syllogism
Capacity for consciously making sense of things
reasoning involves explicit conceptual knowledge regarding inference (metalogical knowledge) and metacognitive awareness of, and control over, inference
Reason
In mathematics, a statement that has been proven
Mathematical Logic. Oxford University Press. Hunter, Geoffrey (1996) [1971]. Metalogic: An Introduction to the Metatheory of Standard First-Order Logic. University
Theorem
Term in linguistics
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Autonomy_of_syntax
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
Steps in reasoning
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Inference
In logic, a statement which is always true
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Tautology_(logic)
Rule of replacement in propositional logic
Q\Leftrightarrow \neg P\lor Q,} where " ⇔ {\displaystyle \Leftrightarrow } " is a metalogical symbol representing "can be replaced in a proof with", P and Q are any
Material implication (rule of inference)
Material_implication_(rule_of_inference)
NP-complete variant of the Boolean satisfiability problem
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
1-in-3-SAT
Subfield of mathematics
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Mathematical_logic
Method in formal logic
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Condensed_detachment
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
List of mathematical logic topics
List_of_mathematical_logic_topics
Rule of replacement in propositional logic
R)\Leftrightarrow (P\to (Q\to R))} Where " ⇔ {\displaystyle \Leftrightarrow } " is a metalogical symbol representing "can be replaced in a proof with." In strict terminology
Exportation_(logic)
Book by Walter Benjamin
of History, Metaphysics, Post-metaphysics, Theology, After-theology, Metalogic, History, Kabbalah, Nihilism, Contra-Nihilism, Epistemology, Criticism
The Origin of German Tragic Drama
The_Origin_of_German_Tragic_Drama
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
XOR-SAT
Framework for studying interactive computational tasks through logic
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Computability_logic
Transformations induced by a mathematical group
this category is a Grothendieck topos (in fact, assuming a classical metalogic, this topos will even be Boolean). We can also consider actions of monoids
Group_action
Rule of logical inference
→ Q are statements (or propositions) in a formal language and ⊢ is a metalogical symbol meaning that Q is a syntactic consequence of P and P → Q in some
Modus_ponens
Formal study of linguistic meaning
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Formal semantics (natural language)
Formal_semantics_(natural_language)
Establishment of a theorem using inference from the axioms
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Formal_proof
Theorem in mathematical logic
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Lindström's_theorem
Precisely specified semantic version of a statement
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Logical_form
3-volume treatise on mathematics, 1910–1913
discussion LOGICISM at pp. 43–46. In his section 8.5.4 Groping towards metalogic Grattan-Guinness 2000:454ff discusses the American logicians' critical
Principia_Mathematica
Statement that is true regardless of the truth or falsity of its constituent propositions
Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions
Logical_truth
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