Intuitionistic logic - Intuitionistic logic Intuitionistic logic, or constructivist logic, is the logic used in mathematical intuitionism and other forms of mathematical constructivism. Roughly speaking, 'intuitionism' holds that logic and mathematics are 'constructive' mental activities. That is, they are not analytic activities wherein deep properties of existence are revealed and applied. Instead, logic and mathematics are the application of internally consistent methods to realize more complex mental constructs (really, a kind of game). In a stricter sense, intuitionistic logic can be investigated as a very concrete and formal kind of mathematical logic. While it may be argued whether such a formal calculus really captures the philosophical aspects of intuitionism, it has properties which are also quite useful from a practical point of view. Both notions of the term will.
Combinatory logic - Combinatory logic This article is not about combinatorial logic, a topic in electronics. Combinatory logic is a simplified model of computation, used in computability theory (the study of what can be computed) and proof theory (the study of what can be mathematically proven.) The theory, despite its simplicity, captures many essential features of the nature of computation. Combinatory logic is a variation of the lambda calculus, in which lambda expressions (used to allow for functional abstraction) are replaced by a limited set of primitive functions. Table of contents showTocToggle("show","hide") 1 Summary of the lambda calculus 2 Combinatory calculi 2.1 Combinatory terms 2.2 Examples of combinators 2.3 Completeness of the S-K basis 2.3.1 Conversion of a lambda term to an equivalent combinatorial term 2.3.2 Explanation of the T[ ].
Substructural logic - Substructural logic In mathematical logic, in particular in connection with proof theory, a number of substructural logics have been introduced, as systems of propositional calculus that are weaker than the conventional one. They differ in having fewer structural rules available: the concept of structural rule is based on the sequent presentation, rather than the natural deduction formulation. Two of the more significant substructural logics are relevant logic and linear logic. In discussing the sequent calculus, one writes each line of a proof as . Here the structural rules are rules for rewriting the LHS Γ of the sequent, initially conceived of as a string of propositions. The standard interpretation of this string is as conjunction: we expect to read as the sequent notation for (A and B).
Paraconsistent logic - Paraconsistent logic A paraconsistent logic is a logic which gracefully deals with contradictions. In classical logic, if and , i.e. there is some theory which allows you to show both and , then it is possible to prove that every formula (and so its negation ) is true in the proof calculus; a similar model theoretic result can be derived. Classical logic, intuitionistic logic, and indeed most other logics suffer from this problem. Paraconsistent logics must not fall into this trap. The need for paraconsistent logics comes from the nature of human reason; we appear to reason paraconsistently, and perhaps the universe itself is paraconsistent. Some paraconsistent logics: Belief revision systems Many valued logics Relevant logic.
Multi-valued logic - Multi-valued logic Traditionally, logical calculi are bivalent--that is, there are only two possible truth values for any proposition, true and false (which generally correspond to our intuitive notions of truth and falsity). But bivalence is only one possible range of truth values that may be assigned, and other logical systems have been developed with variations on bivalence, or with more than two possible truth-value assignments. In the classical bivalence scheme, true and false are determinate values: a proposition is either true or false (exclusively), and if the proposition does not have one of those values, by definition it must have the other. This is the justification for the Law of excluded middle: P ∨ ¬P (i.e., either the proposition or its negation holds). One point to remember is.
List of topics in logic - List of topics in logic This is a list of topics in logic, by Wikipedia page. See also list of rules of inference. There is a list of paradoxes on the paradox page. There is a list of fallacies on the logical fallacy page. Modern mathematical logic is at the list of mathematical logic topics page. There is a more complete list of logicians. For introductory set theory and other supporting material see the list of basic discrete mathematics topics. =A= Abacus logic -- Abduction (logic) -- Affirming the consequent -- Antecedent -- Antinomy -- Argument form -- Aristotelian logic -- Axiom -- Axiomatic system -- Axiomatization =B= Biconditional elimination -- Biconditional introduction -- Bivalence and related laws -- Boolean algebra =C= Categorial logic -- College logic --.
List of mathematical logic topics - List of mathematical logic topics This is a list of mathematical logic topics, by Wikipedia page. For traditional syllogistic logic, see the list of topics in logic. Clicking on related changes shows a list of most-recent edits of articles to which this page links. This page links to itself in order that recent changes to this page will also be included in related changes. Table of contents showTocToggle("show","hide") 1 Working foundations 2 Model theory 3 Set theory 4 Large cardinals 5 Recursion theory 6 Proof theory 7 Mathematical constructivism Working foundations Peano axioms Mathematical induction Three forms of mathematical induction Structural induction Naive set theory Simple theorems in set theory Power set Empty set Empty function Universe (mathematics) Axiomatization Axiomatic system Mathematical proof Tautology Consistency Arithmetization of analysis.
Intuitionism - of a mathematical statement to be equivalent to its having been proved; what other criteria can there be for truth (an intuitionist would argue) if mathematical objects are merely mental constructions? This means that an intuitionist may not believe that a mathematical statement has the same meaning that a classical mathematician would. For example, to say A or B, to an intuitionist, is to claim that either A or B can be proved. In particular, the law of excluded middle, A or not A, is disallowed since one cannot assume that it is always possible to either prove the statement A or its negation. (See also intuitionistic logic.) Intuitionism also rejects the abstraction of actual infinity; i.e., it does not consider as given objects infinite entities such as the set of.
Hausdorff space - and only if it is both preregular and T0. Conversely, a topological space is preregular if and only if its Kolmogorov quotient is Hausdorff. Examples and nonexamples Almost all spaces encountered in analysis are Hausdorff; most importantly, the real numbers are a Hausdorff space. More generally, all metric spaces are Hausdorff. In fact, many spaces of use in analysis, such as topological groups and topological manifolds, have the Hausdorff condition explicitly stated in their definitions. Pseudometric spaces typically are not Hausdorff, but they are preregular, and their use in analysis is usually only in the construction of Hausdorff gauge spaces. Indeed, when analysts run across a non-Hausdorff space, it is still probably at least preregular, and then they simply replace it with its Kolmogorov quotient, which is Hausdorff. In contrast, non-preregular.
Heyting algebra - are a generalization of Boolean algebras. Heyting algebras model intuitionistic logic, in which the law of excluded middle does not in general hold. Formally, a Heyting algebra is a bounded lattice L such that for all a and b in L there is a greatest element x of L such that a ∧ x ≤ b. This element is called the relative pseudo-complement of a with respect to b, and is denoted a⇒b (or a→b). We write ¬a for a⇒0. Heyting algebras are always distributive; this is sometimes stated as an axiom, but in fact it follows from the existence of relative pseudo-complements. Boolean algebras are those Heyting algebras in which x = ¬¬x for all x, or, equivalently, in which x ∨ ¬x = 1 for all x. In this case, the element a⇒b is equal to ¬a ∨ b. Every topology.
Edmund Husserl - influence Martin Heidegger, Jean-Paul Sartre, and Maurice Merleau-Ponty, among others. (Hermann Weyl's interest in intuitionistic logic and impredicativity, for example, seems to have been as a result of contact with Husserl.) Husserl is best known for his extensive use of the notion that the main characteristic of consciousness is that it is always intentional, i.e. directed at some kind of content ("Inhalt"): consciousness is always "consciousness of something." He borrowed the concept of the intentional from Brentano, as can be seen from the latter's Psychologie vom empirischen Standpunkt (Psychology from an Empirical Standpoint). Further, he asserted that studying the flow of consciousness as directed (the act of noesis) at the perceived phenomena (the noemata) yields knowledge of essential structures in reality. In the last period of his life, Husserl shifted to.
Double negative elimination - Double negative elimination In logic and the propositional calculus, double negative elimination is a rule that states that double negatives can be removed from a proposition without changing its meaning: It's not the case that it's not raining. means the same as: It's raining. Formally: ¬ ¬ A ∴ A Also: ¬ ¬ ¬ A ∴ ¬ A The rule of double negative introduction states the converse, that double negatives can be added without changing the meaning of a proposition. This rule is true in classical logic, but in intuitionistic logic, the statement, It's not the case that it's not raining. is weaker than It's raining.. As a slightly clearer example, It's not unreasonable is slightly less direct than It's reasonable. In naive set theory also we have the.
Andrey Nikolaevich Kolmogorov - in the fields of probability theory and topology. He worked early in his career on intuitionistic logic, and Fourier series. He also worked on turbulence, and classical mechanics; and was a founder of algorithmic complexity theory. Quote: "The theory of probability as mathematical discipline can and should be developed from axioms in exactly the same way as Geometry and Algebra." See also: Kolmogorov axioms Kolmogorov complexity Kolmogorov space Kolmogorov-Smirnov test Kolmogorov-Arnold-Moser theorem Kolmogorov's zero-one law.
Axiom schema of replacement - of replacement In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of replacement is a schema of axioms in Zermelo-Fraenkel set theory. Suppose P is any predicate in two variables that doesn't use the symbol B. Then in the formal language of the Zermelo-Fraenkel axioms, the axiom schema reads: (∀ X, ∃! Y, P(X,Y)) → ∀ A, ∃ B, ∀ C, C ∈ B ↔ (∃ D, D ∈ A ∧ P(D,C)); or in words: If, given any set X, there is a unique set Y such that P holds for X and Y, then, given any set A, there is a set B such that, given any set C, C is a member of B if and only if there.
Background and genesis of topos theory - given. This is partly in terms of historical development, but also to some extent an explanation of differing attitudes to category theory. Table of contents showTocToggle("show","hide") 1 In the school of Grothendieck 2 From pure category theory to categorical logic 3 Position of topos theory 4 Summary In the school of Grothendieck During the latter part of the 1950s, the foundations of algebraic geometry were being rewritten; and it is here that the origins of the topos concept are to be found. At that time the Weil conjectures were an outstanding motivation to research. As we now know, the route towards their proof, and other advances, lay in the construction of etale cohomology. With the benefit of hindsight, it can be said that algebraic geometry had been wrestling with two problems,.
Category theory - Categories, functors and natural transformations were introduced by Samuel Eilenberg and Saunders MacLane in 1945. Initially, the notions were applied in topology, especially algebraic topology, as an important part of the transition from homology (an intuitive and geometric concept) to homology theory, an axiomatic approach. It has been claimed, for example by or on behalf of Ulam, that comparable ideas were current in the later 1930s in the Polish school. Eilenberg/MacLane have said that their goal was to understand natural transformations; in order to do that, functors had to be defined; and to define functors one needed categories. The subsequent development of the theory was powered first by the computational needs of homological algebra; and then by the axiomatic needs of algebraic geometry, the field most resistant to the Russell-Whitehead view.
Counterexample - Counterexample In logic, and especially in its applications to mathematics and philosophy, a counterexample is 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 student is a counterexample to "all students are lazy". Table of contents showTocToggle("show","hide") 1 Proof 2 Uses 2.1 In mathematics 2.2 In philosophy Proof 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.
Constructivist analysis - separable spaces; also, some theorems may need to be approached by approximations. Furthermore, many classical theorems can be stated in ways that are logically equivalent according to classical logic, but not all of these forms will be valid in constructive analysis, which uses intuitionistic logic. Table of contents showTocToggle("show","hide") 1 Attitude of mathematicians 2 Examples 2.1 The intermediate value theorem 2.2 The least upper bound principle and compact sets Attitude of mathematicians Traditionally, mathematicians have been suspicious, if not downright antagonistic, towards mathematical constructivism, largely because of the limitations that it poses for constructive analysis. These views were forcefully expressed by David Hilbert in 1928, when he wrote in Die Grundlagen der Mathematik, "Taking the principle of excluded middle from the mathematician would be the same, say, as proscribing the telescope.
Curry-Howard isomorphism - isomorphism. Starting from the point of view that functional programming supports programming languages that are typed and have higher-order functions, the primary content of the "isomorphism" is to identify that amount of structure with that occurring in type theories of intuitionistic logic. Under the influence of category theory there are a number of heuristic ways of looking at the overall position. A favourite formulation is "propositions as types". This is rather more confusing than the matching statement "proofs as programs": using the idea of category in its very simple form, we can say that if proofs and programs are two kinds of morphisms, then propositions and types are going to be two corresponding kinds of objects. It is suggested that we have two distinct categories, with some sort of functor (translation).
Truth value - Truth value In logic, a truth value, or truth-value, is a value indicating to what extent a statement is true. In classical logic, the only possible truth values are true and false. However, other values are possible in other logics. A simple intuitionistic logic has truth values of true, false, and unknown; fuzzy logic and other forms of multi-valued logic also use more truth values than simply true and false. Algebraically, the set {true,false} forms a simple Boolean algebra. Other Boolean algebras may be used as sets of truth values in multi-valued logic, while intuitionistic logic generalises Boolean algebras to Heyting algebras. In topos theory, the subobject classifier of a topos takes the place of the set of truth values. This nomenclature is perhaps more consonant with usages.
