Second-order logic In logic and mathematics second-order logic is an extension of first-order logic, which itself is an extension of propositional logic.[1] 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, in addition, also quantifies over relations. For example, the second-order sentence Syntax and fragments[edit] A sort of variables that range over sets of individuals. Each of the variables just defined may be universally and/or existentially quantified over, to build up formulas. It's possible to forgo the introduction of function variables in the definition given above (and some authors do this) because an n-ary function variable can be represented by a relation variable of arity n+1 and an appropriate formula for the uniqueness of the "result" in the n+1 argument of the relation. or , where is a first-order formula. , or even as ∃SO. has the form
Quantification In logic, quantification is the binding of a variable ranging over a domain of discourse. The variable thereby becomes bound by an operator called a quantifier. Academic discussion of quantification refers more often to this meaning of the term than the preceding one. Natural language[edit] All known human languages make use of quantification (Wiese 2004). Every glass in my recent order was chipped.Some of the people standing across the river have white armbands.Most of the people I talked to didn't have a clue who the candidates were.A lot of people are smart. The words in italics are quantifiers. The study of quantification in natural languages is much more difficult than the corresponding problem for formal languages. Montague grammar gives a novel formal semantics of natural languages. Logic[edit] In language and logic, quantification is a construct that specifies the quantity of specimens in the domain of discourse that apply to (or satisfy) an open formula. Mathematics[edit] . means
View topic - Universal Constructor Based Spaceship A new age of Game of Life exploration and design! I agree with Nick. This marks an important turning point in Life technology, comparable to the discovery of the Gosper Glider Gun and stable Herschel tracks. What is the range of velocities that can be achieved in this way? Any velocity less than c/2 (by the L1, or Manhattan metric) can be attained by a Universal constructor. For each velocity, what are the smallest spaceships (by various definitions of "smallest")? A spaceship of period n, where n is equal to or greater than a constant, can have diameter of O(sqrt(log(n))). How about rakes, guns, puffers, breeders based on such spaceships, and/or constructing them? Gemini can be trivially modified into a rake, puffer or breeder. Producing a Gemini gun is more difficult, but not totally intractable. How readily can changes in velocity be programmed into such a ship? Gemini can only have a constant velocity, and not a variable velocity. To an arbitrary degree of accuracy, in fact.
Sequent calculus In proof theory and mathematical logic, sequent calculus is a family of formal systems sharing a certain style of inference and certain formal properties. The first sequent calculi, systems LK and LJ, were introduced by Gerhard Gentzen in 1934 as a tool for studying natural deduction in first-order logic (in classical and intuitionistic versions, respectively). Gentzen's so-called "Main Theorem" (Hauptsatz) about LK and LJ was the cut-elimination theorem, a result with far-reaching meta-theoretic consequences, including consistency. Introduction[edit] where is any formula of first-order-logic (or whatever logic the deduction system applies to, e.g., propositional calculus or a higher-order logic or a modal logic). The price paid for the simple syntax of a Hilbert-style system is that complete formal proofs tend to get extremely long. where the 's and are again formulae and ", with a single formula on the right-hand side. such that , etc., are all true, will also be true. and are formulae, and
Interpretation From Wikipedia, the free encyclopedia Interpretation may refer to: Culture[edit] Aesthetic interpretation, an explanation of the meaning of a work of artAllegorical interpretation, an approach that assumes a text should not be interpreted literallyDramatic Interpretation, an event in speech and forensics competitions in which participants perform excerpts from playsHeritage interpretation, communication about the nature and purpose of historical, natural, or cultural phenomenaInterpretation (music), the process of a performer deciding how to perform music that has been previously composedLanguage interpretation, the facilitation of dialogue between parties using different languagesLiterary theory, broad methods for interpreting literature, including historicism, feminism, structuralism, deconstruction Literary criticism, interpretation of particular works of literatureOral interpretation, a dramatic art Law[edit] Math and computing[edit] Media[edit] Neuroscience[edit] Philosophy[edit]
Challenges for Ontology Design Author: Thomas GruberTitle: Grande Challenges for Ontology Design (or is it Vente?)Date: March 1, 2007Type: Invited presentation Citation: Tom Gruber (2007). File: challenges-for-ontology-design.ppt Context: Monthly presentation at international conference call of the Ontolog Community in which participants interact over synchornous and asynchronous communication. Abstract: Why bother with ontology design, particularly when it involves the trouble of collaborating with other people and their peculiar ideas? In this session I will frame a discussion about ontology design, using the model of engineering design that has brought tremendous success to electronic and physical engineering disciplines, and to some extent software engineering. Von Neumann universe The rank of a well-founded set is defined inductively as the smallest ordinal number greater than the ranks of all members of the set.[1] In particular, the rank of the empty set is zero, and every ordinal has a rank equal to itself. The sets in V are divided into a transfinite hierarchy, called the cumulative hierarchy, based on their rank. History[edit] The cumulative type hierarchy, also known as the von Neumann universe, is claimed by Gregory H. Existence and uniqueness of the general transfinite recursive definition of sets was demonstrated in 1928 by von Neumann for both Zermelo-Fraenkel set theory[4] and Neumann's own set theory (which later developed into NBG set theory).[5] In neither of these papers did he apply his transfinite recursive method to construct the universe of all sets. The notation V is not a tribute to the name of von Neumann. Definition[edit] The class V is defined to be the union of all the V-stages: An equivalent definition sets for each ordinal α, where
Domain of discourse The term universe of discourse generally refers to the collection of objects being discussed in a specific discourse. In model-theoretical semantics, a universe of discourse is the set of entities that a model is based on. The concept universe of discourse is generally attributed to Augustus De Morgan (1846) but the name was used for the first time in history by George Boole (1854) on page 42 of his Laws of Thought in a long and incisive passage well worth study. Boole's definition is quoted below. The concept, probably discovered independently by Boole in 1847, played a crucial role in his philosophy of logic especially in his stunning principle of wholistic reference. A database is a model of some aspect of the reality of an organisation. Boole’s 1854 Definition[edit] See also[edit] References[edit] Jump up ^ Corcoran, John.
Natural deduction In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning. This contrasts with the axiomatic systems which instead use axioms as much as possible to express the logical laws of deductive reasoning. Motivation[edit] Natural deduction in its modern form was independently proposed by the German mathematician Gentzen in 1935, in a dissertation delivered to the faculty of mathematical sciences of the university of Göttingen. The term natural deduction (or rather, its German equivalent natürliches Schließen) was coined in that paper: Ich wollte zunächst einmal einen Formalismus aufstellen, der dem wirklichen Schließen möglichst nahe kommt. Gentzen was motivated by a desire to establish the consistency of number theory. In natural deduction, a proposition is deduced from a collection of premises by applying inference rules repeatedly. Judgments and propositions[edit] ". ).
Analytic–synthetic distinction Semantic distinction in philosophy While the distinction was first proposed by Immanuel Kant, it was revised considerably over time, and different philosophers have used the terms in very different ways. Furthermore, some philosophers (starting with Willard Van Orman Quine) have questioned whether there is even a clear distinction to be made between propositions which are analytically true and propositions which are synthetically true.[2] Debates regarding the nature and usefulness of the distinction continue to this day in contemporary philosophy of language.[2] Conceptual containment [edit] The philosopher Immanuel Kant uses the terms "analytic" and "synthetic" to divide propositions into two types. analytic proposition: a proposition whose predicate concept is contained in its subject conceptsynthetic proposition: a proposition whose predicate concept is not contained in its subject concept but related Examples of analytic propositions, on Kant's definition, include: Logical positivists
Independence (mathematical logic) A theory T is independent if each axiom in T is not provable from the remaining axioms in T. A theory for which there is an independent set of axioms is independently axiomatizable. Some authors say that σ is independent of T if T simply cannot prove σ, and do not necessarily assert by this that T cannot refute σ. Many interesting statements in set theory are independent of Zermelo-Fraenkel set theory (ZF). The following statements (none of which have been proved false) cannot be proved in ZFC to be independent of ZFC, even if the added hypothesis is granted that ZFC is consistent. The existence of strongly inaccessible cardinalsThe existence of large cardinalsThe non-existence of Kurepa trees The following statements are inconsistent with the axiom of choice, and therefore with ZFC. Mendelson, Elliott (1997), An Introduction to Mathematical Logic (4th ed.), London: Chapman & Hall, ISBN 978-0-412-80830-2 Monk, J.