Abstract
In the literature on logics of imperfect information it is often stated, incorrectly, that the Game-Theoretical Semantics of Independence-Friendly (IF) quantifiers captures the idea that the players of semantical games are forced to make some moves without knowledge of the moves of other players. We survey here the alternative semantics for IF logic that have been suggested in order to enforce this “epistemic reading” of sentences. We introduce some new proposals, and a more general logical language which distinguishes between “independence from actions” and “independence from strategies”. New semantics for IF logic can be obtained by choosing embeddings of the set of IF sentences into this larger language. We compare all the semantics proposed and their purported game-theoretical justifications, and disprove a few claims that have been made in the literature.
Similar content being viewed by others
Notes
The presence of signalling phenomena reveals that IF logic does not only allow partially ordered quantification; it also allows dependency between quantifiers to be an intransitive relation. Hintikka (1996) attempted to eliminate this feature by means of the implicit assumption that quantifiers of the same type (\(\exists \) or \(\forall \)) be always independent of each other. This stipulation, however, has more unpleasant consequences; with it, IF logic is not anymore a conservative extension of first-order logic (see Janssen 2002).
A formula of the form \(\psi \rightarrow _{/x}\chi \) is presumably interpreted by Hintikka as an abbreviation for \(\lnot \psi \vee _{/\{x\}}\chi \); the (GTS) semantics of \(\vee _{/\{x\}}\) is defined analogously to that of \((\exists v/\{x\})\).
Of course here s(a / v) denotes the assignment which assigns a to v, and s(w) to any variable \(w\in dom(s){\setminus } \{v\}\).
Patterns equivalent to signalling sentences can be obtained in Dependence logic by means of peculiar combinations of dependence atoms and quantifiers, for example in sentences of the form \(\forall x\exists z(=(x,z) \wedge \exists y(=(z,y) \wedge \psi )\); and in Dependence-Friendly logic, by means of quantifier sequences, e.g. the prefix \(\forall x(\exists z\backslash x)(\exists y\backslash z)\). These kinds of patterns allow expressing concepts beyond first-order, such as infinity over any signature (Enderton 1970) and NP-complete problems (Sevenster 2014); actually, any concept definable in existential second-order logic can be expressed by signalling sentences (Barbero et al. 2017).
For sure, it would not be sensible to require the same constraint over the notion of sentence-relevance [also introduced in Barbero (2013)], which is analogous to relevance, but affects individual sentences instead of quantifier prefixes.
For example, we do not want the declaration of independence in \(\exists x(\exists y/\{x\})(x=y \vee x\ne y)\) to affect the truth value of what looks like a validity even under an (intuitive) epistemic reading.
A game has imperfect recall if some of the players in it can forget some of their previous moves or knowledge. In a sense, most IF games are of imperfect recall (see Mann et al. 2011, Theorem 6.23).
This statement must be intended in its de dicto reading: we are talking about knowledge of which strategy function is chosen, not knowledge of the strategy as an object.
Actually, the paper Janssen (2005) of Janssen appeared earlier, but from acknowledgements occurring in his paper, the precedence of Sevenster’s attempts is evident. See also the Bibliographical Note at the end of the paper for even earlier suggestions coming from the works of van Benthem.
In the original text, z is written as a single variable, not a vector; and in the left term of the implication, \(\bar{x}\) took the place of our \(\bar{z}\).
Also known, in the literature, as the set of rationalizable strategies, due to a characterization theorem of Pearce (1984).
As in the previous section, \(u_\exists \) denotes the payoff function of any \(\exists \text {loise}\) player.
We say that a strategy function \(f_{v}\) is potentially winning if it is part of a winning strategy, that is, there is a strategy \(\sigma =(\dots ,f_{v},\dots )\) of \(\exists \text {loise}\) such that, for every strategy \(\tau \) of \(\forall \text {belard}\), \(u_\exists (\sigma ,\tau )=1\).
Actually, Luce and Raiffa argue that using, in this context, single-stage elimination of weakly dominated strategies instead of iterated elimination leads to a more plausible solution concept (what they call solution in the weak sense). This point is not investigated further in the present paper, but surely deserves future attention.
In case some variable x from V is quantified more than once above \((\exists v/V)\) (as may happen in an irregular sentence), we consider it existentially quantified if the quantifier over x which is closest above \((\exists v/V)\) is existential.
The regular sentences of IF logic are those in which variables are not requantified; for example, \(\forall x\exists x\psi \) is not regular. Sentence \(\forall xP(x) \vee \exists xQ(x)\) instead is regular, because the two quantifiers over x do not occur in the same branch of the synctactical tree (but it is not a strongly regular formula, because it contains two quantifiers over the same variable).
References
Barbero, F. (2013). On existential declarations of independence in IF logic. The Review of Symbolic Logic, 6, 254–280.
Barbero, F. (2014). Standard and nonstandard semantics for languages of imperfect information. Ph.D. thesis, Universitá di Torino.
Barbero, F., Hella, L., and Rönnholm, R. (2017). Independence-Friendly logic without Henkin quantification. In To appear in the LNCS Proceedings of WoLLIC 2017.
van Benthem, J. (2004). Probabilistic features in logic games. In J. Symons & D. Kolak (Eds.), Quantifiers, Questions and quantum physics. Essays on the philosophy of Jaakko Hintikka (pp. 189–195). Berlin: Springer.
van Benthem, J. (2014). Logic in games. Cambridge, MA: MIT Press.
Cameron, P., & Hodges, W. (2001). Some combinatorics of imperfect information. Journal of Symbolic Logic, 66, 673–684.
Enderton, H. B. (1970). Finite partially ordered quantifiers. Mathematical Logic Quarterly, 16(8), 393–397.
Grädel, E., & Väänänen, J. (2013). Dependence and independence. Studia Logica, 101, 399–410.
Halpern, J. (1996). On ambiguities in the interpretation of game trees. Games and Economic Behavior, 20, 66–96.
Halpern, J., & Pass, R. (2009). A logical characterization of iterated admissibility. In A. Heifetz (Ed.), TARK ’09 proceedings of the 12th conference on theoretical aspects of rationality and knowledge (pp. 146–155). New York: ACM.
Henkin, L. (1961). Some remarks on infinitely long formulas. In Infinitistic methods, pp. 167–183. Oxford: Pergamon Press.
Hintikka, J. (1996). The Principles of Mathematics Revisited. Cambridge: Cambridge University Press.
Hintikka, J. (2013). Function logic and the theory of computability. APA Newsletter on Philosophy and Computers, 13, 10–19.
Hintikka, J., & Sandu, G. (1989). Informational independence as a semantical phenomenon. In J. E. Fenstad, et al. (Eds.), Logic, methodology and philosophy of science VIII (pp. 571–589). Amsterdam: Elsevier.
Hodges, W. (1997). Compositional semantics for a language of imperfect information. Logic Journal of the IGPL, 5, 539–563.
Hodges, W. (2012). Note on janssen’s Independent choices and the interpretation of IF logic. Working paper.
Humberstone, L. (1987). Critical notice of J. Hintikka and J. Kulas, The game of language. Mind, 96, 99–107.
Jamroga, W., & van der Hoek, W. (2004). Agents that know how to play. Fundamenta Informaticae, 63, 185–219.
Janssen, T. M. V. (2001). On the definition of independence in logic. Workshop on Logic and Games, Helsinki: Technical report.
Janssen, T. M. V. (2002). Independent choices and the interpretation of IF logic. Journal of Logic, Language and Information, 11, 367–387.
Janssen, T. M. V. (2005). Independence friendly logic as a strategic game. In P. Dekker & M. Franke (Eds) Proceedings fifteenth Amsterdam colloquium, pp. 125–130.
Janssen, T. M. V. (2007). Independence and Hintikka games. Technical report, ILLC research report and technical notes series.
Luce, R. D., & Raiffa, H. (1957). Games and decisions: Introduction and critical survey. New York: Wiley.
Mann, A. L., Sandu, G., & Sevenster, M. (2011). Independence-Friendly logic—a game-theoretic approach (Vol. 386)., London mathematical society lecture note series Cambridge: Cambridge University Press.
Moulin, H. (1979). Dominance solvable voting schemes. Econometrica, 47, 1337–1351.
Osborne, M. J., & Rubinstein, A. (1994). A course in game theory. Cambridge: MIT Press.
Pearce, D. G. (1984). Rationalizable strategic behavior and the problem of perfection. Econometrica, 52, 1029–1050.
Sandu, G. (1993). On the logic of informational independence and its applications. Journal of Philosophical Logic, 22, 29–60.
Schelling, T. C. (1960). The strategy of conflict. Cambridge: Harvard University Press.
Sevenster, M. (2007). A strategic perspective on if games. Technical report, ILLC scientific publications, Inst. for Logic, Language and Computation.
Sevenster, M. (2014). Dichotomy result for independence-friendly prefixes of generalized quantifiers. The Journal of Symbolic Logic, 79(04), 1224–1246.
Sher, G. (1990). Ways of branching quantifiers. Linguistics and Philosophy, 13, 393–422.
Väänänen, J. (2007). Dependence logic: A new approach to independence friendly logic (Vol. 70)., London mathematical society student texts Cambridge: Cambridge University Press.
Acknowledgements
This paper was completed under the Academy of Finland Grant No. 286991.
Many thanks are due to the anonymous reviewers, whose suggestions were of great help in improving the readability of the paper and for correcting a few mistakes that were present in an earlier draft.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Barbero, F. Cooperation in Games and Epistemic Readings of Independence-Friendly Sentences. J of Log Lang and Inf 26, 221–260 (2017). https://doi.org/10.1007/s10849-017-9255-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10849-017-9255-1