El reconocimiento del texto como recurso para la extracción de un

Further challenges to the interplay of logic and probabilistic reasoning abound. By way of conclusion, here is a dimension that seems hard to capture in purely qualitative logical terms. A characteristic feature of game- and decision-theoretic reasoning is that beliefs and preferences areentangledin various ways (Liu, 2011). For instance, the crucial notion of expected utility entangles probability, standing for beliefs, with utilities, representing preferences. Players faced with probabilistic uncertainty about the opponent’s present and future actions are often advised to maximize expected

utility (von Neumann and Morgenstern,1944;Savage,1954). Hence, even if one has found qualitative counterparts for probabilistic belief and cardinal utility separately, entanglement poses the additional difficulty of merging these two qualitative analyses in a way that matches what the quantitative side achieves easily by forming some arithmetical combination of both components.

6

Gamification

The main topic of this entry is a logical approach to game theory, bringing classical notions and methods from logic to bear upon games. This project is sometimes called ‘logic of games’. There also is a converse direction of ‘logic as games’, where game theoretic concepts are employed to elucidate basic notions of logic. This section presents a brief discussion on this direction as a natural counterpoint to the main lines of the entry. For an extensive survey see the entries onlogic and games and games, abstraction and completeness.

6.1

Logic games

Many notions in logic have been analyzed in game-theoretic terms

Evaluation gamesThere are well-known two-player games for evaluating a first- order formula ϕ within a given logical model. These games are played between Verifier and Falsifier, who can both test atomic assertions, and specify the value of variables from a given domain (Hintikka,1973). The schedule of the game is deter- mined from the syntactic structure of the formula ϕ. Disjunctions and existential quantifiers require choices for the Verifier, conjunctions and universal quantifiers for the Falsifier, and negations trigger a role switch between the two players. The result is the following match between winning strategies and the ordinary semantic notion of truth in the following sense:

Formulaϕis true in model M under assignmentsiff

the Verifier has a winning strategy in the associated gamegame(M, s, ϕ).

Correspondingly, Falsifier has a winning strategy if the formulaϕis false in the model. Evaluation games turn out to be an extremely flexible tool. By suitably modulating rules and winning conventions, adequate evaluation games can be found for most logical systems. Doing so, however, can be a highly non-trivial task, as witnessed by the intricate infinite ‘parity games’ corresponding to fixed-point logics such as the modalµ-calculus (Venema, 2008). For the present purpose, it should be noted that this style of analysis ties the very logical operations, conjunctions, disjunctions, modal operators, to natural moves in a game. Similarly, the notion of truth is linked to the fundamental game-theoretic notion of a strategy in an extensive form game (cf. Section 2): a complex, structured object which may here be understood as a reason or an explanation for the truth or falsity of the formula.

The links between both perspectives are so close, that valid principles of logic come to express game-theoretic facts. For instance, after a little analysis, the law of excluded middle implies that always either Verifier or Falsifier has a winning strategy. In

other words, logical evaluation games for classical logic are determined in the game- theoretic sense. In fact, This property extends to most games for non-classical logics. Further logic games Logic games exist for many other purposes. Ehrenfeucht- Fra¨ıss´e games serve model comparison (Ehrenfeucht, 1961; Ebbinghaus and Flum, 2005), Lorenzen games perform proof analysis (Kamlah and Lorenzen, 1984) and tableau games execute model construction (Hodges,1985). In each case, strategies in the game match important logical notions. In Lorenzen dialogue games, for instance, winning strategies for the Proponent of a claim correspond to proofs of that claim from premises granted by the Opponent, whereas winning strategies for the Opponent are constructions of counter-models. Thus, proofs and models, two quite distinct notions in logic, co-exist within a single game.

There exists an alternative, game-theoretic way of interpreting these connective re- sults. Suppose the game under study is fixed, and associated with some sort of ‘game board’ representing major features of the game’s general state (think of Chess, though more abstract game boards may occur). Then the above equivalences suggest that winning strategies, i.e. a typical game-theoretic notion defined in terms of the com- plete extensive game tree, is equivalent to a simpler ‘invariant’ that can be defined entirely in terms of some game board associated with the tree’s nodes. Identifying useful such invariants is a well-known art in the analysis of concrete games. In terms of a main theme of this entry, invariants can live at different levels of representation associated with a given class of games.

Game semantics One can view logic games as mere didactic devices analyzing logical notions that were already well-understood. Or, in other terms, as offering a concrete way of teaching logic that draws on game-theoretic intuitions. However, logic games have more to offer. First of all, new logics are suggested by pursuing natural variations in winning conventions, moves, or scheduling within existing logic games. Moreover, viewing logical operations as game constructors suggests a new, refined view on logical constants. Conjunction, for instance, now splits naturally into a sequential and a parallel version. Similar examples of parallelism also exist in logics of computation. Moreover, associating quantifiers with object picking, as in evaluation games, turns quantifiers into special types of atomic games that connect to the following formula by an abstract operation of game composition. The general logic of this abstract composition operation combined with propositional operations of choice and switch has been shown decidable, providing a new decidable core logic inside first-order logic whose existence had not been suspected (van Benthem,2014). Games, hence, can offer a fresh perspective on existing logical systems.

A major source of independent, game-theoretic perspectives on logic is the game semantics of computational logics. In this setting, the status of logic games may change. Rather than being a mere pedagogical or exploratory device, to some, these games are considered the true meaning of logical constants.