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Derivation of equations that describe the time course of change for a system with zero-order input and first-order elimination are presented in the articles Exponential decay and Biological half-life, and in scientific literature. [1] [7] = C t is concentration after time t
First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem. First-order logic is the standard for the formalization of mathematics into axioms, and is studied in the foundations of mathematics.
A variant of Gaussian elimination called Gauss–Jordan elimination can be used for finding the inverse of a matrix, if it exists. If A is an n × n square matrix, then one can use row reduction to compute its inverse matrix, if it exists. First, the n × n identity matrix is augmented to the right of A, forming an n × 2n block matrix [A | I].
Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or quantifiers. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logic is the foundation of first-order logic and higher-order logic.
The corresponding first-order theory is the set of sentences that are actually true of the real numbers. There are several different such theories, with different expressive power, depending on the primitive operations that are allowed to be used in the expression.
First-order logic, a formal logical system used in mathematics, philosophy, linguistics, and computer science; First-order predicate, a predicate that takes only individual(s) constants or variables as argument(s) First-order predicate calculus; First-order theorem provers; First-order theory; Monadic first-order logic
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 in 1934/1935 by Gerhard Gentzen [1] as a tool for studying natural deduction in first-order logic (in classical and intuitionistic versions, respectively).
First-order approximation is the term scientists use for a slightly better answer. [3] Some simplifying assumptions are made, and when a number is needed, an answer with only one significant figure is often given ("the town has 4 × 10 3, or four thousand, residents"). In the case of a first-order approximation, at least one number given is exact.