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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.
In a first-order logic system, additional axioms are required to make inferences about the environment (for example, that a block cannot change position unless it is physically moved). The frame problem is the problem of finding adequate collections of axioms for a viable description of a robot environment.
By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced using logical rules and axioms, so the Entscheidungsproblem can also be viewed as asking for an algorithm to decide whether a given statement is provable using the rules of logic.
First-order logic of equality [21] Provability in intuitionistic propositional logic; Satisfaction in modal logic S4 [21] First-order theory of the natural numbers under the successor operation [21] First-order theory of the natural numbers under the standard order [21] First-order theory of the integers under the standard order [21]
From the beginning of the field it was realized that technology to automate logical inferences could have great potential to solve problems and draw conclusions from facts. Ron Brachman has described first-order logic (FOL) as the metric by which all AI knowledge representation formalisms should be evaluated. First-order logic is a general and ...
The situation calculus is a logic formalism designed for representing and reasoning about dynamical domains. It was first introduced by John McCarthy in 1963. [1] The main version of the situational calculus that is presented in this article is based on that introduced by Ray Reiter in 1991.
There are three common ways of handling this in first-order logic: Use first-order logic with two types. Use ordinary first-order logic, but add a new unary predicate "Set", where "Set(t)" means informally "t is a set". Use ordinary first-order logic, and instead of adding a new predicate to the language, treat "Set(t)" as an abbreviation for ...
For propositional logic, systematically applying the resolution rule acts as a decision procedure for formula unsatisfiability, solving the (complement of the) Boolean satisfiability problem. For first-order logic, resolution can be used as the basis for a semi-algorithm for the unsatisfiability problem of first-order logic, providing a more ...