Bringing together over twenty years of research, this book gives a complete overview of independence-friendly logic. It emphasizes the game-theoretical approach to logic, according to which logical concepts such as truth and falsity are best understood via the notion of semantic games. The book pushes the paradigm of game-theoretical semantics further than the current literature by showing how mixed strategies and equilibria can be used to analyze independence-friendly formulas on finite models. The book is suitable for graduate students and advanced undergraduates who have taken a course on first-order logic. It contains a primer of the necessary background in game theory, numerous examples and full proofs.
Independence-friendly logic is a conservative extension of first-order logic that has the same expressive power as existential second-order logic. In her Ph.D. thesis, Dechesne introduces a variant of independence-friendly logic called IFG logic. We attempt to algebraize IFG logic in the same way that Boolean algebra is the algebra of propositional logic and cylindric algebra is the algebra of first-order logic.We define independence-friendly cylindric set algebras and prove two main results. First, every independence-friendly cylindric set algebra over a structure has an underlying Kleene algebra. Moreover, the class of such underlying Kleene algebras generates the variety of all Kleene algebras. Hence the equational theory of the class of Kleene algebras that underly an independence-friendly cylindric set algebra is finitely axiomatizable. Second, every onedimensional independence-friendly cylindric set algebra over a structure has an underlying monadic Kleene algebra. However, the class of such underlying monadic Kleene algebras does not generate the variety of all monadic Kleene algebras. Finally, we offer a conjecture about which subvariety of monadic Kleene algebras the class of such monadic Kleene algebras does generate.For every s, ψ ∈ Sub(φ), s, ψ ∈ Sub + (φ) if s contains an even number of 0s, and s, ψ ∈ Sub − (φ) if s contains an odd number of 0s.From now on, we will assume that all subformulas are indexed by their position in the subformula tree. This will allow us to distinguish between multiple instances of the same formula that may occur as subformulas of φ. For example, if φ is v 0 = v 1 ∨ /v0 v 0 = v 1 we will distinguish between the left and right disjuncts. 3 Game semanticsDechesne defines her semantic games in extensive form [6], which is standard practice in game theory. It turns out that games in extensive form are more general than is necessary for our purposes. We modify her definition in order to focus on those aspects of the games that are relevant to the present discussion.Definition. Given a first-order signature σ and a formula φ, X ∈ L σ IFG , a structure A is called suitable for φ if A has an interpretation for every non-logical symbol in σ.Definition. If φ, X is a formula and A is a suitable structure, then a valuation for φ, X over A is a function from X to A. Since we are assuming that X has the form {v 0 , . . . , v N −1 }, we will identify valuations with sequences of individuals inDefinition. Let a, b ∈ N A be two valuations, and let J ⊆ N . We say that a and b agree outsideNote that ≈ J is an equivalence relation on N A. Also note that ≈ ∅ is the identity relation and ≈ N is the total relation on N A.Definition. If a ∈ N A, b ∈ A, and n < N , define a(n : b) to be the valuation that is like a except that v n is assigned the value b instead of a n . In other words,We call a(n : b) an n-variant of a.Definition. Let φ be a formula with N variables, let A be a suitable structure, and let V ⊆ N A be a team. The semantic game G(A, φ, V ) is defined as follows. A position of the gam...
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