classical logic

In classical logic, we often discuss the truth values that a formula can take. The values are usually chosen as the members of a Boolean algebra. The meet and join operations in the Boolean algebra are identified with the ∧ and ∨ logical connectives, so that the value of a formula of the form A ∧ B is the meet of the value of A and the value of B in the Boolean algebra. Then we have the useful theorem that a formula is a valid sentence of classical logic if and only if its value is 1 for every v ...

See also:

Intuitionistic logic, Intuitionistic logic - Intuitionistic logic as a paradigm for logical reasoning, Intuitionistic logic - Intuitionistic logic as a formal logical calculus, Intuitionistic logic - Heyting algebra semantics, Intuitionistic logic - Kripke semantics

Read more here: » Intuitionistic logic: Encyclopedia II - Intuitionistic logic - Heyting algebra semantics

The cut-elimination theorem is the central result establishing the significance of the sequent calculus. It was originally proved by Gerhard Gentzen in his landmark paper "Investigations in Logical Deduction" for the systems LJ and LK formalising intuitionistic and classical logic respectively. The cut-elimination theorem states that any judgement that possesses a proof in the sequent calculus that makes use of the cut rule also possesses a cut-free proof ...

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Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. Initially controversial, set theory has come to play the role of a foundational theory in modern mathematics, in the sense of a theory invoked to justify assumptions made in mathematics concerning the existence of mathematical objects (such as numbers or functions) and their properties. Formal versions of set theory also have a foundational role to play as specifying a theoretical ideal of mathematical rig ...

Including:

• Axiomatic set theory - The origins of rigorous set theory
• Axiomatic set theory - Axioms for set theory
• Axiomatic set theory - Independence in ZFC
• Axiomatic set theory - Set theory ZFC foundations for mathematics
• Axiomatic set theory - Well-foundedness and hypersets
• Axiomatic set theory - Objections to set theory

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Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of arguments, although the exact definition of logic is a matter of controversy among philosophers. However the subject is grounded, the task of the logician is the same: to advance an account of valid and fallacious inference to allow ...

Including:

• Logic - Nature of logic
• Logic - Informal formal and symbolic logic
• Logic - Rival conceptions of logic
• Logic - History of logic
• Logic - Relation to other sciences
• Logic - Deductive and inductive reasoning
• Logic - Topics in logic
• Logic - Syllogistic logic
• Logic - Predicate logic
• Logic - Modal logic
• Logic - Deduction and reasoning
• Logic - Mathematical logic
• Logic - Philosophical logic
• Logic - Logic and computation
• Logic - Controversies in logic
• Logic - Bivalence and the law of the excluded middle
• Logic - Implication: strict or material?
• Logic - Tolerating the impossible
• Logic - Is logic empirical?

Read more here: » Logic: Encyclopedia - Logic

In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule, i.e., a specific instance of the falsity of a universal quantification (a "for all" statement). For example, consider the proposition "all students are lazy". Because this statement makes the claim that a certain property (laziness) holds for all students, even a single example of a diligent student will prove it false. Thus, any hard-working st ...

Including:

• Counterexample - Proof
• Counterexample - Uses
• Counterexample - In mathematics
• Counterexample - In philosophy

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In abstract algebra, a Boolean algebra is an algebraic structure (a collection of elements and operations on them obeying defining axioms) that captures essential properties of both set operations and logic operations. Specifically, it deals with the set operations of intersection, union, complement; and the logic operations of AND, OR, NOT. For example, the logical assertion that a statement a and its negation ¬a cannot both be true, parallels the set-theory assertion that a subset Including:

• Boolean algebra - Formal definition
• Boolean algebra - Examples
• Boolean algebra - Order theoretic properties
• Boolean algebra - Principle of duality
• Boolean algebra - Other notation
• Boolean algebra - Homomorphisms and isomorphisms
• Boolean algebra - Boolean rings ideals and filters
• Boolean algebra - Representing Boolean algebras
• Boolean algebra - Axiomatics for Boolean algebras

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In logic, the principle of bivalence states that for any proposition P, either P is true or P is false. This is not to be confused with the law of excluded middle and the law of noncontradiction. See bivalence and related laws for a summary of the differences. In classical logic, the principle of bivalence is equivalent to the result that there are no propositions that are neither true nor false. A proposition P that is neither true nor false is undecidable. In intuitionistic logic, sometimes the truth-value of a ...

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In intuitionistic logic, epistemologically unclear steps in proofs are forbidden. In classical logic, a formula—say, A—asserts that A is true. In intuitionistic logic a formula is only considered to be true if it can be proved. As an example of this difference consider the law of excluded middle. Accepted by classical logic, the law is not accepted by intuitionistic logic because, in a language that allows the formula, it is possible to draw the conclusion that P ∨ ¬P without knowing which of ...

See also:

Intuitionistic logic, Intuitionistic logic - Intuitionistic logic as a paradigm for logical reasoning, Intuitionistic logic - Intuitionistic logic as a formal logical calculus, Intuitionistic logic - Heyting algebra semantics, Intuitionistic logic - Kripke semantics

Read more here: » Intuitionistic logic: Encyclopedia II - Intuitionistic logic - Intuitionistic logic as a paradigm for logical reasoning

From a practical point of view, there is also a strong motivation for using intuitionistic logic. Indeed, if one goes for automated reasoning like in logic programming, then one obviously is not interested in mere statements of existence. A computer program is assumed to compute an answer, not to state that there is one. Thus, in applications, one usually looks for a witness for a given existence assertion. In addition, one may have concerns about a proof system which has a proof for ∃x : P(x ...

See also:

Intuitionistic logic, Intuitionistic logic - Intuitionistic logic as a paradigm for logical reasoning, Intuitionistic logic - Intuitionistic logic as a formal logical calculus, Intuitionistic logic - Heyting algebra semantics, Intuitionistic logic - Kripke semantics

Read more here: » Intuitionistic logic: Encyclopedia II - Intuitionistic logic - Intuitionistic logic as a formal logical calculus

Main article Kripke semantics Building upon his work on semantics of modal logic, Saul Kripke created another semantics for intuitionistic logic, known as Kripke semantics or relational semantics. ...

See also:

Intuitionistic logic, Intuitionistic logic - Intuitionistic logic as a paradigm for logical reasoning, Intuitionistic logic - Intuitionistic logic as a formal logical calculus, Intuitionistic logic - Heyting algebra semantics, Intuitionistic logic - Kripke semantics

Read more here: » Intuitionistic logic: Encyclopedia II - Intuitionistic logic - Kripke semantics

In algebra, the absorption law is an identity between two binary operations, say $ and %. It is valid and fundamental in Boolean algebra, and lattice theory. The absorption law states that a $ (a % b) = a % (a $ b) = a. The interest arises because of the cases where $ and % are meet and join in order theory. There it is easy to see that the law should hold. In particular, for the binary operators ∧ and ∨, which are defined respectively as the logica ...

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In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists. When one assumes that an object does not exist and derives a contradiction from that assumption, one still has not found the object and therefore not proved its existence, according to constructivists. See constructive proof. Constructivism is often confused with intuitionism, but in fact, intuitionism is only one kind of constructivism. Intuitionism maintains that the foundations ...

Including:

• Constructivism mathematics - Constructivist mathematics
• Constructivism mathematics - Example from real analysis
• Constructivism mathematics - Cardinality
• Constructivism mathematics - Attitude of mathematicians
• Constructivism mathematics - Mathematicians who have contributed to constructivism
• Constructivism mathematics - Branches

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Constructivist mathematics use constructivist logic, which is essentially a removal of the law of the excluded middle from classical logic. This is not to say that the law of the excluded middle is denied entirely; special cases of the law will be provable as theorems. It is just that the law is not assumed as an axiom. (The law of non-contradiction, on the other hand, is still valid.) For instance, in Heyting arithmetic, one can prove that for any proposition p which does not contain quantifiers, is a theorem (where See also:

Constructivism mathematics, Constructivism mathematics - Constructivist mathematics, Constructivism mathematics - Example from real analysis, Constructivism mathematics - Cardinality, Constructivism mathematics - Attitude of mathematicians, Constructivism mathematics - Mathematicians who have contributed to constructivism, Constructivism mathematics - Branches

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Kurt Gödel - Childhood. Kurt Gödel was born April 28, 1906, in Brünn (now Brno), Moravia, Austria-Hungary (now the Czech Republic) to Rudolf Gödel, the manager of a textile factory, and Marianne Gödel (née Handschuh). In his German-speaking family, young Kurt was known as Der Herr Warum ("Mr. Why"). He attended German language primary and secondary school in Brno and completed them with honors in 1923. Although Kurt had first excelled in languages, he later became more interested in history and mathe ...

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Kurt Gödel, Kurt Gödel - Short biography, Kurt Gödel - Childhood, Kurt Gödel - Studying in Vienna, Kurt Gödel - Working in Vienna, Kurt Gödel - Visiting the USA, Kurt Gödel - Working in Princeton, Kurt Gödel - Psychological disorder, Kurt Gödel - Death, Kurt Gödel - Legacy, Kurt Gödel - Anecdotes, Kurt Gödel - Important publications, Kurt Gödel - Links and references, Kurt Gödel - Further reading, Kurt Gödel - External links

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There are several motivations for paraconsistent logic, all of which arise out of a dissatisfaction with the consistency of classical logic, which seems to lead to results which are counterintuitive. Paraconsistent logic can be used in modelling belief systems which are inconsistent, and yet from which not anything can be inferred. In standard logics, care has to be taken to not allow such statements as the liar paradox to be formed; paraconsistent logics can be much simplified in that they do not have to excise such statements (thoug ...

See also:

Paraconsistent logic, Paraconsistent logic - Motivations, Paraconsistent logic - Problems, Paraconsistent logic - Approaches, Paraconsistent logic - Sources

Read more here: » Paraconsistent logic: Encyclopedia II - Paraconsistent logic - Motivations

A Boolean algebra is a set A, supplied with two binary operations (logical AND), (logical OR), a unary operation (logical NOT) and two elements 0 (logical FALSE) and 1 (logical TRUE), such that, for all elements a, b and c of set A, the following axioms hold: associativity commutativity absorption distributivity ...

See also:

Boolean algebra, Boolean algebra - Formal definition, Boolean algebra - Examples, Boolean algebra - Order theoretic properties, Boolean algebra - Principle of duality, Boolean algebra - Other notation, Boolean algebra - Homomorphisms and isomorphisms, Boolean algebra - Boolean rings ideals and filters, Boolean algebra - Representing Boolean algebras, Boolean algebra - Axiomatics for Boolean algebras

Read more here: » Boolean algebra: Encyclopedia II - Boolean algebra - Formal definition

A second aspect of the Curry-Howard isomorphism is that a program whose type corresponds to a logical formula is itself analogous to a proof of that formula. Consider the two following functions of λ-calculus: K: λxy.x S: λxyz. (x z (y z)) It can be shown that any function can be created by suitable applications of K and S to each other. (See the combinatory logic article for a proof.) For example, the function B ...

See also:

Curry-Howard, Curry-Howard - Types, Curry-Howard - The type inhabitation problem, Curry-Howard - Intuitionistic logic, Curry-Howard - Hilbert-style proofs, Curry-Howard - Programs are proofs, Curry-Howard - Proof of α → α, Curry-Howard - Proof of β → α → γ → β → γ → α, Curry-Howard - Sequent calculus, Curry-Howard - Point of view of category theory

Read more here: » Curry-Howard: Encyclopedia II - Curry-Howard - Programs are proofs

In terms of symbolic logic, counterexamples work as follows: The proposition to be disproved is of the form FORALL x P(x). The counterexample provides a true statement of the form NOT P(c), where c is the counterexample. Assume that the proposition FORALL x P(x) is true. By universal specification, deduce P(c) from this. Next, form the conjunction P(c) AND NOT P(c). This is a contradiction, proving that our assumption FORALL x ...

See also:

Counterexample, Counterexample - Proof, Counterexample - Uses, Counterexample - In mathematics, Counterexample - In philosophy

Read more here: » Counterexample: Encyclopedia II - Counterexample - Proof

Because of its fundamental role in philosophy, the nature of logic has been the object of intense dispute: it is not possible clearly to delineate the bounds of logic in terms acceptable to all rival viewpoints. Despite that controversy, the study of logic has been very coherent and technically grounded. In this article, we first characterise logic by introducing fundamental ideas about form, then by outlining some schools of thought, as well as by giving a brief overview of logic's history, an account of its relationship to other sciences, and finally, an exposition of some of logic's essential concepts. Logic - I ...

See also:

Logic, Logic - Nature of logic, Logic - Informal formal and symbolic logic, Logic - Rival conceptions of logic, Logic - History of logic, Logic - Relation to other sciences, Logic - Deductive and inductive reasoning, Logic - Topics in logic, Logic - Syllogistic logic, Logic - Predicate logic, Logic - Modal logic, Logic - Deduction and reasoning, Logic - Mathematical logic, Logic - Philosophical logic, Logic - Logic and computation, Logic - Controversies in logic, Logic - Bivalence and the law of the excluded middle, Logic - Implication: strict or material?, Logic - Tolerating the impossible, Logic - Is logic empirical?

Read more here: » Logic: Encyclopedia II - Logic - Nature of logic

This is a complex question and, for simplicity of exposition, we will here consider only vacuous truth as concerns logical implication, i.e., the case when S has the form P ⇒ Q, and P is false. This case strikes many people as odd, and it's not immediately obvious whether all such statements are true, all such statements are false, some are true while others are false, or what. ...

See also:

Vacuous truth, Vacuous truth - Examples, Vacuous truth - Scope of the concept, Vacuous truth - Arguments of the semantic truth of vacuously true logical statements, Vacuous truth - Arguments that at least some vacuously true statements are true, Vacuous truth - Arguments for taking all vacuously true statements to be true, Vacuous truth - Arguments that only some vacuously true statements are true, Vacuous truth - Summary, Vacuous truth - Difficulties with the use of vacuous truth, Vacuous truth - Vacuous truths in mathematics

Read more here: » Vacuous truth: Encyclopedia II - Vacuous truth - Arguments of the semantic truth of vacuously true logical statements