16 Facts About First-order logic

Foundations of first-order logic were developed independently by Gottlob Frege and Charles Sanders Peirce.

Unlike natural languages, such as English, the language of first-order logic is completely formal, so that it can be mechanically determined whether a given expression is well formed.

The terms and formulas of first-order logic are strings of symbols, where all the symbols together form the alphabet of the language.

Tarski and Givant showed that the fragment of first-order logic that has no atomic sentence lying in the scope of more than three quantifiers has the same expressive power as relation algebra.

In general, logical consequence in first-order logic is only semidecidable: if a sentence A logically implies a sentence B then this can be discovered.

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One motivation for the use of first-order logic, rather than higher-order logic, is that first-order logic has many metalogical properties that stronger logics do not have.

Unlike propositional logic, first-order logic is undecidable, provided that the language has at least one predicate of arity at least 2.

Decidable subsets of first-order logic are studied in the framework of description logics.

However, the compactness theorem can be used to show that connected graphs are not an elementary class in first-order logic, and there is no formula f of first-order logic, in the logic of graphs, that expresses the idea that there is a path from x to y Connectedness can be expressed in second-order logic but not with only existential set quantifiers, as enjoys compactness.

Instance, first-order logic is undecidable, meaning a sound, complete and terminating decision algorithm for provability is impossible.

Many-sorted first-order logic allows variables to have different sorts, which have different domains.

Many-sorted first-order logic is often used in the study of second-order arithmetic.

Fixpoint logic extends first-order logic by adding the closure under the least fixed points of positive operators.

Characteristic feature of first-order logic is that individuals can be quantified, but not predicates.

Second-order logic extends first-order logic by adding the latter type of quantification.

The Lowenheim–Skolem theorem and compactness theorem of first-order logic become false when generalized to higher-order logics with full semantics.