In logic and mathematics, secondorder logic is an extension of firstorder logic, which itself is an extension of propositional logic.^{[1]} Secondorder logic is in turn extended by higherorder logic and type theory.
Firstorder logic quantifies only variables that range over individuals (elements of the domain of discourse); secondorder logic, in addition, quantifies over relations. For example, the secondorder 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). Secondorder logic also includes quantification over sets, functions, and other variables (see section below). Both firstorder and secondorder 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.
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Second order logic

Introduction to First Order Logic

PREDICATE LOGIC and QUANTIFIER NEGATION  DISCRETE MATHEMATICS

Optimization: First & Second Order Condition
Transcription
Examples
Firstorder 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)
However, we cannot do the same with the predicate. That is, the following expression:
 ∃P P(b)
is not a sentence of firstorder logic, but this is a legitimate sentence of secondorder logic. Here, P is a predicate variable and is semantically a set of individuals.^{[2]}
As a result, secondorder logic has greater expressive power than firstorder logic. For example, there is no way in firstorder logic to identify the set of all cubes and tetrahedrons. But the existence of this set can be asserted in secondorder logic as:
 ∃P ∀x (Px ↔ (Cube(x) ∨ Tet(x))).
We can then assert properties of this set. For instance, the following says that the set of all cubes and tetrahedrons does not contain any dodecahedrons:
 ∀P (∀x (Px ↔ (Cube(x) ∨ Tet(x))) → ¬ ∃x (Px ∧ Dodec(x))).
Secondorder quantification is especially useful because it gives the ability to express reachability properties. For example, if Parent(x, y) denotes that x is a parent of y, then firstorder logic cannot express the property that x is an ancestor of y. In secondorder logic we can express this by saying that every set of people containing y and closed under the Parent relation contains x:
 ∀P ((Py ∧ ∀a ∀b ((Pb ∧ Parent(a, b)) → Pa)) → Px).
It is notable that while we have variables for predicates in secondorderlogic, we don't have variables for properties of predicates. We cannot say, for example, that there is a property Shape(P) that is true for the predicates P Cube, Tet, and Dodec. This would require thirdorder logic.^{[3]}
Syntax and fragments
The syntax of secondorder logic tells which expressions are well formed formulas. In addition to the syntax of firstorder logic, secondorder 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 firstorder 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 kary relations on the individuals. If R is such a kary relation variable and t_{1},...,t_{k} are firstorder 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 kary function variable and t_{1},...,t_{k} are firstorder terms then the expression f(t_{1},...,t_{k}) is a firstorder 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 secondorder logic, as in firstorder logic, is a wellformed 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 nary 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 secondorder logic (MSO) is a restriction of secondorder 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 secondorder logic without these restrictions is sometimes called full secondorder logic to distinguish it from the monadic version. Monadic secondorder logic is particularly used in the context of Courcelle's theorem, an algorithmic metatheorem in graph theory. The MSO theory of the complete infinite binary tree (S2S) is decidable. By contrast, full second order logic over any infinite set (or MSO logic over for example (,+)) can interpret the true secondorder arithmetic.
Just as in firstorder logic, secondorder logic may include nonlogical symbols in a particular secondorder language. These are restricted, however, in that all terms that they form must be either firstorder terms (which can be substituted for a firstorder variable) or secondorder terms (which can be substituted for a secondorder variable of an appropriate sort).
A formula in secondorder logic is said to be of firstorder (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 secondorder) formula is one additionally having some existential quantifiers over second order variables, i.e. , where is a firstorder formula. The fragment of secondorder logic consisting only of existential secondorder formulas is called existential secondorder logic and abbreviated as ESO, as , or even as ∃SO. The fragment of formulas is defined dually, it is called universal secondorder 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 secondorder arithmetic.)
Semantics
The semantics of secondorder logic establish the meaning of each sentence. Unlike firstorder logic, which has only one standard semantics, there are two different semantics that are commonly used for secondorder logic: standard semantics and Henkin semantics. In each of these semantics, the interpretations of the firstorder quantifiers and the logical connectives are the same as in firstorder logic. Only the ranges of quantifiers over secondorder 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 firstorder variables is established, the meaning of the remaining quantifiers is fixed. It is these semantics that give secondorder logic its expressive power, and they will be assumed for the remainder of this article.
Leon Henkin (1950) defined an alternative kind of semantics for secondorder and higherorder theories, in which the meaning of the higherorder domains is partly determined by an explicit axiomatisation, drawing on type theory, of the properties of the sets or functions ranged over. Henkin semantics is a kind of manysorted firstorder semantics, where there are a class of models of the axioms, instead of the semantics being fixed to just the standard model as in the standard semantics. A model in Henkin semantics will provide a set of sets or set of functions as the interpretation of higherorder domains, which may be a proper subset of all sets or functions of that sort. For his axiomatisation, Henkin proved that Gödel's completeness theorem and compactness theorem, which hold for firstorder logic, carry over to secondorder logic with Henkin semantics. Since also the Skolem–Löwenheim theorems hold for Henkin semantics, Lindström's theorem imports that Henkin models are just disguised firstorder models.^{[4]}
For theories such as secondorder arithmetic, the existence of nonstandard interpretations of higherorder domains isn't just a deficiency of the particular axiomatisation derived from type theory that Henkin used, but a necessary consequence of Gödel's incompleteness theorem: Henkin's axioms can't be supplemented further to ensure the standard interpretation is the only possible model. Henkin semantics are commonly used in the study of secondorder arithmetic.
Jouko Väänänen (2001) argued that the choice between Henkin models and full models for secondorder logic is analogous to the choice between ZFC and V as a basis for set theory: "As with secondorder 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."
Expressive power
Secondorder logic is more expressive than firstorder logic. For example, if the domain is the set of all real numbers, one can assert in firstorder logic the existence of an additive inverse of each real number by writing ∀x ∃y (x + y = 0) but one needs secondorder logic to assert the leastupperbound 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 secondorder 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 firstorder sentences valid in the reals has arbitrarily large models due to the compactness theorem. Thus the leastupperbound property cannot be expressed by any set of sentences in firstorder logic. (In fact, every realclosed field satisfies the same firstorder sentences in the signature as the real numbers.)
In secondorder 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 firstorder logic.
Certain fragments of secondorder logic like ESO are also more expressive than firstorder logic even though they are strictly less expressive than the full secondorder logic. ESO also enjoys translation equivalence with some extensions of firstorder logic that allow nonlinear ordering of quantifier dependencies, like firstorder logic extended with Henkin quantifiers, Hintikka and Sandu's independencefriendly logic, and Väänänen's dependence logic.
Deductive systems
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 secondorder 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 firstorder logic (such as natural deduction) augmented with substitution rules for secondorder terms.^{[5]} This deductive system is commonly used in the study of secondorder arithmetic.
The deductive systems considered by Shapiro (1991) and Henkin (1950) add to the augmented firstorder deductive scheme both comprehension axioms and choice axioms. These axioms are sound for standard secondorder semantics. They are sound for Henkin semantics restricted to Henkin models satisfying the comprehension and choice axioms.^{[6]}
Nonreducibility to firstorder logic
One might attempt to reduce the secondorder theory of the real numbers, with full secondorder semantics, to the firstorder theory in the following way. First expand the domain from the set of all real numbers to a twosorted 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 secondorder become firstorder, with the formerly secondorder quantifiers ranging over the second sort instead. This reduction can be attempted in a onesorted 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 firstorder 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 internal numbers, and some countably infinite collection of sets of internal numbers, whose members we will call "internal sets", such that the domain consisting of internal numbers and internal sets satisfies exactly the same firstorder sentences as are satisfied by the domain of real numbers and sets of real numbers. In particular, it satisfies a sort of leastupperbound axiom that says, in effect:
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 leastupperbound 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 firstorder theory of real numbers and sets of real numbers has many models, some of which are countable. The secondorder 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 secondorder logic. This shows that the secondorder theory of the real numbers cannot be reduced to a firstorder theory, in the sense that the secondorder theory of the real numbers has only one model but the corresponding firstorder theory has many models.
There are more extreme examples showing that secondorder logic with standard semantics is more expressive than firstorder logic. There is a finite secondorder 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 secondorder logic is consistent is extremely subtle.
Additional limitations of secondorder logic are described in the next section.
Metalogical results
It is a corollary of Gödel's incompleteness theorem that there is no deductive system (that is, no notion of provability) for secondorder formulas that simultaneously satisfies these three desired attributes:^{[7]}
 (Soundness) Every provable secondorder sentence is universally valid, i.e., true in all domains under standard semantics.
 (Completeness) Every universally valid secondorder formula, under standard semantics, is provable.
 (Effectiveness) There is a proofchecking algorithm that can correctly decide whether a given sequence of symbols is a proof or not.
This corollary is sometimes expressed by saying that secondorder logic does not admit a complete proof theory. In this respect secondorder logic with standard semantics differs from firstorder logic; Quine (1970, pp. 90–91) pointed to the lack of a complete proof system as a reason for thinking of secondorder logic as not logic, properly speaking.
As mentioned above, Henkin proved that the standard deductive system for firstorder logic is sound, complete, and effective for secondorder 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 secondorder logic. They do hold however for Henkin models.^{[8]}^{: xi }
History and disputed value
Predicate logic was introduced to the mathematical community by C. S. Peirce, who coined the term secondorder 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 firstorder logic—eliminated this problem: sets and properties cannot be quantified over in firstorder logic alone. The nowstandard 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 firstorder 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 firstorder quantification, and this, along with Gödel and Skolem's adherence to firstorder 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 predicatelanguage 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 conceptobject 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?]} secondorder logic has made something of a recovery, buoyed by Boolos' interpretation of secondorder quantification as plural quantification over the same domain of objects as firstorder 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 secondorder 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 secondorder quantification.
Relation to computational complexity
The expressive power of various forms of secondorder 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 A can be represented by a finite structure with domain D = {1,...,n}, unary predicates P_{a} for each a ∈ A, satisfied by those indices i such that w_{i} = a, and additional predicates that serve to uniquely identify which index is which (typically, one takes the graph of the successor function on D or the order relation <, possibly with other arithmetic predicates). Conversely, the Cayley tables of any finite structure (over a finite signature) can be encoded by a finite string.
This identification leads to the following characterizations of variants of secondorder logic over finite structures:
 REG (the regular languages) is the set of languages definable by monadic, secondorder formulas (BüchiElgotTrakhtenbrot theorem, 1960)
 NP is the set of languages definable by existential, secondorder formulas (Fagin's theorem, 1974).
 coNP is the set of languages definable by universal, secondorder formulas.
 PH is the set of languages definable by secondorder formulas.
 PSPACE is the set of languages definable by secondorder formulas with an added transitive closure operator.
 EXPTIME is the set of languages definable by secondorder 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 secondorder logic would not make it any more expressive over finite structures.
See also
 Firstorder logic
 Higherorder logic
 Löwenheim number
 Omega language
 Secondorder propositional logic
 Monadic secondorder logic
Notes
 ^ Shapiro (1991) and Hinman (2005) give complete introductions to the subject, with full definitions.
 ^ ^{a} ^{b} Professor Marc Cohen lecture notes https://faculty.washington.edu/smcohen/120/SecondOrder.pdf
 ^ Väänänen, Jouko (2021), "Secondorder and Higherorder Logic", in Zalta, Edward N. (ed.), The Stanford Encyclopedia of Philosophy (Fall 2021 ed.), Metaphysics Research Lab, Stanford University, retrieved 20220503
 ^ *Mendelson, Elliot (2009). Introduction to Mathematical Logic (hardcover). Discrete Mathematics and Its Applications (5th ed.). Boca Raton: Chapman and Hall/CRC. p. 387. ISBN 9781584888765.
 ^ 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.
References
 Andrews, Peter (2002). An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof (2nd ed.). Kluwer Academic Publishers.
 Boolos, George (1984). "To Be Is To Be a Value of a Variable (or to Be Some Values of Some Variables)". Journal of Philosophy. 81 (8): 430–50. doi:10.2307/2026308. JSTOR 2026308.. Reprinted in Boolos, Logic, Logic and Logic, 1998.
 Henkin, L. (1950). "Completeness in the theory of types". Journal of Symbolic Logic. 15 (2): 81–91. doi:10.2307/2266967. JSTOR 2266967. S2CID 36309665.
 Hinman, P. (2005). Fundamentals of Mathematical Logic. A K Peters. ISBN 1568812620.
 Putnam, Hilary (1982). "Peirce the Logician". Historia Mathematica. 9 (3): 290–301. doi:10.1016/03150860(82)901239.. Reprinted in Putnam, Hilary (1990), Realism with a Human Face, Harvard University Press, pp. 252–260.
 W. V. Quine (1970). Philosophy of Logic. Prentice Hall. ISBN 9780674665637.
 Rossberg, M. (2004). "FirstOrder Logic, SecondOrder Logic, and Completeness" (PDF). In V. Hendricks; et al. (eds.). Firstorder logic revisited. Berlin: LogosVerlag.
 Shapiro, S. (2000) [1991]. Foundations without Foundationalism: A Case for SecondOrder Logic. Oxford: Clarendon Press. ISBN 0198250290.
 Väänänen, J. (2001). "SecondOrder Logic and Foundations of Mathematics" (PDF). Bulletin of Symbolic Logic. 7 (4): 504–520. CiteSeerX 10.1.1.25.5579. doi:10.2307/2687796. JSTOR 2687796. S2CID 7465054.
Further reading
 Grädel, Erich; Kolaitis, Phokion G.; Libkin, Leonid; Maarten, Marx; Spencer, Joel; Vardi, Moshe Y.; Venema, Yde; Weinstein, Scott (2007). Finite model theory and its applications. Texts in Theoretical Computer Science. An EATCS Series. Berlin: SpringerVerlag. ISBN 9783540004288. Zbl 1133.03001.
 Väänänen, Jouko. Edward N. Zalta (ed.). Secondorder and Higherorder Logic. The Stanford Encyclopedia of Philosophy (Fall 2021 Edition).