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An " extensional stance " and restriction to a second-order predicate logic means that a propositional function extended to all individuals such as " All ' x ' are blue " now has to list all of the ' x ' that satisfy ( are true in ) the proposition, listing them in a possibly infinite conjunction: e. g. x < sub > 1 </ sub > V x < sub > 2 </ sub > V.
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extensional and stance
This change from a quasi-intensional stance to a fully extensional stance also restricts predicate logic to the second order, i. e. functions of functions: " We can decide that mathematics is to confine itself to functions of functions which obey the above assumption " ( PM 2nd Edition p. 401, Appendix C ).
extensional and restriction
The Boolean-valued model M < sup > B </ sup > is obtained by applying the above construction inside M. The restriction to transitive models is not serious, as the Mostowski collapsing theorem implies that every " reasonable " ( well-founded, extensional ) model is isomorphic to a transitive one.
extensional and predicate
Datalog distinguishes between extensional and intensional predicate symbols.
While extensional predicate symbols are only defined by facts, intensional predicate symbols are defined only by rules.
In the example above is an intensional predicate symbol, and is extensional.
Predicates may also be defined by facts and rules and therefore neither be purely extensional nor intensional, but any datalog program can be rewritten into an equivalent program without such predicate symbols with duplicate roles.
Another was introduced by Maurice Boffa, with the idea of making foundation fail as badly as it can ( or rather, as extensionality permits ): Boffa's axiom implies that every extensional set-like relation is isomorphic to the elementhood predicate on a transitive class.
extensional and logic
In logic, extensionality, or extensional equality refers to principles that judge objects to be equal if they have the same external properties.
Many of Royce's writings on logic and mathematics are critical of of the extensional logic of Principia Mathematica, by Bertrand Russell and Alfred North Whitehead, and can be read as an alternative to their approach.
Intensional logic is not alone in that: also Gottlob Frege accompanied his ( extensional ) calculus with detailed explanations of the semantical motivations, but the formal foundation of its semantics appeared only in the 20th century.
Thus sometimes similar patterns repeated themselves for the history of development of intensional logic like earlier for that of extensional logic.
Modal logic can be regarded also as the most simple appearance of such studies: it extends extensional logic just with a few sentential functors: these are intensional, and they are interpreted ( in the metarules of semantics ) as quantifying over possible worlds.
It can be contrasted to extensional logic.
extensional and means
For example, an extensional definition of the term " nation of the world " might be given by listing all of the nations of the world, or by giving some other means of recognizing the members of the corresponding class.
extensional and propositional
In extensional Type Theory definitional ( i. e. computational ) equality is not distinguished from propositional equality, which requires proof.
extensional and function
The extensional definition of function equality, discussed above, is commonly used in mathematics.
In case of a so-called extensional functor we can in a sense abstract from the " material " part of its inputs and output, and regard the functor as a function turning directly the extension of its input ( s ) into the extension of its output.
extensional and all
An extensional definition would be a list of all past, present and future prime ministers.
An enumerative definition of a concept or term is an extensional definition that gives an explicit and exhaustive listing of all the objects that fall under the concept or term in question.
This is the opposite approach to the extensional definition, which defines by listing everything that falls under that definition — an extensional definition of bachelor would be a listing of all the unmarried men in the world.
extensional and such
The longer, more shallow northwestern flank displays extensional features, such as large slumps and normal faults.
They are best studied through several other rheological properties which relate stress and strain rate tensors under many different flow conditions, such as oscillatory shear, or extensional flow which are measured using different devices or rheometers.
Alternatively such a fault can be called an extensional fault.
During the late stages of the Alpine Orogeny, as the African Plate collided with Eurasian Plate, Mesozoic extensional structures, such as the Weald Basin of southern England underwent structural inversion.
Other mathematical objects are also constructed in such a way that the intuitive notion of " equality " agrees with set-level extensional equality ; thus, equal ordered pairs have equal elements, and elements of a set which are related by an equivalence relation belong to the same equivalence class.
The best-known such system is the one called classical extensional mereology, hereinafter abbreviated CEM ( other abbreviations are explained below ).
This was a problem for extensional semantics, where the semantics of a word such as red is to be defined as the set of objects having this property.
The area experienced the effects of the early stages of Alpine collision during the Late Cretaceous to Palaeogene, with reverse reactivation of many of the extensional faults, creating the most visible of the structures in the Dorset area, such as the Purbeck Monocline.
Examples of such basinal environments include backarc, forearc, passive margin, epicontinental, and extensional basins.
extensional and All
Perhaps the above can be made clearer by the discussion of classes in Introduction to the 2nd Edition, which disposes of the Axiom of Reducibility and replaces it with the notion: " All functions of functions are extensional " ( PM 1962: xxxix ), i. e.,
extensional and x
In other words, the following extensional equality of functions holds for every x:
extensional and are
Verbs are composed of a root followed by one or more extensional suffixes.
Shear flows are much easier to study and thus much more experimental data are available for shear flows than for extensional flows.
There are acoustic rheometers that employ Stokes ' law for measuring extensional viscosity and volume viscosity.
Fault-bend folds are formed by movement of the hanging wall over a non-planar fault surface and are found associated with both extensional and thrust faults.
In plate tectonics, a divergent boundary or divergent plate boundary ( also known as a constructive boundary or an extensional boundary ) is a linear feature that exists between two tectonic plates that are moving away from each other.
A similar extensional definition is usually employed for relations: two relations are said to be equal if they have the same extensions.
Type-theoretical foundations of mathematics are generally not extensional in this sense, and setoids are commonly used to maintain a difference between intensional equality and a more general equivalence relation ( which generally has poor constructibility or decidability properties ).
Transverse crevasses are the most common crevasse type and they form in a zone of longitudinal extension where the principal stresses are normal to the direction of glacier flow, creating extensional tensile stress.
" The granites are petrologically and geochemically similar to Phanerozoic A-type granitoids, presumably emplaced into noncompressive or extensional environments.
In geology, a rift or chasm is a place where the Earth's crust and lithosphere are being pulled apart and is an example of extensional tectonics.
On the other hand, north-south extensional movements, yet unexplained, are pulling the plate apart, creating normal extensional faults and generating a parallel sequence of horsts and grabens, or rift valleys, running in a north-south direction.
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