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Quantifier elimination is a concept of simplification used in mathematical logic, model theory, and theoretical computer science.Informally, a quantified statement "such that …" can be viewed as a question "When is there an such that …?", and the statement without quantifiers can be viewed as the answer to that question.
This is generally considered one of the most important open questions in mathematics and theoretical computer science as it has far-reaching consequences to other problems in mathematics, to biology, [14] philosophy [15] and to cryptography (see P versus NP problem proof consequences). A common example of an NP problem not known to be in P is ...
Logical consequence is necessary and formal, by way of examples that explain with formal proof and models of interpretation. [1] A sentence is said to be a logical consequence of a set of sentences, for a given language , if and only if , using only logic (i.e., without regard to any personal interpretations of the sentences) the sentence must ...
Monty Hall problem, also known as the Monty Hall paradox: [2] An unintuitive consequence of conditional probability. Necktie paradox: A wager between two people seems to favour them both. Very similar in essence to the Two-envelope paradox. Proebsting's paradox: The Kelly criterion is an often optimal strategy for maximizing profit in the long ...
The rational consequence relation is non-monotonic, and the relation is intended to carry the meaning theta usually implies phi or phi usually follows from theta.In this sense it is more useful for modeling some everyday situations than a monotone consequence relation because the latter relation models facts in a more strict boolean fashion—something either follows under all circumstances or ...
The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of mathematics. Additionally, the subsequent columns contains an informal explanation, a short example, the Unicode location, the name for use in HTML documents, [ 1 ] and the LaTeX symbol.
A second thread in the history of foundations of mathematics involves nonclassical logics and constructive mathematics. The study of constructive mathematics includes many different programs with various definitions of constructive. At the most accommodating end, proofs in ZF set theory that do not use the axiom of choice are called ...
For example, even though material conditionals with false antecedents are vacuously true, the natural language statement "If 8 is odd, then 3 is prime" is typically judged false. Similarly, any material conditional with a true consequent is itself true, but speakers typically reject sentences such as "If I have a penny in my pocket, then Paris ...