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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.
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 ...
In mathematical logic, a first-order language of the real numbers is the set of all well-formed sentences of first-order logic that involve universal and existential quantifiers and logical combinations of equalities and inequalities of expressions over real variables.
For first-order logic, the most important case, it follows from the completeness theorem that the two meanings coincide. [2] In other logics, such as second-order logic, there are syntactically consistent theories that are not satisfiable, such as ω-inconsistent theories.
Model theory has a different scope that encompasses more arbitrary first-order theories, including foundational structures such as models of set theory. From the model-theoretic point of view, structures are the objects used to define the semantics of first-order logic, cf. also Tarski's theory of truth or Tarskian semantics.
Prior to the work of these three, term logic (syllogistic logic) was widely considered adequate for formal deductive reasoning. Inferences in term logic can all be represented in the monadic predicate calculus. For example the argument All dogs are mammals. No mammal is a bird. Thus, no dog is a bird.
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: .
In mathematical logic, a first-order predicate is a predicate that takes only individual(s) constants or variables as argument(s). [1] Compare second-order predicate and higher-order predicate . This is not to be confused with a one-place predicate or monad, which is a predicate that takes only one argument.