Mathematical proofA mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation".
Message Passing InterfaceMessage Passing Interface (MPI) is a standardized and portable message-passing standard designed to function on parallel computing architectures. The MPI standard defines the syntax and semantics of library routines that are useful to a wide range of users writing portable message-passing programs in C, C++, and Fortran. There are several open-source MPI implementations, which fostered the development of a parallel software industry, and encouraged development of portable and scalable large-scale parallel applications.
Proof (truth)A proof is sufficient evidence or a sufficient argument for the truth of a proposition. The concept applies in a variety of disciplines, with both the nature of the evidence or justification and the criteria for sufficiency being area-dependent. In the area of oral and written communication such as conversation, dialog, rhetoric, etc., a proof is a persuasive perlocutionary speech act, which demonstrates the truth of a proposition.
LogicLogic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or logical truths. It studies how conclusions follow from premises due to the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. It examines arguments expressed in natural language while formal logic uses formal language.
Proof theoryProof theory is a major branch 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 according to the axioms and rules of inference of a given logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.
Proof by contradictionIn logic, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Although it is quite freely used in mathematical proofs, not every school of mathematical thought accepts this kind of nonconstructive proof as universally valid. More broadly, proof by contradiction is any form of argument that establishes a statement by arriving at a contradiction, even when the initial assumption is not the negation of the statement to be proved.
Law of excluded middleIn logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradiction, and the law of identity. However, no system of logic is built on just these laws, and none of these laws provides inference rules, such as modus ponens or De Morgan's laws. The law is also known as the law (or principle) of the excluded third, in Latin principium tertii exclusi.
Mathematical inductionMathematical induction is a method for proving that a statement is true for every natural number , that is, that the infinitely many cases all hold. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung (the basis) and that from each rung we can climb up to the next one (the step). A proof by induction consists of two cases.
Philosophy of logicPhilosophy of logic is the area of philosophy that studies the scope and nature of logic. It investigates the philosophical problems raised by logic, such as the presuppositions often implicitly at work in theories of logic and in their application. This involves questions about how logic is to be defined and how different logical systems are connected to each other. It includes the study of the nature of the fundamental concepts used by logic and the relation of logic to other disciplines.
Propositional calculusPropositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives. Propositions that contain no logical connectives are called atomic propositions.
