Ethics cannot be based on the authoritative certainty given by mathematics and logic, or prescribed directly from the empirical findings of science.

The axiom of choice has also been thoroughly studied in the context of constructive mathematics, where non-classical logic is employed.

The technique has been applied in the study of mathematics and logic since before Aristotle ( 384 – 322 B. C.

The Platonist seemed to outweigh the Aristotelian in Alan, but he felt strongly that the divine is all intelligibility and argued this notion through much Aristotelian logic combined with Pythagorean mathematics.

While the roots of formalised logic go back to Aristotele, the end of the 19th and early 20th centuries saw the development of modern logic and formalised mathematics.

His Foundations of Arithmetic, published 1884, expressed ( parts of ) mathematics in formal logic.

* Strict conditional, as used in philosophy, logic, and mathematics.

General category theory, an extension of universal algebra having many new features allowing for semantic flexibility and higher-order logic, came later ; it is now applied throughout mathematics.

Today he is appreciated largely for his contributions to logic, mathematics, philosophy, scientific methodology, and semiotics, and for his founding of pragmatism.

In 1879, Peirce was appointed Lecturer in logic at the new Johns Hopkins University, which was strong in a number of areas that interested him, such as philosophy ( Royce and Dewey did their PhDs at Hopkins ), psychology ( taught by G. Stanley Hall and studied by Joseph Jastrow, who coauthored a landmark empirical study with Peirce ), and mathematics ( taught by J. J. Sylvester, who came to admire Peirce's work on mathematics and logic ).

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.

This activity is performed through the verbal impersonation of the characters by the players, while also employing a variety of social and other useful cognitive skills, such as logic, basic mathematics and imagination.

In ordinary language, i. e. outside of contexts such as formal logic, mathematics and programming, " or " sometimes has the meaning of exclusive disjunction.

In mathematics and logic, the infix operator is usually ∨; in electronics, +; and in programming languages, | or or.

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.

In contrast to real numbers that have the property of varying " smoothly ", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values.

Certain forms treat all knowledge as empirical, while some regard disciplines such as mathematics and logic as exceptions.

Hence for both logic and mathematics, the different formal categories are the objects of study, not the sensible objects themselves.

The problem with the psychological approach to mathematics and logic is that it fails to account for the fact that this approach is about formal categories, and not simply about abstractions from sensibility alone.

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.

In logic and mathematics, a two-place logical operator and, also known as logical conjunction, results in true if both of its operands are true, otherwise the value of false.

The first three treatises form the core of the logical theory stricto sensu: the grammar of the language of logic and the correct rules of reasoning.

In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs.

A formula of propositional logic is said to be satisfiable if logical values can be assigned to its variables in a way that makes the formula true.

Two typical components of a CPU are the arithmetic logic unit ( ALU ), which performs arithmetic and logical operations, and the control unit ( CU ), which extracts instructions from memory and decodes and executes them, calling on the ALU when necessary.

Under the Curry – Howard correspondence, the existence of currying and uncurrying is equivalent to the logical theorem, as tuples ( product type ) corresponds to conjunction in logic, and function type corresponds to implication.

Both the logical complexity ( needing very large logic design and logic verification teams and simulation farms with perhaps thousands of computers ) and the high operating frequencies ( needing large circuit design teams and access to the state-of-the-art fabrication process ) account for the high cost of design for this type of chip.

The truth values of logical formulas usually form a finite set, generally restricted to two values: true and false, but logic can also be continuous-valued, e. g., fuzzy logic.

In propositional logic, disjunction elimination ( sometimes named proof by cases or case analysis ), is the valid argument form and rule of inference that allows one to eliminate a disjunctive statement from a logical proof.

* Tarski's axioms: Alfred Tarski ( 1902 – 1983 ) and his students defined elementary Euclidean geometry as the geometry that can be expressed in first-order logic and does not depend on set theory for its logical basis, in contrast to Hilbert's axioms, which involve point sets.

In this stratum we elaborate a " pure grammar " or a logical syntax, and he would call its rules " laws to prevent non-sense ", which would be similar to what logic calls today " formation rules ".

Husserl also talked about what he called " logic of truth " which consists of the formal laws of possible truth and its modalities, and precedes the third logical third stratum.

The work of both authors was heavily influenced by Kurt Gödel's earlier work on his incompleteness theorem, especially by the method of assigning numbers ( a Gödel numbering ) to logical formulas in order to reduce logic to arithmetic.

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.

Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic.

In logic and related fields such as mathematics and philosophy, if and only if ( shortened iff ) is a biconditional logical connective between statements.

In logic formulae, logical symbols are used instead of these phrases ; see the discussion of notation.

It puts one in the position of asserting or implying that truth or standards of logical consistency are relative to a particular thinker or group and that under some other standard, the position is correct despite its failure to stand up to logic.

The early Wittgenstein was concerned with the logical relationship between propositions and the world, and believed that by providing an account of the logic underlying this relationship he had solved all philosophical problems.

* Lambda is the set of logical axioms in the axiomatic method of logical deduction in first-order logic.

The language ’ s grammar is based on predicate logic, which is why it was named Loglan, an abbreviation for " logical language ".