Higher-order logic
Formal system of logic The term "higher-order logic", abbreviated as HOL, is commonly used to mean higher-order simple predicate logic. Here "simple" indicates that the underlying type theory is the theory of simple types, also called the simple theory of types (see Type theory). Leon Chwistek and Frank P. Quantification scope[edit] First-order logic quantifies only variables that range over individuals; second-order logic, in addition, also quantifies over sets; third-order logic also quantifies over sets of sets, and so on. Higher-order logic is the union of first-, second-, third-, ..., nth-order logic; i.e., higher-order logic admits quantification over sets that are nested arbitrarily deeply. Semantics[edit] There are two possible semantics for higher-order logic. In the standard or full semantics, quantifiers over higher-type objects range over all possible objects of that type. In Henkin semantics, a separate domain is included in each interpretation for each higher-order type.
Structure
Arrangement of interrelated elements in an object/system, or the object/system itself Load-bearing[edit] Buildings, aircraft, skeletons, anthills, beaver dams, bridges and salt domes are all examples of load-bearing structures. The results of construction are divided into buildings and non-building structures, and make up the infrastructure of a human society. The structure elements are combined in structural systems. Load-bearing biological structures such as bones, teeth, shells, and tendons derive their strength from a multilevel hierarchy of structures employing biominerals and proteins, at the bottom of which are collagen fibrils.[4] Biological[edit] In another context, structure can also observed in macromolecules, particularly proteins and nucleic acids.[6] The function of these molecules is determined by their shape as well as their composition, and their structure has multiple levels. Chemical[edit] Chemical structure refers to both molecular geometry and electronic structure.
Set theory
The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known. Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals. History[edit] Mathematical topics typically emerge and evolve through interactions among many researchers. Cantor's work initially polarized the mathematicians of his day. Basic concepts and notation[edit] Some ontology[edit] Sets alone.
Zeroth-order logic
First-order logic without variables or quantifiers
Alfred North Whitehead
English mathematician and philosopher Alfred North Whitehead OM FRS FBA (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He is best known as the defining figure of the philosophical school known as process philosophy,[21] which today has found application to a wide variety of disciplines, including ecology, theology, education, physics, biology, economics, and psychology, among other areas. In his early career Whitehead wrote primarily on mathematics, logic, and physics. Whitehead's process philosophy argues that "there is urgency in coming to see the world as a web of interrelated processes of which we are integral parts, so that all of our choices and actions have consequences for the world around us Life[edit] Childhood, education[edit] Whewell's Court north range at Trinity College, Cambridge. Career[edit] Bertrand Russell in 1907. Toward the end of his time in England, Whitehead turned his attention to philosophy. Move to the US, 1924[edit] God[edit]
Principia Mathematica
✸54.43: "From this proposition it will follow, when arithmetical addition has been defined, that 1 + 1 = 2." —Volume I, 1st edition, page 379 (page 362 in 2nd edition; page 360 in abridged version). (The proof is actually completed in Volume II, 1st edition, page 86, accompanied by the comment, "The above proposition is occasionally useful." The title page of the shortened Principia Mathematica to ✸56 I can remember Bertrand Russell telling me of a horrible dream. Hardy, G. He [Russell] said once, after some contact with the Chinese language, that he was horrified to find that the language of Principia Mathematica was an Indo-European one Littlewood, J. The Principia Mathematica (often abbreviated PM) is a three-volume work on the foundations of mathematics written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. PM is not to be confused with Russell's 1903 The Principles of Mathematics. PM has long been known for its typographical complexity. Cf.
Law of identity
Logic statement In logic, the law of identity states that each thing is identical with itself. It is the first of the historical three laws of thought, along with the law of noncontradiction, and the law of excluded middle. However, few systems of logic are built on just these laws. History[edit] Ancient philosophy[edit] Socrates: With regard to sound and colour, in the first place, do you think this about both: that they both are? It is used explicitly only once in Aristotle, in a proof in the Prior Analytics:[1][2] When A belongs to the whole of B and to C and is affirmed of nothing else, and B also belongs to all C, it is necessary that A and B should be convertible: for since A is said of B and C only, and B is affirmed both of itself and of C, it is clear that B will be said of everything of which A is said, except A itself. Medieval philosophy[edit] Aristotle believed the law of non-contradiction to be the most fundamental law. Modern philosophy[edit] Contemporary philosophy[edit]
Particular
From Wikipedia, the free encyclopedia Concept in metaphysics Overview[edit] Sybil Wolfram[3] writes Particulars include only individuals of a certain kind: as a first approximation individuals with a definite place in space and time, such as persons and material objects or events, or which must be identified through such individuals, like smiles or thoughts. Some terms are used by philosophers with a rough-and-ready idea of their meaning. The term particular is also used as a modern equivalent of the Aristotelian notion of individual substance. See also[edit] References[edit]
Abstract particulars
Abstract particulars are metaphysical entities which are both abstract objects and particulars. Examples[edit] Individual numbers are often classified as abstract particulars because they are neither concrete objects nor universals — they are particular things which do not themselves occur in space or time. History[edit] The concept of "abstract particularity" (German: abstrakte Besonderheit) was introduced in philosophy by G. See also[edit] References[edit] Jump up ^ Georg Wilhelm Friedrich Hegel: The Science of Logic, Cambridge University Press, 2010, p. 609. Further reading[edit]
Trope
Trope or tropes may refer to: Arts and entertainment[edit] Philosophy[edit] Religion[edit] Science and technology[edit] People[edit] Michael Trope (born 1951), American trial lawyer and former sports agent Other uses[edit] Trope of litotes, a rhetorical method of denying a negation See also[edit]
