# First order

## First Order Wird oft zusammen gekauft

The First Order ist eine fiktive autokratische Militärdiktatur im Star Wars-Franchise, die im Film Star Wars: The Force Awakens von eingeführt wurde. Lernen Sie die Übersetzung für 'first-order' in LEOs Englisch ⇔ Deutsch Wörterbuch. Mit Flexionstabellen der verschiedenen Fälle und Zeiten ✓ Aussprache. Chapter 1 Calculi for First Order Logic In this chapter we give a brief description of a few proof calculi for first order predicate logic with function symbols which. Englisch-Deutsch-Übersetzungen für first order im Online-Wörterbuch halmstadtri.se (Deutschwörterbuch). LEGO Star Wars - First Order Star Destroyer bei halmstadtri.se | Günstiger Preis | Kostenloser Versand ab 29€ für ausgewählte Artikel.

Lernen Sie die Übersetzung für 'first-order' in LEOs Englisch ⇔ Deutsch Wörterbuch. Mit Flexionstabellen der verschiedenen Fälle und Zeiten ✓ Aussprache. Englisch-Deutsch-Übersetzungen für first order im Online-Wörterbuch halmstadtri.se (Deutschwörterbuch). second-order hierarchy is closed under first-order quantifications. This means that, for example, if p belongs to level k of the full second-order alternation.The substitution rule demonstrates several common aspects of rules of inference. It is entirely syntactical; one can tell whether it was correctly applied without appeal to any interpretation.

It has syntactically defined limitations on when it can be applied, which must be respected to preserve the correctness of derivations.

Moreover, as is often the case, these limitations are necessary because of interactions between free and bound variables that occur during syntactic manipulations of the formulas involved in the inference rule.

A deduction in a Hilbert-style deductive system is a list of formulas, each of which is a logical axiom , a hypothesis that has been assumed for the derivation at hand, or follows from previous formulas via a rule of inference.

The logical axioms consist of several axiom schemas of logically valid formulas; these encompass a significant amount of propositional logic.

The rules of inference enable the manipulation of quantifiers. Typical Hilbert-style systems have a small number of rules of inference, along with several infinite schemas of logical axioms.

It is common to have only modus ponens and universal generalization as rules of inference. Natural deduction systems resemble Hilbert-style systems in that a deduction is a finite list of formulas.

However, natural deduction systems have no logical axioms; they compensate by adding additional rules of inference that can be used to manipulate the logical connectives in formulas in the proof.

The sequent calculus was developed to study the properties of natural deduction systems. Unlike the methods just described, the derivations in the tableaux method are not lists of formulas.

Instead, a derivation is a tree of formulas. To show that a formula A is provable, the tableaux method attempts to demonstrate that the negation of A is unsatisfiable.

The resolution rule is a single rule of inference that, together with unification , is sound and complete for first-order logic.

As with the tableaux method, a formula is proved by showing that the negation of the formula is unsatisfiable.

Resolution is commonly used in automated theorem proving. The resolution method works only with formulas that are disjunctions of atomic formulas; arbitrary formulas must first be converted to this form through Skolemization.

Many identities can be proved, which establish equivalences between particular formulas. These identities allow for rearranging formulas by moving quantifiers across other connectives, and are useful for putting formulas in prenex normal form.

Some provable identities include:. There are several different conventions for using equality or identity in first-order logic.

The most common convention, known as first-order logic with equality , includes the equality symbol as a primitive logical symbol which is always interpreted as the real equality relation between members of the domain of discourse, such that the "two" given members are the same member.

This approach also adds certain axioms about equality to the deductive system employed. These equality axioms are: [18] : — These are axiom schemas , each of which specifies an infinite set of axioms.

The third schema is known as Leibniz's law , "the principle of substitutivity", "the indiscernibility of identicals", or "the replacement property".

The second schema, involving the function symbol f , is equivalent to a special case of the third schema, using the formula.

An alternate approach considers the equality relation to be a non-logical symbol. This convention is known as first-order logic without equality.

If an equality relation is included in the signature, the axioms of equality must now be added to the theories under consideration, if desired, instead of being considered rules of logic.

The main difference between this method and first-order logic with equality is that an interpretation may now interpret two distinct individuals as "equal" although, by Leibniz's law, these will satisfy exactly the same formulas under any interpretation.

That is, the equality relation may now be interpreted by an arbitrary equivalence relation on the domain of discourse that is congruent with respect to the functions and relations of the interpretation.

In first-order logic with equality, only normal models are considered, and so there is no term for a model other than a normal model.

When first-order logic without equality is studied, it is necessary to amend the statements of results such as the Löwenheim—Skolem theorem so that only normal models are considered.

First-order logic without equality is often employed in the context of second-order arithmetic and other higher-order theories of arithmetic, where the equality relation between sets of natural numbers is usually omitted.

If a theory has a binary formula A x , y which satisfies reflexivity and Leibniz's law, the theory is said to have equality, or to be a theory with equality.

The theory may not have all instances of the above schemas as axioms, but rather as derivable theorems.

For example, in theories with no function symbols and a finite number of relations, it is possible to define equality in terms of the relations, by defining the two terms s and t to be equal if any relation is unchanged by changing s to t in any argument.

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.

These results concern general properties of first-order logic itself, rather than properties of individual theories.

They provide fundamental tools for the construction of models of first-order theories. Gödel's completeness theorem , proved by Kurt Gödel in , establishes that there are sound, complete, effective deductive systems for first-order logic, and thus the first-order logical consequence relation is captured by finite provability.

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

This means that there is no decision procedure that determines whether arbitrary formulas are logically valid. This result was established independently by Alonzo Church and Alan Turing in and , respectively, giving a negative answer to the Entscheidungsproblem posed by David Hilbert and Wilhelm Ackermann in Their proofs demonstrate a connection between the unsolvability of the decision problem for first-order logic and the unsolvability of the halting problem.

There are systems weaker than full first-order logic for which the logical consequence relation is decidable. These include propositional logic and monadic predicate logic , which is first-order logic restricted to unary predicate symbols and no function symbols.

Other logics with no function symbols which are decidable are the guarded fragment of first-order logic, as well as two-variable logic.

The Bernays—Schönfinkel class of first-order formulas is also decidable. Decidable subsets of first-order logic are also studied in the framework of description logics.

One of the earliest results in model theory , it implies that it is not possible to characterize countability or uncountability in a first-order language with a countable signature.

The Löwenheim—Skolem theorem implies that infinite structures cannot be categorically axiomatized in first-order logic. For example, there is no first-order theory whose only model is the real line: any first-order theory with an infinite model also has a model of cardinality larger than the continuum.

Since the real line is infinite, any theory satisfied by the real line is also satisfied by some nonstandard models.

When the Löwenheim—Skolem theorem is applied to first-order set theories, the nonintuitive consequences are known as Skolem's paradox.

The compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model.

This theorem was proved first by Kurt Gödel as a consequence of the completeness theorem, but many additional proofs have been obtained over time.

It is a central tool in model theory, providing a fundamental method for constructing models. The compactness theorem has a limiting effect on which collections of first-order structures are elementary classes.

For example, the compactness theorem implies that any theory that has arbitrarily large finite models has an infinite model. Thus the class of all finite graphs is not an elementary class the same holds for many other algebraic structures.

There are also more subtle limitations of first-order logic that are implied by the compactness theorem. For example, in computer science, many situations can be modeled as a directed graph of states nodes and connections directed edges.

Validating such a system may require showing that no "bad" state can be reached from any "good" state.

Thus one seeks to determine if the good and bad states are in different connected components of the graph.

Per Lindström showed that the metalogical properties just discussed actually characterize first-order logic in the sense that no stronger logic can also have those properties Ebbinghaus and Flum , Chapter XIII.

Lindström defined a class of abstract logical systems, and a rigorous definition of the relative strength of a member of this class.

He established two theorems for systems of this type:. Although first-order logic is sufficient for formalizing much of mathematics, and is commonly used in computer science and other fields, it has certain limitations.

These include limitations on its expressiveness and limitations of the fragments of natural languages that it can describe.

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

The Löwenheim—Skolem theorem shows that if a first-order theory has any infinite model, then it has infinite models of every cardinality.

In particular, no first-order theory with an infinite model can be categorical. Thus there is no first-order theory whose only model has the set of natural numbers as its domain, or whose only model has the set of real numbers as its domain.

Many extensions of first-order logic, including infinitary logics and higher-order logics, are more expressive in the sense that they do permit categorical axiomatizations of the natural numbers or real numbers.

This expressiveness comes at a metalogical cost, however: by Lindström's theorem , the compactness theorem and the downward Löwenheim—Skolem theorem cannot hold in any logic stronger than first-order.

First-order logic is able to formalize many simple quantifier constructions in natural language, such as "every person who lives in Perth lives in Australia".

But there are many more complicated features of natural language that cannot be expressed in single-sorted first-order logic.

There are many variations of first-order logic. Some of these are inessential in the sense that they merely change notation without affecting the semantics.

Others change the expressive power more significantly, by extending the semantics through additional quantifiers or other new logical symbols.

For example, infinitary logics permit formulas of infinite size, and modal logics add symbols for possibility and necessity.

First-order logic can be studied in languages with fewer logical symbols than were described above. Restrictions such as these are useful as a technique to reduce the number of inference rules or axiom schemas in deductive systems, which leads to shorter proofs of metalogical results.

The cost of the restrictions is that it becomes more difficult to express natural-language statements in the formal system at hand, because the logical connectives used in the natural language statements must be replaced by their longer definitions in terms of the restricted collection of logical connectives.

Similarly, derivations in the limited systems may be longer than derivations in systems that include additional connectives. There is thus a trade-off between the ease of working within the formal system and the ease of proving results about the formal system.

It is also possible to restrict the arities of function symbols and predicate symbols, in sufficiently expressive theories.

One can in principle dispense entirely with functions of arity greater than 2 and predicates of arity greater than 1 in theories that include a pairing function.

This is a function of arity 2 that takes pairs of elements of the domain and returns an ordered pair containing them.

It is also sufficient to have two predicate symbols of arity 2 that define projection functions from an ordered pair to its components.

In either case it is necessary that the natural axioms for a pairing function and its projections are satisfied.

Ordinary first-order interpretations have a single domain of discourse over which all quantifiers range.

Many-sorted first-order logic allows variables to have different sorts , which have different domains. This is also called typed first-order logic , and the sorts called types as in data type , but it is not the same as first-order type theory.

Many-sorted first-order logic is often used in the study of second-order arithmetic. When there are only finitely many sorts in a theory, many-sorted first-order logic can be reduced to single-sorted first-order logic.

One can quantify over each sort by using the corresponding predicate symbol to limit the range of quantification.

Infinitary logic allows infinitely long sentences. For example, one may allow a conjunction or disjunction of infinitely many formulas, or quantification over infinitely many variables.

Infinitely long sentences arise in areas of mathematics including topology and model theory. Infinitary logic generalizes first-order logic to allow formulas of infinite length.

The most common way in which formulas can become infinite is through infinite conjunctions and disjunctions.

However, it is also possible to admit generalized signatures in which function and relation symbols are allowed to have infinite arities, or in which quantifiers can bind infinitely many variables.

Because an infinite formula cannot be represented by a finite string, it is necessary to choose some other representation of formulas; the usual representation in this context is a tree.

Thus formulas are, essentially, identified with their parse trees, rather than with the strings being parsed. Fixpoint logic extends first-order logic by adding the closure under the least fixed points of positive operators.

The 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. Other higher-order logics allow quantification over even higher types than second-order logic permits.

These higher types include relations between relations, functions from relations to relations between relations, and other higher-type objects.

Thus the "first" in first-order logic describes the type of objects that can be quantified. Unlike first-order logic, for which only one semantics is studied, there are several possible semantics for second-order logic.

The most commonly employed semantics for second-order and higher-order logic is known as full semantics.

For that reason, differences in behaviour are often explained by differences just in contents of first - order states.

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