Abstract
We connect two different forms of game based semantics: Hintikka’s game for Independence Friendly logic (IF logic) and Giles’s game for Łukasiewicz logic. An interpretation of truth values in [0, 1] as equilibrium values in semantic games of imperfect information emerges for a logic that extends both, Łukasiewicz logic and IF logic. We prove that already on the propositional level all rational truth values can be obtained as equilibrium values.
C.G. Fermüller—Supported by Austrian Science Foundation (FWF) grant I1897-N25 (MoVaQ-MFL).
O. Majer—Supported by Czech Science Foundation grant GF15-34650L.
The original version of this chapter was revised. The affiliation and grant number of the second author have been corrected. The erratum to this chapter is available at DOI: 10.1007/978-3-662-54332-0_19
An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-3-662-54332-0_19
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Notes
- 1.
- 2.
Hintikka uses Myself and Nature as names for the players and Verfier and Falisifer for the two roles.
- 3.
It is more customary to attach the variable assignment to the interpretation instead of to the formula that is to evaluated. For the \(\mathcal H\)-game this does not make any difference. However we will later introduce games, where several formulas are to be evaluated over the same interpretation, but each with respect to a (possibly) different variable assignment.
- 4.
\(\xi [c/x]\) denotes the variable assignment that is like \(\xi \), except for assigning c to x.
- 5.
A t-norm is a commutative and associative function \(\circ :[0,1]^2 \rightarrow [0,1]\) such that \(x\circ 1= x\) and \(x<y\) implies \(x\circ z \le y\circ z\).
- 6.
KZ is sometimes called the ‘weak fragment of Łukasiewicz logic’.
- 7.
The powers of the players of a \(\mathcal G\)-game do not depend on the manner in which the current formula is picked at any state. In more formal presentations of the \(\mathcal G\)-game one may introduce the concepts of a regulation and of so-called internal states in formalizing state transitions. We refer to [5] for details.
- 8.
The idea is that for each atomic formula A there is schematic experimental setup that turns into a concrete experiment if elements of the domain of discourse are assigned to the free variables in A.
- 9.
Giles actually never considered strong conjunction and strong disjunction. For a detailed proof including strong conjunction we refer to [5]. That paper also features a link between the \(\mathcal G\)-game and an analytic proof system for Ł based on hypersequents.
- 10.
As shown in [2] and in [4] one may in fact formulate alternative semantic games for Ł that, like the \(\mathcal H\)-game, keep a single formula in focus at any given state, if either an explicit truth value or a stack of formulas is added. These and related variants of semantic games are discussed in [3], but they hardly are relevant in our context.
- 11.
This is somewhat reminiscent of [11], where an inner language for representing events and an outer, many-valued language for expressing assertions about the probability of such events is combined.
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Fermüller, C.G., Majer, O. (2017). Equilibrium Semantics for IF Logic and Many-Valued Connectives. In: Hansen, H., Murray, S., Sadrzadeh, M., Zeevat, H. (eds) Logic, Language, and Computation. TbiLLC 2015. Lecture Notes in Computer Science(), vol 10148. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-54332-0_16
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