Logic, Thought, and Action 18

Chapter 18 SOME GAMES LOGIC PLAYS∗ Ahti-Veikko Pietarinen University of Helsinki

Abstract

1. 1.1

This paper studies the across-the-board character of game-theoretic semantics (GTS) in coping with various logics, most notably the family of IF (‘independence-friendly’) logics of Hintikka. I will show how both GTS and IF logics may be pushed into new directions by seizing the notion of a semantic game by means of the theory of games. I will conclude with some ensuing issues bordering on the interplay between C.S. Peirce’s pragmaticism and the science of pragmatics.

Introduction What is game-theoretic semantics?

Game-theoretic semantics (GTS) is a semantic theory of rational interaction between two imaginary players, who are playing the roles of the Verifier (V ) and the Falsifier (F ). They undertake to show that a logical or natural-language formula is true (by the actions of the player with the role V ) or false (by F ’s actions). This happens in a model with either partially or completely interpreted non-logical constants, or in a suitable linguistic environment given by collateral actions and the mutually acquired and agreed common ground of players. Formally, GTS agrees with Tarski semantics for traditional first-order logic on complete models, but otherwise their motivations as well as philosophical repercussions are worlds apart.1

∗ Supported by the Academy of Finland (Dialogical and Game Semantics, project no. 101687), this paper represents first results of future collaboration in the context of the projects Science in Context and Proof of the Maison des Sciences de l’Homme du Nord-Pas de Calais, led by Shahid Rahman. 1 The key references are Hintikka, 1973; Hintikka, 1996; Hintikka & Sandu, 1997.

D. Vanderveken (ed.), Logic, Thought & Action, 409–431. 2005 Springer. Printed in The Netherlands.

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Semantic games provide a resourceful theory for logical and linguistic analysis. I will focus on three interrelated issues. First, The rules of semantic games are applicable ‘across the logical board’. Second, the definition of truth is an outgrowth of a game-theoretic notion of a strategy. Third, depending on the language under evaluation, the semantics make generous use of tools from the general theory of games, sometimes putting game-theoretic notions into a novel perspective. I will largely ignore considerations of natural language here, but it is worth noting that, because of such features as non-compositionality, appeal to rational actions, and proximity to graphical and diagrammatic systems, GTS fares both on logical and linguistic fronts of meaning analysis at least as well as the discourse-representation theory of Hans Kamp, or compositional dynamic theories of meaning operating more readily on the syntax/semantics than on the semantics/pragmatics interface.2 In fact, there is a matchless virtue in GTS: its analysis of meaning makes ample use of both the derivational record of past actions and of the multiplicity of possible actions and possible plays not realised in the actual play; all this contributes towards a full-dress context-dependent account of meaning of logical and linguistic expressions and discourse.

1.2

What is IF logic?

The other compartment that I will be discussing here is the family of IF (‘independence-friendly’) logics, suggested by Hintikka, 1996. The term refers to such extensions of traditional logics that accommodate the property of informational independence, which is manifested syntactically by a slash notation and brought out semantically by games of imperfect information. I will outline next the essentials of propositional, first-order and modal IF logics.3 I will not be considering any logical properties of these IF logics here, as my concern is restricted on the relationship between GTS and any one of the IF logics that I come to be defining.

1.2.1 Propositional IF logic. The propositional fragment of IF logic (LIF ) builds up by: (i) If p ∈ PROP, the arity of p is n, and i1 . . . in are indices, then pi1 ...in and ¬pi1 ...in are LIF -formulas, (ii) if ϕ and ψ are LIF -formulas then ϕ ∨ ψ and ϕ ∧ ψ are LIF -formulas, (iii) if ϕ is an LIF -formula then ∀in ϕ and ∃in ϕ are LIF -formulas, (iv) if ϕ is

2 See

Hintikka, 2002; Janasik & Sandu, 2002; Janasik et al., 2003; Pietarinen, 2001. Hintikka, 1996; Hintikka, 2002; Hintikka & Sandu, 1997; Pietarinen, 2001; Sandu & Pietarinen, 2001.

3 See

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an LIF -formula then (∃in /U ) ϕ is an LIF -formula (U is a finite set of / U ). indices, in ∈ The notions of free and bound variables are the same as in firstorder logic. In (∃in /U ) ϕ the indices on the right-hand side of the slash are free. For simplicity, I will omit the clauses for dual prefixes such as (∀in /U ). The models for the language will be of the form M = I M , (pM )p∈PROP , where I M is a set A with a designated individual a, and each pM is a set of finite sequences of indices from I M . Let us set a = Left and A−{a} = Right. The use of quantified indices in enables us simultaneously to distinguish different tokens of sentential connectives and to rightfully hide choices concerning their values. Let us also write ∀i1 (∃i2 /i1 ) pi1 i2 as (p11 (∨/∧) p12 )∧(p21 (∨/∧) p22 ). If we wish to represent by restricted quantifiers ‘unbalanced’ formulas, we would use identities between subformulas to denote coinciding indices. For example, (p1 (∨/∧) p2 ) ∧ p3 , rewritten as ∀i1 (∃i1 /i2 ) pi1 i2 , p21 = p22 is balanced by applying idempotence law in their interpretation, subject to certain qualifications as soon as semantic games are implemented (see the next section).

1.2.2 IF logic with quantifiers. IF first-order logic is created thus: Let Qxψ, Q ∈ {∀, ∃} and φ ◦ ψ, ◦ ∈ {∧, ∨} be Lωω -formulas in the scope of Q1 x1 . . . Qn xn , where A = {x1 . . . xn }. Then the first-order language LIF ωω is formed by: If B ⊆ A, then (Qx/B) ψ and φ (◦/B) ψ are wffs of LIF ωω . / U etc.) means that There are options: to require x ∈ / W (likewise in ∈ the information sets of the corresponding game (to be defined below) are reflexively closed (‘Eyes Open’), and to require that all x ∈ W are in Var(ϕ) (the recursively-defined set of variables of ϕ) of any LIF ωω -formula ϕ in which W occurs means that the formulas are globally contextindependent (and locally context-dependent), in other words the associated game does not make references to constants that do not occur in it (‘Its All in the Game’). If we require that all x ∈ W are in BoundVar(ϕ) (the recursively-defined set of bound variables of ϕ) of any IF formula ϕ in which W occurs, and there are no other free variables than those in W, then ϕ is an IF sentence.

2.

Variety of games — variety of interpretations and implementations

There are differing perspectives as to the reality of the game theoretic component of semantics. My viewpoint is to adopt an implementation in terms of the factual theory of games.

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Semantic games for IF logic

The notion of independence may be investigated either from the point of view of Skolem functions or from the point of view of the induced information structures of the correlated games in the sense of game theory.4 The underlying insight is the same in both cases: the strategies (typically functions) have to be such that they may not only be defined on one previous history of the game, but have to work invariably for multiple such histories, in other words, independently of any particular interpretation of some previous element already encountered. Let Aj (h), j ∈ {V, F } define a set of legitimate actions ai ni=1 , n ∈ ω (a move) for each non-terminal quasi-history (q-history) h ∈ H − Z, from the domain |A| of the structure A, for each j. A q-history is a1 . . . an  ∈ A. The structure of the logical formula uniquely determines the order of the elements in q-histories. Given a q-history hn , j chooses ai ∈ Aj (h), and the game proceeds to hn+1 := hn  ai . A root r has no incoming actions. A play is a finite sequence r, a1 , h1 , a2 . . ., from which V ’s as well as F ’s choices can be singled out. I will dispense with the use of roles and denote players directly by V and F . Further, P : (H − Z) → {V, F } is a player function assigning to every h ∈ H − Z a player j in {V, F } whose turn is to move. A pseudo-IF formula is a subformula ψ of an IF formula ϕ in which W = ∅ and x ∈ W, x ∈ Var(ψ). Let Sub(ϕ) be a recursively defined set of pseudosubformulas of an IF formula ϕ. The labelling function L: H → Sub(ϕ) assigns to each h ∈ H: L(r) = ϕ; for every terminal history h ∈ Z, L(h) is a literal. The components of the game GA = H, L, P, uj  jointly satisfy: if L(h) = ¬ϕ and P (h) = V , then h  ϕ ∈ H, L(h  ϕ) = ϕ, P (h  ϕ) = F ; if L(h) = ¬ϕ and P (h) = F , then h  ϕ ∈ H, L(h  ϕ) = ϕ, P (h  ϕ) = V ; if L(h) = (ψ (∨/W ) θ) or L(h) = (ψ (∧/W ) θ), then h  Left ∈ H, h  Right ∈ H, L(h  Left) = ψ, and L(h  Right) = θ; if L(h) = (ψ (∨/W ) θ), then P (h) = V ; if L(h) = (ψ (∧/W ) θ), then P (h) = F ; if L(h) = (∃x/W ) ϕ or L(h) = (∀x/W ) ϕ, then h  a ∈ H for every a ∈ |A|; if L(h) = (∃x/W ) ϕ, then P (h) = V ; if L(h) = (∀x/W ) ϕ, then P (h) = F . Payoffs are mappings uj (h) → {1, −1}, h ∈ Z. For every h ∈ Z, if L(h) = St1 . . . tm and (A, g) |= St1 . . . tm , then uV (h) = 1 and uF (h) = −1, and if L(h) = St1 . . . tm and (A, g) |= St1 . . . tm , then uV (h) = −1 and uF (h) = 1.

4 For

instance, ∃f1 f2 ∀x1 . . . xn , z1 . . . zm Sx1 . . . xn , z1 . . . zm , f1 (x1 . . . xn ), f2 (z1 . . . zm ) is ∀x1 . . . xn ∃y Sx1 . . . xn , z1 . . . zm , y, w, for some n, m ∈ ω. ∀z1 . . . zm ∃w

the Skolem normal form of

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A history is now qualified to be a prefix of the play-sequence with labels terminating at a q-history hn ∈ Z, n > 0. By a history I customarily mean a finite pre-sequence of (L(r), a1 , L(h1 ), a2 . . . L(hn )), hn ∈ Z. Histories are thus labelled with the subformulas of the formula under evaluation. A set of plays is a game frame. A set of plays with payoffs assigned to the terminating histories gives rise to a game G(ϕ, A, g) with a τ IF structure A for ϕ ∈ LIF ωω and to G(ψ, M ) for ψ ∈ L . A deterministic strategy has for all h ∈ H − Z and ai ∈ A probability fj (h)(ai ) ∈ {0, 1}. Actions are prescribed by a deterministic strategy fj : P −1 ({j}) → (2A − ∅), fj (h) ∈ A(h). Next, Ij is information partition of P −1 ({j}) such that for all h, h ∈ i Sj , h  a ∈ H if and only if h  a ∈ H, a ∈ A. In case W = ∅, the only sets in Ij are singletons. As to G(ϕ, A, g) and G(φ, M ), it is required that if h, h ∈ Sij then fj (h) = fj (h ). In the terminology of extensive games, strategies are defined on the components ofthe information partition, viz. non information n i i sets Sj . A V -partition is i=1 SV and an F -partition i=1 SFi . Let hj be a quasi-history produced by fj . Then fj is winning for j if for every hj ∈ H, uj (hj ) = 1. A truth (resp. falsity) is defined as an existence of a winning strategy for the player who initiated G(ϕ, A, g) or G(φ, M ) as V (resp. F ). There are plenty of histories not on the winning (equilibrium) path and would for that reason be assigned a zero probability. In the definition of truth it suffices to take into account only a subset of strategies that does not lead to such histories. Considerable work has been done in game theory to make solution concepts work for all positions.

2.2

Concurrency vs. sequentiality

The above imperfect-information games are sequential in the sense that there is a left-linear order of moves from the first component onwards captured by the system of game rules described above. The linear order of moves means that at each position, at most one player makes a choice. Accordingly, games in which more than one player may choose at any position are concurrent. I will call this sense of concurrency informational. It follows from the above definitions that the semantic games for IF logic are in this sense not concurrent. However, an alternative sense of concurrency is that it is informationally independent moves that count as concurrent. Thus, the histories that pass through an information set (see below) prescribe a concurrent move at this information set with respect to the sets through which

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these histories have passed at a lesser depth. This sense may or may not induce temporal considerations. I will call this sense actional concurrency. The semantic games for IF logic are in this sense concurrent; they involve independent actions. The linear order of moves (sequentiality) thus means that there are singleton information sets at all histories. Such games are associated with the slash-free fragment of IF logic. An IF formula ϕ associated with a truly concurrent game is one in which no variable xn in Q xn or an index in in Q in , Q ∈ {∀, ∃} fails to occur in some set W deeper in ϕ. A typical implementation of semantics for an IF formula lies between sequential and truly concurrent game.

2.3

Teams that communicate

Different IF formulas give rise to structurally different streams of information. For instance, one may hold that semantic games are ones of perfect information in the sense that the two players V and F do not lose knowledge about positions and the history of the game, and in which there is a difference in the communication of information, not between V and F but between members that constitute these two players, viewed as teams with agents M j = {mj1 . . . mjn }, j ∈ {V, F }, n ∈ ω. Another point worth noting is that, since we deal with finite formulas, the length of communication and the amount of information transmitted is limited by the length, organisation and type of the components the formula contains. Precisely which information is picked, and thus the content of information that may need to be revealed to others, is left for the players to decide. In team games, the available information concerning past actions is restricted for individual members. I will consider both the cases in which there is communication (coordination) between the members of the team and the cases in which there is no such communication. If the team members are not allowed to communicate, each member chooses ai ∈ A(h) independently of others’ decisions. Player’s choice sequence in any particular play is made by M1j ⊆ M j . Why such games? Consider an LIF ωω -formula ∀x∃y(∃z/x) Sxyz. This is correlated with non-communicating team games, in which it is assumed that existential quantifiers are explicitly slashed for other existentially quantifiers variables occurring further out in the formula. Otherwise they are dependent, as in ∀x∃y(∃z/xy) Sxyz. Such formulas are not the only ones coming together with non-communication. For instance, in ∀x∃y(∃z/y) Sxyz, the V -team consists of {mV1 , mV2 }, in which neither mV1 nor mV2 passes on the information they have derived from higher up.

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Even a two-stage game between V and F may need {m1 , m2 }, as in ∀x (S1 x (∨/x) S2 x). Here V should not get the value of x, but since it goes with the disjuncts, we take mV1 to receive this value while being blocked from communicating it to mV2 , who gets to choose at L(h) = (S1 x (∨/x) S2 x). However, consider also φ = ∀x∀y∃z (S1 x (∨/x) S2 yz). The truth of φ is revealed in terms of extended Skolem normal form: ∃f ∀x∀y ((S1 x ∧ f (y) = 0) ∨ (S2 yz ∧ f (y) = 0)). Yet, if we consider — move by move — what is going on in G(φ, A, g), when individuals have been distributed for x, y after the first two moves by F at r and its successors, does not V get this illicit information by observing the labels attached to histories when the fourth move of disjunction is planned? To circumvent this, I will assume that variables in the labelled subformulas carry no instantiated values, in other words, the players do not observe assignments. Ditto for y in (∃z/y) and x in (∨/x). Accordingly, semantic games may be viewed in their extensive forms, because in order to make φ true, V needs to know the value of x when choosing for z, and she needs to know the value of y when making a decision that would lead to either S1 x or to S1 yz. To know the values can symbolically be represented only by instantiating constants to corresponding variables, including those on the left-hand side of the slashes, which amounts to a pseudo-Skolem normal form representing only one particular play of the game with respective instantiations. For in extensive forms, all values, hidden and public, are included in histories. Players may also try to recover the identity of the information set they are at by looking at the available choices also in cases in which there is a possibility that some of the histories within an information set are terminal. Propositionally, an example of this is (p1 (∨/∧) p2 ) ∧ p3 . A different implementation is produced if the coordinating player gets to decide which choices are actually put forward among those proposed by his or her agents. Such strategies are two-tiered: first, memberspecific strategies delineate actions, and second, F ’s or V ’s coordinating strategies pick from the team-internal, private choice sets so induced. Yet another possibility is that a predetermined set of suboptimal agents proposes the actions, the weighed average of such (possibly randomised) actions being elected as the player’s preferred, representative choice. This is related to the concept of bounded rationality popularised in interactive decision theory of agents. Further, it makes the ‘small worlds’ doctrine, according to which agents are able to preview only fragments of domains and states of the game, better understood.

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Teams that do not communicate

If the team members are allowed to communicate, we get games that are associated with such IF formulas in which existentially (universally) quantified variables may depend on other existentially (universally) quantifier variables, as in ∀x∃y(∃z/x) Sxyz. The amount of intra-team communication determines the extent to which such dependence is realised. Such incestuous dependence creates channels by which a player may elicit additional information. For instance, although the existentially quantified variable z does not depend on the universally quantified variable x, the second member mV2 of the V -team choosing a value for z gets to hear the value F chose for x via the other member, mV1 , who chose a value for y, the choice of which being dependent on the choice of the value for x, for the sole reason that mV1 ’s communication to mV2 is not specifically blocked. The strategies in communication games differ from non-communicative ones in that their input includes the choices of action in the histories by the other members of the team, which is not permitted in the noncommunicative case.

2.5

Two ways of losing information

Let  be a partial order on the tree structure of extensive-form games. A game satisfies non-absentmindedness, if for any h, h ∈ H, h, h ∈ Sji , i ∈ {V, F }: if h  h then h = h . Let a depth d(Q) of a quantifier and a connective be defined inductively in a standard way. All semantic games for IF formulas ϕ described here satisfy non-absentmindedness, because every Q has a unique depth d(Q), and hence every subformula of ϕ has a unique position in the game given by its labelling. For any two subformulas of ϕ at h, h ∈ Sji , h  h and h  h. Let Z(h) be a set of plays that pass through any h ∈ H, if h becomes a subsequence of any h ∈ Z(h). Likewise, let Z(ai ) be a set of plays that pass through an action ai ∈ A or a sequence of actions ai ni=1 , if ai ∈ h ∈ Z(ai ). Define a precedence relation