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In logic and deductive reasoning, an argument is sound if it is both valid in form and has no false premises. [1] Soundness has a related meaning in mathematical logic, wherein a formal system of logic is sound if and only if every well-formed formula that can be proven in the system is logically valid with respect to the logical semantics of the system.
Semantic completeness is the converse of soundness for formal systems. A formal system is complete with respect to tautologousness or "semantically complete" when all its tautologies are theorems, whereas a formal system is "sound" when all theorems are tautologies (that is, they are semantically valid formulas: formulas that are true under every interpretation of the language of the system ...
Given a goal clause, represented as the negation of a problem to be solved : with selected literal , and an input definite clause: . whose positive literal (atom) unifies with the atom of the selected literal , SLD resolution derives another goal clause, in which the selected literal is replaced by the negative literals of the input clause and the unifying substitution is applied:
Logic programming languages are designed specifically to express facts using logical formulas and to draw inferences from these facts. For example, Prolog is a logic programming language based on predicate logic. [170] Computer scientists also apply concepts from logic to problems in computing. The works of Claude Shannon were influential in ...
Logic programming is a programming, database and knowledge representation paradigm based on formal logic. A logic program is a set of sentences in logical form, representing knowledge about some problem domain. Computation is performed by applying logical reasoning to that knowledge, to solve problems in the domain.
In first-order logic, resolution condenses the traditional syllogisms of logical inference down to a single rule. To understand how resolution works, consider the following example syllogism of term logic: All Greeks are Europeans. Homer is a Greek. Therefore, Homer is a European. Or, more generally: .
Hoare logic (also known as Floyd–Hoare logic or Hoare rules) is a formal system with a set of logical rules for reasoning rigorously about the correctness of computer programs. It was proposed in 1969 by the British computer scientist and logician Tony Hoare , and subsequently refined by Hoare and other researchers. [ 1 ]
The soundness of the logic had been a difficult problem, and in fact a counterexample of John Reynolds had shown the unsoundness of an earlier, unpublished version of the logic; the issue raised by Reynolds's example is described briefly in O'Hearn's paper, and more thoroughly in Brookes's.