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It is also possible to define game semantics for first-order logic , but aside from requiring the axiom of choice , game semantics agree with Tarskian semantics for first-order logic, so game semantics will not be elaborated herein.
The most common way of specifying an interpretation especially in mathematics is to specify a structure also called a model ; see below. The structure consists of a domain of discourse D and an interpretation function I mapping non-logical symbols to predicates, functions, and constants.
The domain of discourse D is a nonempty set of "objects" of some kind. For example, one can take D to be the set of integers. The truth value of this formula changes depending on whether x and y denote the same individual.
The following rules are used to make this assignment:. Next, each formula is assigned a truth value. The inductive definition used to make this assignment is called the T-schema. If a formula does not contain free variables, and so is a sentence, then the initial variable assignment does not affect its truth value. There is a second common approach to defining truth values that does not rely on variable assignment functions.
Instead, given an interpretation M , one first adds to the signature a collection of constant symbols, one for each element of the domain of discourse in M ; say that for each d in the domain the constant symbol c d is fixed. The interpretation is extended so that each new constant symbol is assigned to its corresponding element of the domain.
One now defines truth for quantified formulas syntactically, as follows:. This alternate approach gives exactly the same truth values to all sentences as the approach via variable assignments. A sentence is satisfiable if there is some interpretation under which it is true. Satisfiability of formulas with free variables is more complicated, because an interpretation on its own does not determine the truth value of such a formula.
The most common convention is that a formula with free variables is said to be satisfied by an interpretation if the formula remains true regardless which individuals from the domain of discourse are assigned to its free variables. This has the same effect as saying that a formula is satisfied if and only if its universal closure is satisfied.
A formula is logically valid or simply valid if it is true in every interpretation. An alternate approach to the semantics of first-order logic proceeds via abstract algebra. This approach generalizes the Lindenbaum—Tarski algebras of propositional logic. There are three ways of eliminating quantified variables from first-order logic that do not involve replacing quantifiers with other variable binding term operators:.
These algebras are all lattices that properly extend the two-element Boolean algebra. 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.
They also prove that first-order logic with a primitive ordered pair is equivalent to a relation algebra with two ordered pair projection functions. A first-order theory of a particular signature is a set of axioms , which are sentences consisting of symbols from that signature.
The set of axioms is often finite or recursively enumerable , in which case the theory is called effective. Some authors require theories to also include all logical consequences of the axioms. The axioms are considered to hold within the theory and from them other sentences that hold within the theory can be derived.
A first-order structure that satisfies all sentences in a given theory is said to be a model of the theory. An elementary class is the set of all structures satisfying a particular theory.
These classes are a main subject of study in model theory. Many theories have an intended interpretation , a certain model that is kept in mind when studying the theory. For example, the intended interpretation of Peano arithmetic consists of the usual natural numbers with their usual operations. A theory is consistent if it is not possible to prove a contradiction from the axioms of the theory.
A theory is complete if, for every formula in its signature, either that formula or its negation is a logical consequence of the axioms of the theory. The definition above requires that the domain of discourse of any interpretation must be nonempty. There are settings, such as inclusive logic , where empty domains are permitted. Moreover, if a class of algebraic structures includes an empty structure for example, there is an empty poset , that class can only be an elementary class in first-order logic if empty domains are permitted or the empty structure is removed from the class.
Thus, when the empty domain is permitted, it must often be treated as a special case. Most authors, however, simply exclude the empty domain by definition. A deductive system is used to demonstrate, on a purely syntactic basis, that one formula is a logical consequence of another formula. There are many such systems for first-order logic, including Hilbert-style deductive systems , natural deduction , the sequent calculus , the tableaux method , and resolution.
These share the common property that a deduction is a finite syntactic object; the format of this object, and the way it is constructed, vary widely. These finite deductions themselves are often called derivations in proof theory. They are also often called proofs , but are completely formalized unlike natural-language mathematical proofs. A deductive system is sound if any formula that can be derived in the system is logically valid.
Conversely, a deductive system is complete if every logically valid formula is derivable. All of the systems discussed in this article are both sound and complete. They also share the property that it is possible to effectively verify that a purportedly valid deduction is actually a deduction; such deduction systems are called effective. A key property of deductive systems is that they are purely syntactic, so that derivations can be verified without considering any interpretation.
Thus a sound argument is correct in every possible interpretation of the language, regardless of whether that interpretation is about mathematics, economics, or some other area. 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 for example, by searching for a proof until one is found, using some effective, sound, complete proof system.
However, if A does not logically imply B, this does not mean that A logically implies the negation of B. There is no effective procedure that, given formulas A and B, always correctly decides whether A logically implies B. A rule of inference states that, given a particular formula or set of formulas with a certain property as a hypothesis, another specific formula or set of formulas can be derived as a conclusion.
The rule is sound or truth-preserving if it preserves validity in the sense that whenever any interpretation satisfies the hypothesis, that interpretation also satisfies the conclusion. For example, one common rule of inference is the rule of substitution.
The problem is that the free variable x of t became bound during the substitution. 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 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.
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. 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. 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.
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. 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.
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.
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. 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. 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".
Hence, first-order logic is used as a basis for knowledge representation languages , such as FO. Still, there are complicated features of natural language that cannot be expressed in 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. With your permission we and our partners may use precise geolocation data and identification through device scanning.
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