Team Logic and Second-Order Logic∗

Juha Kontinen† Ville Nurmi‡

Abstract

Team logic is a new logic, introduced by Väänänen [12], extend-ing dependence logic by classical negation. Dependence logic addsto first-order logic atomic formulas expressing functional dependenceof variables on each other. It is known that on the level of sentencesdependence logic and team logic are equivalent with existential second-order logic and full second-order logic, respectively. In this article weshow that, in a sense that we make explicit, team logic and second-order logic are also equivalent with respect to open formulas. A similarearlier result relating open formulas of dependence logic to the negativefragment of existential second-order logic was proved in [8].

1 IntroductionTeam Logic is a new logic arising from the so-called Dependence Logic [12] byadding classical negation. Dependence logic adds the concept of dependenceto first-order logic by means of new atomic formulas

=(x1, . . . , xn, y) (1)

the meaning of which is that the values of x1, . . . , xn completely determinethe values of y. This is made precise in dependence logic by basing semanticsin the concept of a set of assignments satisfying a formula (rather than anindividual assignment satisfying a formula, as in first-order logic). Such sets∗The first author was supported by grant 127661 of the Academy of Finland and the

European Science Foundation Eurocores programme LogICCC [FP002 - Logic for Inter-action (LINT)] through grant 129208 of the Academy of Finland. The second author wassupported by the MALJA Graduate school in Mathematical logic.†Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, FI-

00014 University of Helsinki, Finland. Tel: +358-9-19151314, Fax: +358-9-19151400,email: juha.kontinen@helsinki.fi‡Department of Mathematics and Statistics, University of Helsinki, Finland

1

are called teams. The idea of team based semantics is due to Hodges [6]. Ateam X is said to satisfy the formula (1) if any two assignments s and s′

from X that agree about x1, . . . , xn also agree about y. If we think of X asa database, this is the concept of functional dependence, studied extensivelyin database theory.

We review briefly the short history of team logic. In first-order logic theorder in which quantifiers are written determines the mutual dependencerelations between variables. For example, in

∀x0∃x1∀x2∃x3φ

the variable x1 depends on x0, and the variable x3 depends on both x0 andx2. In dependence logic we write down explicitly the dependence relationsbetween variables and by so doing make it possible to express dependenciesnot otherwise expressible in first-order logic.

The first step in this direction was taken by Henkin [3] with his partiallyordered quantifiers, e.g. (

∀x0 ∃x1

∀x2 ∃x3

)φ,

where x1 depends only on x0 and x3 depends only on x2. The second stepwas taken by Hintikka and Sandu [5, 4], who introduced the slash-notation

∀x0∃x1∀x2∃x3/∀x0φ,

where ∃x3/∀x0 means that x3 is "independent" of x0 in the sense that a choicefor the value of x3 should not depend on what the value of x0 is. The observa-tion of Hintikka and Sandu was that we can add slashed quantifiers ∃x3/∀x0

coherently to first-order logic if we give up some of the classical propertiesof negation, most notably the Law of Excluded Middle. They called theirlogic Independence Friendly Logic (IF). In [12] Väänänen takes the furtherstep of writing down explicitly the mutual dependence relationships betweenvariables. Thus he writes

∀x0∃x1∀x2∃x3(=(x2, x3) ∧ φ)

to indicate that x3 depends on x2 only. The new atomic formula =(x2, x3) hasthe explicit meaning that x3 depends on x2 and on nothing else. This resultsin a logic Väänänen calls Dependence Logic. It is equivalent in expressivepower to the logic of Hintikka and Sandu in the sense that there are truth-preserving translations from one to the other. Both have the same expressivepower on the level of sentences being equivalent to the existential fragmentof second-order logic.

2

The negation ¬ of dependence logic does not satisfy the law of excludedmiddle and is therefore not the classical Boolean negation. This is clearlymanifested by the existence of non-determined sentences φ in dependencelogic. In such cases, the failure ofM |= φ does not implyM |= ¬φ. Hintikka[4] introduced extended independence friendly logic by taking the Booleanclosure of his independence friendly logic. We take the further action ofmaking classical negation ∼ one of the logical operations on a par with otherpropositional operations and quantifiers. This yields an extension of theBoolean closure of dependence logic. The new logic is called Team Logic in[12].

While we define team logic we have to restrict the negation ¬ of depen-dence logic. The game-theoretic intuition behind ¬φ is that it says somethingabout "the other player." The introduction of ∼ unfortunately ruins the ba-sic game-theoretic intuition, and there is no "other player" anymore. If φ isa formula of dependence logic, then ∼φ has the meaning "player II does nothave a winning strategy," but it is not clear what the meaning of ¬∼φ wouldbe.

Just as a formula φ of first-order logic with free variables x1, . . . , xn definesan n-ary relation in a model M, i.e., the set of n-tuples (a1, . . . , an) thatsatisfy φ in M, a formula φ of team logic defines a set of n-ary relations,namely the set of teams that satisfies φ. It was shown in [8] that a setX 6= ∅ of n-ary relations is defined in this sense by a formula φ(x1, . . . , xn)of dependence logic if and only if X is definable in existential second-orderlogic by a formula Φ(P ) with an n-ary predicate symbol P occuring negativelyonly. The goal of this paper is to prove a similar result for team logic, namely,we show that a set X of n-ary relations is defined in this sense by a formulaφ(x1, . . . , xn) of team logic if and only if X is definable in second-order logicby a formula Φ(P ) with an n-ary predicate symbol P .

2 PreliminariesIn this section we define dependence logic and team logic and recall some oftheir properties.

The syntax of dependence logic and team logic extend the syntax of first-order logic by new atomic (dependence) formulas of the form

=(t1, . . . , tn), (2)

where t1, . . . , tn are terms. The intuitive meaning of the dependence formula(2) is that the value of the term tn is determined by the values of the terms

3

t1, . . . , tn−1. As singular cases we have

=(),

which we take to be universally true, and

=(t),

which declares that the value of the term t depends on nothing, i.e., is con-stant.

In order to define the semantics of dependence logic and team logic, wefirst need to define the concept of a team. LetM be a model with domainM . Assignments ofM are finite mappings from variables intoM . The valueof a term t in an assignment s is denoted by 〈t〉s or 〈t〉f,s if t contains somefunction variables whose interpretations are given by f .

If s is an assignment, x a variable, and a ∈ M , then s(x 7→ a) denotesthe assignment (with domain dom(s)∪ {x}) which agrees with s everywhereexcept that it maps x to a.

LetM be a set and {x1, . . . , xk} a finite (possibly empty) set of variables.A team X of M with domain {x1, . . . , xk} is any set of assignments fromthe variables {x1, . . . , xk} into the set M . In particular, there are two teamswith the empty domain, namely, the empty team ∅ and the full team {∅}.We denote by Rel(X) the k-ary relation of M corresponding to X

Rel(X) = {(s(x1), . . . , s(xk)) : s ∈ X}.

Definition 2.1. Let X be a team of M , a ∈ M and F : X → M . Thefollowing operations on teams will be used:

X(x 7→ a) := {s(x 7→ a) : s ∈ X}X(x 7→ F ) := {s(x 7→ F (s)) : s ∈ X}X(x 7→M) := {s(x 7→ a) : s ∈ X and a ∈M}

It might help one’s intuition to note that the first two operations canonly decrease the cardinality of a team, whereas the third operation canpotentially multiply team’s cardinality by the factor of |M |.

We are now ready to define the syntax and semantics of dependence logicand team logic. Since team logic is the closure of dependence logic underclassical negation, we will first give simultaneously the syntax and semanticsof team logic and then indicate how dependence logic can be obtained as afragment of team logic.

4

Definition 2.2. Let L be a vocabulary,M a L-model and X a team of M .We define the satisfaction relation M, X |= φ for L-formulas of team logicin the following way. Below, we assume that the domain of X contains atleast the variables which appear free in φ. Also, the case of atomic formulas(R ∈ L ∪ {=}) refers to the usual truth definition of first-order logic.

M, X |= Rt1, . . . , tk ⇐⇒ for all s ∈ X: M, s |= Rt1, . . . , tk

M, X |= ¬Rt1, . . . , tk ⇐⇒ for all s ∈ X: M, s |= ¬Rt1, . . . , tkM, X |= =(t1, . . . , tk, u) ⇐⇒ there is f such that for all s ∈ X:

〈u〉s = f(〈t1〉s, . . . , 〈tk〉s)M, X |= ¬=(t1, . . . , tk, u) ⇐⇒ X = ∅M, X |= ∼φ ⇐⇒ M, X 6|= φ

M, X |= φ ∨ ψ ⇐⇒ M, X |= φ orM, X |= ψ

M, X |= φ ∧ ψ ⇐⇒ M, X |= φ andM, X |= ψ

M, X |= φ⊗ ψ ⇐⇒ there is Y, Z ⊆ X such that Y ∪ Z = X andM, Y |= φ andM, Z |= ψ

M, X |= φ⊕ ψ ⇐⇒ for all Y, Z ⊆ X: if Y ∪ Z = X thenM, Y |= φ orM, Z |= ψ

M, X |= ∃xφ ⇐⇒ there is F : X →M such thatM, X(x 7→ F ) |= φ

M, X |= ∀xφ ⇐⇒ for all F : X →M : M, X(x 7→ F ) |= φ

M, X |= !xφ ⇐⇒ M, X(x 7→M) |= φ

Finally, a sentence φ is true in a modelM ifM, {∅} |= φ.

As already mentioned, team logic is the extension of dependence logicby classical negation ∼. Dependence logic can be acquired as the fragmentof team logic allowing only atomic formulas and their negations (¬) and theconnectives ∧, ⊗, ∃ and !. In the context of dependence logic, the connectives⊗ and ! are usually denoted by ∨ and ∀. However, in team logic classicaldisjunction ∨ and the connectives ⊕ and ∀ arise as the duals of ∧, ⊗ and ∃with respect to ∼. Therefore, in this paper we think of dependence logic asthe fragment of team logic using only the connectives ∧, ⊗, ∃ and !.

It is instructive to note that also first-order logic can be embedded intoteam logic (and dependence logic) in a canonical way. The canonical trans-lation of a first-order formula φ (in negation normal-form) into a team logic

5

formula φ∗ goes as follows.

(Rt1, . . . , tk)∗ := Rt1, . . . , tk (¬φ)∗ := ¬φ∗

(φ ∨ ψ)∗ := φ∗ ⊗ ψ∗ (φ ∧ ψ)∗ := φ∗ ∧ ψ∗ (3)(∃xφ)∗ := ∃xφ∗ (∀xφ)∗ := !xφ∗

It is easy to verify that this translation satisfies

M, X |= φ∗ ⇐⇒ ∀s ∈ X : M, s |= φ,

from which it follows, for any sentence φ, that φ and φ∗ are logically equiva-lent.

3 BackgroundIn this section we discuss the expressive power of dependence logic and teamlogic and formalize the goal of this paper. We begin with dependence logic.

It is known that, on the level of sentences, the expressive power of depen-dence logic coincides with that of existential second-order logic (Σ1

1):

Theorem 3.1. For every sentence φ of dependence logic there is a sentenceΦ of Σ1

1 such that

For all modelsM: M, {∅} |= φ ⇐⇒ M |= Φ. (4)

Conversely, for every sentence Φ of Σ11 there is a sentence φ of dependence

logic such that (4) holds.

Proof. Using the method of [14, 2] (See Theorems 6.2 and 6.15 in [12]).

Theorem 3.1 does not tell us, on the face of it, anything about the formulasof dependence logic with free variables. The following basic fact (Fact 11.1in [6], see Proposition 3.10 in [12]) shows that the formulas of dependencelogic can only define downward monotone properties of teams.

Proposition 3.2 (Downward closure). Let ϕ be a formula of dependencelogic,M a model, and Y ⊆ X teams. ThenM, X |= ϕ impliesM, Y |= ϕ.

Another basic property of formulas of dependence logic is the following(see Lemma 3.9 in [12]):

Proposition 3.3. For every formula φ of dependence logic and every modelM: M, ∅ |= φ.

6

Note that a formula φ(x1, . . . , xn) of dependence logic defines, in a modelM, a set of n-ary relations Rel(X) ⊆ Mn, namely, the set of teams X withdomain {x1, . . . , xn} that satisfy φ inM. The translation from dependencelogic into Σ1

1 (see Theorem 3.4 below) implies that if a set X of n-ary relationsis defined in this sense by a formula φ(x1, . . . , xn) of dependence logic then Xis definable by a Σ1

1-sentence Φ(P ), where P is n-ary and is only allowed tooccur negatively in Φ(P ). Note that, on the semantical side, the assumptionthat P can only occur negatively corresponds to downward monotonicity.

Theorem 3.4. Let L be a vocabulary and φ(v1, . . . , vr) a L-formula of de-pendence logic with free variables v1, . . . , vr. Then there is a L∪{R}-sentenceψ of Σ1

1, in which R appears only negatively, such that for all modelsM andteams X with domain {v1, . . . , vr}:

M, X |= φ ⇐⇒ M,Rel(X) |= ψ.

Proof. See Theorem 6.2 in [12].

In [8] it was shown that also the converse holds.

Theorem 3.5. Let L be a vocabulary and R /∈ L. Then for every L ∪ {R}-sentence ψ of Σ1

1, in which R appears only negatively, there is a formulaφ(v1, . . . , vr) of dependence logic such that, for allM and X,

M, X |= φ ⇐⇒ M,Rel(X) |= ψ ∨ ∀x¬R(x).

Proof. See Theorem 4.10 in [8]. Note that the disjunct ∀x¬R(x) is neededon the right because, by Proposition 3.3, the empty team X = ∅ satisfiesall formulas of dependence logic but ψ need not always be true in the caseRel(X) = ∅ [7].

Our goal in this article is to characterize definable sets of teams, i.e., setsof the form,

{X :M, X |= φ},in team logic. It was shown in [9] that on the level of sentences the expressivepower of team logic corresponds to full second-order logic (SO).

Theorem 3.6. For every sentence φ of team logic there is a sentence Φ ofsecond-order logic such that

For all modelsM: M, {∅} |= φ ⇐⇒ M |= Φ. (5)

Conversely, for every sentence Φ of second-order logic there is a sentence φof team logic such that (5) holds.

In this article we generalize Theorem 3.6 from sentences to open formulasanalogously to Theorem 3.5. This result will then also imply a characteriza-tion for the definable sets of teams in team logic (see Corollary 5.5).

7

4 ExamplesIn this section we illustrate the question studied in this paper with examples.

Example 4.1. Let L be a vocabulary,M a L-model, and X 6= ∅ a family ofsets of r-tuples of M . By Theorems 3.4 and 3.5 the following are equivalent:

1. X = {Rel(X) :M, X |= ψ(v1, . . . , vr)} for some L-formula ψ(v1, . . . , vr)of dependence logic.

2. X = {Y :M, Y |= φ(R)} for some sentence φ ∈ Σ11[L ∪ {R}], in which

R occurs only negatively.

Since team logic is closed under boolean operations, it follows that thefamily of sets X of teams over M, which can be defined in team logic, isclosed under boolean operations and thus properly contains the family of Xwhich can be defined in dependence logic.

In the following example we show that very simple formulas can be usedto express properties of teams which cannot be expressed by any formula ofdependence logic.

Example 4.2. Let φ(x) and ψ(x, y) be the formulas

ψ(x, y) := ∼=(x, y)

φ(x) := ∼¬=(x).

LetM be a model and X is a team of M with domain {x, y}. It is easy toverify that

M, X |= ψ(x, y) iff ∃s, s′ ∈ X(s(x) = s′(x) ∧ s(y) 6= s′(y)).

On the other hand, By Definition 2.2, the formula ¬=(x) is only satisfied bythe empty team, hence for allM and X:

M, X |= φ(x) iff X 6= ∅.

Note that the formula ψ(x, y) does not satisfy the downward closure, asdefined in Proposition 3.2, and therefore it cannot be expressed in dependencelogic. On the other hand, the formula φ(x) is not satisfied by the empty teamhence, by Proposition 3.3, it cannot be expressed in dependence logic.

The next example shows that team logic is closed under interesting oper-ations which can be used to define new properties of teams from any propertywhich is already definable (see Examples 8.14 and 8.15 in [12]).

8

Example 4.3. Suppose that ψ(x) is a formula of team logic. There areformulas ϕ(x) and θ(x) of team logic such that for all modelsM and teamsX:

M, X |= ϕ(x) ⇐⇒ ∀Y ⊆ X(M, Y |= ψ(x))

M, X |= θ(x) ⇐⇒ ∀Y ⊆ X∃Z ⊆ Y (M, Z |= ψ(x)).

In our last example we indicate that team logic has interesting applica-tions also outside of logic.

Example 4.4. Recall that Arrow’s theorem in social choice theory showsthat no voting system, with three or more discrete options to choose from,can convert the ranked preferences of individuals into a community-wideranking which would meet a certain set of reasonable criteria. Equivalently,if a social welfare function satisfies a certain set of reasonable criteria with atleast three discrete options to choose from then it is a dictatorship, i.e., thereis a single person whose preferences determine the preferences of the socialwelfare function. Arrow’s Theorem can be interpreted as a valid sentence inteam logic [13]. Without going to the details we note the logical form of thecorresponding sentence is

!x∀y(∼ (φ ∧=(t, v)) ∨ (N∨

i=1

(ti = v))),

where φ is a quantifier-free first-order formula.

5 The main resultIn this section we characterize the expressive power of open formulas of teamlogic.

The following theorem (Theorem 8.13 in [12]) gives an upper-bound forthe solution.

Theorem 5.1. Let L be a vocabulary and φ(v1, . . . , vr) a L-formula of teamlogic with free variables v1, . . . , vr. Then there is a L∪{R}-sentence ψ, whereR is r-ary, of second-order logic such that for all models M and teams Xwith domain {v1, . . . , vr}:

M, X |= φ ⇐⇒ M,Rel(X) |= ψ.

We will next show that the converse of Theorem 5.1 also holds. In otherwords, we will show that the analogue of Theorem 3.5 holds also for team

9

logic. Just like in the case of dependence logic, the result can be shown tohold with respect to non-empty teams only. In team logic, not all formulasare satisfied by the empty team but the empty team still has the followingvery strong property which is analogous to Theorem 3.3.

Let L be vocabulary. We define an auxiliary function V mapping L-formulas of team logic into the set {0, 1} as follows:

• For an atomic formula φ, V (φ) = V (¬φ) = 1,

• V (∼φ) = 1− V (φ),

• V (φ ∨ ψ) = V (φ⊕ ψ) = max{V (φ), V (ψ)},

• V (φ ∧ ψ) = V (φ⊗ ψ) = min{V (φ), V (ψ)},

• V (∃φ) = V (∀φ) = V (!φ) = V (φ).

Theorem 5.2. For every formula φ of team logic one of the following holds:

i) M, ∅ |= φ for allM

ii) M, ∅ 6|= φ for allM

and φ satisfies (i) iff V (φ) = 1 (and (ii) iff V (φ) = 0).

Proof. The claim is proved using induction on the construction of φ. Atomicformulas φ and their negations ¬φ satisfy (i) by Definition 2.2. The othercases of the induction are straightforward.

It is worth noting that the proof of Theorem 5.2 also shows that, given aformula φ, the question whether (i) or (ii) holds can be decided by lookingat the syntactic structure of φ, that is, whether V (φ) = 1.

In our proof we will be using the fact that formulas of second-order logiccan be transformed to the so-called Skolem Normal Form [10] (see [11]).

Lemma 5.3. Every formula of second-order logic is equivalent to a formulaof the form

∃f 11 . . . ∃f 1

n ∀f 21 . . . ∀f 2

n . . . ∃fp1 . . . ∃fp

n ∀x1 . . . ∀xn θ,

where θ is a quantifier-free formula.

10

Proof. The Skolem Normal Form for second-order logic follows directly fromthe corresponding normal form of Σ1

1-formulas (see, e.g., Theorem 6.12 in[12]): It is well known that every formula φ of second-order logic can betransformed into Prenex Normal Form

∃f 11 . . . ∃f 1

m ∀f 21 . . . ∀f 2

m . . . ∃fp1 . . . ∃fp

m ψ, (6)

where ψ is a first-order formula. We transform the Σ11-subformula

∃fp1 . . . ∃fp

m ψ,

of (6) into Skolem Normal Form

∃fp1 . . . ∃fp

n ∀x1 . . . ∀xn θ,

where n ≥ m, and θ is quantifier-free. By adding dummy function variables,we obtain a Skolem Normal Form for φ:

∃f 11 . . . ∃f 1

n ∀f 21 . . . ∀f 2

n . . . ∃fp1 . . . ∃fp

n ∀x1 . . . ∀xn θ.

Theorem 5.4. Let L be a vocabulary, R r-ary predicate symbol not in L,and φ an L∪{R}-sentence of second-order logic. Then there is an L-formulaof team logic ψ(v1, . . . , vr) such that for all modelsM and teams X 6= ∅ withdomain {v1, . . . , vr}:

M, X |= ψ ⇐⇒ M,Rel(X) |= φ.

Proof. We first transform φ in several steps into a useful normal form andthen directly define its translation into team logic.

Step 1. By Lemma 5.3, we can replace φ by a logically equivalent formula φ∗in Skolem Normal Form:

φ∗ := ∃f 11 . . . ∃f 1

n ∀f 21 . . . ∀f 2

n . . . ∃fp1 . . . ∃fp

n ∀x1 . . . ∀xn θ′,

where θ′ is quantifier-free.

Step 2. Using the transformations defined in the proof of Theorem 6.15 in [12],we may assume that θ′ is in conjunctive normal form, all occurencesof the relation variable R in θ′ are of the form Rx, where x denotesthe sequence x1, . . . , xr, and finally each function variable f i

j occurs inθ′ only in occurrences of the term tij := f i

jui,j1 , . . . , u

i,jk(i,j), where each

ui,jk ∈ {x1, . . . , xn}.

11

Step 3. We transform φ∗ into a logically equivalent form ζ

ζ :=A

j≤n

f 1j

B

j≤n

f 2j . . .

A

j≤n

fpj

B

j≤n

xj θ,

where n is possibly increased for the sake of obtaining two new functionvariables fp

n−1 and fpn which we shall simply call f1 and f2, and

θ := (Rx ∨ ¬(f1x = f2x)) ∧ (¬Rx ∨ f1x = f2x) ∧ θ′,

where we replace the occurrences of Rx in θ′ by f1x = f2x. Note that θis in conjunctive normal form just like θ′ and there are only one positiveand one negative occurrence of Rx in θ, both occurrences in their ownconjuncts.

Let ψ be the team logic formula

ψ :=C

i≤n

xi

A

j≤n

y1j

B

j≤n

y2j . . .

A

j≤n

ypj

(∼∧

i≤p evenj≤n

χij ∨( ∧i≤p odd

j≤n

χij ∧ θ∗

)),

where θ∗ is the canonical translation of θ into team logic (see translation (3))with the following replacements of terms and subformulas;

tij 7→ yij

¬Rx 7→⊗i≤r

¬(vi = xi)

Rx 7→B

i≤r

zi

(∨i≤r

∼=(zi) ∨ ∼⊗i≤r

¬(vi = zi) ∨⊗i≤r

¬(xi = zi)),

where vi and zi, for i ≤ r, are new variables, and χij for i ≤ p and j ≤ n we

define as

χij := =

(ui,j

1 , . . . , ui,jk(i,j), y

ij

).

We shall call ypn−1 simply y1 and yp

n we call y2, according to the notation off1 and f2.

LetX be a team with domain {v1, . . . , vr}. Assume first thatM,Rel(X) |=ζ. We will showM, X |= ψ. FromM,Rel(X) |= ζ we get that alternately foreach odd i ≤ p we can pick some particular sequence f i

1, . . . , fin of functions

such that whichever sequence we pick for each even i ≤ p, it always resultsin M,Rel(X), (f i

j)i,j |= ∀x1 . . . ∀xnθ. We can translate this same function

12

picking strategy to the side of team logic; for each i and j, let F ij map as-

signments like F ij (s) = f i

j

(s(ui,j

1 ), . . . , s(ui,jk(i,j))

). If we can show that team

Y , where

Y := X(xi 7→M)i≤n(yij 7→ F i

j )i≤p, j≤n, (7)

satisfies the conditional subformula of ψ, we get thatM, X |= ψ. The teamY satisfies χi

j for each i and j. Therefore, in order for X to satisfy ψ, Yshould satisfy θ∗, i.e. Y should satisfy each of the conjuncts in θ∗. Note that,for each s ∈ Y , 〈yi

j〉s = 〈tij〉fij ,s.

Note that in essence we are proving the powerful claim that, for arbitraryinterpretations of the function variables f i

j for all i and j, it holds that

M,Rel(X), (f ij)i,j |= ∀x1 . . . ∀xnθ ⇐⇒ M, Y |= θ∗, (8)

where Y extends X as defined in (7).There are three types of conjuncts in θ∗. We will show that Y satisfies

conjuncts of all the three types. This is the implication from left to right in(8).

• To see that Y satisfies the formula that replaced Rx ∨ ¬(f1x = f2x),consider any s ∈ Y . From M,Rel(X), (f i

j)i,j, s |= θ we get eitherM,Rel(X), s |= Rx orM, (f i

j)i,j, s |= ¬(f1x = f2x). We can then splitY = Y1 ∪ Y2, where Y1 = {s ∈ Y : (s(x1), . . . , s(xr)) ∈ Rel(X)} andY2 = Y \Y1 such that s(y1) 6= s(y2) holds for all s ∈ Y2. ThusM, Y2 |=¬(y1 = y2). We also have that, for all s ∈ Y1 and all (a1, . . . , ar) ∈Rel(X) there is some s′ ∈ Y1 such that s′(vi) = ai for all i ≤ r ands′(xi) = s(xi) for all i ≤ n.

Let (F1, . . . , Fr) be a sequence of successive supplement functions for Y1

and consider Z := Y1(zi 7→ Fi)i≤r. If some Fi is not a constant function,thenM, Z |= ∼=(zi) and thusM, Z |=

∨i≤r∼=(zi). Otherwise Z =

Y1(zi 7→ ai)i≤r for some (a1, . . . , ar) ∈ M r. If for all s ∈ Z there isi ≤ r such that s(xi) 6= ai, thenM, Z |=

⊗i≤r ¬(xi = zi). Otherwise

there is s ∈ Z such that s(xi) = ai for all i ≤ r. Then there issome s′ ∈ Z, where s′(vi) = s(xi) for all i ≤ r. Since we must haves′(zi) = s(zi) = ai, for all i ≤ r, it holds that s′(vi) = s(xi) = ai = s′(zi)for all i ≤ r, whence M, Z |= ∼

⊗i≤r ¬(vi = zi). This shows that Y

satisfies the conjunct.

• To see M, Y |=⊗

i≤r ¬(vi = xi) ⊗ y1 = y2, consider any s ∈ Y . Asabove, we get from M,Rel(X), (f i

j)i,j, s |= θ either M,Rel(X), s |=¬Rx or M, (f i

j)i,j, s |= f1x = f2x. We can then split Y = Y1 ∪ Y2,

13

where Y1 = {s ∈ Y : (s(x1), . . . , s(xr)) 6∈ Rel(X)} and Y2 = {s ∈Y : s(y1) = s(y2)}. Then M, Y1 |=

⊗i≤r ¬(vi = xi) because for each

s ∈ Y1, (s(v1), . . . , s(vr)) ∈ Rel(X). ClearlyM, Y2 |= y1 = y2.

• To seeM, Y |=⊗

i≤q α∗i , where each αi is an atomic formula where R

does not occur, simply split Y =⋃

i≤q Yi such that each Yi = {s ∈ Y :

M, (f ij)i,j, s |= αi}. ThenM, Yi |= α∗i for each i ≤ q.

For the other direction, assumeM, X |= ψ. We will showM,Rel(X) |=ζ. From M, X |= ψ we get M, Y |= θ∗, where Y is as in (7) for certainsequences of supplement functions F i

j . We translate them into functions f ij

by setting for each sequence of a1, . . . , ak(i,j) ∈M , f ij(a1, . . . , ak(i,j)) = F i

j (s),where s(ui,j

k ) = ak for each k ≤ k(i, j). Then each f ij is well defined because

M, Y |= χij for each i and j. If we can show that M,Rel(X), (f i

j)i,j |=∀x1 . . . ∀xnθ, we get M,Rel(X) |= ζ. To this end, let s be an assignmentthat interprets each xi, i ≤ n. There are (possibly several) extensions of sin Y that we will be referring to. To make things simple, we can think thats itself is one of those extensions in Y—the only difference is that s theninterprets the additional variables v1, . . . , vr, y

11, . . . , y

pn that do not occur in

θ. Note that 〈yij〉s = 〈tij〉f

ij ,s.

We will next show the implication from right to left in (8), i.e. we willshow it for the three kinds of conjuncts in θ for an arbitrary s.

• To seeM,Rel(X), (f ij)i,j, s |= Rx∨¬(f1x = f2x), assumeM, (f i

j)i,j, s |=f1x = f2x. Then note that fromM, Y |= θ∗ we get a split Y = Y1 ∪ Y2

such that

M, Y1 |=B

i≤r

zi

(∨i≤r

∼=(zi) ∨ ∼⊗i≤r

¬(vi = zi) ∨⊗i≤r

¬(xi = zi))

(9)

andM, Y2 |= ¬(y1 = y2). Because s(y1) = s(y2), we have s ∈ Y1. Now,for all (a1, . . . , ar) ∈M r, if ai = s(xi), for i ≤ r, then there is some s′ ∈Y1 such that ai = s′(vi) for all i ≤ r (this follows from the assumptionthat (9) holds). But we know that (s′(v1), . . . , s

′(vr)) ∈ Rel(X), whichis what we wanted.

• To seeM,Rel(X), (f ij)i,j, s |= ¬Rx∨f1x = f2x, assumeM,Rel(X), s |=

Rx. Then note that from M, Y |= θ∗ we get a split Y = Y1 ∪ Y2

such that M, Y1 |=⊗

i≤r ¬(vi = xi) and M, Y2 |= y1 = y2. Con-sider s′ := s(vi 7→ s(xi))i≤r. Then s′ ∈ Y because s ∈ Y and(s(x1), . . . , s(xr)) ∈ Rel(X). Because s′(vi) = s(xi) for all i ≤ r, wehave s′ ∈ Y2, whence s′(y1) = s′(y2), i.e.,M, (f i

j)i,j, s |= f1x = f2x, aswe wanted.

14

• It is left to showM,Rel(X), (f ij)i,j, s |=

∨i≤q αi, where no αi mentions

R. FromM, Y |= θ∗ we get a split Y =⋃

i≤q Yi such thatM, Yi |= α∗ifor each i ≤ q. Because s ∈ Yi for some i, we haveM, s |= αi.

Theorem 5.4 can be also localized to a fixed model analogously to Example4.1.

Corollary 5.5. Let L be a vocabulary,M a L-model, and X a family of setsof r-tuples of M . Then the following are equivalent:

1. X = {Rel(X) :M, X |= ψ(v1, . . . , vr)} for some L-formula ψ(v1, . . . , vr)of team logic.

2. X = {Y :M, Y |= φ(R)} for some sentence φ ∈ SO[L ∪ {R}].

6 ApplicationsIn this section we discuss some consequences of Theorems 5.2 and 5.4 andpresent an open problem. We begin by showing that, by Theorem 5.2, thesemantics of sentences of team logic is essentially bivalent.

Remark 6.1. Let φ be a sentence of team logic andM a model. Now one ofthe four possibilities hold:

1. M, ∅ |= φ andM, {∅} |= φ,

2. M, ∅ |= φ andM, {∅} 6|= φ,

3. M, ∅ 6|= φ andM, {∅} |= φ,

4. M, ∅ 6|= φ andM, {∅} 6|= φ.

These four cases correspond to the four dependence values of team logicdefined in [12] (See Definition 8.6, Table 8.14, and Figure 8.1). Theoretically,we could view any one of the four conditions above as a criterion for truth.By Theorem 5.2, M, ∅ |= φ holds if and only if M, ∅ |= φ holds for all M,and furthermore, this question can be decided by looking at the syntacticstructure of φ. Therefore, we see that the only non-trivial question is whetherthe team {∅} satisfies φ inM.

15

Let us then discuss an application of Theorem 5.4. Theorem 5.4 showsthat team logic is equivalent to full second-order logic in a strong way. In[8] an analogous semantical "completeness" result of dependence logic withrepect to (negative) Σ1

1 was used to argue that extending dependence logicby certain new connectives does not increase the expressive power of de-pendence logic. Theorem 5.4 allows us to reason analogously about teamlogic. The motivation for this line of study arises from the fact that thereare many natural and interesting connectives that can be used with logicshaving semantics based on teams (see, e.g., [1]).

We will next take an example of a new connective ↪→ that could be addedto team logic. Define the connective ↪→ by the clause:

M, X |= φ ↪→ ψ ⇐⇒ for all Y ⊆ X, ifM, Y |= φ and for all Zsuch that Y ( Z ⊆ X it holds thatM, Z 6|= φ, thenM, Y |= ψ.

Denote by TL(↪→) the extension of team logic by ↪→. It is straightforwardto verify that the connective ↪→ can be expressed in second-order logic. Inparticular, the proof of Theorem 5.1 can be extended to cover also the caseof ↪→. Now, by Theorem 5.4 (and the fact that TL(↪→) also satisfies theanalogue of Theorem 5.2) we can actually translate any formula of the logicTL(↪→) into an equivalent team logic formula. The question remaining openis whether there is a compositional way of doing this translation which doesnot use the translation to second-order logic and Theorem 5.4.

Acknowledgments

We are grateful to Jouko Väänänen for valuable comments and suggestionsin all the stages of writing this article.

References[1] S. Abramsky and J. Väänänen. From IF to BI. Synthese, 167(2):207–

230, 2009.

[2] H. B. Enderton. Finite partially-ordered quantifiers. Z. Math. LogikGrundlagen Math., 16:393–397, 1970.

[3] L. Henkin. Some remarks on infinitely long formulas. In InfinitisticMethods (Proc. Sympos. Foundations of Math., Warsaw, 1959), pages167–183. Pergamon, Oxford, 1961.

[4] J. Hintikka. The Principles of Mathematics Revisited. Cambridge Uni-versity Press, Cambridge, 1996.

16

[5] J. Hintikka and G. Sandu. Informational independence as a semanticalphenomenon. In Logic, methodology and philosophy of science, VIII(Moscow, 1987), volume 126 of Stud. Logic Found. Math., pages 571–589. North-Holland, Amsterdam, 1989.

[6] W. Hodges. Compositional semantics for a language of imperfect infor-mation. Log. J. IGPL, 5(4):539–563 (electronic), 1997.

[7] J. Kontinen and J. Väänänen. Erratum: On definability in dependencelogic. To appear in Journal of Logic, Language and Information.

[8] J. Kontinen and J. Väänänen. On definability in dependence logic. Jour-nal of Logic, Language and Information, 18(3):317–332, 2009.

[9] V. Nurmi. Dependence Logic: Investigations into Higher-Order Seman-tics Defined on Teams. PhD thesis, University of Helsinki, 2009.

[10] T. Skolem. Logisch-kombinatorische Untersuchungen über die Erfüll-barkeit oder Beweisbarkeit mathematischer Sätze nebst einem Theoremeüber dichte Mengen. Skrifter utgit av Videnskappsselskapet i Kristiania,1920.

[11] T. Skolem. Selected works in logic. Edited by Jens Erik Fenstad. Uni-versitetsforlaget, Oslo, 1970.

[12] J. Väänänen. Dependence logic: A New Approach to IndependenceFriendly Logic, volume 70 of London Mathematical Society StudentTexts. Cambridge University Press, Cambridge, 2007.

[13] J. Väänänen. personal communication, 2009.

[14] W. J. Walkoe, Jr. Finite partially-ordered quantification. J. SymbolicLogic, 35:535–555, 1970.

17