** If S is a set of sentences of first-order logic and B is a consistent subset of S, then B is included in a set that is maximal among consistent subsets of S. The special case where S is the set of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal theorem ; see the section " Weaker forms " below.

** Gödel's completeness theorem for first-order logic: every consistent set of first-order sentences has a completion.

The Löwenheim-Skolem theorems tell us that if we restrict ourselves to first-order logic, any axiom system for the reals admits other models, including both models that are smaller than the reals and models that are larger.

In contrast, other, more systematic algorithms achieved, at least theoretically, completeness for first-order logic.

A basic objective of the first normal form defined by Codd in 1970 was to permit data to be queried and manipulated using a " universal data sub-language " grounded in first-order logic.

Properties definable in first-order logic that an equivalence relation may or may not possess include:

* Tarski's axioms: Alfred Tarski ( 1902 – 1983 ) and his students defined elementary Euclidean geometry as the geometry that can be expressed in first-order logic and does not depend on set theory for its logical basis, in contrast to Hilbert's axioms, which involve point sets.

The asks for an algorithm that takes as input a statement of a first-order logic ( possibly with a finite number of axioms beyond the usual axioms of first-order logic ) and answers " Yes " or " No " according to whether the statement is universally valid, i. e., valid in every structure satisfying the axioms.

By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms, so the can also be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic.

Alphabets can also be infinite ; e. g. first-order logic is often expressed using an alphabet which, besides symbols such as ∧, ¬, ∀ and parentheses, contains infinitely many elements x < sub > 0 </ sub >, x < sub > 1 </ sub >, x < sub > 2 </ sub >, … that play the role of variables.

It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic ( a less precise term ).

A theory about some topic is usually first-order logic together with: a specified domain of discourse over which the quantified variables range, finitely many functions which map from that domain into it, finitely many predicates defined on that domain, and a recursive set of axioms which are believed to hold for those things.

Sometimes " theory " is understood in a more formal sense, which is just a set of sentences in first-order logic.

The adjective " first-order " distinguishes first-order logic from higher-order logic in which there are predicates having predicates or functions as arguments, or in which one or both of predicate quantifiers or function quantifiers are permitted.

There are many deductive systems for first-order logic that are sound ( all provable statements are true ) and complete ( all true statements are provable ).

A weaker first-order system called Peano arithmetic is obtained by explicitly adding the addition and multiplication operation symbols and replacing the second-order induction axiom with a first-order axiom schema.

One such consequence is the existence of uncountable models of true arithmetic, which satisfy every first-order induction axiom but have non-inductive subsets.

The axiom system, written in first-order logic, has an infinite number of axioms because an infinite axiom schema is used.

Tarski's axioms, due to Alfred Tarski, are an axiom set for the substantial fragment of Euclidean geometry, called " elementary ," that is formulable in first-order logic with identity, and requiring no set theory.

Boolos argued that if one reads the second-order variables in monadic second-order logic plurally, then second-order logic can be interpreted as having no ontological commitment to entities other than those over which the first-order variables range.

The axiom of choice has also been thoroughly studied in the context of constructive mathematics, where non-classical logic is employed.

Thus, an axiom is an elementary basis for a formal logic system that together with the rules of inference define a deductive system.

Categorical axiom systems for these structures can be obtained in stronger logics such as second-order logic.

With the omission of the law of the excluded middle as an axiom, the remaining logical system has an existence property which classical logic does not: whenever is proven constructively, then in fact is proven constructively for ( at least ) one particular, often called a witness.

However, Principia Mathematica required, in addition to the basic axioms of type theory, three further axioms that seemed to not be true as mere matters of logic, namely the axiom of infinity, the axiom of choice, and the axiom of reducibility.

It turned out, though, that the well-ordering theorem is equivalent to the axiom of choice, in the sense that either one together with the Zermelo – Fraenkel axioms is sufficient to prove the other, in first order logic.

) In second order logic, however, the well-ordering theorem is strictly stronger than the axiom of choice: from the well-ordering theorem one may deduce the axiom of choice, but from the axiom of choice one cannot deduce the well-ordering theorem.

Define Naive Set Theory ( NST ) as the theory of predicate logic with a binary predicate and the following axiom schema of unrestricted comprehension:

Even Russell said that this axiom did not really belong to logic.

The axiom M by itself is not canonical ( Goldblatt, 1991 ), but the combined logic S4. 1 ( in fact, even K4. 1 ) is canonical.

It has the axioms of basic fuzzy logic plus an axiom of double negation, and its models correspond to MV-algebras.

In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo – Fraenkel set theory.

The axiom given above assumes that equality is a primitive symbol in predicate logic.

In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo – Fraenkel set theory.

In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of specification, axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension, is a schema of axioms in Zermelo – Fraenkel set theory.

Each step needed to fetch, decode and execute the machine instructions ( including any operand address calculations, reads and writes ) was controlled directly by combinatorial logic and rather minimal sequential state machine circuitry.

Any person in the present day who wishes to learn logic will be wasting his time if he reads Aristotle or any of his disciples.

The logic, depending on the formulation, reads roughly as follows:

We believe the logic of the book to be unanswerable, its laws fully deduced ", the rest of the sentence in the review reads " and the whole, considered as a play of metaphysical subtlety, absolutely complete ; and yet we venture to predict that its conclusions will not be accepted as probable by one in ten thousand readers.

In this superstitious logic, if a menstruating woman touches a tree it will never again bear fruit ; if she consumes milk the cow will not give anymore milk ; if she reads a book about Saraswati, the goddess of education, she will become angry ; if she touches a man, he will be ill.

In the formal language of the Zermelo – Fraenkel axioms, the axiom reads:

To avoid this consequence, we can modify the axiom of extensionality to apply only to nonempty sets, so that it reads:

In the formal language of the Zermelo – Fraenkel axioms, the axiom reads:

In the formal language of the Zermelo – Fraenkel axioms, the axiom reads:

In the formal language of the Zermelo – Fraenkel axioms, the axiom reads:

In the formal language of the Zermelo – Fraenkel axioms, the axiom reads:

In the formal language of the Zermelo – Fraenkel axioms, the axiom reads: