independendly-friendly logic: dependence and independence of quantifiers in logic

Independendly-Friendly Logic: Dependence andIndependence of Quantifiers in Logic

Gabriel Sandu*University of Helsinki

Abstract

Independence-Friendly logic (IF-logic) introduced by Hintikka and Sandu (1989) studies patternsof dependence and independence of quantifiers which exceed those found in ordinary first-orderlogic. The present survey focuses on the game-theoretical interpretation of IF-logic, includingconnections to solution concepts (equilibria in mixed strategies) in classical game theory, but weshall also present its compositional interpretation together with its connections to notions ofdependence and dependence between terms.

1. Dependence and Independence of Quantifiers

1.1. INTRODUCTION

In a seminal paper Goldfarb points out that ‘‘The connection between quantifiers andchoice functions or, more precisely, between quantifier-dependence and choice functions,is the heart of how classical logicians in the twenties viewed the nature of quantification.’’(Goldfarb 1979, p. 357). A typical exemplification of this phenomenon is the so-calledepsilon-delta definition of continuous functions developed by Cauchy and his followers.

A function f is said to be continuous at the point x0 if given any � > 0, one can choosed > 0 so that for all y, when x0 is within distance d from y, then f(x0) is within distance� from f(y). This last statement is rendered in the mathematical symbolism by

jx0 � yj < d! jf ðx0Þ � f ðyÞj < �:

After pushing the conditions on the quantifiers into the matrix of the formula, andquantifying over the points x0, the definition of a continuous function takes the form

8x08�9d8yRðx0; �; d; yÞ:The order of the quantifiers makes explicit the fact that the choice of d depends on �

(and obviously on the point x0). It sometimes turns out that one can find a d whichworks no matter what x0 is. If this is so, then the function f is said to be uniformly continu-ous. In order to represent uniform continuity in the logical symbolism, one has to be ableto represent the fact that the choice of d depends on � but is independent of x0. In thepresent case this can be done by simply rearranging the logical scopes of the quantifiers

8�9d8x08yRðx0; �; d; yÞ;

but this is not always so. In connection to this,Terence Tao remarks:

… I recently came across the phenomenon of nonfirstorderisability in mathematical logic: thereare perfectly meaningful and useful statements in mathematics which cannot be phrased withinthe confines of first order logic (combined with the language of set theory, or any other stan-

Philosophy Compass 7/10 (2012): 691–711, 10.1111/j.1747-9991.2012.00511.x

ª 2012 The AuthorPhilosophy Compass ª 2012 Blackwell Publishing Ltd

dard mathematical theory); one must use a more powerful language such as second order logicinstead. This phenomenon is very well known among logicians, but I hadn’t learned about ituntil very recently, and had naively assumed that first order logic sufficed for ‘everyday’ usageof mathematics. (Terence Tao 2007)

Tao first considers the examples

1. For every x, there exists a y depending on x such that B(x,y) is true2. For every x, there exists a y independent of x such that B(x,y) is true where

B(x,y) is a binary relation on two objects x and y.Both can be rendered in first-order logic as

8x9yBðx; yÞand

9y8xBðx; yÞrespectively. But, he remarks, things become more complicated when four quantifiersand a ternary relation Q(x,x¢,y,y¢) are involved. We can express the statement

3. For every x and x¢, there exists a y depending only on x and a y¢ depending on x andx¢ such that Q(x,x¢,y,y) is true

by8x9y8x09y0Qðx; x0; y; y0Þ

and the statement4. For every x and x¢, there exists a y depending only on x and x¢ and a y¢ depending onx¢ such that Q(x,x¢,y,y) is true

by8x09y08x9yQðx; x0; y; y0Þ

but we cannot always express in first-order logic the statement.5. For every x and x¢, there exists a y depending only on x and a y¢ depending only onx¢ such that Q(x,x¢,y,y) is true.

Independence-Friendly Logic (IF logic) was introduced by Hintikka and Sandu (1989) toexpress arbitrary patterns of dependence and independence of quantifiers, including theones discussed above. It extends ordinary first-order logic with quantifiers and connec-tives of the form

ð9x=W Þ; ð8x=W Þ; ð_=W Þ; ð^=W Þ

where W is a finite set of variables. Intuitively, the idea is that e.g. ($x/W) is indepen-dent of the quantifiers which bind the variables in W and dependent on all the othersquantifiers and connectives superordinate to it. Thus the notion of uniformly continuousfunction will be rendered in IF logic by

8x08�ð9d=fx0gÞ8yRðx0; �; d; yÞ:The ordinary quantifiers and connectives turn out to be particular cases in which

W ¼ Ø.IF logic and its variants have been studied intensively during the last two decades. One

could distinguish two lines of research:

692 Independence Friendly Logic

ª 2012 The Author Philosophy Compass 7/10 (2012): 691–711, 10.1111/j.1747-9991.2012.00511.xPhilosophy Compass ª 2012 Blackwell Publishing Ltd

Dependence and independence of quantifiers. One line of research, exploredextensively in Hintikka (1996); Hintikka and Sandu (1997); Mann et al. (2011) has focusedon the analysis of quantifier dependence, independence and choice functions in a game-theoretical setting. In a nuttshell, the idea is to analyze quantifiers independence in terms ofimperfect information in two player, win-loss, extensive games. This interpretation is veryclose to the one informally proposed by Tao to interpret the sequence of the four quantifi-ers in (5): the two existential quantifiers are to be thought of as two players such that ‘‘theydo not both know x and they do not both know x¢.’’ This research line has led to an inter-esting connection between notions studied by logicians (truth, expressive power) and issuesin classical game-theory, such as strategies, indeterminacy, signaling, equilibria, etc.

Dependence and independence of terms. The second line of research, more popu-lar among computer scientists and mathematical logicians (Vaananen 2007; Gradel andVaananen forthcoming; Abramsky and Vaananen 2009) has been stimulated by Hod-ges’compositional interpretation of IF logic (Hodges 1997). It has replaced dependenceand independence between quantifiers as a game-theoretical phenomenon with variousnotions of dependence and independence between terms, analyzed by compositionalmethods which generalize Tarski’s semantical interpretations for classical first-order lan-guages.

In this survey we shall trace both developments favouring, however, the first researchline. It is perhaps worth mentioning right away that at the level of sentences all thesevariants are equivalent.

2. Game-Theoretical Semantics for First-Order Languages

We fix an first order language in a vocabulary L. An L-structure M is defined in theusual way: In addition to its universe M, it contains an individual cA 2 M for each con-stant symbol c, a function f A:Mn fi M for each function symbol f of arity n, and a rela-tion RM ˝ Mn for each relation symbol R of arity n. We take an assignment s in M to bea function whose domain is a finite set of variables, and values in M. If s is an assignmentin M, and a 2 M, s(xi/a) denotes the assignment with domain dom(s)[{xi} defined by:

sðxi=aÞðxjÞ ¼sðxjÞ; if i 6¼ j

a; if i ¼ j

�

We use s,s¢,… to stand for assignments. We denote by Ø the empty assignment.With each formula u (in negation normal form), structure M and assignment s in M, a

semantical game GðM; s;uÞ is associated, played by Eloise ($) and Abelard ("). Here are therules of the game. They correspond roughly to Hintikka’s original formulation in Hint-ikka (1983).

•The game has reached the position (w,s¢), with u an atomic formula or its negation (i.e.a literal): No move takes place. If M; s0 � w, then $ wins right away; otherwise " wins.

•The game has reached the position (w�h,s¢): $ chooses v 2 {w,h}, and the game con-tinues from the position (v,s¢).

•The game has reached the position ðw ^ h; s0Þ: 8 chooses v 2 {w,h} and the gamecontinues from the position (v,s¢).

•The game has reached the position ($xw,s¢): $ chooses a 2 M, and the game continuesfrom the position (w,s¢(x/a)).

•The game has reached the position ("xw,s¢): " chooses a 2 M, and the game contin-ues from the position (w,s¢(x/a)).

Independence Friendly Logic 693


Thus each play (maximal history) of the game GðM; s;uÞ is a finite sequence of positionsstarting with the pair (u,s), and ending up with the pair (w,s¢), where w is a literal (anatomic subformula or its negation) which is a subformula of u, and s¢ is an extension ofthe initial assignment s. It is customary to represent a semantical game in extensive normalform, that is, in a tree form, where the initial history is ((u,s)) and each maximal branchrepresents a possible play of the game. It should be obvious that when the universe M ofthe structure M is infinite, then the game GðM; s;uÞ has an infinite number of plays,but each of them is of finite length. Notice that from each sequence h of positionsreached in a play of the game, an assignment sh can be extracted: it is the assigment of the lastposition of the sequence h. For each play h, if the terminal formula of h is w we have:

If M; sh � w; then 9 wins the play h

If M; sh2w; then 8 wins the play h:

For technical reasons we prefer to identify the choices promted by the quantifier $x or"x with ordered pairs (x,a), instead of individuals.

An example might help at this point. Let u be the sentence "x$yx ¼ y and M be thetwo element structure M ¼ fa; bg. We let w denote $yx ¼ y. The initial history is((u,Ø)). " has two possible choices, either (x,a) or (x,b) which lead, respectively, to twopossible histories

ha ¼ ððu;[Þ; ðw; fðx; aÞgÞÞhb ¼ ððu;[Þ; ðw; fðx; bÞgÞÞ:

Each of them is a choice point for $. We end up with four possible plays of the game:

haa ¼ ððu;[Þ; ðw; fðx; aÞgÞ; ðx ¼ y; fðx; aÞ; ðy; aÞgÞÞhab ¼ ððu;[Þ; ðw; fðx; aÞgÞ; ðx ¼ y; fðx; aÞ; ðy; bÞgÞÞhba ¼ ððu;[Þ; ðw; fðx; bÞgÞ; ðx ¼ y; fðx; bÞ; ðy; aÞgÞÞhbb ¼ ððu;[Þ; ðw; fðx; bÞgÞ; ðx ¼ y; fðx; bÞ; ðy; bÞgÞÞ:

Obviously $ wins the plays haa and hbb and " wins the other two.Now when we say that e.g., $ has two choices, (y,a), and (y,b), we assume she follows

a (deterministic) strategy, a function r which gives her the ‘‘right’’ choice. And the sameholds for ". More exactly, a strategy for a player p in the game GðM; s;uÞ is standardlydefined as a function rp which for every sequence h where p is to move, gives p a choicerp(h) according to the rules of the game. The strategy rp is winning if p wins every playwhere p follows rp. We leave the notion ‘‘player p follows the strategy rp in the play h’’undefined, and prefer an example to the formal definition.

Returning to our example, a strategy for $ is any function r which gives her a choicer(ha) and a choice r(hb). Obviously there are four such strategies for $, but if r is to be awinning one, then $ must win both plays where she uses r:

haa ¼ ððu;[Þ; ðw; fðx; aÞgÞ; ðx ¼ y; fðx; aÞ; rðhaÞgÞÞhbb ¼ ððu;[Þ; ðw; fðx; bÞgÞ; ðx ¼ y; fðx; bÞ; rðhbÞgÞÞ:

There is only one winning strategy

rðhaÞ ¼ ðy; aÞ and rðhbÞ ¼ ðy; bÞ:



On the other side, a strategy for " is a function s defined on the initial history (u,Ø).None of the two possible strategies s1(u,Ø) ¼ (x,x) and s2(u,Ø) ¼ (x,b) is winning. "follows the first one in haa and hab, but he looses the former; and he follows the strategy(x,b) in hba and hbb, but he looses the latter.

It is interesting to compare the above game with the one associated with the sentence$x"y:x ¼ y played on the same structure M. Like the previous one, this game has fourplays h

0aa , h

0ab, h

0ba, and h

0bb, which are identical to haa,hab,hba, and hbb, respectively, except

for the order of the moves, which are reversed, and for the terminal formula which isnot any longer x ¼ y, but its negation. Therefore the payoffs are reversed too: $ wins h

0ab

and h0ba and " wins the other two. So in the second game it is " who has a winning

strategy s: just copy Eloise’s winning strategy from the first game. More exactly

sðhaÞ ¼ ðy; aÞ and sðhbÞ ¼ ðy; bÞ:

Let u be a formula, M a structure and s an assignment in M whose domain includes theset of free variables of u. We define M; s �þGTS u, u is game-theoretically true in M rel-ative to s, and symmetrically M; s ��GTS u:

M; s �þGTS u iff there is a winning strategy for Eloise in GðM; s;uÞM; s ��GTS u iff there is a winning strategy for Abelard in GðM; s;uÞ:

When u is a sentence, and s is the empty assignment, we write M �þGTS u wheneverM;[ �þGTS u, and say that u is true in M. Symmetrically we write M ��GTS u wheneverM;[ ��GTS u, and say that u is false in M.

2.1. DETERMINACY

Technically speaking, each semantical game GðM; s;uÞ may be presented as what in classi-cal game theory is known as a two-person, 1-sum, win-loss, extensive game of perfect information(cf. Osborne and Rubinstein 1994). These games have a tree structure, with the initial posi-tion as the root, and the maximal branches of the tree as the possible plays of the game.There are only two players, say $ and " and only two payoffs, 1 and 0 so that for each playh, one player gets the payoff 1 (wins the play) iff the other player gets the payoff 0 (loses it).More exactly, there are two payoff functions u$ and u" such that for each play h we have

u9ðhÞ ¼ 1() u8ðhÞ ¼ 0; and u9ðhÞ ¼ 0() u8ðhÞ ¼ 1:This obvious reformulation has the advantage that we can use the following well knownresult in classical game theory:

Theorem 1 (Zermelo’s theorem) Every 1-sum, win-loss finite game of perfect informa-tion is determined: exactly one of the players has a winning strategy in the game.

As a consequence we see that the principle of bivalence in first-order logic –M �þGTS u or M ��GTS u for every first-order formula u, structure M and assignment s– is a consequence of the principle of determinacy of games.

2.2. GAME-THEORETICAL NEGATION

The case of negation occurring infront of complex formulas requires some additionalstructure. We assume that at the beginning of each game, $ has the role of verifier and "that of falsifier. Each occurrence of the negation sign indicates that the two players switch



roles (hence the dual role of negation). The rules for the connectives and quantifiers, andthe rules for winning and loosing a play are restated accordingly:

•Disjunctions and existential quantifiers prompt moves by the player who is the verifier;conjunctions and universal quantifiers are decision points for the player who is the falsi-fier.•The rules of winning and losing are restated: if the atomic formula reached at the endof the play is satisfied by the current assignment, the player who is the verifier wins;otherwise the falsifier wins.

For illustration, in the game associated with the sentence :$x"y:x ¼ y and the two ele-ment structure M ¼ fa; bg, the players first switch roles; after that the verifier (") hasthe first move, followed by a move by the falsifier ($). Then the players change rolesagain which means that $ wins every play in which her own choice matches the one of". Obviously $ has a winning strategy: choose the same individual as ".

This example should convince the reader that for any formula u, structure M and as-sigment s, we have:

M; s �þGTS :u iff M; s ��GTS u

M; s ��GTS :u iff M; s �þGTS u:

We also notice that the introduction of negation does not take one outside the class oftwo-person, 1-sum, win-loss, extensive game of perfect information. Therefore the principle ofdeterminacy of games still holds. But then it follows that the game-theoretical negation isclassical negation, i.e., for any first-order formula u, structure M and assigment s,

M; s �þGTS :u()M; s2þGTSu:

Indeed, suppose that M; s �þGTS :u. Hence M; s2þGTSu, that is, " has a winning strategyfor GðM; s;uÞ. But then $ cannot have one, because the game is strictly competitive.Conversely, if $ does not have a winning strategy for GðM; s;uÞ, then by Zermelo’s the-orem, " must have one.

Hodges (1983) and Mann et al. (2011) show how one can recover Tarski’s composi-tional interpretation from the game-theoretical interpretation:

M; s �þGTS :u iff M; s2þGTS

u

M; s �þGTS u _ w iff M; s �þGTS u or M; s �þGTS w

M; s �þGTS u ^ w iff M; s �þGTS u and M; s �þGTS w

M; s �þGTS 9xu iff M; sðx=aÞ �þGTS u; for some a 2 M

M; s �þGTS 8xu iff M; sðx=aÞ �þGTS u; for every a 2 M :

3. Game-Theoretical Semantics for IF First-Order Languages: Uniformity

We consider here only IF formulas with independent quantifiers (and not independentconnectives). In semantical games with imperfect information, the rules for ($x/W) and("x/W) are identical to the rules for the standard quantifiers. But the information availableto the players when W „ Ø is imperfect: the players do not ‘‘know’’ the values for the



variables in W. In the context of extensive games, we model the players’ imperfect infor-mation in the following way. Consider two possible playes h,h¢ of the game which havereached the stage where the next move is prompted by ($x/W). Then the correspondingassignments sh, sh¢ have the same domain of variables, dom(sh) ¼ dom(sh¢). When sh and s0hagree on all the variables in their domain except those in W, that is, for every variablex 2 dom(sh) ) W: sh(x) ¼ sh¢(x), we write sh �9W sh0 and say that h,h¢ are W)equivalentfor Eloise. Notice that sh �9 sh0 iff sh ¼ sh¢.

A strategy for $ is, as in the earlier case, a function r defined on all histories h where it is$’s turn to move. But we now require r to be W-uniform, that is, to satisfy the condition

If sh �9W sh0 ; then rðhÞ ¼ rðh0Þ:

As usual, r is winning if $ wins every play in which she uses r. Uniform strategies forAbelard are defined analogously.

For illustration it may be useful to compare the games associated with the sentences"x$yx ¼ y and uMP:"x($y/{x})x ¼ y played on the structure M ¼ fa; bg. These twogames have the same histories with the same payoffs. But in the second game the historiesha and hb are {x})equivalent for $, given that the assignments sha

¼ {(x,a)} and shb¼

{(x,b)}) trivially coincide on the set of variables Ø ¼ {x} ) {x}. By the uniformity con-dition, any strategy r for $ must also be such that r(ha) ¼ r(hb).

There are obviously two such strategies, none of which is winning. For if r(ha) ¼r(hb) ¼ (y,a), then $ will not win the play hba; and if r(ha) ¼ r(hb) ¼ (y,b), she will notwin the play hab. So there is no winning strategy for $ and we conclude that

M2þGTS

uMP . On the other side, there is no winning strategy for " either: his strategies

are exactly those he has in the game of perfect information "x$yx ¼ y, and we already

observed that none of them is winning. Hence M2�GTS

uMP .

IF first-order languages have greater expressive power than ordinary first-order lan-guages. To take one example, the IF sentence uinf

9w8xð9y=fwgÞð9z=fw; xgÞðx ¼ z ^ w 6¼ yÞdefines (Dedekind) infinity. That is for any structure M : M �þGTS uinf iff there is a totalfunction f:M fi M which is an injection and whose range is not the entire universe M.

It can also be shown (Mann et al. 2011) that every IF sentence is truth-equivalent toan IF sentence of the form

8x1. . .8xkð9xkþ1=Wkþ1Þ. . .ð9xkþn=WkþnÞwwhere w is a quantifier-free first-order formula. Results adapted from Walkoe (1970) andEnderton (1970) show that every second-order existential formula (R1

1� formula) isequivalent to an IF formula having the above form. In the appendix to Hintikka (1996),Sandu (1996) uses this result to show that IF logic defines its own truth-predicate. Thatis, there is an IF sentence W(x) in the language of arithmetics with x as its only free vari-able so that for any IF sentence u in the same language, we have

N�GTSu iff N�WðpuqÞ

where N is the standard structure of arithmetic.Hintikka (1996) uses the greater expressive power of IF logic over ordinary first-order

logic to argue for its superiority in foundational studies. He also uses the definability of the



truth-predicate in IF logic to argue that we are not any longer committed to the infinitehierarchy of meta-languages in Tarski’s style. Feferman (2006) is a critical evaluation of thefirst claim, whereas Bozon and de Rouilhan (2006) contains a critical evaluation of the sec-ond claim together with a comparison between the definability of truth in IF logic andKripke’s theory of truth. Suffice it to mention at this point that the game-theoretical nega-tion in IF logic is not any longer classical negation (except for its ordinary first-order sub-fragment). In this context it is also worth mentioning Burgess note in (Burgess 2003) to theeffect that negation in IF logic is not a semantical operation. This sends us to the next topic.

3.1. INDETERMINACY AND SIGNALING

uMP is our first example of an indeterminate IF sentence in the structure M ¼ fa; bg. Infact it is easy to see that uMP is indeterminate on every structure (we exclude 1-elementstructures). GðM;[;uMPÞ corresponds to a well studied game in classical game theory:Matching Pennies (hence the notation uMP). In this game " chooses to show the Head orthe Tail of a coin, and $ tries to guess "’s choices without seeing them. Successfulmatching is a win for $ and a loss for "; otherwise win and loss are reversed.

Hodges (1997) was the first one to notice that the insertion of a dummy quantifier $zin uMP has the effect that the resulting sentence h

8x9zð9y=fxgÞx ¼ y

is valid (true in all structures). The dummy quantifier $z allows $ to ‘‘store’’ or ‘‘copy’’the choices of Abelard so that they are available later on. A winning strategy for $ in thenew game amounts essentially to this: first copy the choices of "; then repeat one’s ownchoices. $ uses this strategy in the histories (we reduce the description of histories to theircorresponding assignments)

ððx; aÞ; ðz; aÞ; ðy; aÞÞððx; bÞ; ðz; bÞ; ðy; bÞÞ

that she both wins.The job done by the dummy quantifier in the example above is known as signaling, a

common phenomenon in games of imperfect information. It has been thouroughly stud-ied in IF logic by Caicedo et al., (2009). There are two ways to block signaling and pre-vent $ to have a winning strategy in our example. We can prevent her from seeing thevalue of x both times

8xð9z=fxgÞð9y=fxgÞx ¼ y

or we can prevent her from seeing the value of z in her second move

8x9zð9y=fx; zgÞx ¼ y:There is an alternative way to interpret Hodges’ sentence "x$z($y/{x})x ¼ y: we thinkof Eloise as consisting of a team of two players, the first one, $z signaling to her team-made $y the value of x. With this interpretation in mind we can consider a variation ofthe same sentence

8x9zð9y=fxgÞ½SðxÞ ! ðRðzÞ ^ RðyÞ ^ y ¼ xÞ� ðð�ÞÞintended to say that the first existential player signals the state x to the second existentialplayer whose response is to identify x. Here we read ‘R(z)’ as ‘z is a signal’ and ‘R(y)’ as‘y is a response’.



This variation over Hodges’ sentence nicely connects to Lewis’ signaling games and hisaccount of conventions (Lewis 1969). Lewis’ communication situations involves a commu-nicator (C) and an audience (A). C observes one of several states m which he tries to com-municate or ‘‘signal’’ to A, who does not see m. After receiving the signal s, A performsone of several alternative actions, called responses. Every situation m has a correspondingresponse b(m) that C and A agree is the best response to take when m holds. Lewis arguesthat a word acquires its meaning in virtue of its role in the solution to various signalingproblems.

Let S be a set of situations or states of affairs, R a set of signals, and R a set ofresponses. Let b:S fi R the function that maps each situation to its best response. Cemploys an encoding f:S fi R to choose a signal for every situation. A employs a func-tion g:R fi R to decide which action to perform in response to the signal it receives. Asignaling system is a pair (f,g) of encoding and decoding functions such that their composi-tion g • f ¼ b.

The sentence (*) can be slightly modified to express a Lewisian signaling situation:

8x9zð9y=fxgÞ½SðxÞ ! ðRðzÞ ^ RðyÞ ^ y ¼ bðxÞÞ�:

Let us denote it by usig. It can be shown that when M is a suitable model for usig, in thesense that the number of states equals the number of signals (as in Lewis’ original inter-pretation), then M�þusig if and only if there is signaling system (f,g), with f representingthe strategy of C and g the decoding stratefy of A.

4. A Compositional Interpretation of IF First-Order Languages

Cameron and Hodges (2001) call a semantics for IF languages an adequate semantics, ifthere is a function which assigns to each pair ðu;MÞ, where u is an IF formula and M isa structure whose vocabulary includes that of u, an interpretation =u=M

which has thefollowing two properties:

(a) There is a distinguished value, call it TRUE, such that if u is an IF sentence, then=u=M ¼ TRUE if and only if M�þGTSu:(b) Suppose u, w are IF formulas, v is an IF formula, and M is a structure whose vocab-ulary includes that of u, w and v. If =u=M ¼ =w=M

and v¢ comes from v by replacingan occurrence of u in v by an occurrence of w, then =v=M ¼ TRUE iff=v0=M ¼ TRUE:

(a) says that an adequate semantics agrees with the GTS semantics on IF sentences. (b) is aweaker form of the principle of compositionality. The question is now whether an adequatesemantics for IF languages exists. There are two main results here that we quickly describe.

First, there is a positive result given by Hodges (1997). A team X in a structure M is aset of assignments in M that share the same domain. The basic semantical notion isM;X �Tr u: the team X satisfies the IF formula u in the structure M, where the freevariables of u are included in the common domain of X. The basic clauses are:

H1. For literals w, M;X �Tr w iff all the assignments in X satisfy w in M (in the classicalsense).H2. M;X �Tr w _ h iff there are subteams Y,Z˝X such that Y[Z ¼ X and M;Y �Tr wand M;Z �Tr h.



H3. M;X �Tr w ^ h iff M;X �Tr w and M;X �Tr h.H4. M;X �Tr ð9x=W Þw iff there is a function f:X fi M such that f is W-uniform andthe team X[x,f] which is formed by extending each assignment s in X with (x,f(s)) satisfiesw in M:H5. M;X �Tr ð8x=W Þw iff the team X[x,M] formed by extending each assignment s inX with (x,a), for every a 2 M, satisfies w in M:

It is easy to see that the empty team Ø always satisfies an IF formula in every structure,i.e. M;[ �Tr u always holds. When u is a sentence, and M; f[g �Tr u, we call u truein M.

We consider the Matching Pennies sentence uMP ¼ "x($y/{x})x ¼ y to illustratehow the team semantics works. Let w be the formula ($y/{x})x ¼ y, and M ¼ fa; bg.Let

sa ¼ fðx; aÞg saa ¼ fðx; aÞ; ðy; aÞgsb ¼ fðx; bÞg sbb ¼ fðx; bÞ; ðy; bÞg:

We claim that M; fsa; sbg2Trð9y=fxgÞx ¼ y. First, observe that trivially: sa �9[ sb. ForM; fsa; sbg�Trð9y=fxgÞx ¼ y to hold, there must exist an {x})uniform function f suchthat the team {sa,sb}[x,f] satisfies x ¼ y in M: But any {x}) uniform function f must sat-isfy also the condition f(sa) ¼ f(sb) which is impossible. This together with the clause forthe universal quantifier entails M; fg2Tr8xð9y=fxgÞx ¼ y. We see that the composi-tional interpretation agrees with the game-theoretical interpretation.

The meaning uM of an arbitrary IF formula u in the structure M is defined as

=u=MTr ¼ fX : X is a team in M; and M;X�þTrug:

After identifying TRUE with {{/}}, we may check that

=u=MTr ¼ TRUE iff M; f[g�þTru:

Mann et al. (2011, Section 3.5) gives a detailed proof of

M; f[g�þTru if and only if M;[�þGTSu

This establishes (a). On the other side, M;X�þTru has been defined in a compositionalway, hence (b) is also satisfied.

The second result is an impossibility result due to Cameron and Hodges (2001). Itshows that there is no adequate interpretation =u=M

such that =u=Mis a set of assign-

ments. It follows that the interpretation

=u=MGTS ¼ fs : s is an assignment in M and M; s�þGTSug:

which is adequate for ordinary first-order languages, is not adequate for IF languages.

5. Quantifier Rules in IF Logic

5.1. HIDDEN VARIABLES AND VACUOUS QUANTIFIERS

The contrast between "x($y/{x})x ¼ y and "x$z($y/{x})x ¼ y discussed above showshow vacuous quantifiers, which are harmless in ordinary first-order logic, can affect themeaning of IF formulas by creating possibilities of signaling. In the context of compositional



semantics, the example shows an interesting interplay between the context principle (onlyin the context of a sentence does a formula have a meaning) and the principle of composi-tionality. On a structure M ¼ fa; bg and as part of "x($y/{x})x ¼ y , the atomic for-mula x ¼ y is interpreted by teams in x and y as shown in the previous section. On theother side, as part of "x$z($y/{x})x ¼ y, the same atomic formula is interpreted byteams in x,y and z (even if at this stage the variable z does not explicitly appear in theformula). Let

sa ¼ fðx; aÞg saa ¼ fðx; aÞ; ðz; aÞg saaa ¼ fðx; aÞ; ðy; aÞ; ðz; aÞgsb ¼ fðx; bÞg sbb ¼ fðx; bÞ; ðz; bÞg sbbb ¼ fðx; aÞ; ðy; bÞ; ðz; bÞg:

Obviously we have

M ; fsaaa; sbbbg�x ¼ y:

When g is a function such that g(saa) ¼ a and g(sbb) ¼ b, then {saa,sbb}[y,g] ¼ {saaa,sbbb}. Inaddition g is (trivially) uniform in {x}. Hence

M ; fsaa; sbbg½y; g��x ¼ y

which together with clause (H4) entails

M ; fsaa; sbbg�ð9y=fxgÞx ¼ y:

Let f be a function such that f(sa) ¼ a and f(sb) ¼ b. Then {sa,sb}[z,f] ¼ {saa,sbb} whichtogether with the previous claim entails that

M ; fsa; sbg�9zð9y=fxgÞx ¼ y:

But {sa,sb} ¼ {/}[x,M] from which we infer

M ; f/g�8x9zð9y=fxgÞx ¼ y:This example shows that one has to be careful in manipulating IF open formulas. Whenassessing the equivalence between such formulas one must take into account their con-text: any variable that has been assigned a value may affect the strategies of the playerseven it does not occur in the formulas. For this reason, the logical equivalence betweenIF formulas has to be relativized to a context U which will be taken to be a finite set ofvariables. Let u,w, and v be arbitrary IF formulas, and W,V˝U be finite sets of variablessuch that U contains the free variables of u,w, and v. Then we have the following defini-tions:

•u�þUw (u truth entails w relative to U), if for every structure M and team X withdomain U we have

M ;X�þu implies M ;X�þw:

•The definition of u��Uw (u falsity entails w relative to U) is completely analogue.•u�Uw entails w relative to U), if

u�þUw and u��Uw:

•u �þU w (u is truth equivalent to w relative to U), if



u�þUw and w�þUu:

•The definition of u ��U w (u is falsity equivalent to w relative to U), if

u��Uw and w��Uu:

•Finally u ” U w (u is equivalent to w relative to U), if

u �þU w and w ��U u:

For sentences the context U is empty.

Returning to the vacuous quantifiers question, it was shown (Caicedo et al.) that oneway to prevent a vacuous quantifier from altering the meaning of an IF formula is to hidethe value of the vacuously quantified variable. That is, if x occurs neither in u nor in U,then

u �U ðQx=W Þu=xwhere u/x is the IF formula which results from u by making any quantifier independentfrom x.

Another way to prevent a vacuous quantifier from affecting the meaning of an IF for-mula is to make its slash set so large that the vacuously quantified variable cannot storeany useful information. That is, if x occurs neither in u nor in U we have:

u �U ðQx=UÞu

5.2. INTERCHANGE OF QUANTIFIERS AND QUANTIFIER EXTRACTION

In ordinary first-order logic like quantifiers commute, because they are moves for thesame player. Also $x"yu entails "y$xu because in the former game Eloise has moreinformation than in the former. With IF logic we can specify the values upon which agiven quantifier depends irrespective of its position in the formula. We can also switchany two adjacent quantifiers by adjusting their slash sets like in the example

8xð9y=fxgÞu � 9yð8x=fygÞu(assuming u does not contain other free variables than x and y). The general principle is(Caicedo et al.): If x and y are distinct varaibles not in U, then

ðQx=V ÞðQ0y=W [ fxgÞu �U ðQ0y=W ÞðQx=V [ fygÞufor Q,Q¢ 2 {$,"}.

In ordinary first-order logic one may move a formula w into the scope of a quantifieras long as the quantified variable does not occur free in w:

Qxu � w � Qxðu � wÞ:This allows us to pull all the quantifiers to the front of a first-order formula, putting it inprenex normal form.



In IF logic quantifier extraction is more delicate because of signaling. There are mainlytwo types of problems. A player may be able to use x to signal the value of a hiddenvaribale as in the following example. Let M ¼ fa; bg and

s0a ¼ fðz; aÞg s0aa ¼ fðx; aÞ; ðz; aÞgs0b ¼ fðz; bÞg s0bb ¼ fðx; bÞ; ðz; bÞg:

Then M; fs0a; s0bg2þ9xðx 6¼ xÞ _ ð9y=fzgÞy ¼ z because the left disjunt, being a contra-diction, is not satisfiable, and the right one is neither satisfied nor disatisfied by the teamfs0a; s0bg. But we do have

M; fs0a; s0bg�þ9x½x 6¼ xÞ _ ð9y=fzgÞy ¼ z�because

M; fs0aa; s0bbg�þð9y=fzgÞy ¼ z:

The second kind of problem arises when x has already a value and assigning it a new oneerases information that was previously stored there. Here is an example from Caicedoet al. The structure and the assignments are as above. Then

M; fs0aa; s0bbg�þ8xðx 6¼ xÞ _ ð9y=fzgÞy ¼ z

but

M; fs0aa; s0bbg�þ8x½x 6¼ xÞ _ ð9y=fzgÞy ¼ z

because the universal quantifier erases the information about z that was stored in x.We solve the first problem by adding x to all the slashed sets in w thereby prevent-

ing it from affecting the players’ strategies for w. We solve the second problem byexcluding x from the domain of the teams used to evaluate the formulas. The mainresult is:

•For every IF formulas u and w and x not occurring neither in w nor in U:

ðQx=W Þu � w �U ðQx=QÞ½u � w=fxg�:

Now the way is free for a prenex normal form theorem as shown in Mann et al. (2011).

5.3. IMPERFECT RECALL

The extensive games of imperfect information which interpret IF formulas are games ofimperfect recall. A player p has action recall if p always remember his or her own moves,and p has knowledge memory if p never forgets information he once knew. If p has bothaction recall and knowledge memory then p is said to have perfect recall. If p lacks atleast one of this property then p is said to have imperfect recall. For instance, in the sen-tence

8x9zð9y=fxgÞx ¼ y

Eloise does not have knowledge memory because she knows the value of x when shechooses the value of z, but not when she chooses the value of y. On the other side, Elo-ise has knowledge memory in the game



8xð9y=fxgÞ9zð9u=fygÞRðx; y; z; uÞbut she does not have action recall because she forgets her own earlier move when shechooses a value for u.

The following theorem nicely connects perfect recall (playability of games) to first-order logic.

Theorem 2 (Sevenster 2006; Mann et al. 2011) Every IF sentence for which Eloise(Abelard) has perfect recall is truth (falsity) equivalent to a first-order sentence.

As a corollary we get a connection between expressive power and imperfect recall: ifan an IF sentence is not truth (falsity) equivalent to an ordinary first-order sentence, thenEloise (Abelard) has imperfect recall.

6. Dependence and Independence Between Terms

6.1. DEPENDENCE LOGIC

Hodges’ compositional interpretation of IF logic has stimulated several developments.Vaananen (2007) replaces the dependence and independence of quantifiers of IF logicwith dependence between terms. In this new setting the IF sentence

8x8zð9y=fzgÞð9w=fx; ygÞRðx; z; y;wÞis represented by

8x8z9y9wð¼ ðx; yÞ^ ¼ ðz;wÞ ^ Rðx; z; y;wÞÞ:

The novel elements are atomic formulas of the form ‘¼(x,y)’ with the intended interpre-tation: y (functionally) depends on x. More generally the dependence atoms have theform ¼ ð x!; y!Þ ¼ ð x!; y!Þ, where x!; y! are finite sequences of variables. The intendedinterpretation is

x! functionally determines y!:These languages called D-languages are given a compositional interpretation in Hodges’style. More exactly, the clauses (H1)–(H3) remain intact, and the clauses for the standardquantifiers are reformulated as

H4*. M;X �Tr 9xw iff there is a function f:X fi A such that M;X ½x; f � �Tr w:H5*. M;X �Tr 8xw iff M;X½x;M � �Tr w.

The clause for the dependence atoms is

•M;X �¼ ð x!; y!Þ iff for every assignments s,s¢ in X: if s and s¢ agree on x!, then theyagree on y!:

Dependence logic (D-logic) is the extension of first-order with dependence atoms. Itsmodel-theoretic properties are studied in Vaananen (2007). The notion of functionaldependence encoded in the dependence atoms ¼ ð x!; y!Þ turns out to be tailor made toexpress functional dependence in data base theory. In fact under the present interpretation



it can be shown that the dependence formulas validate the so-called Armstrong’s axiomsfor functional dependence in data-base theory:

1. ¼ ð x!; x!Þ : anything is functionally dependent on itself.2. If ¼ ð x!; y!Þ and x! z!, then ¼ ð z!; y!Þ: functional dependence is preserved byincreasing input data.3. If u! is a permutation of x! and z! is a permutation of y!, then ¼ ð u!; z!Þ: functionaldependence is invariant under permutations.4. If ¼ ð x!; y!Þ and ¼ ð y!; z!Þ, then ¼ ð x!; z!Þ: functional dependence is transitive.

Here are few properties of dependence logic:

•Downward closure. For every D-formula u, structure M and teams X,Y:

if M;X �Tr u and Y X ; then M;Y �Tr u:

•Greater expressive power than first-order logic. One can define (Dedekind) infinity in D-logic, i.e. for every structure M; the D-sentence

9z8x8x09y9y0ð¼ ðx; yÞ^ ¼ ðx0; y0Þ ^ ðx ¼ x0 $ y ¼ y0Þ ^ y 6¼ zÞis true in M iff M is infinite.

•Equivalence to second-order existential logic. For every D-sentence u there is a R11� sentence

u* such that for every structure M :

M; f[g �Tr u if and only if M�u�:

Conversely, for every R11� sentence u* there is a D-sentence u such that for every struc-

ture M


•Normal form. Every D-sentence is truth-equivalent to a D-sentence of the form

8x1. . .8xn9y1. . .9ymh

where h is a conjunction of dependence atoms and quantifier free first-order formulas.•Equivalence between D-logic and IF logic. We pointed out earlier that every IF sentence istruth-equivalent to a sentence of the form

8x1. . .8xkð9xkþ1=Wkþ1Þ. . .ð9xkþn=WkþnÞwwhere w is a quantifier-free first-order formula. This fact together with the normal formproperty of D-logic offers a straightforward way to see that on the level of sentences, IFsentences and D-sentences are equivalent: We erase the sets Wk+1,…,Wk+n and add theappropriate dependence formulas to w. In the other direction, we delete the dependenceatoms and express the appropriate dependence relations as slashed variables.

6.2. INDEPENDENCE LOGIC

Gradel and Vaananen (forthcoming) replace dependence atoms with independence atomsof the form x ^ y: x and y are independent of each other. For the interpretation theauthors draw a comparison with probability theory, which has its own concept of indepen-



dence: roughly, two random variables are independendent if observing one does not affectthe (conditional) probability of the other. Adapting this idea to the present case, they pro-pose that variables x and y are independent if observing one does not limit in any way thevalue of the other. When this interpretation is relativized to teams X we get the following:

•M;X � x ? y iff for every assignments s,s¢ in X there is an assignment s¢¢ in X suchthat:

s00ðyÞ ¼ s0ðyÞ and s00ðxÞ ¼ sðxÞ:

A moment reflection shows that this clause says that the value of s(x) alone for someassignment s 2 X is not enough for determining the value of s(y). The reason is thatthere might be s¢ 2 X such that s¢(y) „ s(y). But then there must exist also s¢¢ 2 Xsuch that

s00ðyÞ ¼ s0ðyÞ and s00ðxÞ ¼ sðxÞ:

This has the effect that s(x) cannot functionally determine s(y), for there is anotherelement s¢(y) „ s(y) such that s(x) (¼s¢¢(x)) correlates also with s¢(y). The table

Table 1.

x y

s a1 b1

s¢ a2 b2

s¢¢ a1 b2

where X ¼ {s,s¢,s¢¢} illustrates the situation M;X�x ? y. On the basis of the assign-ments s,s¢ one might think that relative to the team {s,s¢}, y functionally depends on x.But the assignments s¢¢ witnesses that relative to the team X we have x’y.

It is immediate from the definition that independence is symmetric: if x ^ y theny ^ x. The independence x ^ y turns out to be a special case of a more general notionx!?

z! y! : the variables y! are independent of the variables x! when the variables z!remain constant. The semantical clause is a generalization of the previous one:

M;X � x!?z! y! iff for every assignments s,s¢ in X which agree on z! there is an

assignment s¢¢ in X such that:

s00ð z!Þ ¼ sð z!Þ; s00ð y!Þ ¼ s0ð y!Þ; and s00ð x!Þ ¼ sð x!Þ:

Independence logic, I-logic, is the extension of first-order logic with independence atoms.In Gradel and Vaananen (forthcoming) it is shown that D-logic is contained in I-logic.The authors also show that at the level of sentences, I-Logic is as expressive as second-order existential logic. That is, for every I-sentence u there is a R1

1� sentence u* suchthat for every structure M:


As I-logic is as expressive on sentences as D-logic, which, as mentioned above, is asexpressive on sentences as IF- logic, it follows that all three logics are equivalent inexpressive power on sentences:



I � D � IF:

7. Beyond Indeterminacy

7.1. STRATEGIC GAMES

We have used the Matching Pennies sentence uMP ¼ "x($y/x)x ¼ y as our favouriteexample of indeterminate IF sentence. There is also its relative, the Inversed Matching Pen-nies uIMP ¼ "x($y/x)x „ y. Both are indeterminate on all structures. There is neverthe-less a basic difference between the two which is not captured by the GTS account but isvisible if we reason probabilistically. The probability that Eloise picks up the same elementas Abelard, decreases, and the probability that she picks up a distinct element, increases asthe cardinality of the underlying structure increases. The situation is described in the tablebelow

Table 2.

Cardinality of M uMP uIMP

1 1 02 1

212

3 13

23..

. ... ..

.

n 1n

n�1n

The advantage of the game-theoretical account of independence is that it may bedeveloped to account for these intuitions. The idea, mentioned in Blass and Gurevich(1986), is due to Ajtai. We sketch it here following Sevenster (2006).

Recall that in the semantical game GðM;[;uMPÞ with M ¼ fa; bg, the set S$ of Elo-ise’s strategies consists of two functions r1,r2:

r1ðhaÞ ¼ r1ðhbÞ ¼ ðy; aÞr2ðhaÞ ¼ r2ðhbÞ ¼ ðy; bÞ:

Likewise the set S" of Abelard’s strategies consists of the functions s1 and s2:

s1ðu;[Þ ¼ ðx; aÞs2ðu;[Þ ¼ ðx; bÞ:

It is convenient to identify r1 with a, r2 with b, s1 with a, and s2 with b. ThusS9 ¼ S8 ¼ M. Now instead of considering the semantic game GðM;[;uMPÞ, we con-sider its strategic version CðM;[;uMPÞ ¼ ðS9; S8; u9; u8Þ displayed in the table below

Table 3.

a b

a (1,0) (0,1)b (0,1) (1,0)



The rows represent S$, the columns represent S", and each pair (x,y) represents thepayoffs u$,u" of the players for the corresponding pair of strategies computed accordingto the the principle:

u9ðr; sÞ ¼1 if h is a win for Eloise in GðM;[;uMPÞ0 if h is a win for Abelard in GðM;[;uMPÞ

�

u8ðr; sÞ ¼ 1� u9ðr; sÞ:

For instance, when Eloise plays her strategy a and Abelard plays her strategy a the resultinghistory in GðM;[;uMPÞ is (a,a) which is a win for Eloise and a loss for Abelard. Hencethe payoffs pair (1,0), that is, u$(a,a) ¼ 1 and u"(a,a) ¼ 0, etc.

Here is a similar representation for the strategic game CðM;[;uIMPÞ:

Table 4.

a b

a (0,1) (1,0)b (1,0) (0,1)

7.2. STRATEGIC EQUILIBRIA

Unlike extensive games which have a sequential character, representable in a tree formwhich shows the informational state of each player at every possible decision point, in astrategic game we abstract from the sequential structure of the extensive game and thinkof the players choosing their strategies simultaneously. Once the strategies are chosen, thegame ends and the players receive automatically their payoffs. The players’ strategic rea-soning is not represented in the matrix of the game but is encoded in what is known inclassical game theory as solution concepts. Their purpose is to model what is rational for theplayers to do taking into account all the possible strategies of the other player. One ofthe most studied solution concepts is that of strategic equilibrium. We illustrate this notionfor the game CðM;[;uMPÞ.

The pair (r¢,s¢) where r¢ 2 S$ and s¢ 2 S" is an equilibrium in pure strategies inCðM;[;uMPÞ if none of the players is in a better off position by a unilateral deviationfrom it. In other words, (r¢,s¢) is an equilibrium if

(i) for every strategy r in S$: u$(r¢,s¢) ‡ u$(r,s¢).(ii) for every strategy s in S": u"(r¢,s¢) ‡ u"(r¢,s).

Obviously in the strategic game CðM;[;uMPÞ there is no equilibrium because no matterwhich strategies the players follow, one of them will be able to unilaterally improve hisor her payoff. For instance, if both players choose a, then Abelard is better off by playingb, etc.

To overcome the inexistence of equilibria in pure strategies, game theorists move tomixed strategies. A mixed strategy l for player p (p is either $ or ") is a probability distribu-tion over Sp, that is, a function l: Sp fi [0,…,1] such that

Pr 2 Sp

l(r) ¼ 1. If l is amixed strategy over S$ and m is a mixed strategy over S", then the probability that Eloisefollows r 2 S$ is l(r), and the probability that Abelard follows s 2 S" is m(s). Since we



assume that the players choose their strategies independently of each other, the probabilitythat Eloise follows r and that Abelard follows s is simply l(r)m(s).

The notion of payoffs of the two players for a pair (r,s) of strategies is now replaced withthe notion of expected utility. Let D(S$) (D(S")) denote the set of mixed strategies of $ (")over S$ (S"), and let l 2 D(S$), and m 2 D(S"). The expected utility U$(l,m) for Eloise ofthe pair (l,m) of mixed strategies is the weighted average of the payoffs $ receives whenthe players follow a pair of pure strategies. More exactly:

U9ðl; mÞ ¼Xr2s9

Xs2s9

lðrÞmðsÞu9ðr; sÞ:

The expected utility for Abelard is defined analogously. Given that C is a one-sum game,it follows that U"(l,m) ¼ 1)U$(l,m).

In the Matching Pennies let $ choose a with probability l(a) ¼ p and b with probabilityl(b) ¼ 1 ) p. Likewise, let " choose a with probability m(a) ¼ q, and b with probabilitym(b) ¼ 1 ) q. Then the expected utility for $ is:

U9ðl; mÞ ¼Xr2s9

Xs2s9

lðrÞmðsÞu9ðr; sÞ

¼ lðaÞmðaÞu9ða; aÞ þ lðaÞmðbÞu9ða; bÞþ lðbÞmðaÞu9ðb; aÞ þ lðbÞmðbÞu9ðb; bÞ¼ pqþ ð1� pÞð1� qÞ

For example if p ¼ 12

and q ¼ 12, then U9ðl; mÞ ¼ 1

2. Since Matching Pennies is a 1-sum

game, we have U8ðl; mÞ ¼ 1� 12¼ 1

2.

The notion of equilibrium in pure strategies extends quite naturally to mixed strate-gies.The pair (l¢,m¢) where l¢ 2 D(S$) and m¢ 2 D(S") is an equilibrium if and only if

(iii) U$(l¢,m¢) ‡ U$(l,m¢), for every l 2 D(S$).(iv) U"(l¢,m¢) ‡ U"(l¢,m), for every m 2 D(S").

If in the Matching Pennies l¢ is the mixed strategy such that l0ðaÞ ¼ l0ðbÞ ¼ 12, then

U9ðl0; mÞ ¼1

4þ 1

4¼ 1

2¼ U8ðl0; mÞ

for every mixed strategy m in D(S"). Similarly if m¢ is the mixed strategy such thatm0ðaÞ ¼ m0ðbÞ ¼ 1

2, then

U9ðl; m0Þ ¼1

2¼ U8ðl; m0Þ

for every mixed strategy l in D(S$). This shows that (l¢,m¢) is an equilibrium.In a similar way, one can show that when the cardinality of the underlying structure

M is 3, then the pair (l*,m*) of mixed strategies wich assigns to each pure strategy theprobability 1

3is also an equilibrium such that U9ðl�; m�Þ ¼ 1

3. For the Inversed Matching

Pennies, the uniform mixed strategies pair is an equilibrium also for any structure of finitecardinality. The expected utitlity returned to Eloise by such an equilibrium is n�1

n, where

n is the cardinality of the underlying structure. Thus the expected utilities returned toEloise by the equilibria in the Matching Pennies and the Inversed Matching Pennies gamescorrespond to the values in the table.



7.3. EQUILIBRIUM SEMANTICS: FROM TRUTH AND FALSITY TO PROBABILISTIC VALUES

In order to extend the above considerations to a systematic theory which covers any IFstrategic game CðM;[;uÞ we have to ensure that

•There always exists an equilibrium in mixed strategies in any IF strategic gameCðM;[;uÞ.

•If there is more than one, all the equilibria return the same expected utility to Eloise.•Game-theoretical truth and falsity (in the sense of M �þGTS u and M ��GTS u) are thelimiting cases of the new semantical values for CðM;[;uÞ.

The first condition is nothing else than von Neumann’s Minimax theorem which, adaptedto the present framework, states that every finite 2-person, 1-sum strategic game has anequilibrium in mixed strategies. The second condition follows from the first. The thirdcondition is easily derivable from the other two and definitions. Together they allow usto define the value of any IF sentence u on a finite structure M as the expected utilityreturned to Eloise by any of the equlibria pair (l,m) in the strategic game CðM;[;uÞ.The systematic approach, initiated in Sevenster (2006), has been generalized in Sevensterand Sandu (2010) and in Mann et al. (2011). Sandu (2011) is a good survey of these top-ics. Galliani and Sandu (forthcoming) contain some interesting applications of the equilib-rium semantics to some traditional epistemic puzzles (Monty Hall) and a connection togame-semantics.

Short Biography

Gabriel Sandu’s research is located at the intersection of logic, game theory and philoso-phy of language. He has authored or co-authored papers in these areas for Synthese, Jour-nal of Philosophical Logic, Journal of Philosophy, Annals of Pure and Applied Logic,The Oxford Handbook of Compositionality, and the Cambridge Encyclopedia of Lan-guage Sciences. His last book Independence-Friendly Logic: A Game-theoretic Approach(Cambridge, 2011), co-authored with Allen Mann and Merlijn Sevenster, applies solutionconcepts from classical game theory to the semantic analysis of first-order languages, theresult being an ‘‘equilibrium semantics’’. Current research develops this framework andapplies it to issues in formal epistemology. He was trained in economics at The Academyof Economic Sciences, Bucharest, and in philosophy at the department of philosophy ofthe University of Helsinki, where he took a doctoral degree with Jaakko Hintikka. Hebecame a professor of theoretical philosophy at the same university. He has also beenDirector of Research at CNRS, Paris, and professor of philosophy at Paris 1, Sorbonne.

Note

* Correspondence: University of Helsinki, 24 Unioninkatu 40 A, Helsinki, Finland 00140. Email: [email protected].

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independendly-friendly logic: dependence and independence of quantifiers in logic

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