* Lawrence C. Paulson of the University of Cambridge, work on higher-order logic system, co-developer of the Isabelle Theorem Prover

General category theory, an extension of universal algebra having many new features allowing for semantic flexibility and higher-order logic, came later ; it is now applied throughout mathematics.

The adjective " first-order " distinguishes first-order logic from higher-order logic in which there are predicates having predicates or functions as arguments, or in which one or both of predicate quantifiers or function quantifiers are permitted.

Second-order logic, for example, does not have a completeness theorem for its standard semantics ( but does have the completeness property for Henkin semantics ), and the same is true of all higher-order logics.

The history of the problem of vagueness is traced, from the first Sorites Paradox to contemporary attempts to deal with higher-order vagueness such as many-valued logic, supervaluationism, and fuzzy logic.

Alonzo Church, inventor of the lambda calculus, developed a higher-order logic commonly called Church's Theory of Types, in order to avoid the Kleene – Rosser paradox afflicting the original pure lambda calculus.

This library implements an abstract data type of proven theorems so that new objects of this type can only be created using the functions in the library which correspond to inference rules in higher-order logic.

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 Lawvere programme was to write higher-order logic in terms of category theory.

Second-order logic is in turn extended by higher-order logic and type theory.

* Theorem Proving System, an automated theorem proving system for first-order and higher-order logic

Henkin originally proved the completeness of Church's higher-order logic, and then observed that the same methods of proof could be applied to first-order logic.

Henkin's proof for higher-order logic uses a variant of the standard semantics.

In mathematics and logic, a higher-order logic is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and a stronger semantics.

The term " higher-order logic ", abbreviated as HOL, is commonly used to mean higher order simple predicate logic.

* Filter ( higher-order function ), a higher-order function that processes a data structure ( typically a list ) in some order to produce a new data structure containing exactly those elements of the original data structure for which a given predicate returns the boolean value true

Compare second-order predicate and higher-order predicate.

Compare higher-order predicate.

Its syntax was based on PDDL, but it had a much more elaborate type system, which allowed users to make use of higher-order constructs such as explicit λ-expressions allowing for efficient type-inference ( i. e. not only domain objects had types ( level 0 types ), but also the functions / fluents defined above these objects had types in the form of arbitrary mappings ( level 1 types ), which could be generic, so their parameters ( the domain and range of the generic mapping ) could be defined with variables, which could have an even higher level type ( level 2 type ) not to speak of that the mappings could be arbitrary, i. e. the domain or range of a function ( e. g. predicate, numeric fluent ) could be any level 0 / 1 / 2 type.

Constraints with one, two, or more variables are called unary, binary, or higher-order constraints.

The two higher-order judgements, the hypothetical and the general,, correspond to the ordinary and dependent function space, respectively.

For proof systems based on higher-order typed lambda calculus through a Curry – Howard isomorphism, cut elimination algorithms correspond to the strong normalization property ( every proof term reduces in a finite number of step into a normal form ).

It does have a notion of " generator ", which amounts to a function accepting a function as an argument, and, since it is an assembly-level language, code can be used as data, so IPL can be regarded as having higher-order functions.

In calculus, an example of a higher-order function is the differential operator, which returns the derivative of a function.

Later in his life, Greenberg proposed that nearly all of the language families of northern Eurasia belong to a single higher-order family, which he called Eurasiatic.

The N-terminal tail of histone H4, on the other hand, has a region of highly basic amino acids ( 16-25 ), which, in the crystal structure, forms an interaction with the highly acidic surface region of a H2A-H2B dimer of another nucleosome, being potentially relevant for the higher-order structure of nucleosomes.

Although the latter is higher-order, viewing positive cones as maximal prepositive cones provides a larger context in which field orderings are extremal partial orderings.

A formalism has been worked out for those piezoelectric crystals, for which the polarization is of the crystal-field induced type, that allows for the calculation of piezoelectrical coefficients from electrostatic lattice constants or higher-order Madelung constants.

Shen Kuo was also known for hydraulic clockworks, as he invented a new overflow-tank clepsydra which had more efficient higher-order interpolation instead of linear interpolation in calibrating the measure of time.

* Dorsolateral prefrontal cortex, which decides which voluntary movements to make according to higher-order instructions, rules, and self-generated thoughts.

A short comparison with pulse-width modulation shows that a 1-bit DAC with a simple first-order integrator would have to run at 3 THz ( which is physically unrealizable ) to achieve 24 meaningful bits of resolution, requiring a higher-order low-pass filter in the noise-shaping loop.

In this example, would be the known solution to the exactly solvable initial problem and, ... represent the higher-order terms which may be found iteratively by some systematic procedure.

While it was easy to maintain that this was the value of goods consumed by the end user ( lower-order goods ), it was harder to make this case for higher-order goods which had no end user and merely went into the making of lower-order goods.

* Constructive cultures, in which members are encouraged to interact with people and approach tasks in ways that help them meet their higher-order satisfaction needs.

In the untyped lambda calculus, all functions are higher-order ; in a typed lambda calculus, from which most functional programming languages are derived, higher-order functions are values with types of the form.

Another very common kind of higher-order function in those languages which support them are sorting functions which take a comparison function as a parameter, allowing the programmer to separate the sorting algorithm from the comparisons of the items being sorted.

This Python program contains the higher-order function g () which takes a function as its first argument and returns a number.

*" Alerter system " ( in higher-order systems in which the switch activates to sound an alarm rather than deactivates to disable the higher-order system )

By preserving the value, only one division operation is needed and the higher-order statistics can thus be calculated for little incremental cost.

The distinction between the two is subtle: " higher-order " describes a mathematical concept of functions that operate on other functions, while " first-class " is a computer science term that describes programming language entities that have no restriction on their use ( thus first-class functions can appear anywhere in the program that other first-class entities like numbers can, including as arguments to other functions and as their return values ).

It is possible to produce sound deductive systems for higher-order logics, but no such system can be complete.

These higher-order averages are more difficult to measure, but can yield additional information about molecular ordering.

Several such chunks can then be combined into a higher-order chunk, thereby forming a hierarchy of chunks.

In elliptic coordinates, one can write the higher-order modes using Ince polynomials.

* Another form of Bilingual Education is a type of Dual Language program that has students study in two different ways: 1 ) A variety of academic subjects are taught in the students ' second language, with specially trained bilingual teachers who can understand students when they ask questions in their native language, but always answer in the second language ; and 2 ) Native language literacy classes improve students ' writing and higher-order language skills in their first language.

Other binding operators, like the summation sign, can be thought of as higher-order functions applying to a function.

Subleq2 can also be used to synthesize higher-order instructions, although it generally requires more operations for a given task.

To first order, a self-balancing unicycle can be considered as a non-linear control system similar to that of a two-dimensional inverted pendulum with a unicycle cart at its base: however, there are many higher-order effects involved in modelling the full system.

Since Haskell supports higher-order functions, functions can be passed as arguments.

We can find the higher-order deviations by a similar procedure, though the calculations become quite tedious with our current formulation.

for the derivative can be applied recursively to get higher-order derivatives.

However, recent research in the Netherlands ( involving 19 grapheme-color synesthetes ) demonstrated that synesthesia can be the result of either direct bottom-up cross-activation from grapheme processing areas within the fusiform gyrus, or indirectly via higher-order parietal areas.

While the net magnetic field produced by the system can also have higher-order multipole components, those will drop off with distance more rapidly, so that only the dipolar component will dominate the magnetic field of the system at distances far away from it.

In computer science, a function type ( also arrow type or exponential ) is the type of a variable or parameter to which a function has or can be assigned or the result type of a higher-order function returning a function.

At high pumping levels in long fibers, higher-order Raman spectra can be generated by using the Raman spectrum as a new starting point, thereby building a chain of new spectra with decreasing amplitude.

Shopping malls also tend to serve a wider ( regional ) public and require higher-order infrastructure such as highway access and can have floorspaces in excess of a million square feet ( ca.

He posits that the constitution must create a " higher-order law " in which human rights and constitutional fundamentals in a democracy can be protected by the courts against the abuses of government.

All groups that do not contain several higher-order axes ( order 3 or more ) can be arranged in a table:

Agent-based models can explain the emergence of higher-order patterns — network structures of terrorist organizations and the Internet, power-law distributions in the sizes of traffic jams, wars, and stock-market crashes, and social segregation that persists despite populations of tolerant people.