On Semantic Gamification Ignacio Ojea Quintana Columbia University Abstract. The purpose of this essay is to study the extent in which the semantics for different logical systems can be represented game theoretically. I will begin by considering different definitions of what it means to gamify a semantics, and show completeness and limitative results. In particular, I will argue that under a proper definition of gamification, all finitely algebraizable logics can be gamified, as well as some infinitely algebraizable ones (like ukasiewicz) and some non-algebraizable (like intuitionistic and van Fraassen supervaluation logic). 1 Introduction 1.1 Logic gamification The present work builds on the well established work on game semantics for classical logic developed by Jaakko Hintikka [5, 6], and independently by Rohit Parikh [13]. It is embedded in a research line that seeks to build formal connections between logic and game theory, systematically developed in van Benthem [18]. Its contribution amounts extending some of those results to non-classical logics, and to provide an answer to the general question of which semantics can be represented game-theoretically. In the past few years there have been several developments in game semantics for many valued logics, for example by Fermüller [2, 3]. In particular there are several well studied applications on games in Lukasiewicz-style (fuzzy) logics; Mundici [11] provides an alternative semantics for finite-valued Lukasiewicz logics in terms of Ulam's games and Cintula & Majer [1] develop an approach similar to what is going to be done here. Here I will briefly discuss the differences between my approach and Fermüller's, and provide a justification for my account. But the central issue in this essay is to clearly define and discuss what it means for a semantics to be gamifiable, and to show that under an appropriate definition (a) all finitely-algebraizable logics are gamifiable, and (b) some nonalgebraizable logics are gamifiable. 1.2 The basic case Perfect information games in extensive form are trees whose nodes are possible states and turns for the players, arrows from a node to its children represent the available moves or actions that the player has at that node. A strategy for a player i is a function that assigns a move at each node corresponding to a turn for player i. In terminal nodes payoffs are assigned for the players. In Evaluation Games for Classical Propositional logic, two players V and F (for Verifier and Falsifier) dispute over the truth value of a formula φ of a language L in some model M. To avoid unnecesary complications, I assume that L is Classical a Propositional language. It is possible to assign a game GMφ to each pair ă φ,M ą of formulas and Classical Propositional models in the following way: – If φ " p for atomic sentence p, then GMp is a single (terminal) node tree in which V wins if M ( p and F wins otherwise. – If φ " α, then GMφ is G M α , with turns and win-lose markings reversed. – If φ " α_ β, then GMφ is a tree that starts with a node which has G M α and GMβ as its only children and that it is a turn for V. The basic idea is that she decides with which subformula to continue the game. – Finally, if φ " α ^ β, then GMφ is a tree that starts with a node which has GMα and G M β as its only children and that it is a turn for F. The basic idea is that he decides with which subformula to continue the game. The point of such assignment is that the following bridging result holds: Proposition 1 (Success for Classical Propositional Logic). For all formulas φ and Propositional models M: V has a winning strategy in GMφ if and only if M ( φ. The well-known result is due to Hintikka and it generalizes to first order logic. A first intuitive definition of what it means for a semantic to be gamifiable can be generalized from this result. Definition 1 (Semantic gamification intuitive). We start with a logic L and a semantics S for that logic that assigns truth values to the formulas in the language of L. We say S is intuitively gamifiable if there is a game theoretic representation GS and a game-theoretic condition (expressed using a solution concept) C such that for all formulas φ in the language, S assigns certain truth v to φ if and only if the condition C applies to the game theoretic representation of the formula GSφ . 1.3 Structure of the essay In the next section I will discuss gamification for finitely algebraizable logics. In particular, I will present a hierarchy of notions of gamification and show the extent to which those logics can be gamified. Also, I will discuss some of the philosophical and technical aspects of those definitions. In section three I will discuss non-finitely algebraizable logics, in particular intuitionistic and supervaluationist. The purpose of this section is to show that game semantics can be viewed as more general than the standard approaches to logics for semantics. The last section includes concluding remarks on the significance of the results. 2 Finitely algebraizable logics 2.1 Logical Matrices Logical Matrices were first introduced by Lukasiewicz and Tarski [9] in the 1920's as a general concept that was implicitly used in the work of other logicians. The reader can refer to [4, 16] for a more advanced treatment than the one given here. The basic idea is a generalization of the Boolean Algebra underlying truth values in Classical Logic. Formulas are assigned truth values in the domain of the algebra and connectives are interpreted as the algebraic operations over those truth values. Given a propositional language L, a L  matrix is a pair ă A, F ą where A is an algebra of type-L with universe A, and F Ď A; where F is the set of designated values. An assignment h is an homomorphism from the algebra of formulas Fm to the algebra A of the same L-type [h P HompFm,Aq]. Here the elements of the algebra serve as the truth-values of the semantics. One of the key features here is compositionality. h starts by assigning elements of A to the set Var of propositional variables and can be extended to all of L by interpreting operations in the language as operations in the algebra: – hppiq " ai, where pi P V ar and ai P A. – hp φq " Ahpφq. – hpφ   ψq " hpφq  A hpψq, for any diadic connective  . The notion of model is the same as before. A logic L in the language L is said to be complete relative to a class of L-matrices M, if all the elements of M are models of L and for every Γ Y tφu Ď Fm such that Γ &L φ there is a matrix ă A, F ąP M and h P HompFm,Aq such that hrΓ s Ď F but hpφq R F . If this is the case, then it is said that M is a matrix semantics for L, or that M is strongly characteristic for L. In particular, if M is a singleton with matrix M, then M is the characteristic matrix of S. Definition 2 (Finitely algebraizable). A logic L is finitely algebraizable if it is complete relative to a class of finite L-matrices M. 2.2 Games In this work we are interested in a very restricted class of games: two player perfect information extensive games of finite depth [and in almost all cases, strictly competitive or zero-sum games]. As before the players are V and F. We will introduce the basic notions following [10, 12]; where the reader should turn for a more elaborate presentation. An extensive game model is a tree G "ă S,R, turn,V ą with a set of statenodes S and a family R of binary transition relations for the available moves, pointing from parent to daughter nodes. R is assumed here to be well-founded1 in 1 Since R is well founded, branches of the trees have only finite depth. that there is no infinite sequence ă a1, a2, ... ą of nodes such that ă ai, aì1 ąP M for all i P N . turn is a function that assigns players to non-terminal nodes, indicating the player whose turn it is. V is a function that assigns utility values for players at all terminal nodes, but possibly also to any other node.2 A strategy for player i is a function si that assigns at each of i's turns one of the available actions. A mixed strategy for a player i is a function σi : Si Ñ r0, 1s which assigns a probability σipsiq ě 0 to each pure strategy si P Si, satisfying that ř siPSi σipsiq " 1. Given a set of players I " t1, ..., nu, a pure strategy profile is an n-tuple ă s1, ..., sn ą where each si is a pure strategy for player i. Each pure strategy profile is associated with a terminal node in the game model, the one that would be reached if players played the strategy in the profile. Furthermore, given a pure strategy profile ă s1, ..., sn ą, Vipă s1, ..., sn ąq "df Vipaq, where a is the terminal node of that strategy profile and Vi is the utility for any player i. The payoff of a (possibly mixed) strategy profile ă σ1, ..., σn ą, Vipă σ1, ..., σn ąq " ř ăsąPSrσ1ps1q...σ1psnqsVipă s1, ..., sn ąq. The solution concept that we will be using in almost all cases is that of Nash Equilibrium: A strategy profile ă σ1, ..., σn ą is a Nash equilibrium if and only if for any player i P t1, ..., nu and any strategy σ1i ‰ σi for that player, Vipă σ1, ..., σi, ..., σn ąq ě Vipă σ1, ..., σ1i, ..., σn ąq. The insight behind Nash Equilibrium is that unilateral deviation is not profitable. Once the strategy profile is reached, no player has an incentive to change strategies given the other player's strategic choices are fixed. Yet a particular subset of the Nash Equilibra will be used here, namely those obtained by the Backward Induction procedure. 2.3 Strong gamification When evaluation games for Classical Propositional logic were introduced before, there was an implicit function game that assigned extensive game trees of the ones just presented to formulas in L in some model M; so that φ in M got assigned to GMφ . This way of presenting the evaluation games followed van Benthem in [17] and Parikh in [13, 14]. The simple generalization proposed here requires us to drop the model dependence of the function, so that each formula φ P L gets an game form, a tree Gφ in all which terminal nodes ă pi ą corresponding to atomic sentences pi have no assigned payoffs for the players. We later define V in a way that assigns members of the relevant (non-Classical) matrix to terminal nodes but can be extended to other nodes. In general, given a game G, VpGq is the payoff that Verifier gets in the (relevant) equilibria of G.3 In detail, state-nodes of the trees are members of S and are denoted here with tuples ă φ ą, where φ P L. It is useful to reformulate the definition of gamepφq " Gφ: 2 A further assumptions is that there is complete and perfect information. 3 For the games considered, it is not hard to show existence of equilibria as well as uniqueness of payoff under all equilibria. – Gpi is a single node tree ă pi ą, which can be seen as a test or payoff gaining game. – G φ is Gφ with turns reversed, replacing each terminal atomic node ă pi ą by a node ă pi ą and vice versa. Also, formulas ă φ ą in game nodes are syntactically dualized, interchanging conjunctions and disjunctions. – Gφ_ψ is the disjoint union of two game trees Gφ and Gψ put under a common root node ă φ_ ψ ą that is a turn for V. – Gφ^ψ is the disjoint union of two game trees Gφ and Gψ put under a common root node ă φ^ ψ ą that is a turn for F. It is worth noticing that in this definition Gφ is generated solely from the syntactic structure of φ. Let us go back briefly to the alethic or Model-theoretic approach to logic. Part of the gist of it is that we are able to model our natural or intuitive understanding of the connectives that appear in the formula algebra Fm with operations in our modeling algebra A. This was captured by the fact that for any assignment (homomorphism) h : Fm Ñ A, hp φq " Ahpφq and hpφ   ψq " hpφq  A hpψq. Thus the algebra can successfully represent the alethic structure that we want it to embody. In the pragmatic or game-theoretic approach to logic, we want to have a relation of the same sort between the games and some underlying algebra. This will be captured by analogous principles: – VpGpiq " ai, where gi is an atomic game and ai P A. – VpG φq " AVpGφq. – VpGφ ψq " VpGφq  A VpGψq for any diadic connective  . Furthermore, in principle nothing ensures that the algebraic operation will coincide with our strategic intuitions and theories about how games are resolved (i.e. its equilibria). Conversely, it should not be clear prima facie that concepts in game theory and game structures function the same way as algebraic transformations. Yet, at least for some algebraic structures we know that the relation holds. The overall project is then: Fm Games A gamepφq " Gφ game hpφq h VpGφq V Given a formula algebra Fm, an underlying algebra A and h P HompFm,Aq, the central purpose of evaluation games is to provide a translation function game and a payoff function Vh so that for any formula φ P Fm: hpφq " Vh   gamepφq Definition 3 (Strong Semantic Gamification). Begin with a logic L and a semantics S for that logic. We say S is strongly gamifiable if for each formula φ in the language and each assignment hS there is (a) a game theoretic translation Gφ, (b) a payoff assignment Vh to Gφ that is defined in terms of h and (c) a game-theoretic condition (solution concept) C such that: For all formulas φ, an assignment hS assigns certain truth v to φ if and only if the condition C applies to the game Gφ with payoffs determined by Vh. I will now overview a few results that show that some many valued logics are strongly gamifiable. Since the precise cases considered here are not the main focus of the present essay, I will only provide a superficial presentation of each case. Nevertheless, the reader is invited to read some of the proofs in the appendix to get a gist of the basic techniques used here. 2.4 Strong Kleene We start with Kleene's 3-valued system developed in [7, 8] because generalizing evaluation games for it is straightforward. The set of truth values is K " t1, 12 , 0u, where "1" codes truth, "0" codes falsity and " 12" codes undefined. The operations K3 ,_K3 ,^K3 are defined in analogy to Classical logic: (a) K3x " 1  x, (b) x_K3 y " maxtx, yu and (c) x^K3 y " mintx, yu. In order to develop evaluation games for Kleene's 3-valued system I need, given a homomorphism h, a translation function G (or game) and an evaluation function Vh that assigns payoffs to terminal nodes of those trees and values to complex game trees using a solution concept. The translation function is the same as for the classical case. The valuation function Vh needs first to assign members of K " t1, 12 , 0u to the terminal nodes ă pi ą and ă pi ą such that the payoff of both players are specified. VhpGφq is the payoff that Verifier gets in the equilibria of the game Gφ; and since I am considering strictly competitive games, the payoff that Falsifier gets will be 1-VhpGpiq. The valuation for terminal nodes is: – VhpGpiq " hppiq. Hence Verifier gets hppiq and Falsifier gets 1-hppiq. – VhpGpiq " 1  hppiq. Hence Verifier gets 1  hppiq and Falsifier gets hppiq. Proposition 2 (Success for Strong Kleene). Given the matrix K3 and arbitrary assignment h, for all formulas φ P LK: hpφq " x if and only if VhpGφq " x [i.e. in all the Nash Equilibria in Gφ Verifier gets a payoff of x]. The proof of the proposition is included in the appendix. Furthermore, the observant reader might have noticed that there is nothing essential in the fact that only three truth-values were considered. What is crucial is that the truth values are linearly ordered and the (Kleene) operations correspond to max, min and dualization. Then any logic of this form, with finite or infinite truth values, can be modeled analogously with a strictly-competitive two player game. 2.5 Gamification and some results A natural question is whether all finitely algebraizable logics can be strongly gamified. I do not have an answer to this. Nevertheless, it is possible to show that all finitely algebraizable logics can be gamified, under a weaker notion of gamification. Definition 4 (Semantic Gamification). Begin with a logic L and a semantics S for that logic. We say S is gamifiable if for each formula φ in the language and each assignment hS there is a game Gφh whose structure and payoffs depend on h and (b) a game-theoretic condition (solution concept) C such that: For all formulas φ, an assignment hS assigns certain truth v to φ if and only if the condition C applies to the game Gφh. The crucial difference here is that formulas are not mapped to game forms, but rather the mapping goes from formulas and assignments to completely specified games. In particular, the same formula can be mapped to different games under different assignments (and of course different matrices). The strategy adopted here to show that all finitely algebraizable logics can be gamified is indirect. The first step consists in showing that Post logics are gamifiable. The second, to argue that this is sufficient given the truth-functional completeness of those logics. 2.6 Post and truth-functional completeness In 1921 [15] Emil Post presented a finitely many valued logic and showed that it is truth-functionally complete i.e. that all truth functions f : An Ñ A are expressible in terms of the truth functions corresponding to the connectives provided by that logic. Post's intepretation of the disjunction and conjunction is similar to that of Strong Kleene and Lukasiewicz, max and min respectively. The most salient feature of Post logic is Post's negation „; so let L„ by L augmented with that connective. Its interpretation -when A " t0, ..., nu is the following: – hp„ φq " hpφq   1pmod n` 1q. Proposition 3 (Success for Post). For every formula φ in the language of Post, truth value v and assignment h: hpφq " v if and only if in all the Nash Equilibria in Ghφ Verifier gets a payoff of v. The proof of this result, although innelegant and tedious, is included in the Appendix. Any matrix M "ă A,F ą with finite universe A can be represented in a Post matrix of size |A|, making use of the fact that it is truth-functionally complete. This is done in two steps. First, by corresponding each truth value in the matrix with a truth value in the Post logic. Second, the interpretation that matrix M gives to each connective is nothing more than a truth function; which by truth functional completeness can be captured in the Post logic by some composition of Post connectives. In a nutshell, providing a game semantic for Post logic is virtually the same as providing a game semantic for any finite matrix. A similar result was given by Fermüller [3] in 2013, but with a different approach. There Fermüller associates games with signed formulas, which capture the idea that Verifier asserts a certain truth value for the formula at hand. For example, the expression 'v : φ' stands for Verifier's claim that the formula φ has truth value v in the relevant assignment. His basic idea is to have winlose games in which Verifier makes the assertion that φ has certain truth value and Falsifier contests that assertion. In this way, his result are also expressed in terms of winning strategies, rather than Nash Equilibria or Backwards Induction solutions. This is, h assigns v to φ if and only if Verifier has a winning strategy in the game corresponding to v : φ. 3 General Gamification So far the focus of the paper has been on finitely-algebraizable logics, but what about other kinds of logics? Allow me to slightly generalize the definition of semantic gamification so that formulas in a semantics are represented by a set of games, rather than a single game. Definition 5 (General Semantic Gamification). Begin with a logic L and a semantics S for that logic. We say S is gamifiable if for each formula φ in the language and each assignment hS there is a set of games Gφh, each of whose structure and payoffs depend on h and (b) a game-theoretic condition (solution concept) C such that: For all formulas φ, an assignment hS assigns certain truth v to φ if and only if the condition C applies to all the games in Gφh. Under this simple generalization, it is not hard to show that some nonalgebraizable logics, such as Intuitionistic and Supervaluationistic, are gamifiable in general. 3.1 Supervaluationist Supervaluationist logic was developed by Van Fraassen [19, 20] to treat issues of vagueness while satisfying some classical logic principles. The basic idea to evaluate a formula is to start with a partial assignment with three truth values and consider all the classical extensions of that assignment. If the formula is true in all its classical extensions, then it is true in the supervaluation; similarly for falsehood. If the formula is true in some extensions and false in others, then it gets an intermediate value. More formally, an initial truth-value assignment is any function h such that for hppiq P t0, 1 2 , 1u for all propositional variables pi and that is extended to all formulas using the Strong Kleene compositional rules. A classical extension h1 to a initial truth-value assignment h [h ď h1] is a function such that (a) h1ppiq P t0, 1u for all propositional variables pi and extends to all formulas as expected, and (b) for all pi P V ar, if hppiq P t0, 1u, then hppiq " h 1ppiq. A supervaluation induced by an assignment h is a function fh such that for all φ P L: (a) fhpφq " 1 if and only if for all classical extensions h1 of h, h1pφq " 1; (b) fhpφq " 0 if and only if for all classical extensions h 1 of h, h1pφq " 0; and (c) fhpφq " 1 2 otherwise. One interesting aspect of supervaluatinist logic is that it is not compositional. For example, if hpφq " hpψq " 12 , then fhpφ_ ψq " 1 if ψ " φ but fhpφ_ ψq " 1 2 if φ " p1 and ψ " p2. The basic idea of gamifying supervaluationist logic involves mapping each formula and assignment pair pφ, hq to a set of classical games, namely those classical games that correspond to the classical extensions of h. Proposition 4 (Success for Supervaluation). Given an arbitrary assignment h and a supervaluational semantics fh, for all formulas φ: (a) fhpφq " 1 if and only if V has a winning strategy in every game in Ghφ, (b) fhpφq " 0 if and only if F has a winning strategy in every game in Ghφ, and (c) fhpφq " 1 2 otherwise. 3.2 Intuitionistic Logic Intuitionistic logic requires no introduction, and I will presume the reader is familiar with the Kripke semantics for intuitionistic logic. The only subtelty that is involved in gamifying the Kripke semantics for intuitionistic logic is that given a structure K of partially ordered nodes, the translation function associates to each formula-node pair pφ, kq a game Gpφ,kq. Once again, the shape of the game depends on the structure provided by the Krike frame. The basic idea is that games represent what is for the formula to be true in that node. As an example, consider the usual clauses for the conditional and negation: – A node k forces φÑ ψ if, for every k1 ě k, if k1 forces φ then k1 forces ψ. – A node k forces φ if, for no k1 ě k does k1 forces φ. Then the translation functions are the following: – The game corresponding to pφ Ñ ψ, kq, GpφÑψ,kq has a root node that is a move for Falsifier whose children are the games Gp„φ_ψ,k1q 4, for all k1 ě k. – The game corresponding to p φ, kq, Gp φ,kq has a root node that is a move for Falsifier whose children are the games Gpφ,k1q, for all k 1 ě k. Here Gpφ,k1q is just like Gpφ,k1q but with roles and payoffs switched [just like in the classical negation clause]. The next obvious step is to match, for each formula φ the set of all games Gpφ,kq for all k in the Kripke structure K. In this way we obtain G K φ , the set of games corresponding to φ in K. Proposition 5 (Success for Intuitionism). Given an arbitrary Kripke structure K, for all formulas φ: (a) K " 1 if and only if V has a winning strategy in every game in GKφ . 4 Here „ is just classical negation. As far as I know, Proposition 7 is new although it is a natural application of dynamic reasoning. 4 Conclusion and discussion To gamify a semantics means, intuitively, to provide a game-theoretic representation of it. The purpose of this essay was to clarify different notions of gamification and to study the extension to which different propositional logical systems can be gamifiable. I presented three notions of gamification weak, basic and general. I argued that several finitely-algebraizable logics strongly gamifiable, but it is still open whether all of them are. In the next section I presented a result that shows that all finitely-algebraizable logics are gamifiable. The last section shows that even non-algebraizable and non-compositional logics are easily gamifiable if we relax the condition of uniqueness and allow formulas to be represented as sets of games. So far, I have not provided any philosophical account of what we learn about a semantics by knowing if it is, or not, gamifiable. That was not the main purpose of the essay, but a few words are worth saying. Hintikka's original motivation to provide a game semantics had to do more with the pragmatic nature of assertion, or the meaning of conditionals, than with purely logical concerns. The purpose here was to advance an approach to logic that is neither semantic nor syntactic, but rather pragmatic. Valuation functions for formulas usually express the truth values that formulas have under some assignment or model a la Tarski, so that -in generalthe truth value of a compound expression depends in some way on the truth value of its components. When providing game-theoretic semantics, I intended to avoid alethic considerations and ideas and substitute them by instrumental, pragmatic or operational concepts. The hope is that furthering this approach will provide us with more insights about the relation between Theoretical Reason captured in our logical systems and Practical Rationality captured in our game and decision theoretic ideas. For a logical system to be gamifiable, then, means that its Theoretical import can be captured strategically. To conclude, two questions and potential lines of research emerge from here. To begin, it would be interesting to answer whether all algebraizable logics can be strongly gamified. Furthermore, the converse problem for weak gamification is also interesting: Given a class of two-player games closed under some operations and with payoffs in a set V, whether there is a language L closed under some operations and a matrix-semantics A with assignment h such that (a) there is a function that translates games into formulas and (b) the (BI, Nash, etc.) solutions of the game correspond in some way to the value that its corresponding formula gets in h.5 5 Notice here that nothing secures uniqueness of solutions for these games, so solving this problem might require generalizing the presented definition of matrix algebra in some way. A Appendix: Proofs Success for Strong Kleene. The proof of this proposition is by induction, and it is analogous to the traditional proof of Proposition 1. The atomic case is trivial and guaranteed by the definition of V in labeled and unlabeled terminal nodes [i.e. literals]. For complex expressions, assuming by Induction Hypothesis that Proposition 2 holds for all the subformulas, we need to ensure that: (a) VpG φq " K3VpGφq; (b) VpGφ_ψq " VpGφq _K3 VpGψq; and (c) VpGφ^ψq " VpGφq ^K3 VpGψq. In the case of binary connectives we get the identity easily, since the Nash Equilibria are obtained by the Backward Induction procedure and the player's payoffs are such that Verifier prefers maximizing between VpGφq and VpGψq, and Falsifier prefers minimizing between those two alternatives. The case for negation requires an observation: Observation 1 (Mirroring of Pure Strategies and NE). If sV is a pure strategy for Verifier in Gφ, then sV is a pure strategy for Falsifier in G φ [and vice versa]. Furthermore, given some payoff assignment V to the terminal nodes, if ă sV , sF ą is a Nash Equilibrium in Gφ, then ă sF , sV ą is a Nash Equilibrium in G φ. A short proof of the observation is the following. Any pure strategy profile ă sV , sF ą in Gφ is associated with a terminal node, the one that is reached by the path indicated by the strategies, with some payoffs px, 1 xq for Verifier and Falsifier respectively. Also, ă sF , sV ą in G φ leads to the same node, but now with payoffs p1 x, xq. Notice that in G φ there was a turn switch; so ă sF , sV ą is the profile where Verifier plays sF and Falsifier plays sV . If the second profile ă sF , sV ą in game G φ is not a Nash Equilibrium, then at least one player, say Falsifier, can change the strategy to s1V so that ă sF , s 1 V ą terminates in a node with payoff y ą x for him [leaving sF fixed]. But then Verifier can change her strategy in Gφ to also obtain a better payoff. Obviously, the argument is symmetric, and hence one profile is a Nash Equilibrium if and only if the other is. With this observation, we get that VpG φq " 1   VpGφq, which is what we needed. Success for Post. It is easy to see that Post's negation does not satisfy De Morgan's properties in general. Yet, a weaker (partial) form of De Morgan's is satisfied: Observation 2 (Partial De Morgan's for Post's Logic). – hp„ pφ_ ψqq " " hp„ φ^ „ ψq if hpφq " 0 ‰ hpψq or hpφq " 0 ‰ hpψq hp„ φ_ „ ψq if otherwise – hp„ pφ^ ψqq " " hp„ φ_ „ ψq if hpφq " 0 ‰ hpψq or hpφq " 0 ‰ hpψq hp„ φ^ „ ψq if otherwise For simplicity, I avoid this proof here. Providing a game semantics for Post requires a slight change in methodology. In all the cases presented here, the function game : L„ Ñ A was defined independently of the evaluation function V for the games. In this way we had game forms. For disjunctions and conjunctions, nothing different is needed. Yet, in order to provide a game that corresponds to a formula that involves Post's negation, it is necessary to look at the truth values of the components. Notice then that for formulas φ not involving Post negation, we can rely in the success lemmas already shown, so that hpφq " V   gamepφq. Hence in when defining game we can use this fact. Se we only need to show that the success holds for formulas that involve the negation. gamep„ φq is gamepφq transformed inductively in the following way: For terminal nodes, we need to generalize the x function that we had before because now 'bars' are cummulative. So we have a function in the exponent that tracks the amount of times terminal nodes were negated. Non negated terminal nodes ă pi ą are now replaced by ă p 0 i ą. Give gamepφq, we start by replacing each terminal node ă pki ą with ă p k`1 i ą. As expected, we stipulate that VpGpk i q " rhppiq kspmod n`1q " hp„ ... „ ppiqq where the negation is iterated k times. If ă ψ ą is a non-terminal node and has children ă α ą and ă β ą, and if hpαq " n ‰ hpβq or hpβq " n ‰ hpαq: turnpă ψ ąq " " V if turnpă ψ ąq " F F if turnpă ψ ąq " V Otherwise, turns are not changed. The only thing that is really needed now is to show the case for Post-negated formulas, i.e. that Vpgamep„ φqq " rVpgamepφqq 1spmod n`1q. In order to do this we need an observation. Notice that the tree corresponding to gamep„ φq and the tree coresponding to gamepφq are the same, but the games are different in that turn assignments to non-terminal nodes and exponential markings in terminal nodes might have changed. Observation 3. Given trees gamep„ φq and gamepφq, for any node ă ψ ą φ in the former corresponding to a node ă ψ ąφ in the latter we have that V   gamepψ φq " rV   gamepψφq   1spmod n` 1q. The proof is by (Backwards) Induction. If ă pki ą is terminal, then the observation follows by definition. Say ă ψ ą is a non-terminal node and has children ă α ą and ă β ą such that V   gamepα φq " rV   gamepαφq   1spmod n` 1q and V   gamepβ φq " rV   gamepβφq   1spmod n` 1q. Case 1: V   gamepα φq " n ‰ V   gamepβ φq or V   gamepβ φq " n ‰ V   gamepα φq. Suppose the former, the latter case is symmetrical. By inductive hypothesis, V   gamepαφq " 0 ‰ V   gamepβφq. If turnpă α ąφq " V, V   gamepψφq " maxtV  gamepαφq,V  gamepβφqu " V  gamepβφq. Also, by definition, turnpă α ą φq " F, and then V gamepψ φq " mintV gamepα φq,V  gamepβ φqu " mintn,V gamepβφq 1pmod n`1qu " V gamepβφq 1pmod n` 1q. So V   gamepψ φq " rV   gamepψφq   1spmod n` 1q. If turnpă α ąφq " F, V   gamepψφq " mintV   gamepαφq,V   gamepβφqu " 0 " V   gamepαφq. Also, by definition, turnpă α ą φq " V, and then V   gamepψ φq " maxtV   gamepα φq,V gamepβ φqu " maxtn,V gamepβ φqu " n " V gamepα φq " rV   gamepαφq   1spmod n` 1q. Case 2: Otherwise. Either (i) V   gamepα φq " V   gamepβ φq " n or (ii) V   gamepα φq ‰ n ‰ V   gamepβ φq. This proof is left to the reader for its simplicity. Success for Supervaluationism. The main consideration here is to translate each formula and assignment pair pφ, hq to a set of classical games. This is done in two steps. First, map each formula φ to its game form without any specified payoffs, using the original translation method. Second, consider in order the propositional letters pi that appears in the game form and the value that h assigns to pi. If hppiq P t0, 1u, then assign the corresponding payoff to pi and move to the next. If hppiq " 1 2 , then split the game into two games, one in which the payoff of pi is (1,0) and another in which it is (0,1). Continue the procedure with all the games that were generated in the steps before. It should be clear to the reader that the set of games obtained are the games corresponding to all of the classical extensions of h. Then the proposition follows by mere definition. Success for Intuitionism. Begin with a Kripke structure K of partially ordered nodes tkiuiPI . Recall that this proof is just for the propositional case; nothing conceptually different is added for the first order case. There is an atomic forcing relation defined for all nodes k such that for all propositional letters pi, either k forces pi [i.e. k makes pi true, or the forcing relation is not defined for that node and propositional letter. This atomic forcing relation is subject to the constrain that if k ď k1 and k forces pi, then k 1 forces pi. The extensio of the forcing relation to all formulas is the following: (a) A node k forces φ^ψ if it forces φ and ψ; (b) A node k hforces φ_ψ if it forces φ or ψ; (c) A node k forces φÑ ψ if, for every k1 ě k, if k1 forces φ then k1 forces ψ; and (d) A node k forces φ if, for no k1 ě k does k1 forces φ. Given a formula and a node k in a Kripke structure K, the translation functions are the following: (a) The game corresponding to ppi, kq is a one node game with payoffs (1,0) if k forces pi and (0,1) otherwise; (b) The game corresponding to pφ ^ ψ, kq, Gpφ^ψ,kq, consists of a root node for Falsifier with two subgames,Gpφ,kq and Gpψ,kq; (c) The one corresponding to disjunction is as expected; (d) The game corresponding to pφ Ñ ψ, kq, GpφÑψ,kq, has a root node that is a move for Falsifier whose children are the games Gp„φ_ψ,k1q, for all k1 ě k; and (e) The game corresponding to p φ, kq, Gp φ,kq has a root node that is a move for Falsifier whose children are the games Gpφ,k1q, for all k 1 ě k. Here Gpφ,k1q is just like Gpφ,k1q but with roles and payoffs switched [just like in the classical negation clause]. To show the result is sufficies to show that for any pair pφ, kq, k forces φ if and only if Verifier has a winning strategy in Gpφ,kq. The atomic case is trivial. So are the cases for conjunction, disjunction and classical negation. This is just the same proof as for classical logic. The case for the conditional is slightly more complicated. It is worth noticing is that here „ refers to classical negation, and hence „ φ_ψ is just code for the material conditional. We begin with pφÑ ψ, kq, and suppose k forces φ Ñ ψ. Then, in a nutshell, for every k1 ě k, k1 forces „ φ _ ψ. But then, by inductive hypothesis, Verifier has a winning strategy in every such Gp„φ_ψ,k1q. So it has a winning strategy in GpφÑψ,kq. Suppose Verifier has a winning strategy in GpφÑψ,kq. Then whichever choice Falsifier makes at the root node, Verifier still has a winning strategy. That means that for all k1 ě k, Verifier has a winning strategy in Gp„φ_ψ,k1q. By inductive hypothesis this just means that for every k1 ě k, if k1 forces φ then k1 forces ψ. Consider p φ, kq. Suppose k forces φ. Then there is no k1 ě k such that k1 forces φ. The game Gp φ,kq has a root node that is a move for Falsifier whose children are the games Gpφ,k1q, for all k 1 ě k. Now, by (a minor extension of) Observation 2 Mirroring of pure strategies and Nash Equilibria -, we know that for all such k1 Falsifier has a winning strategy in Gpφ,k1q if and only if Verifier has a winning strategy in Gpφ,k1q, and viceversa. We also know, by inductive hypothesis, that Verifier has a winning strategy in Gpφ,k1q if and only if k 1 forces φ. Since k forces φ, there is no k1 ě k such that Verifier has a winning strategy in Gpφ,k1q. But all of the Gpφ,k1q are two-player perfect information games, and therefore are determined. Ergo, Falsifier has a winning strategy in all those Gpφ,k1q. By Observation 2, Verifier has a winning strategy in all the Gpφ,k1q. Hence, whichever move Falsifier makes in the root node of Gp φ,kq, it leads to a game won by Verifier. To conclude, Verifier has a winning strategy for Gp φ,kq. Now for the converse. 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