With the Prior Analytics, Aristotle is credited with the earliest study of formal logic, and his conception of it was the dominant form of Western logic until 19th century advances in mathematical logic.

Introduction to mathematical logic.

Russell and Whitehead thought they could derive all mathematical truth using axioms and inference rules of formal logic, in principle opening up the process to automatisation.

The actual mathematical operation for each instruction is performed by a subunit of the CPU known as the arithmetic logic unit or ALU.

In any case, this article follows ISO 31-11 and the standard convention in mathematical logic, which make 0 a natural number .</ ref >

In mathematics, particularly theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers or the computable reals, are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm.

Hilbert is known as one of the founders of proof theory and mathematical logic, as well as for being among the first to distinguish between mathematics and metamathematics.

One who had to leave Germany, Paul Bernays, had collaborated with Hilbert in mathematical logic, and co-authored with him the important book Grundlagen der Mathematik ( which eventually appeared in two volumes, in 1934 and 1939 ).

Hilbert's work had started logic on this course of clarification ; the need to understand Gödel's work then led to the development of recursion theory and then mathematical logic as an autonomous discipline in the 1930s.

The study of mathematical proof is particularly important in logic, and has applications to automated theorem proving and formal verification of software.

In 1970, a novel result in mathematical logic known as Matiyasevich's theorem settled the problem negatively: in general Diophantine problems are unsolvable.

In mathematical logic, there are two quantifiers, " some " and " all ", though as Brentano ( 1838 – 1917 ) pointed out, we can make do with just one quantifier and negation.

According to Husserl, this view of logic and mathematics accounted for the objectivity of a series of mathematical developments of his time, such as n-dimensional manifolds ( both Euclidean and non-Euclidean ), Hermann Grassmann's theory of extensions, William Rowan Hamilton's Hamiltonians, Sophus Lie's theory of transformation groups, and Cantor's set theory.

This is the case of the Mycin and Dendral expert systems, and of, for example, fuzzy logic, predicate logic ( Prolog ), symbolic logic and mathematical logic.

Logical empiricism ( aka logical positivism or neopositivism ) was an early 20th century attempt to synthesize the essential ideas of British empiricism ( e. g. a strong emphasis on sensory experience as the basis for knowledge ) with certain insights from mathematical logic that had been developed by Gottlob Frege and Ludwig Wittgenstein.

A finite-state machine ( FSM ) or finite-state automaton ( plural: automata ), or simply a state machine, is a mathematical model of computation used to design both computer programs and sequential logic circuits.

In logic and the foundations of mathematics, formal languages are used to represent the syntax of axiomatic systems, and mathematical formalism is the philosophy that all of mathematics can be reduced to the syntactic manipulation of formal languages in this way.

Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic.

First-order predicate calculus is commonly used as a mathematical basis for these systems, to avoid excessive complexity.

IST is an extension of Zermelo-Fraenkel set theory ( ZF ) in that alongside the basic binary membership relation, it introduces a new unary predicate standard which can be applied to elements of the mathematical universe together with some axioms for reasoning with this new predicate.

In the mathematical model, reasoning about such data is done in two-valued predicate logic, meaning there are two possible evaluations for each proposition: either true or false ( and in particular no third value such as unknown, or not applicable, either of which are often associated with the concept of NULL ).

In mathematical logic, a second-order predicate is a predicate that takes a first-order predicate as an argument.

Z is based on the standard mathematical notation used in axiomatic set theory, lambda calculus, and first-order predicate logic.

While Hilbert's tenth problem is not a formal mathematical statement as such the nearly universal acceptance of the ( philosophical ) identification of a decision algorithm with a total computable predicate allows us to use the MRDP theorem to conclude the tenth problem is unsolvable.

So, let x be any one particular friend of Peter, X the set of all Peter's friends, P ( x ) be the predicate ( mathematical logic ) " x likes to dance ", and lastly Q ( x ) the predicate " x likes to go to the beach ".

In mathematical logic, stratification is any consistent assignment of numbers to predicate symbols guaranteeing that a unique formal interpretation of a logical theory exists.

* Any mathematical expression that does not use the new predicate standard explicitly or implicitly is an internal formula.

This is the most widespread approach, and is based on the idea that the meaning of the various parts of the propositions are given by the possible ways we can give a recursively specified group of interpretation functions from them to some predefined mathematical domains: an interpretation of first-order predicate logic is given by a mapping from terms to a universe of individuals, and a mapping from propositions to the truth values " true " and " false ".

The study of basic mathematical logic such as propositional logic and predicate logic ( normally in conjunction with set theory ) is considered an important theoretical underpinning to any undergraduate computer science course.

In mathematical logic, Kalmár proved that certain classes of formulas of the first order predicate calculus were decidable.

: This article is a technical mathematical article in the area of predicate logic.

In mathematical logic, a sentence of a predicate logic is a boolean-valued well formed formula with no free variables.

A mathematical formula is nothing more than a pattern for solving a specific problem.

The equation is used for the mathematical process of solving the problem.

However, it is essential that the various mathematical symbols used in the equations be understood so that the mathematical processes can be done properly and in their correct order.

We devote a chapter to the binomial distribution not only because it is a mathematical model for an enormous variety of real life phenomena, but also because it has important properties that recur in many other probability models.

It has become painfully clear that the very attempt to make the language of social research free of values by erecting mathematical and physical models, is itself a conditioned response to a world which pays a premium price for technological manipulation.

A mathematical block diagram for the leveling system is shown in Fig. 7-2.

So, too, is the mathematical competence of a college graduate who has majored in mathematics.

The importance of this 5 can largely be explained by the natural mathematical properties of the middle number and its special relationship to all the rest of the numbers -- quite apart from any numerological considerations, which is to say, any symbolic meaning arbitrarily assigned to it.

it is also their mathematical mean, since it is equal to half the sum of every opposing pair, all of which equal 10.

The study of altruism was the initial impetus behind George R. Price's development of the Price equation, which is a mathematical equation used to study genetic evolution.

The working principle of a yupana is unknown, but in 2001 an explanation of the mathematical basis of these instruments was proposed by Italian mathematician Nicolino De Pasquale.

The abacus teaches mathematical skills that can never be replaced with talking calculators and is an important learning tool for blind students.

One motivation for this use is that a number of generally accepted mathematical results, such as Tychonoff's theorem, require the axiom of choice for their proofs.

However, that particular case is a theorem of Zermelo – Fraenkel set theory without the axiom of choice ( ZF ); it is easily proved by mathematical induction.

The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts.

In mathematics, an automorphism is an isomorphism from a mathematical object to itself.

In the context of abstract algebra, for example, a mathematical object is an algebraic structure such as a group, ring, or vector space.

In mathematical notation, this is:

An atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom.

Acoustic theory is the field relating to mathematical description of sound waves.

The mathematical equation for an ideal gas undergoing a reversible ( i. e., no entropy generation ) adiabatic process is