In logic and mathematics **second-order logic** is an extension of first-order logic, which itself is an extension of propositional logic.^{ [1] } Second-order logic is in turn extended by higher-order logic and type theory.

- Examples
- Syntax and fragments
- Semantics
- Expressive power
- Deductive systems
- Non-reducibility to first-order logic
- Metalogical results
- History and disputed value
- Relation to computational complexity
- See also
- Notes
- References
- Further reading

First-order logic quantifies only variables that range over individuals (elements of the domain of discourse); second-order logic, in addition, also quantifies over relations. For example, the second-order sentence says that for every formula *P*, and every individual *x*, either *Px* is true or not(*Px*) is true (this is the law of excluded middle). Second-order logic also includes quantification over sets, functions, and other variables (see section below). Both first-order and second-order logic use the idea of a domain of discourse (often called simply the "domain" or the "universe"). The domain is a set over which individual elements may be quantified.

First-order logic can quantify over individuals, but not over properties. That is, we can take an atomic sentence like Cube(*b*) and obtain a quantified sentence by replacing the name with a variable and attaching a quantifier:^{ [2] }

- ∃
*x*Cube(*x*)

But we cannot do the same with the predicate. That is, the following expression:

- ∃P P(
*b*)

is not a sentence of first-order logic. But this is a legitimate sentence of second-order logic.^{ [2] }

As a result, second-order logic has much more “expressive power” than first-order logic does. For example, there is no way in first-order logic to say that *a* and *b* have some property in common; but in second-order logic this would be expressed as

- ∃P (P(
*a*) ∧ P(*b*)).

Suppose we would like to say that *a* and *b* have the same shape. The best we could do in first-order logic is something like this:

- (Cube(
*a*) ∧ Cube(*b*)) ∨ (Tet(*a*) ∧ Tet(*b*)) ∨ (Dodec(*a*) ∧ Dodec(*b*))

If the only shapes are cube, tetrahedron, and dodecahedron, for *a* and *b* to have the same shape is for them either to be both cubes, both tetrahedra, or both dodecahedra. But this first-order logic sentence doesn’t seem to mean quite the same thing as the English sentence it is translating—for example, it doesn’t say anything about the fact that it is shape that *a* and *b* have in common.^{ [2] }

In second-order logic, by contrast, we could add a predicate Shape that is true of precisely the properties corresponding to the predicates Cube, Tet, and Dodec. That is,

- Shape(Cube) ∧ Shape(Tet) ∧ Shape(Dodec)

So we could write:

- ∃P (Shape(P) ∧ P(a) ∧ P(
*b*))

And this is true exactly when *a* and *b* are both cubes, both tetrahedra, or both dodecahedra. So in second-order logic we can express the idea of *same shape* using identity and the second-order predicate Shape; we can do without the special predicate SameShape.^{ [2] }

Similarly, we can express the claim that no object has every shape in a way that brings out the quantifier in *every shape*:

- ¬∃
*x*∀P(Shape(P) → P(*x*))

In first-order logic a block is said to be one of the following: a cube, a tetrahedron, or a dodecahedron:^{ [3] }^{:258}

- ¬∃
*x*(Cube(*x*) ∧ Tet(*x*) ∧ Dodec(*x*))

The syntax of second-order logic tells which expressions are well formed formulas. In addition to the syntax of first-order logic, second-order logic includes many new **sorts** (sometimes called **types**) of variables. These are:

- A sort of variables that range over sets of individuals. If
*S*is a variable of this sort and*t*is a first-order term then the expression*t*∈*S*(also written*S*(*t*), or*St*to save parentheses) is an atomic formula. Sets of individuals can also be viewed as unary relations on the domain. - For each natural number
*k*there is a sort of variables that ranges over all*k*-ary relations on the individuals. If*R*is such a*k*-ary relation variable and*t*_{1},...,*t*_{k}are first-order terms then the expression*R*(*t*_{1},...,*t*_{k}) is an atomic formula. - For each natural number
*k*there is a sort of variables that ranges over all functions taking*k*elements of the domain and returning a single element of the domain. If*f*is such a*k*-ary function variable and*t*_{1},...,*t*_{k}are first-order terms then the expression*f*(*t*_{1},...,*t*_{k}) is a first-order term.

Each of the variables just defined may be universally and/or existentially quantified over, to build up formulas. Thus there are many kinds of quantifiers, two for each sort of variables. A **sentence** in second-order logic, as in first-order logic, is a well-formed formula with no free variables (of any sort).

It's possible to forgo the introduction of function variables in the definition given above (and some authors do this) because an *n*-ary function variable can be represented by a relation variable of arity *n*+1 and an appropriate formula for the uniqueness of the "result" in the *n*+1 argument of the relation. (Shapiro 2000, p. 63)

** Monadic second-order logic ** (MSO) is a restriction of second-order logic in which only quantification over unary relations (i.e. sets) is allowed. Quantification over functions, owing to the equivalence to relations as described above, is thus also not allowed. The second-order logic without these restrictions is sometimes called **full second-order logic** to distinguish it from the monadic version. Monadic second-order logic is particularly used in the context of Courcelle's theorem, an algorithmic meta-theorem in graph theory.

Just as in first-order logic, second-order logic may include non-logical symbols in a particular second-order language. These are restricted, however, in that all terms that they form must be either first-order terms (which can be substituted for a first-order variable) or second-order terms (which can be substituted for a second-order variable of an appropriate sort).

A formula in second-order logic is said to be of first-order (and sometimes denoted or ) if its quantifiers (which may be universal or existential) range only over variables of first order, although it may have free variables of second order. A (existential second-order) formula is one additionally having some existential quantifiers over second order variables, i.e. , where is a first-order formula. The fragment of second-order logic consisting only of existential second-order formulas is called **existential second-order logic** and abbreviated as ESO, as , or even as ∃SO. The fragment of formulas is defined dually, it is called universal second-order logic. More expressive fragments are defined for any *k* > 0 by mutual recursion: has the form , where is a formula, and similar, has the form , where is a formula. (See analytical hierarchy for the analogous construction of second-order arithmetic.)

The semantics of second-order logic establish the meaning of each sentence. Unlike first-order logic, which has only one standard semantics, there are two different semantics that are commonly used for second-order logic: **standard semantics** and **Henkin semantics**. In each of these semantics, the interpretations of the first-order quantifiers and the logical connectives are the same as in first-order logic. Only the ranges of quantifiers over second-order variables differ in the two types of semantics (Väänänen 2001).

In standard semantics, also called full semantics, the quantifiers range over *all* sets or functions of the appropriate sort. Thus once the domain of the first-order variables is established, the meaning of the remaining quantifiers is fixed. It is these semantics that give second-order logic its expressive power, and they will be assumed for the remainder of this article.

In Henkin semantics, each sort of second-order variable has a particular domain of its own to range over, which may be a proper subset of all sets or functions of that sort. Leon Henkin (1950) defined these semantics and proved that Gödel's completeness theorem and compactness theorem, which hold for first-order logic, carry over to second-order logic with Henkin semantics. This is because Henkin semantics are almost identical to many-sorted first-order semantics, where additional sorts of variables are added to simulate the new variables of second-order logic. Second-order logic with Henkin semantics is not more expressive than first-order logic. Henkin semantics are commonly used in the study of second-order arithmetic.

Jouko Väänänen (2001) argued that the choice between Henkin models and full models for second-order logic is analogous to the choice between ZFC and * V * as a basis for set theory: "As with second-order logic, we cannot really choose whether we axiomatize mathematics using *V* or ZFC. The result is the same in both cases, as ZFC *is* the best attempt so far to use *V* as an axiomatization of mathematics."

Second-order logic is more expressive than first-order logic. For example, if the domain is the set of all real numbers, one can assert in first-order logic the existence of an additive inverse of each real number by writing ∀*x* ∃*y* (*x* + *y* = 0) but one needs second-order logic to assert the least-upper-bound property for sets of real numbers, which states that every bounded, nonempty set of real numbers has a supremum. If the domain is the set of all real numbers, the following second-order sentence (split over two lines) expresses the least upper bound property:

- (∀ A) ([(∃
*w*) (*w*∈ A)∧(∃*z*)(∀*u*)(*u*∈ A →*u*≤*z*)] - :→(∃
*x*)(∀*y*)([(∀*w*)(*w*∈ A →*w*≤*x*)] ∧ [(∀*u*)(*u*∈ A →*u*≤*y*)] →*x*≤*y*))

This formula is a direct formalization of "every nonempty, bounded set A has a least upper bound." It can be shown that any ordered field that satisfies this property is isomorphic to the real number field. On the other hand, the set of first-order sentences valid in the reals has arbitrarily large models due to the compactness theorem. Thus the least-upper-bound property cannot be expressed by any set of sentences in first-order logic. (In fact, every real-closed field satisfies the same first-order sentences in the signature as the real numbers.)

In second-order logic, it is possible to write formal sentences that say "the domain is finite" or "the domain is of countable cardinality." To say that the domain is finite, use the sentence that says that every surjective function from the domain to itself is injective. To say that the domain has countable cardinality, use the sentence that says that there is a bijection between every two infinite subsets of the domain. It follows from the compactness theorem and the upward Löwenheim–Skolem theorem that it is not possible to characterize finiteness or countability, respectively, in first-order logic.

Certain fragments of second-order logic like ESO are also more expressive than first-order logic even though they are strictly less expressive than the full second-order logic. ESO also enjoys translation equivalence with some extensions of first-order logic that allow non-linear ordering of quantifier dependencies, like first-order logic extended with Henkin quantifiers, Hintikka and Sandu's independence-friendly logic, and Väänänen's dependence logic.

A deductive system for a logic is a set of inference rules and logical axioms that determine which sequences of formulas constitute valid proofs. Several deductive systems can be used for second-order logic, although none can be complete for the standard semantics (see below). Each of these systems is sound, which means any sentence they can be used to prove is logically valid in the appropriate semantics.

The weakest deductive system that can be used consists of a standard deductive system for first-order logic (such as natural deduction) augmented with substitution rules for second-order terms.^{ [4] } This deductive system is commonly used in the study of second-order arithmetic.

The deductive systems considered by Shapiro (1991) and Henkin (1950) add to the augmented first-order deductive scheme both comprehension axioms and choice axioms. These axioms are sound for standard second-order semantics. They are sound for Henkin semantics restricted to Henkin models satisfying the comprehension and choice axioms.^{ [5] }

One might attempt to reduce the second-order theory of the real numbers, with full second-order semantics, to the first-order theory in the following way. First expand the domain from the set of all real numbers to a two-sorted domain, with the second sort containing all *sets of* real numbers. Add a new binary predicate to the language: the membership relation. Then sentences that were second-order become first-order, with the formerly second-order quantifiers ranging over the second sort instead. This reduction can be attempted in a one-sorted theory by adding unary predicates that tell whether an element is a number or a set, and taking the domain to be the union of the set of real numbers and the power set of the real numbers.

But notice that the domain was asserted to include * all* sets of real numbers. That requirement cannot be reduced to a first-order sentence, as the Löwenheim–Skolem theorem shows. That theorem implies that there is some countably infinite subset of the real numbers, whose members we will call

Every nonempty *internal* set that has an *internal* upper bound has a least *internal* upper bound.

Countability of the set of all internal numbers (in conjunction with the fact that those form a densely ordered set) implies that that set does not satisfy the full least-upper-bound axiom. Countability of the set of all *internal* sets implies that it is not the set of *all* subsets of the set of all *internal* numbers (since Cantor's theorem implies that the set of all subsets of a countably infinite set is an uncountably infinite set). This construction is closely related to Skolem's paradox.

Thus the first-order theory of real numbers and sets of real numbers has many models, some of which are countable. The second-order theory of the real numbers has only one model, however. This follows from the classical theorem that there is only one Archimedean complete ordered field, along with the fact that all the axioms of an Archimedean complete ordered field are expressible in second-order logic. This shows that the second-order theory of the real numbers cannot be reduced to a first-order theory, in the sense that the second-order theory of the real numbers has only one model but the corresponding first-order theory has many models.

There are more extreme examples showing that second-order logic with standard semantics is more expressive than first-order logic. There is a finite second-order theory whose only model is the real numbers if the continuum hypothesis holds and that has no model if the continuum hypothesis does not hold (cf. Shapiro 2000, p. 105). This theory consists of a finite theory characterizing the real numbers as a complete Archimedean ordered field plus an axiom saying that the domain is of the first uncountable cardinality. This example illustrates that the question of whether a sentence in second-order logic is consistent is extremely subtle.

Additional limitations of second-order logic are described in the next section.

It is a corollary of Gödel's incompleteness theorem that there is no deductive system (that is, no notion of *provability*) for second-order formulas that simultaneously satisfies these three desired attributes:^{ [6] }

- (Soundness) Every provable second-order sentence is universally valid, i.e., true in all domains under standard semantics.
- (Completeness) Every universally valid second-order formula, under standard semantics, is provable.
- (Effectiveness) There is a proof-checking algorithm that can correctly decide whether a given sequence of symbols is a proof or not.

This corollary is sometimes expressed by saying that second-order logic does not admit a complete proof theory. In this respect second-order logic with standard semantics differs from first-order logic; Quine (1970, pp. 90–91) pointed to the lack of a complete proof system as a reason for thinking of second-order logic as not *logic*, properly speaking.

As mentioned above, Henkin proved that the standard deductive system for first-order logic is sound, complete, and effective for second-order logic with Henkin semantics, and the deductive system with comprehension and choice principles is sound, complete, and effective for Henkin semantics using only models that satisfy these principles.

The compactness theorem and the Löwenheim–Skolem theorem do not hold for full models of second-order logic. They do hold however for Henkin models.^{ [7] }^{:xi}

Predicate logic was introduced to the mathematical community by C. S. Peirce, who coined the term *second-order logic* and whose notation is most similar to the modern form (Putnam 1982). However, today most students of logic are more familiar with the works of Frege, who published his work several years prior to Peirce but whose works remained less known until Bertrand Russell and Alfred North Whitehead made them famous. Frege used different variables to distinguish quantification over objects from quantification over properties and sets; but he did not see himself as doing two different kinds of logic. After the discovery of Russell's paradox it was realized that something was wrong with his system. Eventually logicians found that restricting Frege's logic in various ways—to what is now called first-order logic—eliminated this problem: sets and properties cannot be quantified over in first-order logic alone. The now-standard hierarchy of orders of logics dates from this time.

It was found that set theory could be formulated as an axiomatized system within the apparatus of first-order logic (at the cost of several kinds of completeness, but nothing so bad as Russell's paradox), and this was done (see Zermelo–Fraenkel set theory), as sets are vital for mathematics. Arithmetic, mereology, and a variety of other powerful logical theories could be formulated axiomatically without appeal to any more logical apparatus than first-order quantification, and this, along with Gödel and Skolem's adherence to first-order logic, led to a general decline in work in second (or any higher) order logic.^{[ citation needed ]}

This rejection was actively advanced by some logicians, most notably W. V. Quine. Quine advanced the view^{[ citation needed ]} that in predicate-language sentences like *Fx* the "*x*" is to be thought of as a variable or name denoting an object and hence can be quantified over, as in "For all things, it is the case that . . ." but the "*F*" is to be thought of as an *abbreviation* for an incomplete sentence, not the name of an object (not even of an abstract object like a property). For example, it might mean " . . . is a dog." But it makes no sense to think we can quantify over something like this. (Such a position is quite consistent with Frege's own arguments on the concept-object distinction). So to use a predicate as a variable is to have it occupy the place of a name, which only individual variables should occupy. This reasoning has been rejected by George Boolos.^{[ citation needed ]}

In recent years^{[ when? ]} second-order logic has made something of a recovery, buoyed by Boolos' interpretation of second-order quantification as plural quantification over the same domain of objects as first-order quantification (Boolos 1984). Boolos furthermore points to the claimed nonfirstorderizability of sentences such as "Some critics admire only each other" and "Some of Fianchetto's men went into the warehouse unaccompanied by anyone else", which he argues can only be expressed by the full force of second-order quantification. However, generalized quantification and partially ordered (or branching) quantification may suffice to express a certain class of purportedly nonfirstorderizable sentences as well and these do not appeal to second-order quantification.

The expressive power of various forms of second-order logic on finite structures is intimately tied to computational complexity theory. The field of descriptive complexity studies which computational complexity classes can be characterized by the power of the logic needed to express languages (sets of finite strings) in them. A string *w* = *w*_{1}···*w _{n}* in a finite alphabet

This identification leads to the following characterizations of variants of second-order logic over finite structures:

- REG (the set of regular languages) is definable by monadic, second-order formulas (Büchi's theorem, 1960)
- NP is the set of languages definable by existential, second-order formulas (Fagin's theorem, 1974).
- co-NP is the set of languages definable by universal, second-order formulas.
- PH is the set of languages definable by second-order formulas.
- PSPACE is the set of languages definable by second-order formulas with an added transitive closure operator.
- EXPTIME is the set of languages definable by second-order formulas with an added least fixed point operator.

Relationships among these classes directly impact the relative expressiveness of the logics over finite structures; for example, if **PH** = **PSPACE**, then adding a transitive closure operator to second-order logic would not make it any more expressive over finite structures.

- ↑ Shapiro (1991) and Hinman (2005) give complete introductions to the subject, with full definitions.
- 1 2 3 4 Professor Marc Cohen lecture notes https://faculty.washington.edu/smcohen/120/SecondOrder.pdf
- ↑ Stapleton, G., Howse, J., & Lee, J. M., eds.,
*Diagrammatic Representation and Inference: 5th International Conference, Diagrams 2008*(Berlin/Heidelberg: Springer, 2008), p. 258. - ↑ Such a system is used without comment by Hinman (2005).
- ↑ These are the models originally studied by Henkin (1950).
- ↑ The proof of this corollary is that a sound, complete, and effective deduction system for standard semantics could be used to produce a recursively enumerable completion of Peano arithmetic, which Gödel's theorem shows cannot exist.
- ↑ Manzano, M.,
*Model Theory*, trans. Ruy J. G. B. de Queiroz (Oxford: Clarendon Press, 1999), p. xi.

An **axiom**, **postulate** or **assumption** is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek *axíōma* 'that which is thought worthy or fit' or 'that which commends itself as evident.'

Informally, a **definable real number** is a real number that can be uniquely specified by its description. The description may be expressed as a construction or as a formula of a formal language. For example, the positive square root of 2, , can be defined as the unique positive solution to the equation , and it can be constructed with a compass and straightedge.

**First-order logic**—also known as **predicate logic**, **quantificational logic**, and **first-order predicate calculus**—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man", where "there exists*"* is a quantifier, while *x* is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic.

**Gödel's completeness theorem** is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic.

In mathematics, more precisely in mathematical logic, **model theory** is the study of the relationship between formal theories, and their models, taken as interpretations that satisfy the sentences of that theory. The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954. Since the 1970s, the subject has been shaped decisively by Saharon Shelah's stability theory. The relative emphasis placed on the class of models of a theory as opposed to the class of definable sets within a model fluctuated in the history of the subject, and the two directions are summarised by the pithy characterisations from 1973 and 1997 respectively:

In classical deductive logic, a **consistent** theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if it has a model, i.e., there exists an interpretation under which all formulas in the theory are true. This is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term *satisfiable* is used instead. The syntactic definition states a theory is consistent if there is no formula such that both and its negation are elements of the set of consequences of . Let be a set of closed sentences and the set of closed sentences provable from under some formal deductive system. The set of axioms is **consistent** when for no formula .

**Leon Albert Henkin** was one of the most important logicians and mathematicians of the 20th century. His works played a strong role in the development of logic, particularly in the theory of types. He was an active scholar at the University of California, Berkeley, where he made great contributions as a researcher, teacher, as well as in administrative positions. At this university he directed, together with Alfred Tarski, the *Group in Logic and the Methodology of Science*, from which many important logicians and philosophers emerged. He had a strong sense of social commitment and was a passionate defensor of his pacifist and progressive ideas. He took part in many social projects aimed at teaching mathematics, as well as projects aimed at supporting women's and minority groups to pursue careers in mathematics and related fields. A lover of dance and literature, he appreciated life in all its facets: art, culture, science and, above all, the warmth of human relations. He is remembered by his students for his great kindness, as well as for his academic and teaching excellence.

**Metalogic** is the study of the metatheory of logic. Whereas *logic* studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems. Logic concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derived *about* the languages and systems that are used to express truths.

A **formal system** is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the **logical calculus** of the formal system. A formal system is essentially an "axiomatic system".

In the foundations of mathematics, **von Neumann–Bernays–Gödel set theory** (**NBG**) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel-Choice set theory (ZFC). NBG introduces the notion of class, which is a collection of sets defined by a formula whose quantifiers range only over sets. NBG can define classes that are larger than sets, such as the class of all sets and the class of all ordinals. Morse–Kelley set theory (MK) allows classes to be defined by formulas whose quantifiers range over classes. NBG is finitely axiomatizable, while ZFC and MK are not.

In mathematics and logic, a **higher-order logic** is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics. Higher-order logics with their standard semantics are more expressive, but their model-theoretic properties are less well-behaved than those of first-order logic.

In mathematical logic, an **axiom schema** generalizes the notion of axiom.

**Independence-friendly logic** is an extension of classical first-order logic (FOL) by means of slashed quantifiers of the form and , where is a finite set of variables. The intended reading of is "there is a which is functionally independent from the variables in ". IF logic allows one to express more general patterns of dependence between variables than those which are implicit in first-order logic. This greater level of generality leads to an actual increase in expressive power; the set of IF sentences can characterize the same classes of structures as existential second-order logic. For example, it can express branching quantifier sentences, such as the formula which expresses infinity in the empty signature; this cannot be done in FOL. Therefore, first-order logic cannot, in general, express this pattern of dependency, in which depends *only* on and , and depends *only* on and . IF logic is more general than branching quantifiers, for example in that it can express dependencies that are not transitive, such as in the quantifier prefix , which expresses that depends on , and depends on , but does not depend on .

In mathematical logic, a **theory** is a set of sentences in a formal language. In most scenarios, a deductive system is first understood from context, after which an element of a theory is then called a theorem of the theory. In many deductive systems there is usually a subset that is called "the set of axioms" of the theory , in which case the deductive system is also called an "axiomatic system". By definition, every axiom is automatically a theorem. A **first-order theory** is a set of first-order sentences (theorems) recursively obtained by the inference rules of the system applied to the set of axioms.

In mathematical logic, **second-order arithmetic** is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative to axiomatic set theory as a foundation for much, but not all, of mathematics.

In mathematical logic, an **ω-consistent****theory** is a theory that is not only (syntactically) consistent, but also avoids proving certain infinite combinations of sentences that are intuitively contradictory. The name is due to Kurt Gödel, who introduced the concept in the course of proving the incompleteness theorem.

**Logic** is the formal science of using reason and is considered a branch of both philosophy and mathematics and to a lesser extent computer science. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct and incorrect inferences. Logicians study the criteria for the evaluation of arguments.

An **interpretation** is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation. The general study of interpretations of formal languages is called formal semantics.

In logic, a **quantifier** is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier in the first order formula expresses that everything in the domain satisfies the property denoted by . On the other hand, the existential quantifier in the formula expresses that there is something in the domain which satisfies that property. A formula where a quantifier takes widest scope is called a quantified formula. A quantified formula must contain a bound variable and a subformula specifying a property of the referent of that variable.

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