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  2. First-order logic - Wikipedia

    en.wikipedia.org/wiki/First-order_logic

    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.

  3. Logic - Wikipedia

    en.wikipedia.org/wiki/Logic

    First-order logic includes the same propositional connectives as propositional logic but differs from it because it articulates the internal structure of propositions. This happens through devices such as singular terms, which refer to particular objects, predicates , which refer to properties and relations, and quantifiers, which treat notions ...

  4. Propositional calculus - Wikipedia

    en.wikipedia.org/wiki/Propositional_calculus

    In this sense, propositional logic is the foundation of first-order logic and higher-order logic. Propositional logic is typically studied with a formal language, [c] in which propositions are represented by letters, which are called propositional variables. These are then used, together with symbols for connectives, to make propositional formula.

  5. List of axiomatic systems in logic - Wikipedia

    en.wikipedia.org/wiki/List_of_axiomatic_systems...

    Classical propositional calculus is the standard propositional logic. Its intended semantics is bivalent and its main property is that it is strongly complete, otherwise said that whenever a formula semantically follows from a set of premises, it also follows from that set syntactically. Many different equivalent complete axiom systems have ...

  6. List of logic symbols - Wikipedia

    en.wikipedia.org/wiki/List_of_logic_symbols

    propositional logic, Boolean algebra, first-order logic ⊤ {\displaystyle \top } denotes a proposition that is always true. The proposition ⊤ ∨ P {\displaystyle \top \lor P} is always true since at least one of the two is unconditionally true.

  7. Predicate (mathematical logic) - Wikipedia

    en.wikipedia.org/wiki/Predicate_(mathematical_logic)

    In propositional logic, atomic formulas are sometimes regarded as zero-place predicates. [1] In a sense, these are nullary (i.e. 0-arity) predicates. In first-order logic, a predicate forms an atomic formula when applied to an appropriate number of terms.

  8. Circumscription (logic) - Wikipedia

    en.wikipedia.org/wiki/Circumscription_(logic)

    The role of variables in propositional logic (something that can be true or false) is played in first-order logic by predicates. Namely, a propositional formula can be expressed in first-order logic by replacing each propositional variable with a predicate of zero arity (i.e., a predicate with no arguments). Therefore, minimization is done on ...

  9. Completeness (logic) - Wikipedia

    en.wikipedia.org/wiki/Completeness_(logic)

    Examples of refutation-complete systems include: SLD resolution on Horn clauses, superposition on equational clausal first-order logic, Robinson's resolution on clause sets. [3] The latter is not strongly complete: e.g. { a } ⊨ a ∨ b {\displaystyle \{a\}\models a\lor b} holds even in the propositional subset of first-order logic, but a ∨ ...