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Constructive proof. In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existence proof or pure existence theorem), which proves the existence of a particular ...
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of intuitionistic logic do not assume the law of the excluded middle and double negation ...
(Constructive proofs of the specific example above are not hard to produce; for example = and = are both easily shown to be irrational, and =; a proof allowed by intuitionists). By non-constructive Davis means that "a proof that there actually are mathematic entities satisfying certain conditions would not have to provide a method to ...
A non-constructive proof might show a solution exists without specifying either an algorithm to obtain it or a specific bound. Even if the proof is constructive, showing an explicit bounding polynomial and algorithmic details, if the polynomial is not very low-order the algorithm might not be sufficiently efficient in practice.
Existential generalization / instantiation. Constructive dilemma[1][2][3] is a valid rule of inference of propositional logic. It is the inference that, if P implies Q and R implies S and either P or R is true, then either Q or S has to be true. In sum, if two conditionals are true and at least one of their antecedents is, then at least one of ...
Proof theory is a major branch [1] of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as lists, boxed lists, or trees, which are constructed ...
Gödel's completeness theorem. The formula (∀ x. R (x, x)) → (∀ x ∃ y. R (x, y)) holds in all structures (only the simplest 8 are shown left). By Gödel's completeness result, it must hence have a natural deduction proof (shown right). Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a ...
An axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field. [2] [3] [4] This means the following: The real numbers form a set, commonly denoted , containing two distinguished elements denoted 0 and 1, and on which are defined two binary operations and one binary relation; the operations are called addition and multiplication of real ...