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  • Axiom of regularityFrom Wikipedia, the free encyclopedia

  • Contents

    1 Axiom of pairing 11.1 Formal statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Non-independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Generalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Another alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

    2 Axiom of regularity 42.1 Elementary implications of regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

    2.1.1 No set is an element of itself . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.2 No innite descending sequence of sets exists . . . . . . . . . . . . . . . . . . . . . . . . 42.1.3 Simpler set-theoretic denition of the ordered pair . . . . . . . . . . . . . . . . . . . . . . 52.1.4 Every set has an ordinal rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.5 For every two sets, only one can be an element of the other . . . . . . . . . . . . . . . . . 5

    2.2 The axiom of dependent choice and no innite descending sequence of sets implies regularity . . . . 52.3 Regularity and the rest of ZF(C) axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Regularity and Russells paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.5 Regularity, the cumulative hierarchy, and types . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.6 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

    2.8.1 Primary sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

    3 First-order logic 93.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

    3.2.1 Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2.2 Formation rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2.3 Free and bound variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

    3.3 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

    i

  • ii CONTENTS

    3.3.1 First-order structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3.2 Evaluation of truth values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3.3 Validity, satisability, and logical consequence . . . . . . . . . . . . . . . . . . . . . . . . 163.3.4 Algebraizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3.5 First-order theories, models, and elementary classes . . . . . . . . . . . . . . . . . . . . . 173.3.6 Empty domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

    3.4 Deductive systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.4.1 Rules of inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.4.2 Hilbert-style systems and natural deduction . . . . . . . . . . . . . . . . . . . . . . . . . . 183.4.3 Sequent calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4.4 Tableaux method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4.5 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4.6 Provable identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

    3.5 Equality and its axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.5.1 First-order logic without equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.5.2 Dening equality within a theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

    3.6 Metalogical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.6.1 Completeness and undecidability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.6.2 The LwenheimSkolem theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.6.3 The compactness theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.6.4 Lindstrms theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

    3.7 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.7.1 Expressiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.7.2 Formalizing natural languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

    3.8 Restrictions, extensions, and variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.8.1 Restricted languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.8.2 Many-sorted logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.8.3 Additional quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.8.4 Innitary logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.8.5 Non-classical and modal logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.8.6 Fixpoint logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.8.7 Higher-order logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

    3.9 Automated theorem proving and formal methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.10 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.13 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

    4 ZermeloFraenkel set theory 304.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

    4.2.1 1. Axiom of extensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

  • CONTENTS iii

    4.2.2 2. Axiom of regularity (also called the Axiom of foundation) . . . . . . . . . . . . . . . . 314.2.3 3. Axiom schema of specication (also called the axiom schema of separation or of restricted

    comprehension) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2.4 4. Axiom of pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.2.5 5. Axiom of union . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.2.6 6. Axiom schema of replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2.7 7. Axiom of innity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2.8 8. Axiom of power set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2.9 9. Well-ordering theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

    4.3 Motivation via the cumulative hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.4 Metamathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

    4.4.1 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.5 Criticisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.9 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 39

    4.9.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.9.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.9.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

  • Chapter 1

    Axiom of pairing

    In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom ofpairing is one of the axioms of ZermeloFraenkel set theory.

    1.1 Formal statementIn the formal language of the ZermeloFraenkel axioms, the axiom reads:

    8A 8B 9C 8D [D 2 C () (D = A _D = B)]or in words:

    Given any set A and any set B, there is a set C such that, given any set D, D is a member of C if and onlyif D is equal to A or D is equal to B.

    or in simpler words:

    Given two sets, there is a set whose members are exactly the two given sets.

    1.2 InterpretationWhat the axiom is really saying is that, given two sets A and B, we can nd a set C whose members are precisely Aand B. We can use the axiom of extensionality to show that this set C is unique. We call the set C the pair of A andB, and denote it {A,B}. Thus the essence of the axiom is:

    Any two sets have a pair.

    {A,A} is abbreviated {A}, called the singleton containing A. Note that a singleton is a special case of a pair.The axiom of pairing also allows for the denition of ordered pairs. For any sets a and b , the ordered pair is denedby the following:

    (a; b) = ffag; fa; bgg:Note that this denition satises the condition

    (a; b) = (c; d) () a = c ^ b = d:

    1

  • 2 CHAPTER 1. AXIOM OF PAIRING

    Ordered n-tuples can be dened recursively as follows:

    (a1; : : : ; an) = ((a1; : : : ; an1); an):

    1.3 Non-independenceThe axiom of pairing is generally considered uncontroversial, and it or an equivalent appears in just about any alterna-tive axiomatization of set theory. Nevertheless, in the standard formulation of the ZermeloFraenkel set theory, theaxiom of pairing follows from the axiom schema of replacement applied to any given set with two or more elements,and thus it is sometimes omitted. The existence of such a set with two elements, such as { {}, { {} } }, can bededuced either from the axiom of empty set and the axiom of power set or from the axiom of innity.

    1.4 GeneralisationTogether with the axiom of empty set, the axiom of pairing can be generalised to the following schema:

    8A1 : : : 8An 9C 8D [D 2 C () (D = A1 _ _D = An)]that is:

    Given any nite number of sets A1 through An, there is a set C whose members are precisely A1 throughAn.

    This set C is again unique by the axiom of extension, and is denoted {A1,...,An}.Of course, we can't refer to a nite number of sets rigorously without already having in our hands a (nite) set towhich the sets in question belong. Thus, this is not a single statement but instead a schema, with a separate statementfor each natural number n.

    The case n = 1 is the axiom of pairing with A = A1 and B = A1. The case n = 2 is the axiom of pairing with A = A1 and B = A2. The cases n > 2 can be proved using the axiom of pairing and the axiom of union multiple times.

    For example, to prove the case n = 3, use the axiom of pairing three times, to produce the pair {A1,A2}, the singleton{A3}, and then the pair {{A1,A2},{A3}}. The axiom of union then produces the desired result, {A1,A2,A3}. Wecan extend this schema to include n=0 if we interpret that case as the axiom of empty set.Thus, one may use this as an axiom schema in the place of the axioms of empty set and pairing. Normally, however,one uses the axioms of empty set and pairing separately, and then proves this as a theorem schema. Note that adoptingthis as an axiom schema will not replace the axiom of union, which is still needed for other situations.

    1.5 Another alternativeAnother axiom which implies the axiom of pairing in the presence of the axiom of empty set is

    8A 8B 9C 8D [D 2 C () (D 2 A _D = B)]Using {} for A and x for B, we get {x} for C. Then use {x} for A and y for B, getting {x,y} for C. One may continuein this fashion to build up any nite set. And this could be used to generate all hereditarily nite sets without usingthe axiom of union.

  • 1.6. REFERENCES 3

    1.6 References Paul Halmos,Naive set theory. Princeton, NJ: D. VanNostrandCompany, 1960. Reprinted by Springer-Verlag,New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition).

    Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN3-540-44085-2.

    Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0-444-86839-9.

  • Chapter 2

    Axiom of regularity

    In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of ZermeloFraenkelset theory that states that every non-empty set A contains an element that is disjoint from A. In rst-order logic theaxiom reads:

    8x (x 6= ?! 9y 2 x (y \ x = ?))

    The axiom implies that no set is an element of itself, and that there is no innite sequence (an) such that ai+1 is anelement of ai for all i. With the axiom of dependent choice (which is a weakened form of the axiom of choice), thisresult can be reversed: if there are no such innite sequences, then the axiom of regularity is true. Hence, the axiomof regularity is equivalent, given the axiom of dependent choice, to the alternative axiom that there are no downwardinnite membership chains.The axiom of regularity was introduced by von Neumann (1925); it was adopted in a formulation closer to the onefound in contemporary textbooks by Zermelo (1930). Virtually all results in the branches of mathematics basedon set theory hold even in the absence of regularity; see chapter 3 of Kunen (1980). However, regularity makessome properties of ordinals easier to prove; and it not only allows induction to be done on well-ordered sets but alsoon proper classes that are well-founded relational structures such as the lexicographical ordering on f(n; )jn 2! ^ ordinal an is g :Given the other axioms of ZermeloFraenkel set theory, the axiom of regularity is equivalent to the axiom of induc-tion. The axiom of induction tends to be used in place of the axiom of regularity in intuitionistic theories (ones thatdo not accept the law of the excluded middle), where the two axioms are not equivalent.In addition to omitting the axiom of regularity, non-standard set theories have indeed postulated the existence of setsthat are elements of themselves.

    2.1 Elementary implications of regularity

    2.1.1 No set is an element of itself

    Let A be a set, and apply the axiom of regularity to {A}, which is a set by the axiom of pairing. We see that theremust be an element of {A} which is disjoint from {A}. Since the only element of {A} is A, it must be that A is disjointfrom {A}. So, since A {A}, we cannot have A A (by the denition of disjoint).

    2.1.2 No innite descending sequence of sets exists

    Suppose, to the contrary, that there is a function, f, on the natural numbers with f(n+1) an element of f(n) for eachn. Dene S = {f(n): n a natural number}, the range of f, which can be seen to be a set from the axiom schema ofreplacement. Applying the axiom of regularity to S, let B be an element of S which is disjoint from S. By the denitionof S, B must be f(k) for some natural number k. However, we are given that f(k) contains f(k+1) which is also an

    4

  • 2.2. THEAXIOMOFDEPENDENTCHOICEANDNO INFINITEDESCENDING SEQUENCEOF SETS IMPLIES REGULARITY5

    element of S. So f(k+1) is in the intersection of f(k) and S. This contradicts the fact that they are disjoint sets. Sinceour supposition led to a contradiction, there must not be any such function, f.The nonexistence of a set containing itself can be seen as a special case where the sequence is innite and constant.Notice that this argument only applies to functions f that can be represented as sets as opposed to undenable classes.The hereditarily nite sets, V, satisfy the axiom of regularity (and all other axioms of ZFC except the axiom ofinnity). So if one forms a non-trivial ultrapower of V, then it will also satisfy the axiom of regularity. The resultingmodel will contain elements, called non-standard natural numbers, that satisfy the denition of natural numbers inthat model but are not really natural numbers. They are fake natural numbers which are larger than any actualnatural number. This model will contain innite descending sequences of elements. For example, suppose n is anon-standard natural number, then (n 1) 2 n and (n 2) 2 (n 1) , and so on. For any actual natural numberk, (n k 1) 2 (n k) . This is an unending descending sequence of elements. But this sequence is not denablein the model and thus not a set. So no contradiction to regularity can be proved.

    2.1.3 Simpler set-theoretic denition of the ordered pairThe axiom of regularity enables dening the ordered pair (a,b) as {a,{a,b}}. See ordered pair for specics. Thisdenition eliminates one pair of braces from the canonical Kuratowski denition (a,b) = {{a},{a,b}}.

    2.1.4 Every set has an ordinal rankThis was actually the original form of von Neumanns axiomatization.

    2.1.5 For every two sets, only one can be an element of the otherLet X and Y be sets. Then apply the axiom of regularity to the set {X,Y}. We see there must be an element of {X,Y}which is also disjoint from it. It must be either X or Y. By the denition of disjoint then, we must have either Y is notan element of X or vice versa.

    2.2 The axiom of dependent choice and no innite descending sequence ofsets implies regularity

    Let the non-empty set S be a counter-example to the axiom of regularity; that is, every element of S has a non-emptyintersection with S. We dene a binary relation R on S by aRb :, b 2 S \ a , which is entire by assumption. Thus,by the axiom of dependent choice, there is some sequence (an) in S satisfying anRan+1 for all n in N. As this is aninnite descending chain, we arrive at a contradiction and so, no such S exists.

    2.3 Regularity and the rest of ZF(C) axiomsRegularity was shown to be relatively consistent with the rest of ZF by von Neumann (1929), meaning that if ZFwithout regularity is consistent, then ZF (with regularity) is also consistent. For his proof in modern notation seeVaught (2001, 10.1) for instance.The axiom of regularity was also shown to be independent from the other axioms of ZF(C), assuming they areconsistent. The result was announced by Paul Bernays in 1941, although he did not publish a proof until 1954. Theproof involves (and led to the study of) Rieger-Bernays permutation models (or method), which were used for otherproofs of independence for non-well-founded systems (Rathjen 2004, p. 193 and Forster 2003, pp. 210212).

    2.4 Regularity and Russells paradoxNaive set theory (the axiom schema of unrestricted comprehension and the axiom of extensionality) is inconsistent dueto Russells paradox. Set theorists have avoided that contradiction by replacing the axiom schema of comprehension

  • 6 CHAPTER 2. AXIOM OF REGULARITY

    with the much weaker axiom schema of separation. However, this makes set theory too weak. So some of thepower of comprehension was added back via the other existence axioms of ZF set theory (pairing, union, powerset,replacement, and innity) which may be regarded as special cases of comprehension. So far, these axioms do notseem to lead to any contradiction. Subsequently, the axiom of choice and the axiom of regularity were added toexclude models with some undesirable properties. These two axioms are known to be relatively consistent.In the presence of the axiom schema of separation, Russells paradox becomes a proof that there is no set of all sets.The axiom of regularity (with the axiom of pairing) also prohibits such a universal set, however this prohibition isredundant when added to the rest of ZF. If the ZF axioms without regularity were already inconsistent, then addingregularity would not make them consistent.The existence of Quine atoms (sets that satisfy the formula equation x = {x}, i.e. have themselves as their only ele-ments) is consistent with the theory obtained by removing the axiom of regularity fromZFC. Various non-wellfoundedset theories allow safe circular sets, such as Quine atoms, without becoming inconsistent by means of Russellsparadox.(Rieger 2011, pp. 175,178)

    2.5 Regularity, the cumulative hierarchy, and typesIn ZF it can be proven that the classS V (see cumulative hierarchy) is equal to the class of all sets. This statementis even equivalent to the axiom of regularity (if we work in ZF with this axiom omitted). From any model which doesnot satisfy axiom of regularity, a model which satises it can be constructed by taking only sets inS V .Herbert Enderton (1977, p. 206) wrote that The idea of rank is a descendant of Russells concept of type". Com-paring ZF with type theory, Alasdair Urquhart wrote that Zermelos system has the notational advantage of notcontaining any explicitly typed variables, although in fact it can be seen as having an implicit type structure built intoit, at least if the axiom of regularity is included. The details of this implicit typing are spelled out in [Zermelo 1930],and again in a well-known article of George Boolos [Boolos 1971]. Urquhart (2003, p. 305)Dana Scott (1974) went further and claimed that:

    The truth is that there is only one satisfactory way of avoiding the paradoxes: namely, the use of someform of the theory of types. That was at the basis of both Russells and Zermelos intuitions. Indeed thebest way to regard Zermelos theory is as a simplication and extension of Russells. (We mean Russellssimple theory of types, of course.) The simplication was to make the types cumulative. Thus mixing oftypes is easier and annoying repetitions are avoided. Once the later types are allowed to accumulate theearlier ones, we can then easily imagine extending the types into the transnitejust how far we want togo must necessarily be left open. Now Russell made his types explicit in his notation and Zermelo leftthem implicit. [emphasis in original]

    In the same paper, Scott shows that an axiomatic system based on the inherent properties of the cumulative hierarchyturns out to be equivalent to ZF, including regularity. (Lvy 2002, p. 73)

    2.6 HistoryThe concept of well-foundedness and rank of a set were both introduced by Dmitry Mirimano (1917) cf. Lvy(2002, p. 68) and Hallett (1986, 4.4, esp. p. 186, 188). Mirimano called a set x regular (French: ordinaire) ifevery descending chain x x1 x2 ... is nite. Mirimano however did not consider his notion of regularity (andwell-foundedness) as an axiom to be observed by all sets (Halbeisen 2012, pp. 6263); in later papers Mirimanoalso explored what are now called non-well-founded sets (extraordinaire in Mirimanos terminology) (Sangiorgi2011, pp. 1719, 26).According to Adam Rieger, von Neumann (1925) describes non-well-founded sets as superuous (on p. 404 invan Heijenoort 's translation) and in the same publication von Neumann gives an axiom (p. 412 in translation) whichexcludes some, but not all, non-well-founded sets (Rieger 2011, p. 179). In a subsequent publication, von Neumann(1928) gave the following axiom (rendered in modern notation by A. Rieger):

    8x (x 6= ; ! 9y 2 x (y \ x = ;))

  • 2.7. SEE ALSO 7

    2.7 See also Non-well-founded set theory

    2.8 References Jech, Thomas (2003), Set Theory: The Third Millennium Edition, Revised and Expanded, Springer, ISBN 3-540-44085-2

    Kunen, Kenneth (1980), Set Theory: An Introduction to Independence Proofs, Elsevier, ISBN 0-444-86839-9 Boolos, George (1971), The iterative conception of set, Journal of Philosophy 68: 215231, doi:10.2307/2025204reprinted in Boolos, George (1998), Logic, Logic and Logic, Harvard University Press, pp. 1329

    Enderton, Herbert B. (1977), Elements of Set Theory, Academic Press Urquhart, Alasdair (2003), The Theory of Types, in Grin, Nicholas, The Cambridge Companion to BertrandRussell, Cambridge University Press

    Halbeisen, Lorenz J. (2012), Combinatorial Set Theory: With a Gentle Introduction to Forcing, Springer Sangiorgi, Davide (2011), Origins of bisimulation and coinduction, in Sangiorgi, Davide; Rutten, Jan, Ad-vanced Topics in Bisimulation and Coinduction, Cambridge University Press

    Lvy, Azriel (2002) [rst published in 1979], Basic set theory, Dover Publications, ISBN 0-486-42079-5 Hallett, Michael (1996) [rst published 1984], Cantorian set theory and limitation of size, Oxford UniversityPress, ISBN 0-19-853283-0

    Rathjen, M. (2004), Predicativity, Circularity, and Anti-Foundation, in Link, Godehard, One Hundred Yearsof Russell s Paradox: Mathematics, Logic, Philosophy (PDF), Walter de Gruyter, ISBN 978-3-11-019968-0

    Forster, T. (2003), Logic, induction and sets, Cambridge University Press Rieger, Adam (2011), Paradox, ZF, and the Axiom of Foundation, in David DeVidi, Michael Hallett, PeterClark, Logic, Mathematics, Philosophy, Vintage Enthusiasms. Essays in Honour of John L. Bell., pp. 171187,doi:10.1007/978-94-007-0214-1_9, ISBN 978-94-007-0213-4

    Vaught, Robert L. (2001), Set Theory: An Introduction (2nd ed.), Springer, ISBN 978-0-8176-4256-3

    2.8.1 Primary sources Mirimano, D. (1917), Les antinomies de Russell et de Burali-Forti et le probleme fondamental de la theoriedes ensembles, L'Enseignement Mathmatique 19: 3752

    von Neumann, J. (1925), Eine axiomatiserung der Mengenlehre, Journal fr die reine und angewandte Math-ematik 154: 219240; translation in van Heijenoort, Jean (1967), From Frege to Gdel: A Source Book inMathematical Logic, 18791931, pp. 393413

    von Neumann, J. (1928), "ber die Denition durch transnite Induktion und verwandte Fragen der allge-meinen Mengenlehre, Mathematische Annalen 99: 373391, doi:10.1007/BF01459102

    von Neumann, J. (1929), Uber eine Widerspruchfreiheitsfrage in der axiomatischen Mengenlehre, Journalfur die reine und angewandte Mathematik 160: 227241, doi:10.1515/crll.1929.160.227

    Zermelo, Ernst (1930), "ber Grenzzahlen und Mengenbereiche. Neue Untersuchungen ber die Grundlagender Mengenlehre. (PDF), Fundamenta Mathematicae 16: 2947; translation in Ewald, W.B., ed. (1996),From Kant to Hilbert: A Source Book in the Foundations of Mathematics Vol. 2, Clarendon Press, pp. 121933

    Bernays, P. (1941), A system of axiomatic set theory. Part II, The Journal of Symbolic Logic 6: 117,doi:10.2307/2267281

  • 8 CHAPTER 2. AXIOM OF REGULARITY

    Bernays, P. (1954), A system of axiomatic set theory. Part VII, The Journal of Symbolic Logic 19: 8196,doi:10.2307/2268864

    Riegger, L. (1957), A contribution to Gdels axiomatic set theory (PDF), Czechoslovak Mathematical Jour-nal 7: 323357

    Scott, D. (1974), Axiomatizing set theory,Axiomatic set theory. Proceedings of Symposia in PureMathematicsVolume 13, Part II, pp. 207214

    2.9 External links http://www.trinity.edu/cbrown/topics_in_logic/sets/sets.html contains an informative description of the axiomof regularity under the section on Zermelo-Fraenkel set theory.

    Axiom of Foundation at PlanetMath.org.

  • Chapter 3

    First-order logic

    First-order logic is a formal system used in mathematics, philosophy, linguistics, and computer science. It is alsoknown as rst-order predicate calculus, the lower predicate calculus, quantication theory, and predicate logic.First-order logic uses quantied variables over (non-logical) objects. This distinguishes it from propositional logicwhich does not use quantiers.A theory about some topic is usually rst-order logic together with a specied domain of discourse over which thequantied variables range, nitelymany functions whichmap from that domain into it, nitelymany predicates denedon that domain, and a recursive set of axioms which are believed to hold for those things. Sometimes theory isunderstood in a more formal sense, which is just a set of sentences in rst-order logic.The adjective rst-order distinguishes rst-order logic from higher-order logic in which there are predicates havingpredicates or functions as arguments, or in which one or both of predicate quantiers or function quantiers arepermitted.[1] In rst-order theories, predicates are often associated with sets. In interpreted higher-order theories,predicates may be interpreted as sets of sets.There are many deductive systems for rst-order logic that are sound (all provable statements are true in all models)and complete (all statements which are true in all models are provable). Although the logical consequence relation isonly semidecidable, much progress has been made in automated theorem proving in rst-order logic. First-order logicalso satises several metalogical theorems that make it amenable to analysis in proof theory, such as the LwenheimSkolem theorem and the compactness theorem.First-order logic is the standard for the formalization of mathematics into axioms and is studied in the foundationsof mathematics. Mathematical theories, such as number theory and set theory, have been formalized into rst-orderaxiom schemas such as Peano arithmetic and ZermeloFraenkel set theory (ZF) respectively.No rst-order theory, however, has the strength to describe uniquely a structure with an innite domain, such as thenatural numbers or the real line. A uniquely describing, i.e. categorical, axiom system for such a structure can beobtained in stronger logics such as second-order logic.For a history of rst-order logic and how it came to dominate formal logic, see Jos Ferreirs (2001).

    3.1 IntroductionWhile propositional logic deals with simple declarative propositions, rst-order logic additionally covers predicatesand quantication.A predicate takes an entity or entities in the domain of discourse as input and outputs either True or False. Considerthe two sentences Socrates is a philosopher and Plato is a philosopher. In propositional logic, these sentencesare viewed as being unrelated and are denoted, for example, by p and q. However, the predicate is a philosopheroccurs in both sentences which have a common structure of "a is a philosopher. The variable a is instantiated asSocrates in the rst sentence and is instantiated as Plato in the second sentence. The use of predicates, such asis a philosopher in this example, distinguishes rst-order logic from propositional logic.Predicates can be compared. Consider, for example, the rst-order formula if a is a philosopher, then a is a scholar.This formula is a conditional statement with "a is a philosopher as hypothesis and "a is a scholar as conclusion.

    9

  • 10 CHAPTER 3. FIRST-ORDER LOGIC

    The truth of this formula depends on which object is denoted by a, and on the interpretations of the predicates is aphilosopher and is a scholar.Variables can be quantied over. The variable a in the previous formula can be quantied over, for instance, in therst-order sentence For every a, if a is a philosopher, then a is a scholar. The universal quantier for every inthis sentence expresses the idea that the claim if a is a philosopher, then a is a scholar holds for all choices of a.The negation of the sentence For every a, if a is a philosopher, then a is a scholar is logically equivalent to thesentence There exists a such that a is a philosopher and a is not a scholar. The existential quantier there existsexpresses the idea that the claim "a is a philosopher and a is not a scholar holds for some choice of a.The predicates is a philosopher and is a scholar each take a single variable. Predicates can take several variables.In the rst-order sentence Socrates is the teacher of Plato, the predicate is the teacher of takes two variables.To interpret a rst-order formula, one species what each predicate means and the entities that can instantiate thepredicated variables. These entities form the domain of discourse or universe, which is usually required to be anonempty set. Given that the interpretation with the domain of discourse as consisting of all human beings and thepredicate is a philosopher understood as have written the Republic, the sentence There exists a such that a is aphilosopher is seen as being true, as witnessed by Plato.

    3.2 SyntaxThere are two key parts of rst-order logic. The syntax determines which collections of symbols are legal expressionsin rst-order logic, while the semantics determine the meanings behind these expressions.

    3.2.1 AlphabetUnlike natural languages, such as English, the language of rst-order logic is completely formal, so that it can bemechanically determined whether a given expression is legal. There are two key types of legal expressions: terms,which intuitively represent objects, and formulas, which intuitively express predicates that can be true or false. Theterms and formulas of rst-order logic are strings of symbols which together form the alphabet of the language. Aswith all formal languages, the nature of the symbols themselves is outside the scope of formal logic; they are oftenregarded simply as letters and punctuation symbols.It is common to divide the symbols of the alphabet into logical symbols, which always have the same meaning, andnon-logical symbols, whose meaning varies by interpretation. For example, the logical symbol ^ always representsand"; it is never interpreted as or. On the other hand, a non-logical predicate symbol such as Phil(x) could beinterpreted to mean "x is a philosopher, "x is a man named Philip, or any other unary predicate, depending on theinterpretation at hand.

    Logical symbols

    There are several logical symbols in the alphabet, which vary by author but usually include:

    The quantier symbols and The logical connectives: for conjunction, for disjunction, for implication, for biconditional, fornegation. Occasionally other logical connective symbols are included. Some authors use Cpq, instead of ,and Epq, instead of , especially in contexts where is used for other purposes. Moreover, the horseshoe may replace ; the triple-bar may replace ; a tilde (~), Np, or Fpq, may replace ; ||, or Apq may replace; and &, Kpq, or the middle dot, , may replace , especially if these symbols are not available for technicalreasons. (Note: the aforementioned symbols Cpq, Epq, Np, Apq, and Kpq are used in Polish notation.)

    Parentheses, brackets, and other punctuation symbols. The choice of such symbols varies depending on context. An innite set of variables, often denoted by lowercase letters at the end of the alphabet x, y, z, . Subscriptsare often used to distinguish variables: x0, x1, x2, .

    An equality symbol (sometimes, identity symbol) =; see the section on equality below.

  • 3.2. SYNTAX 11

    It should be noted that not all of these symbols are required only one of the quantiers, negation and conjunc-tion, variables, brackets and equality suce. There are numerous minor variations that may dene additional logicalsymbols:

    Sometimes the truth constants T, Vpq, or , for true and F, Opq, or , for false are included. Without anysuch logical operators of valence 0, these two constants can only be expressed using quantiers.

    Sometimes additional logical connectives are included, such as the Sheer stroke, Dpq (NAND), and exclusiveor, Jpq.

    Non-logical symbols

    The non-logical symbols represent predicates (relations), functions and constants on the domain of discourse. It usedto be standard practice to use a xed, innite set of non-logical symbols for all purposes. A more recent practice isto use dierent non-logical symbols according to the application one has in mind. Therefore it has become necessaryto name the set of all non-logical symbols used in a particular application. This choice is made via a signature.[2]

    The traditional approach is to have only one, innite, set of non-logical symbols (one signature) for all applications.Consequently, under the traditional approach there is only one language of rst-order logic.[3] This approach is stillcommon, especially in philosophically oriented books.

    1. For every integer n 0 there is a collection of n-ary, or n-place, predicate symbols. Because they representrelations between n elements, they are also called relation symbols. For each arity n we have an innite supplyof them:

    Pn0, Pn1, Pn2, Pn3,

    2. For every integer n 0 there are innitely many n-ary function symbols:

    f n0, f n1, f n2, f n3,

    In contemporary mathematical logic, the signature varies by application. Typical signatures in mathematics are {1,} or just {} for groups, or {0, 1, +, ,

  • 12 CHAPTER 3. FIRST-ORDER LOGIC

    3.2.2 Formation rules

    The formation rules dene the terms and formulas of rst order logic. When terms and formulas are representedas strings of symbols, these rules can be used to write a formal grammar for terms and formulas. These rules aregenerally context-free (each production has a single symbol on the left side), except that the set of symbols may beallowed to be innite and there may be many start symbols, for example the variables in the case of terms.

    Terms

    The set of terms is inductively dened by the following rules:

    1. Variables. Any variable is a term.

    2. Functions. Any expression f(t1,...,tn) of n arguments (where each argument ti is a term and f is a functionsymbol of valence n) is a term. In particular, symbols denoting individual constants are 0-ary function symbols,and are thus terms.

    Only expressions which can be obtained by nitely many applications of rules 1 and 2 are terms. For example, noexpression involving a predicate symbol is a term.

    Formulas

    The set of formulas (also called well-formed formulas [4] or ws) is inductively dened by the following rules:

    1. Predicate symbols. If P is an n-ary predicate symbol and t1, ..., tn are terms then P(t1,...,t) is a formula.

    2. Equality. If the equality symbol is considered part of logic, and t1 and t2 are terms, then t1 = t2 is a formula.

    3. Negation. If is a formula, then : is a formula.

    4. Binary connectives. If and are formulas, then (! ) is a formula. Similar rules apply to other binarylogical connectives.

    5. Quantiers. If is a formula and x is a variable, then 8x' (for all x, ' holds) and 9x' (there exists x suchthat ' ) are formulas.

    Only expressions which can be obtained by nitely many applications of rules 15 are formulas. The formulas ob-tained from the rst two rules are said to be atomic formulas.For example,

    8x8y(P (f(x))! :(P (x)! Q(f(y); x; z)))

    is a formula, if f is a unary function symbol, P a unary predicate symbol, and Q a ternary predicate symbol. On theother hand, 8xx! is not a formula, although it is a string of symbols from the alphabet.The role of the parentheses in the denition is to ensure that any formula can only be obtained in one way by followingthe inductive denition (in other words, there is a unique parse tree for each formula). This property is known asunique readability of formulas. There are many conventions for where parentheses are used in formulas. Forexample, some authors use colons or full stops instead of parentheses, or change the places in which parentheses areinserted. Each authors particular denition must be accompanied by a proof of unique readability.This denition of a formula does not support dening an if-then-else function ite(c, a, b), where c is a conditionexpressed as a formula, that would return a if c is true, and b if it is false. This is because both predicates andfunctions can only accept terms as parameters, but the rst parameter is a formula. Some languages built on rst-orderlogic, such as SMT-LIB 2.0, add this.[5]

  • 3.2. SYNTAX 13

    Notational conventions

    For convenience, conventions have been developed about the precedence of the logical operators, to avoid the needto write parentheses in some cases. These rules are similar to the order of operations in arithmetic. A commonconvention is:

    : is evaluated rst

    ^ and _ are evaluated next

    Quantiers are evaluated next

    ! is evaluated last.

    Moreover, extra punctuation not required by the denition may be inserted to make formulas easier to read. Thus theformula

    (:8xP (x)! 9x:P (x))

    might be written as

    (:[8xP (x)])! 9x[:P (x)]:

    In some elds, it is common to use inx notation for binary relations and functions, instead of the prex notationdened above. For example, in arithmetic, one typically writes 2 + 2 = 4 instead of "=(+(2,2),4)". It is common toregard formulas in inx notation as abbreviations for the corresponding formulas in prex notation.The denitions above use inx notation for binary connectives such as ! . A less common convention is Polishnotation, in which one writes! , ^ , and so on in front of their arguments rather than between them. This conventionallows all punctuation symbols to be discarded. Polish notation is compact and elegant, but rarely used in practicebecause it is hard for humans to read it. In Polish notation, the formula

    8x8y(P (f(x))! :(P (x)! Q(f(y); x; z)))

    becomes "xyPfx PxQfyxz.

    3.2.3 Free and bound variables

    Main article: Free variables and bound variables

    In a formula, a variable may occur free or bound. Intuitively, a variable is free in a formula if it is not quantied: in8y P (x; y) , variable x is free while y is bound. The free and bound variables of a formula are dened inductively asfollows.

    1. Atomic formulas. If is an atomic formula then x is free in if and only if x occurs in . Moreover, thereare no bound variables in any atomic formula.

    2. Negation. x is free in : if and only if x is free in . x is bound in : if and only if x is bound in .

    3. Binary connectives. x is free in (! ) if and only if x is free in either or . x is bound in (! ) if andonly if x is bound in either or . The same rule applies to any other binary connective in place of! .

    4. Quantiers. x is free in 8 y if and only if x is free in and x is a dierent symbol from y. Also, x is boundin 8 y if and only if x is y or x is bound in . The same rule holds with 9 in place of 8 .

  • 14 CHAPTER 3. FIRST-ORDER LOGIC

    For example, in 8 x 8 y (P(x)! Q(x,f(x),z)), x and y are bound variables, z is a free variable, andw is neither becauseit does not occur in the formula.Free and bound variables of a formula need not be disjoint sets: x is both free and bound in P (x)! 8xQ(x) .Freeness and boundness can be also specialized to specic occurrences of variables in a formula. For example, inP (x) ! 8xQ(x) , the rst occurrence of x is free while the second is bound. In other words, the x in P (x) is freewhile the x in 8xQ(x) is bound.A formula in rst-order logic with no free variables is called a rst-order sentence. These are the formulas that willhave well-dened truth values under an interpretation. For example, whether a formula such as Phil(x) is true mustdepend on what x represents. But the sentence 9xPhil(x) will be either true or false in a given interpretation.

    3.2.4 ExamplesOrdered abelian groups

    In mathematics the language of ordered abelian groups has one constant symbol 0, one unary function symbol , onebinary function symbol +, and one binary relation symbol . Then:

    The expressions +(x, y) and +(x, +(y, (z))) are terms. These are usually written as x + y and x + y z. The expressions +(x, y) = 0 and (+(x, +(y, (z))), +(x, y)) are atomic formulas.

    These are usually written as x + y = 0 and x + y z x + y.

    The expression (8x8y(+(x; y); z) ! 8x8y+(x; y) = 0) is a formula, which is usually written as8x8y(x+ y z)! 8x8y(x+ y = 0):

    Loving relation

    English sentences like everyone loves someone can be formalized by rst-order logic formulas like xy L(x,y).This is accomplished by abbreviating the relation "x loves y" by L(x,y). Using just the two quantiers and andthe loving relation symbol L, but no logical connectives and no function symbols (including constants), formulas with8 dierent meanings can be built. The following diagrams show models for each of them, assuming that there areexactly ve individuals a,...,e who can love (vertical axis) and be loved (horizontal axis). A small red box at row x andcolumn y indicates L(x,y). Only for the formulas 9 and 10 is the model unique, all other formulas may be satised byseveral models.Each model, represented by a logical matrix, satises the formulas in its caption in a minimal way, i.e. whiteningany red cell in any matrix would make it non-satisfying the corresponding formula. For example, formula 1 is alsosatised by the matrices at 3, 6, and 10, but not by those at 2, 4, 5, and 7. Conversely, the matrix shown at 6 satises1, 2, 5, 6, 7, and 8, but not 3, 4, 9, and 10.Some formulas imply others, i.e. all matrices satisfying the antecedent (LHS) also satisfy the conclusion (RHS) ofthe implication e.g. formula 3 implies formula 1, i.e.: each matrix fullling formula 3 also fullls formula 1, butnot vice versa (see the Hasse diagram for this ordering relation). In contrast, only some matrices,[6] which satisfyformula 2, happen to satisfy also formula 5, whereas others,[7] also satisfying formula 2, do not; therefore formula 5is not a logical consequence of formula 2.The sequence of the quantiers is important! So it is instructive to distinguish formulas 1: x y L(y,x), and 3: xy L(x,y). In both cases everyone is loved; but in the rst case everyone (x) is loved by someone (y), in the secondcase everyone (y) is loved by just exactly one person (x).

    3.3 SemanticsAn interpretation of a rst-order language assigns a denotation to all non-logical constants in that language. It alsodetermines a domain of discourse that species the range of the quantiers. The result is that each term is assigned anobject that it represents, and each sentence is assigned a truth value. In this way, an interpretation provides semantic

  • 3.3. SEMANTICS 15

    meaning to the terms and formulas of the language. The study of the interpretations of formal languages is calledformal semantics. What follows is a description of the standard or Tarskian semantics for rst-order logic. (It is alsopossible to dene game semantics for rst-order logic, but aside from requiring the axiom of choice, game semanticsagree with Tarskian semantics for rst-order logic, so game semantics will not be elaborated herein.)The domain of discourseD is a nonempty set of objects of some kind. Intuitively, a rst-order formula is a statementabout these objects; for example, 9xP (x) states the existence of an object x such that the predicate P is true wherereferred to it. The domain of discourse is the set of considered objects. For example, one can takeD to be the set ofinteger numbers.The interpretation of a function symbol is a function. For example, if the domain of discourse consists of integers, afunction symbol f of arity 2 can be interpreted as the function that gives the sum of its arguments. In other words,the symbol f is associated with the function I(f) which, in this interpretation, is addition.The interpretation of a constant symbol is a function from the one-element setD0 toD, which can be simply identiedwith an object in D. For example, an interpretation may assign the value I(c) = 10 to the constant symbol c .The interpretation of an n-ary predicate symbol is a set of n-tuples of elements of the domain of discourse. Thismeans that, given an interpretation, a predicate symbol, and n elements of the domain of discourse, one can tellwhether the predicate is true of those elements according to the given interpretation. For example, an interpretationI(P) of a binary predicate symbol P may be the set of pairs of integers such that the rst one is less than the second.According to this interpretation, the predicate P would be true if its rst argument is less than the second.

    3.3.1 First-order structures

    Main article: Structure (mathematical logic)

    The most common way of specifying an interpretation (especially in mathematics) is to specify a structure (alsocalled a model; see below). The structure consists of a nonempty set D that forms the domain of discourse and aninterpretation I of the non-logical terms of the signature. This interpretation is itself a function:

    Each function symbol f of arity n is assigned a function I(f) fromDn toD . In particular, each constant symbolof the signature is assigned an individual in the domain of discourse.

    Each predicate symbol P of arity n is assigned a relation I(P) overDn or, equivalently, a function fromDn toftrue; falseg . Thus each predicate symbol is interpreted by a Boolean-valued function on D.

    3.3.2 Evaluation of truth values

    A formula evaluates to true or false given an interpretation, and a variable assignment that associates an elementof the domain of discourse with each variable. The reason that a variable assignment is required is to give meaningsto formulas with free variables, such as y = x . The truth value of this formula changes depending on whether x andy denote the same individual.First, the variable assignment can be extended to all terms of the language, with the result that each term maps toa single element of the domain of discourse. The following rules are used to make this assignment:

    1. Variables. Each variable x evaluates to (x)

    2. Functions. Given terms t1; : : : ; tn that have been evaluated to elements d1; : : : ; dn of the domain of discourse,and a n-ary function symbol f, the term f(t1; : : : ; tn) evaluates to (I(f))(d1; : : : ; dn) .

    Next, each formula is assigned a truth value. The inductive denition used to make this assignment is called theT-schema.

    1. Atomic formulas (1). A formula P (t1; : : : ; tn) is associated the value true or false depending on whetherhv1; : : : ; vni 2 I(P ) , where v1; : : : ; vn are the evaluation of the terms t1; : : : ; tn and I(P ) is the interpreta-tion of P , which by assumption is a subset of Dn .

  • 16 CHAPTER 3. FIRST-ORDER LOGIC

    2. Atomic formulas (2). A formula t1 = t2 is assigned true if t1 and t2 evaluate to the same object of the domainof discourse (see the section on equality below).

    3. Logical connectives. A formula in the form : , ! , etc. is evaluated according to the truth table forthe connective in question, as in propositional logic.

    4. Existential quantiers. A formula 9x(x) is true according to M and if there exists an evaluation 0 ofthe variables that only diers from regarding the evaluation of x and such that is true according to theinterpretation M and the variable assignment 0 . This formal denition captures the idea that 9x(x) is trueif and only if there is a way to choose a value for x such that (x) is satised.

    5. Universal quantiers. A formula 8x(x) is true according toM and if (x) is true for every pair composedby the interpretationM and some variable assignment0 that diers from only on the value of x. This capturesthe idea that 8x(x) is true if every possible choice of a value for x causes (x) to be true.

    If a formula does not contain free variables, and so is a sentence, then the initial variable assignment does not aectits truth value. In other words, a sentence is true according to M and if and only if it is true according to M andevery other variable assignment 0 .There is a second common approach to dening truth values that does not rely on variable assignment functions.Instead, given an interpretation M, one rst adds to the signature a collection of constant symbols, one for eachelement of the domain of discourse in M; say that for each d in the domain the constant symbol cd is xed. Theinterpretation is extended so that each new constant symbol is assigned to its corresponding element of the domain.One now denes truth for quantied formulas syntactically, as follows:

    1. Existential quantiers (alternate). A formula 9x(x) is true according toM if there is some d in the domainof discourse such that (cd) holds. Here (cd) is the result of substituting cd for every free occurrence of x in.

    2. Universal quantiers (alternate). A formula 8x(x) is true according toM if, for every d in the domain ofdiscourse, (cd) is true according to M.

    This alternate approach gives exactly the same truth values to all sentences as the approach via variable assignments.

    3.3.3 Validity, satisability, and logical consequenceSee also: Satisability

    If a sentence evaluates to True under a given interpretationM, one says thatM satises ; this is denotedM '. A sentence is satisable if there is some interpretation under which it is true.Satisability of formulas with free variables is more complicated, because an interpretation on its own does notdetermine the truth value of such a formula. The most common convention is that a formula with free variables issaid to be satised by an interpretation if the formula remains true regardless which individuals from the domain ofdiscourse are assigned to its free variables. This has the same eect as saying that a formula is satised if and only ifits universal closure is satised.A formula is logically valid (or simply valid) if it is true in every interpretation. These formulas play a role similarto tautologies in propositional logic.A formula is a logical consequence of a formula if every interpretation that makes true also makes true. Inthis case one says that is logically implied by .

    3.3.4 AlgebraizationsAn alternate approach to the semantics of rst-order logic proceeds via abstract algebra. This approach generalizesthe LindenbaumTarski algebras of propositional logic. There are three ways of eliminating quantied variables fromrst-order logic that do not involve replacing quantiers with other variable binding term operators:

    Cylindric algebra, by Alfred Tarski and his coworkers;

  • 3.3. SEMANTICS 17

    Polyadic algebra, by Paul Halmos; Predicate functor logic, mainly due to Willard Quine.

    These algebras are all lattices that properly extend the two-element Boolean algebra.Tarski and Givant (1987) showed that the fragment of rst-order logic that has no atomic sentence lying in the scopeof more than three quantiers has the same expressive power as relation algebra. This fragment is of great interestbecause it suces for Peano arithmetic and most axiomatic set theory, including the canonical ZFC. They also provethat rst-order logic with a primitive ordered pair is equivalent to a relation algebra with two ordered pair projectionfunctions.

    3.3.5 First-order theories, models, and elementary classesA rst-order theory of a particular signature is a set of axioms, which are sentences consisting of symbols from thatsignature. The set of axioms is often nite or recursively enumerable, in which case the theory is called eective.Some authors require theories to also include all logical consequences of the axioms. The axioms are considered tohold within the theory and from them other sentences that hold within the theory can be derived.A rst-order structure that satises all sentences in a given theory is said to be amodel of the theory. An elementaryclass is the set of all structures satisfying a particular theory. These classes are a main subject of study in modeltheory.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 usualoperations. However, the LwenheimSkolem theorem shows that most rst-order theories will also have other,nonstandard models.A theory is consistent if it is not possible to prove a contradiction from the axioms of the theory. A theory is completeif, for every formula in its signature, either that formula or its negation is a logical consequence of the axioms of thetheory. Gdels incompleteness theorem shows that eective rst-order theories that include a sucient portion ofthe theory of the natural numbers can never be both consistent and complete.For more information on this subject see List of rst-order theories and Theory (mathematical logic)

    3.3.6 Empty domainsMain article: Empty domain

    The denition above requires that the domain of discourse of any interpretation must be a nonempty set. There aresettings, such as inclusive logic, where empty domains are permitted. Moreover, if a class of algebraic structuresincludes an empty structure (for example, there is an empty poset), that class can only be an elementary class inrst-order logic if empty domains are permitted or the empty structure is removed from the class.There are several diculties with empty domains, however:

    Many common rules of inference are only valid when the domain of discourse is required to be nonempty. Oneexample is the rule stating that _9x implies 9x(_ ) when x is not a free variable in . This rule, whichis used to put formulas into prenex normal form, is sound in nonempty domains, but unsound if the emptydomain is permitted.

    The denition of truth in an interpretation that uses a variable assignment function cannot work with emptydomains, because there are no variable assignment functions whose range is empty. (Similarly, one cannotassign interpretations to constant symbols.) This truth denition requires that one must select a variable as-signment function ( above) before truth values for even atomic formulas can be dened. Then the truth valueof a sentence is dened to be its truth value under any variable assignment, and it is proved that this truthvalue does not depend on which assignment is chosen. This technique does not work if there are no assignmentfunctions at all; it must be changed to accommodate empty domains.

    Thus, when the empty domain is permitted, it must often be treated as a special case. Most authors, however, simplyexclude the empty domain by denition.

  • 18 CHAPTER 3. FIRST-ORDER LOGIC

    3.4 Deductive systemsA deductive system is used to demonstrate, on a purely syntactic basis, that one formula is a logical consequenceof another formula. There are many such systems for rst-order logic, including Hilbert-style deductive systems,natural deduction, the sequent calculus, the tableaux method, and resolution. These share the common property thata deduction is a nite syntactic object; the format of this object, and the way it is constructed, vary widely. Thesenite deductions themselves are often called derivations in proof theory. They are also often called proofs, but arecompletely 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 deductivesystem is complete if every logically valid formula is derivable. All of the systems discussed in this article are bothsound and complete. They also share the property that it is possible to eectively verify that a purportedly validdeduction is actually a deduction; such deduction systems are called eective.A key property of deductive systems is that they are purely syntactic, so that derivations can be veried withoutconsidering any interpretation. Thus a sound argument is correct in every possible interpretation of the language,regardless whether that interpretation is about mathematics, economics, or some other area.In general, logical consequence in rst-order logic is only semidecidable: if a sentence A logically implies a sentenceB then this can be discovered (for example, by searching for a proof until one is found, using some eective, sound,complete proof system). However, if A does not logically imply B, this does not mean that A logically implies thenegation of B. There is no eective procedure that, given formulas A and B, always correctly decides whether Alogically implies B.

    3.4.1 Rules of inference

    Further information: List of rules of inference

    A rule of inference states that, given a particular formula (or set of formulas) with a certain property as a hypothesis,another specic 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 satises the hypothesis, that interpretation alsosatises the conclusion.For example, one common rule of inference is the rule of substitution. If t is a term and is a formula possiblycontaining the variable x, then [t/x] (often denoted [x/t]) is the result of replacing all free instances of x by t in. The substitution rule states that for any and any term t, one can conclude [t/x] from provided that no freevariable of t becomes bound during the substitution process. (If some free variable of t becomes bound, then tosubstitute t for x it is rst necessary to change the bound variables of to dier from the free variables of t.)To see why the restriction on bound variables is necessary, consider the logically valid formula given by 9x(x = y), in the signature of (0,1,+,,=) of arithmetic. If t is the term x + 1, the formula [t/y] is 9x(x = x+1) , which willbe false in many interpretations. The problem is that the free variable x of t became bound during the substitution.The intended replacement can be obtained by renaming the bound variable x of to something else, say z, so thatthe formula after substitution is 9z(z = x+ 1) , which is again logically valid.The substitution rule demonstrates several common aspects of rules of inference. It is entirely syntactical; one cantell whether it was correctly applied without appeal to any interpretation. It has (syntactically dened) limitations onwhen it can be applied, which must be respected to preserve the correctness of derivations. Moreover, as is oftenthe case, these limitations are necessary because of interactions between free and bound variables that occur duringsyntactic manipulations of the formulas involved in the inference rule.

    3.4.2 Hilbert-style systems and natural deduction

    A deduction in a Hilbert-style deductive system is a list of formulas, each of which is a logical axiom, a hypothesisthat has been assumed for the derivation at hand, or follows from previous formulas via a rule of inference. Thelogical axioms consist of several axiom schemas of logically valid formulas; these encompass a signicant amount ofpropositional logic. The rules of inference enable the manipulation of quantiers. Typical Hilbert-style systems havea small number of rules of inference, along with several innite schemas of logical axioms. It is common to have onlymodus ponens and universal generalization as rules of inference.

  • 3.4. DEDUCTIVE SYSTEMS 19

    Natural deduction systems resemble Hilbert-style systems in that a deduction is a nite list of formulas. However,natural deduction systems have no logical axioms; they compensate by adding additional rules of inference that canbe used to manipulate the logical connectives in formulas in the proof.

    3.4.3 Sequent calculus

    Further information: Sequent calculus

    The sequent calculus was developed to study the properties of natural deduction systems. Instead of working withone formula at a time, it uses sequents, which are expressions of the form

    A1; : : : ; An ` B1; : : : ; Bk;

    where A1, ..., An, B1, ..., Bk are formulas and the turnstile symbol ` is used as punctuation to separate the two halves.Intuitively, a sequent expresses the idea that (A1 ^ ^An) implies (B1 _ _Bk) .

    3.4.4 Tableaux method

    Further information: Method of analytic tableaux

    Unlike the methods just described, the derivations in the tableaux method are not lists of formulas. Instead, a deriva-tion is a tree of formulas. To show that a formula A is provable, the tableaux method attempts to demonstrate thatthe negation of A is unsatisable. The tree of the derivation has :A at its root; the tree branches in a way that reectsthe structure of the formula. For example, to show that C _ D is unsatisable requires showing that C and D areeach unsatisable; this corresponds to a branching point in the tree with parent C _D and children C and D.

    3.4.5 Resolution

    The resolution rule is a single rule of inference that, together with unication, is sound and complete for rst-orderlogic. As with the tableaux method, a formula is proved by showing that the negation of the formula is unsatisable.Resolution is commonly used in automated theorem proving.The resolutionmethod works only with formulas that are disjunctions of atomic formulas; arbitrary formulas must rstbe converted to this form through Skolemization. The resolution rule states that from the hypothesesA1_ _Ak_Cand B1 _ _Bl _ :C , the conclusion A1 _ _Ak _B1 _ _Bl can be obtained.

    3.4.6 Provable identities

    The following sentences can be called identities because the main connective in each is the biconditional.

    :8xP (x), 9x:P (x):9xP (x), 8x:P (x)8x 8y P (x; y), 8y 8xP (x; y)9x 9y P (x; y), 9y 9xP (x; y)8xP (x) ^ 8xQ(x), 8x (P (x) ^Q(x))9xP (x) _ 9xQ(x), 9x (P (x) _Q(x))P ^ 9xQ(x), 9x (P ^Q(x)) (where x must not occur free in P )P _ 8xQ(x), 8x (P _Q(x)) (where x must not occur free in P )

  • 20 CHAPTER 3. FIRST-ORDER LOGIC

    3.5 Equality and its axiomsThere are several dierent conventions for using equality (or identity) in rst-order logic. The most common con-vention, known as rst-order logic with equality, includes the equality symbol as a primitive logical symbol whichis always interpreted as the real equality relation between members of the domain of discourse, such that the twogiven members are the same member. This approach also adds certain axioms about equality to the deductive systememployed. These equality axioms are:

    1. Reexivity. For each variable x, x = x.2. Substitution for functions. For all variables x and y, and any function symbol f,

    x = y f(...,x,...) = f(...,y,...).3. Substitution for formulas. For any variables x and y and any formula (x), if ' is obtained by replacing any

    number of free occurrences of x in with y, such that these remain free occurrences of y, thenx = y ( ').

    These are axiom schemas, each of which species an innite set of axioms. The third schema is known as Leibnizslaw, the principle of substitutivity, the indiscernibility of identicals, or the replacement property. The secondschema, involving the function symbol f, is (equivalent to) a special case of the third schema, using the formula

    x = y (f(...,x,...) = z f(...,y,...) = z).

    Many other properties of equality are consequences of the axioms above, for example:

    1. Symmetry. If x = y then y = x.2. Transitivity. If x = y and y = z then x = z.

    3.5.1 First-order logic without equalityAn alternate approach considers the equality relation to be a non-logical symbol. This convention is known as rst-order logic without equality. If an equality relation is included in the signature, the axioms of equality must now beadded to the theories under consideration, if desired, instead of being considered rules of logic. The main dierencebetween this method and rst-order logic with equality is that an interpretation may now interpret two distinct indi-viduals as equal (although, by Leibnizs 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 discoursethat is congruent with respect to the functions and relations of the interpretation.When this second convention is followed, the term normal model is used to refer to an interpretation where nodistinct individuals a and b satisfy a = b. In rst-order logic with equality, only normal models are considered, andso there is no term for a model other than a normal model. When rst-order logic without equality is studied, it isnecessary to amend the statements of results such as the LwenheimSkolem theorem so that only normal modelsare considered.First-order logic without equality is often employed in the context of second-order arithmetic and other higher-ordertheories of arithmetic, where the equality relation between sets of natural numbers is usually omitted.

    3.5.2 Dening equality within a theoryIf a theory has a binary formula A(x,y) which satises reexivity and Leibnizs 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 asderivable theorems. For example, in theories with no function symbols and a nite number of relations, it is possibleto dene equality in terms of the relations, by dening the two terms s and t to be equal if any relation is unchangedby changing s to t in any argument.Some theories allow other ad hoc denitions of equality:

  • 3.6. METALOGICAL PROPERTIES 21

    In the theory of partial orders with one relation symbol , one could dene s = t to be an abbreviation for s t^ t s.

    In set theory with one relation 2 , one may dene s = t to be an abbreviation for 8 x (s 2 x $ t 2 x) ^ 8x (x 2 s $ x 2 t). This denition of equality then automatically satises the axioms for equality. In thiscase, one should replace the usual axiom of extensionality, 8x8y[8z(z 2 x , z 2 y) ) x = y] , by8x8y[8z(z 2 x, z 2 y)) 8z(x 2 z , y 2 z)] , i.e. if x and y have the same elements, then they belongto the same sets.

    3.6 Metalogical properties

    One motivation for the use of rst-order logic, rather than higher-order logic, is that rst-order logic has manymetalogical properties that stronger logics do not have. These results concern general properties of rst-order logicitself, rather than properties of individual theories. They provide fundamental tools for the construction of modelsof rst-order theories.

    3.6.1 Completeness and undecidability

    Gdels completeness theorem, proved by Kurt Gdel in 1929, establishes that there are sound, complete, eectivedeductive systems for rst-order logic, and thus the rst-order logical consequence relation is captured by nite prov-ability. Naively, the statement that a formula logically implies a formula depends on every model of ; thesemodels will in general be of arbitrarily large cardinality, and so logical consequence cannot be eectively veried bychecking every model. However, it is possible to enumerate all nite derivations and search for a derivation of from. If is logically implied by , such a derivation will eventually be found. Thus rst-order logical consequence issemidecidable: it is possible to make an eective enumeration of all pairs of sentences (,) such that is a logicalconsequence of .Unlike propositional logic, rst-order logic is undecidable (although semidecidable), provided that the language hasat least one predicate of arity at least 2 (other than equality). This means that there is no decision procedure thatdetermines whether arbitrary formulas are logically valid. This result was established independently byAlonzo Churchand Alan Turing in 1936 and 1937, respectively, giving a negative answer to the Entscheidungsproblem posed byDavid Hilbert in 1928. Their proofs demonstrate a connection between the unsolvability of the decision problem forrst-order logic and the unsolvability of the halting problem.There are systems weaker than full rst-order logic for which the logical consequence relation is decidable. Theseinclude propositional logic andmonadic predicate logic, which is rst-order logic restricted to unary predicate symbolsand no function symbols. Other logics with no function symbols which are decidable are the guarded fragment ofrst-order logic, as well as two-variable logic. The BernaysSchnnkel class of rst-order formulas is also decidable.Decidable subsets of rst-order logic are also studied in the framework of description logics.

    3.6.2 The LwenheimSkolem theorem

    The LwenheimSkolem theorem shows that if a rst-order theory of cardinality has an innite model, then it hasmodels of every innite cardinality greater than or equal to . One of the earliest results in model theory, it impliesthat it is not possible to characterize countability or uncountability in a rst-order language. That is, there is norst-order formula (x) such that an arbitrary structure M satises if and only if the domain of discourse of M iscountable (or, in the second case, uncountable).The LwenheimSkolem theorem implies that innite structures cannot be categorically axiomatized in rst-orderlogic. For example, there is no rst-order theory whose only model is the real line: any rst-order theory with aninnite model also has a model of cardinality larger than the continuum. Since the real line is innite, any theorysatised by the real line is also satised by some nonstandard models. When the LwenheimSkolem theorem isapplied to rst-order set theories, the nonintuitive consequences are known as Skolems paradox.

  • 22 CHAPTER 3. FIRST-ORDER LOGIC

    3.6.3 The compactness theoremThe compactness theorem states that a set of rst-order sentences has a model if and only if every nite subset of ithas a model. This implies that if a formula is a logical consequence of an innite set of rst-order axioms, then itis a logical consequence of some nite number of those axioms. This theorem was proved rst by Kurt Gdel as aconsequence of the completeness theorem, but many additional proofs have been obtained over time. It is a centraltool in model theory, providing a fundamental method for constructing models.The compactness theorem has a limiting eect on which collections of rst-order structures are elementary classes.For example, the compactness theorem implies that any theory that has arbitrarily large nite models has an in-nite model. Thus the class of all nite graphs is not an elementary class (the same holds for many other algebraicstructures).There are also more subtle limitations of rst-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 (directededges). Validating such a system may require showing that no bad state can be reached from any good state. Thusone seeks to determine if the good and bad states are in dierent connected components of the graph. However, thecompactness theorem can be used to show that connected graphs are not an elementary class in rst-order logic,and there is no formula (x,y) of rst-order logic, in the signature of graphs, that expresses the idea that there is apath from x to y. Connectedness can be expressed in second-order logic, however, but not with only existential setquantiers, as 11 also enjoys compactness.

    3.6.4 Lindstrms theoremMain article: Lindstrms theorem

    Per Lindstrm showed that the metalogical properties just discussed actually characterize rst-order logic in the sensethat no stronger logic can also have those properties (Ebbinghaus and Flum 1994, Chapter XIII). Lindstrm deneda class of abstract logical systems, and a rigorous denition of the relative strength of a member of this class. Heestablished two theorems for systems of this type:

    A logical system satisfying Lindstrms denition that contains rst-order logic and satises both the LwenheimSkolem theorem and the compactness theorem must be equivalent to rst-order logic.

    A logical system satisfying Lindstrms denition that has a semidecidable logical consequence relation andsatises the LwenheimSkolem theorem must be equivalent to rst-order logic.

    3.7 LimitationsAlthough rst-order logic is sucient for formalizing much of mathematics, and is commonly used in computerscience and other elds, it has certain limitations. These include limitations on its expressiveness and limitations ofthe fragments of natural languages that it can describe.For instance, rst-order logic is undecidable, meaning a sound, complete and terminating decision algorithm is im-possible. This has led to the study of interesting decidable fragments such as C2, rst-order logic with two variablesand the counting quantiers 9n and 9n (these quantiers are, respectively, there exists at least n" and there existsat most n") (Horrocks 2010).

    3.7.1 ExpressivenessThe LwenheimSkolem theorem shows that if a rst-order theory has any innite model, then it has innite modelsof every cardinality. In particular, no rst-order theory with an innite model can be categorical. Thus there isno rst-order theory whose only model has the set of natural numbers as its domain, or whose only model has theset of real numbers as its domain. Many extensions of rst-order logic, including innitary logics and higher-orderlogics, are more expressive in the sense that they do permit categorical axiomatizations of the natural numbers orreal numbers. This expressiveness comes at a metalogical cost, however: by Lindstrms theorem, the compactnesstheorem and the downward LwenheimSkolem theorem cannot hold in any logic stronger than rst-order.

  • 3.8. RESTRICTIONS, EXTENSIONS, AND VARIATIONS 23

    3.7.2 Formalizing natural languages

    First-order logic is able to formalize many simple quantier constructions in natural language, such as every personwho lives in Perth lives in Australia. But there are many more complicated features of natural language that cannotbe expressed in (single-sorted) rst-order logic. Any logical system which is appropriate as an instrument for theanalysis of natural language needs a much richer structure than rst-order predicate logic (Gamut 1991, p. 75).

    3.8 Restrictions, extensions, and variationsThere are many variations of rst-order logic. Some of these are inessential in the sense that they merely changenotation without aecting the semantics. Others change the expressive power more signicantly, by extending thesemantics through additional quantiers or other new logical symbols. For example, innitary logics permit formulasof innite size, and modal logics add symbols for possibility and necessity.

    3.8.1 Restricted languages

    First-order logic can be studied in languages with fewer logical symbols than were described above.

    Because 9x(x) can be expressed as :8x:(x) , and 8x(x) can be expressed as :9x:(x) , either of thetwo quantiers 9 and 8 can be dropped.

    Since _ can be expressed as :(:^: ) and ^ can be expressed as :(:_: ) , either _ or ^ canbe dropped. In other words, it is sucient to have : and _ , or : and ^ , as the only logical connectives.

    Similarly, it is sucient to have only: and! as logical connectives, or to have only the Sheer stroke (NAND)or the Peirce arrow (NOR) operator.

    It is possible to entirely avoid function symbols and constant symbols, rewriting them via predicate symbolsin an appropriate way. For example, instead of using a constant symbol 0 one may use a predicate 0(x)(interpreted as x = 0 ), and replace every predicate such as P (0; y) with 8x (0(x) ! P (x; y)) . Afunction such as f(x1; x2; :::; xn) will similarly be replaced by a predicate F (x1; x2; :::; xn; y) interpretedas y = f(x1; x2; :::; xn) . This change requires adding additional axioms to the theory at hand, so thatinterpretations of the predicate symbols used have the correct semantics.

    Restrictions such as these are useful as a technique to reduce the number of inference rules or axiom schemas indeductive systems, which leads to shorter proofs of metalogical results. The cost of the restrictions is that it becomesmore dicult to express natural-language statements in the formal system at hand, because the logical connectivesused in the natural language statements must be replaced by their (longer) denitions in terms of the restricted col-lection of logical connectives. Similarly, derivations in the limited systems may be longer than derivations in systemsthat include additional connectives. There is thus a trade-o between the ease of working within the formal systemand the ease of proving results about the formal system.It is also possible to restrict the arities of function symbols and predicate symbols, in suciently expressive theories.One can in principle dispense entirely with functions of arity greater than 2 and predicates of arity greater than 1in theories that include a pairing function. This is a function of arity 2 that takes pairs of elements of the domainand returns an ordered pair containing them. It is also sucient to have two predicate symbols of arity 2 that deneprojection functions from an ordered pair to its components. In either case it is necessary that the natural axioms fora pairing function and its projections are satised.

    3.8.2 Many-sorted logic

    Ordinary rst-order interpretations have a single domain of discourse over which all quantiers range. Many-sortedrst-order logic allows variables to have dierent sorts, which have dierent domains. This is also called typed rst-order logic, and the sorts called types (as in data type), but it is not the same as rst-order type theory. Many-sortedrst-order logic is often used in the study of second-order arithmetic.

  • 24 CHAPTER 3. FIRST-ORDER LOGIC

    When there are only nitely many sorts in a theory, many-sorted rst-order logic can be reduced to single-sorted rst-order logic. One introduces into the single-sorted theory a unary predicate symbol for each sort in the many-sortedtheory, and adds an axiom saying that these unary predicates partition the domain of discourse. For example, if thereare two sorts, one adds predicate symbols P1(x) and P2(x) and the axiom

    8x(P1(x) _ P2(x)) ^ :9x(P1(x) ^ P2(x))Then the elements satisfying P1 are thought of as elements of the rst sort, and elements satisfying P2 as elementsof the second sort. One can quantify over each sort by using the corresponding predicate symbol to limit the rangeof quantication. For example, to say there is an element of the rst sort satisfying formula (x), one writes

    9x(P1(x) ^ (x))

    3.8.3 Additional quantiersAdditional quantiers can be added to rst-order logic.

    Sometimes it is useful to say that "P(x) holds for exactly one x", which can be expressed as 9! x P(x). Thisnotation, called uniqueness quantication, may be taken to abbreviate a formula such as 9 x (P(x) ^8 y (P(y)! (x = y))).

    First-order logic with extra quantiers has new quantiers Qx,..., with meanings such as there are many xsuch that .... Also see branching quantiers and the plural quantiers of George Boolos and others.

    Bounded quantiers are often used in the study of set theory or arithmetic.

    3.8.4 Innitary logicsMain article: Innitary logic

    Innitary logic allows innitely long sentences. For example, one may allow a conjunction or disjunction of innitelymany formulas, or quantication over innitely many variables. Innitely long sentences arise in areas of mathematicsincluding topology and model theory.Innitary logic generalizes rst-order logic to allow formulas of innite length. The most common way in whichformulas can become innite is through innite conjunctions and disjunctions. However, it is also possible to admitgeneralized signatures in which function and relation symbols are allowed to have innite arities, or in which quantierscan bind innitely many variables. Because an innite formula cannot be represented by a nite string, it is necessaryto choose some other representation of formulas; the usual representation in this context is a tree. Thus formulas are,essentially, identied with their parse trees, rather than with the strings being parsed.The most commonly studied innitary logics are denoted L, where and are each either cardinal numbersor the symbol . In this notation, ordinary rst-order logic is L. In the logic L, arbitrary conjunctions ordisjunctions are allowed when building formulas, and there is an unlimited supply of variables. More generally, thelogic that permits conjunctions or disjunctions with less than constituents is known as L. For example, L1permits countable conjunctions and disjunctions.The set of free variables in a formula of L can have any cardinality strictly less than , yet only nitely many ofthem can be in the scope of any quantier when a formula appears as a subformula of another.[8] In other innitarylogics, a subformula may be in the scope of innitely many quantiers. For example, in L, a single universal or ex-istential quantier may bind arbitrarily many variables simultaneously. Similarly, the logic L permits simultaneousquantication over fewer than variables, as well as conjunctions and disjunctions of size less than .

    3.8.5 Non-classical and modal logics Intuitionistic rst-order logic uses intuitionistic rather than classical propositional calculus; for example, need not be equivalent to .

  • 3.9. AUTOMATED THEOREM PROVING AND FORMAL METHODS 25

    First-order modal logic allows one to describe other possible worlds as well as this contingently true worldwhich we inhabit. In some versions, the set of possible worlds varies depending on which possible world oneinhabits. Modal logic has extra modal operators with meanings which can be characterized informally as, forexample it is necessary that " (true in all possible worlds) and it is possible that " (true in some possibleworld). With standard rst-order logic we have a single domain and each predicate is assigned one extension.With rst-order modal logic we have a domain function that assigns each possible world its own domain, so thateach predicate gets an extension only relative to these possible worlds. This allows us to model cases where,for example, Alex is a Philosopher, but might have been a Mathematician, and might not have existed at all.In the rst possible world P(a) is true, in the second P(a) is false, and in the third possible world there is no ain the domain at all.

    rst-order fuzzy logics are rst-order extensions of propositional fuzzy logics rather than classical propositionalcalculus.

    3.8.6 Fixpoint logicFixpoint logic extends rst-order logic by adding the closure under the least xed points of positive operators.[9]

    3.8.7 Higher-order logicsMain article: Higher-order logic

    The characteristic feature of rst-order logic is that individuals can be quantied, but not predicates. Thus

    9a(Phil(a))is a legal rst-order formula, but

    9Phil(Phil(a))is not, in most formalizations of rst-order logic. Second-order logic extends rst-order logic by adding the lattertype of quantication. Other higher-order logics allow quantication over even higher types than second-order logicpermits. These higher types include relations between relations, functions from relations to relations between relations,and other higher-type objects. Thus the rst in rst-order logic describes the type of objects that can be quantied.Unlike rst-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 fullsemantics. The combination of additional quantiers and the full semantics for these quantiers makes higher-orderlogic stronger than rst-order logic. In particular, the (semantic) logical consequence relation for second-order andhigher-order logic is not semidecidable; there is no eective deduction system for second-order logic that is soundand complete under full semantics.Second-order logic with full semantics is more expressive than rst-order logic. For example, it is possible to createaxiom systems in second-order logic that uniquely characterize the natural numbers and the real line. The cost ofthis expressiveness is that second-order and higher-order logics have fewer attractive metalogical properties than rst-order logic. For example, the LwenheimSkolem theorem and compactness theorem of rst-order logic becomefalse when generalized to higher-order logics with

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