# math.GLOSSARY

## GLOSSARY

* [Abstract machine](#abstract-machine)
* [Abstraction](#abstraction)
* [Accident](#accident)
* [Affine type system](#affine-type-system)
* [Algebra](#algebra)
* [Analytic-synthetic distinction](#analytic-synthetic-distinction)
* [Argumentation theory](#argumentation-theory)
* [Axiology](#axiology)
* [Axiom](#axiom)
* [Church-Turing thesis](#church-turing-thesis)
* [Conjecture](#conjecture)
* [Dedekindian](#dedekindian)
* [Defeasible inference](#defeasible-inference)
* [Defeasible reasoning](#defeasible-reasoning)
* [Entscheidungsproblem](#entscheidungsproblem)
* [Equiconsistent](#equiconsistent)
* [Essence](#essence)
* [Extensionality](#extensionality)
* [Fallacy](#fallacy)
* [First-order logic](#first-order-logic)
* [Formal logic](#formal-logic)
* [Fuzzy logic](#fuzzy-logic)
* [Generalization](#generalization)
* [iff](#iff)
* [Informal logic](#informal-logic)
* [Laws of Thought](#laws-of-thought)
* [Linear Type System](#linear-type-system)
* [Logical Connectives](#logical-connectives)
* [Logical Consequence](#logical-consequence)
* [Logical Form](#logical-form)
* [Logical Framework](#logical-framework)
* [Logical Truth](#logical-truth)
* [Mathematical logic](#mathematical-logic)
* [Mathematical object](#mathematical-object)
* [Metalogic](#metalogic)
* [Metamathematics](#metamathematics)
* [Metatheorem](#metatheorem)
* [Naïve theory](#naïve-theory)
* [Necessity and sufficiency](#necessity-and-sufficiency)
* [Non-classical logic](#non-classical-logic)
* [Noumenon](#noumenon)
* [Ordered type system](#ordered-type-system)
* [Phenomenon](#phenomenon)
* [Philosophical logic](#philosophical-logic)
* [Philosophy of logic](#philosophy-of-logic)
* [Predicate](#predicate)
* [Premise](#premise)
* [Primitive notion](#primitive-notion)
* [Proof by exhaustion](#proof-by-exhaustion)
* [Propositional logic](#propositional-logic)
* [Relevant type system](#relevant-type-system)
* [Second-order Logic](#second-order-logic)
* [Statement](#statement)
* [Strict Conditional](#strict-conditional)
* [Substitution](#substitution)
* [Substructural Type System](#substructural-type-system)
* [Syllogisms](#syllogisms)
* [Symbolic Logic](#symbolic-logic)
* [Syntax](#syntax)
* [Theorem](#theorem)
* [Truth Value](#truth-value)
* [Turing Completeness](#turing-completeness)
* [Turing Machine](#turing-machine)
* [Universal Turing machine](#universal-turing-machine)
* [Validity](#validity)
* [Weakening](#weakening)

### Abel Prize

Abel Prize, award granted annually for research in mathematics, in commemoration of the 19th-century Norwegian mathematician Niels Henrik Abel. The Niels Henrik Abel Memorial Fund was established in 2002, and it is administered by the Norwegian Ministry of Education and Research.

### Abel's test

In mathematical analysis, Abel's test is used for determining if an infinite series converges to some finite value. The test is named for the Norwegian mathematician Niels Henrik Abel (1802-29).

### Absolute value

Absolute value, Measure of the magnitude of a real number, complex number, or vector. Geometrically, the absolute value represents (absolute) displacement from the origin (or zero) and is therefore always nonnegative. If a real number a is positive or zero, its absolute value is itself.

### Abstract machine

An abstract machine is a theoretical model of a computer hardware or software system used in automata theory.

### Abstraction

Abstraction is the process in which the previously studied and well-understood details (of the studied object or concept) are going through the deliberate simplification as the process of understanding moves from the lower to higher levels, toward the totality. In math, abstraction is the process of extracting the underlying *essence* of a *mathematical concept*, then *generalizing* it so it has wider utility.

### Accident

In philosophy, *essence* is the property that makes an entity what it fundamentally is, which it has by necessity, without which it loses its identity. Essence is contrasted with **accident** - a property that the entity has contingently, without which it still retains its identity.

### Affine type system

Affine type systems allow exchange and weakening, but not contraction: every (occurrence of a) variable is used at most once. Affine types are a version of linear types allowing a resource to be unused. An affine resource can only be used once, while a linear resource must be used once.

### Algebra

1. Algebra, branch of mathematics in which arithmetical operations and formal manipulations are applied to abstract symbols rather than specific numbers.
2. An algebra (or algebraic structure) is a mathematical structure consisting of a carrier set that is endowed with additional structure(s) and axioms.

### Algebraic equation

Algebraic equation is a statement of equality between two expressions, formulated by applying algebraic operations (like add, sub, mul, div, pow, root, etc.) to a set of variables.

### Algebraic geometry

Algebraic geometry, study of the geometric properties of solutions to polynomial equations, including solutions in dimensions beyond three. Solutions in two dimensions are covered in plane analytic geometry, and solutions in two dimensions are covered in solid analytic geometry.

### Algebraic number

Algebraic number, real number for which there exists a polynomial equation with integer coefficients such that the given real number is a solution. Algebraic numbers include real, some irrational, and complex numbers of the form pi+q.

### Algebraic topology

Algebraic topology uses algebraic structures to study transformations of geometric objects. It uses functions (maps) to represent continuous transformations.

### Algorithm

Algorithm is a systematic procedure that produces, in a finite number of steps, the answer to a question (or the solution of a problem). The name derives from the Latin translation, "Algoritmi de numero Indorum", of the 9th-century Muslim mathematician al-Khwarizmi's treatise on arithmetic.

### Aliquot sum

An aliquot sum of an integer is the sum of its factors, excluding that integer.

### Analytic proposition

The truth value of analytic propositions depends on their meaning, unlike synthetic propositions', that depend on how their meaning relates to the world.

### Analytic-synthetic distinction

The analytic-synthetic distinction (which is a dichotomy) is a semantic distinction, used primarily in philosophy to distinguish propositions into two types, analytic and synthetic propositions. Analytic propositions are true by virtue of their meaning, while synthetic propositions are true by how their meaning relates to the world.

### Analysis

Analysis, a branch of mathematics that deals with continuous change and with certain general types of processes that have emerged from the study of continuous change, such as limits, differentiation, and integration.

### Analysis of algorithms

Analysis of algorithms is a CS discipline that aids in the development of effective programs. Analysis of algorithms provides proof of the correctness of algorithms, allows for the accurate prediction of program performance, and can be used as a measure of computational complexity.

### Analytic geometry

Analytic geometry uses algebraic symbolism and methods to represent and solve problems in geometry. The importance of analytic geometry is that it establishes a correspondence between geometric curves and algebraic equations.

### Analytical Engine

Analytical Engine is generally considered the first computer. It was designed and partly built by the English inventor Charles Babbage in the 19th century (he worked on it until his death in 1871).

### Argumentation theory

Argumentation theory is the interdisciplinary study of how conclusions can be reached through logical reasoning i.e. claims based, soundly or not, on premises. It includes the arts and sciences of civil debate, dialogue, conversation, and persuasion.

### Axiology

Axiology is the philosophical study of value. It is either the collective term for ethics and aesthetics, philosophical fields that depend crucially on notions of worth, or the foundation for these fields, and thus similar to value theory and meta-ethics.

### Axiom

An axiom or **postulate** is a statement that is taken to be true that serves as an initial premise or starting point for further reasoning and arguments. In classical logics, an axiom is an evident and well-established statement that needs no further proof. In modern logics, an axiom is a premise for reasoning.

## B

### Church-Turing thesis

In computability theory, the Church-Turing thesis is a hypothesis about the nature of computable functions. It states that a function on the natural numbers is computable by a man following an algorithm, ignoring resource limitations, iff it is computable by a Turing machine.

### Class

A class is a set of sets that can be unambiguously defined by a property all member sets have in common.

### Conjecture

An unproved statement that is believed true is called a conjecture. To be considered a conjecture, a statement must usually be proposed publicly, at which point the name of the proponent may be attached to the conjecture, as with Goldbach's conjecture. Other famous conjectures include the Collatz conjecture and the Riemann hypothesis. On the other hand, Fermat's Last Theorem has always been known by that name, even before it was proved; it was never known as "Fermat's conjecture".

### Dedekindian

Richard Dedekind (1831-1916) is known as the last universal matematician, having dabbled successfully in every branch of math known at his time. Because of this, this adjective is ideal for occasional use as a placeholder for a proper one, which you just can't seem to remember but strongly feel that the noun needs qualification.

### Defeasible inference

Defeasible inferences is a kind of inference in which reasoners draw tentative conclusions, reserving the right to retract their conclusions based on further evidence.

### Defeasible reasoning

Defeasible reasoning is rationally compelling but deductively invalid kind of reasoning.

### Entscheidungsproblem

"Decision problem" is a challenge posed by David Hilbert in 1928, that asks the ant, Chai Doong, what is his problem, esse. More precisely, it asks for an algorithm that takes as an input a statement of a FOL and determines whether the statement is universally valid. By the completeness theorem of FOL, a statement is universally valid iff it can be deduced from the axioms, so the decision problem can also be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic. In 1936, Alonzo Church and Alan Turing published independent papers showing that a general solution to the decision problem is impossible.

### Equiconsistent

Two theories are equiconsistent if the consistency of one theory implies the consistency of the other theory, and vice versa.

### Essence

In philosophy, **essence** is the property that makes an entity what it fundamentally is, which it has by necessity, without which it loses its identity. Essence is contrasted with *accident* - a property that the entity has contingently, without which it still retains its identity.

### Extensionality

Extensionality refers to principles that judge objects to be equal if they have the same external properties, as opposed to **intensionality**, which is concerned with whether the internal definitions of objects are the same. There are many predicates that are intensionaly different but extensionally identical. For example, the expressions, 2+4 and 2\*3, are extensionally equal (from outside), but intensionality different (from within).

### Fallacy

In reasoning to argue a claim, a fallacy is reasoning that is evaluated as logically incorrect and that undermines the logical validity of the argument and permits its recognition as unsound.

### First-order logic

While propositional logic deals with simple declarative propositions, first-order logic additionally covers predicates and quantification. Predicate logic uses quantified variables over objects and allows the use of sentences that contain variables, so rather than propositions such as "Socrates is a man" one can have expressions in the form "there exists X such that X is Socrates and X is a man" and "there exists" is a quantifier while "X" is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations. In first-order theories, predicates are often associated with sets.

### Formal logic

Formal logic is the study of inference with purely formal content. An inference possesses a purely formal content if it can be expressed as a particular application of a wholly abstract rule i.e. a rule that is not about any particular thing or property.

### Fuzzy logic

Fuzzy logic is a form of many-valued logic in which the truth values of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely false. By contrast, in Boolean logic, the truth values of variables may only be the integer values 0 or 1.

### Generalization

A generalization is the formulation of general concepts from specific instances by abstracting common properties. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteristics shared by those elements (thus creating a conceptual model). As such, they are the essential basis of all valid deductive inferences. The process of verification is necessary to determine whether a generalization holds true for any given situation.

## H

### iff

Bijection is a logic connective, sometimes read as "if and only if", used to mark two statements as logically equivalent, symbolically denoted as $$p \iff q$$. Two statements are logically equivalent if the former implies the latter, $$p \to q$$, and if the latter implies the former, $$q \to p$$.

### Independence

In mathematical logic, independence is the unprovability of a sentence from other sentences.

### Informal logic

Informal logic is the study of natural language arguments. The study of fallacies is an important branch of informal logic. Since much informal argument is not strictly speaking deductive, on some conceptions of logic, informal logic is not logic at all.

## J

## K

### Laws of Thought

Laws of thought are the 3 fundamental laws, *the law of identity*, *the law of non-contradiction* and *the law of excluded middle*, often considered as the basis of rational discourse itself. These rules have been known and accepted in logic for centuries, until the modern logicians placed them under detailed scrutiny, which resulted in inventions of new kinds of logic that exercised strict control regarding the three fundamental laws.

### Linear Type System

Linear type systems allow exchange, but neither weakening nor contraction: every variable is used exactly once. Linear types ensures that objects are used exactly once, allowing the system to safely deallocate an object after its use. Linear types can be used to model heap-based memory allocation.

### Logical Connectives

Logical connectives (logical operators) are symbols or words used to connect sentences in a grammatically valid way, such that the value of the compound sentence produced depends only on that of the original sentences and on the meaning of the connective. Common logical connectives are negation, conjunction, disjunction, biconditional, implication.

### Logical Consequence

Logical consequence (also **entailment**) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically follows from one or more statements. A valid logical argument is one in which the conclusion is entailed by the premises, because the conclusion is the consequence of the premises.

### Logical Form

A logical form of a syntactic expression is a precisely-specified semantic version of that expression in a formal system. Informally, the logical form attempts to formalize a possibly ambiguous statement into a statement with a precise, unambiguous logical interpretation with respect to a formal system.

The concept of logical form is central to logic: the validity of an argument is determined by its logical form, not by its content. A logical form of a syntactic expression is a precisely specified semantic version of that expression in a formal system. It is the formalization of possibly ambiguous statements from a natural language into, often symbolic, statements with precise unambiguous interpretation - their logical form.

### Logical Framework

Logical Framework provides a means to define a logic as a signature in a higher-order type theory in such a way that provability of a formula in the original logic reduces to a type inhabitation problem in the framework type theory.

### Logical Truth

Logical truth is one of the most fundamental concepts in logic, and there are different theories on its nature. A logical truth is a statement which is true, and remains true under all reinterpretations of its components other than its logical constants.

### Magnitude

In mathematics, magnitude is the size of a mathematical object, a property which determines whether the object is larger or smaller than other objects of the same kind. More formally, an object's magnitude is the displayed result of an ordering (or ranking) of the class of objects to which it belongs.

### Mathematical logic

Mathematical logic is an extension of symbolic logic into other areas, in particular to the study of model theory, proof theory, set theory, and computability (recursion) theory.

## Mathematical notation

Mathematical notation is a system of symbolic representations of mathematical objects and ideas. Mathematical notations are used in mathematics, the physical sciences, engineering, and economics. Mathematical notations include relatively simple symbolic representations, such as the numbers 0, 1 and 2; function symbols such as sin; operator symbols such as "+"; conceptual symbols such as lim and dy/dx; equations and variables; and complex diagrammatic notations such as Penrose graphical notation and Coxeter-Dynkin diagrams.

### Mathematical object

An abstract object that can be formally defined, and which one may do deductive reasoning and mathematical proofs with.

### Metalogic

Metalogic is the study of the metatheory of logic. Whereas logic studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems.

### Metamathematics

Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories.

### Metatheorem

A metatheorem is a statement about a formal system proven in a metalanguage. Unlike theorems proved within a given formal system, a metatheorem is proved within a metatheory, and may reference concepts that are present in the metatheory but not the object theory.

### Naïve theory

A theory qualified with the adjective "naïve" is an unformalized theory, presented using natural language.

### Necessity and sufficiency

In logic, necessity and sufficiency are terms used to describe an implicational relationship between statements. The assertion that one statement is a necessary and sufficient condition of another means that the former statement is true if and only if the latter is true.

### Non-classical logic

Non-classical logics (and sometimes alternative logics) are formal systems that differ in a significant way from standard logical systems such as propositional and predicate logic. There are several ways in which this is done, including by way of extensions, deviations, and variations.

### Noumenon

In metaphysics, the **noumenon** is a posited object or event that exists independently of human sense or perception. The term noumenon is generally used in relation to the term *phenomenon*, which refers to anything that can be apprehended by, or is an object of, the senses.

### Ordered type system

Ordered type systems discard exchange, contraction, and weakening: every variable is used exactly once in the order it was introduced. Ordered types can be used to model stack-based memory allocation (contrast with linear types which can be used to model heap-based memory allocation). Without the exchange property, an object may only be used when at the top of the modelled stack, after which it is popped off resulting in every variable being used exactly once in the order it was introduced.

### Phenomenon

In metaphysics, the *noumenon* is a posited object or event that exists independently of human sense or perception. The term noumenon is generally used in relation to the term **phenomenon**, which refers to anything that can be apprehended by, or is an object of, the senses.

### Philosophical logic

Philosophical logic refers to those areas of philosophy in which recognized methods of logic have traditionally been used to solve or advance the discussion of philosophical problems. Among these, Sybil Wolfram highlights the study of argument, meaning, and truth, while Colin McGinn presents identity, existence, predication, necessity and truth as the main topics of his book on the subject.Philosophical logic also addresses extensions and alternatives to traditional, "classical" logic known as "non-classical" logics.

### Philosophy of logic

Following the developments in formal logic with symbolic logic in the late nineteenth century and mathematical logic in the twentieth, topics traditionally treated by logic not being part of formal logic have tended to be termed either philosophy of logic or philosophical logic if no longer simply logic. Compared to the history of logic the demarcation between philosophy of logic and philosophical logic is of recent coinage and not always entirely clear.

### Predicate

A predicate takes an entity or entities in the domain of discourse as input while outputs are either True or False. Informally, a predicate is a statement that may be true or false depending on the values of its variables. For example, predicates can be used in set builder notation to indicate set membership: a predicate P(x) will be true or false, depending on whether x belongs to a set.

### Premise

A premise is a statement that an argument claims will induce or justify a conclusion. A premise is an assumption that something is true.

### Primitive notion

In mathematics, a primitive notion is an undefined concept, not defined in terms of previously defined concepts, to be taken for granted. It lacks a proof, and in that regard it's analogous to an axiom of a formal system (axioms don't require proof). Sometimes the primitive notions cannot be avoided because we need to start somewhere lest regress into downward spiral of definitions, forever defining concept in terms of previously defined concept, which also need definition in terms of previous ones, and so on, ad nauseum. Sometimes a concept just doesn't have a formal definition (e.g. "set").

### Proof by exhaustion

Proof by exhaustion (aka proof by cases, proof by case analysis, complete induction, brute force method) is a method of mathematical proof in which the statement to be proved is split into a finite number of cases or sets of equivalent cases and each type of case is checked to see if the proposition in question holds. This is a method of direct proof.

### Proper class

In von Neumann-Bernays-Gödel (NBG) set theory, proper class is an entity that is not a member of another entity.

## Propositions

Propositions are declarative sentences that have a truth value.

### Propositional logic

Propositional (sentential) logic is based on propositions and argument flow. Compound propositions are formed by connecting propositions by logical connectives. The propositions without logical connectives are called atomic propositions. Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or quantifiers. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logic is the foundation of first-order logic and higher-order logic.

## Q

### Relevant type system

Relevant type systems allow exchange and contraction, but not weakening: every variable is used at least once.

### Second-order Logic

Second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory. First-order logic quantifies only variables that range over individuals (elements of the domain of discourse), second-order logic additionally quantifies over relations.

### Statement

In logic, the term statement is variously understood to mean either: (a) a meaningful declarative sentence that is true or false, or (b) the assertion that is made by a true or false declarative sentence.In the latter case, a statement is distinct from a sentence in that a sentence is only one formulation of a statement, whereas there may be many other formulations expressing the same statement.

### Strict Conditional

In logic, a strict conditional (strict implication) is a conditional governed by a modal operator, that is, a logical connective of modal logic. It is logically equivalent to the material conditional of classical logic, combined with the necessity operator from modal logic.

### Substitution

Substitution is a fundamental concept in logic. A substitution is a syntactic transformation on formal expressions.

### Substructural Type System

Substructural type systems are a family of type systems analogous to substructural logics where one or more of the structural rules are absent or only allowed under controlled circumstances. Such systems are useful for constraining access to system resources such as files, locks and memory by keeping track of changes of state that occur and preventing invalid states. Substructural type systems: linear, affine, relevant, ordered.

### Syllogisms

A syllogism is a kind of logical argument where a quantified statement of a specific form (the conclusion) is inferred from two other quantified statements (the premises). It is a logical argument where one starts with premises and reaches a conclusion. A syllogism is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two or more propositions that are asserted or assumed to be true. In its earliest form, defined by Aristotle, from the combination of a general statement (the major premise) and a specific statement (the minor premise), a conclusion is deduced.

### Symbolic Logic

Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference. It is often divided into two main branches, propositional and predicate logic.

### Syntax

In logic, syntax is anything having to do with formal languages or formal systems without regard to any interpretation or meaning given to them. Syntax is concerned with the rules used for constructing, or transforming the symbols and words of a language, as contrasted with the semantics of a language which is concerned with its meaning.

### Synthetic proposition

The truth value of synthetic propositions depends on how their meaning relates to the world.

### Theorem

A theorem is a statement that has been proven on the basis of previously established statements (other theorems) and generally accepted statements (axioms). It is a logical consequence of the axioms.

### Truth Value

Truth or logical value is a value indicating the relation of a proposition to truth. A proposition has a truth value if it can be evaluated to true or false.

### Turing Completeness

Turing completeness is the ability for a system of instructions to simulate a Turing machine. A programming language that is Turing complete is theoretically capable of expressing all tasks accomplishable by computers. Almost all programming languages are Turing complete.

### Turing Machine

A Turing machine (TM) is a mathematical model of computation that defines an abstract machine, which manipulates symbols on a strip of tape according to a table of rules. Given any algorithm, a TM capable of simulating that algorithm's logic can be constructed. The Turing machine was "invented" in 1936 by Alan Turing.

### Universal Turing machine

A Turing machine that is able to simulate any other TM is called a universal Turing machine.

### Validity

In logic, an argument is valid iff it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false. It is not required that a valid argument have premises that are actually true, but to have premises that, if they were true, would guarantee the truth of the argument's conclusion.

### Weakening

The weakening rule, one of the structural rules that operate directly on the structure of a deduction system, states that the hypotheses or conclusion of a sequent may be extended with additional members and still remain valid.
