Independence-friendly cylindric set algebras

Abstract

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...

Compositional Natural Language Semantics using Independence Friendly Logic or Dependence Logic

“…This incorrect result is a consequence of the meaning assigned to the de re reading of believe that a stranger crippled John's cow: in (19) it is formalized by =(x) that the stranger does not depend on anything, and thus ∃x is impeded for any influence of other variables. This shows that DL is not suitable for obtaining the de re meaning of (20) and (23) in a compositional way. For the de dicto reading there is no problem because then =(x) does not occur, hence x may depend on z.…”

“…The de dicto reading of (20) can be produced in an analogous way, but then in all stages of the derivation the variable x for the stranger has to depend on the world under consideration. So for IF the ∃x /w must be replaced by ∃x, and for DL the = (x) must be removed.…”

“…In [19] a meaning preserving translation is defined between formulas of IF and DL for the case that only a fixed finite set of variables is considered; the same restriction is used by Mann [20]. This restriction is understandable because it makes the mathematical theory easier.…”

Perfect IFG-Formulas

“…By two simultaneous inductions on the subformulas of φ. A full proof using the present notation can be found in [11,Theorem 1.32]. …”

“…We introduced IFG-cylindric set algebras in [10,11] as a way to study the algebra of IFG logic. Recall from [5, p. 2] or from [4, Definition 4.3.4 on p. 154] that the universe of Cs N (A), the N -dimensional cylindric set algebra over A, consists of the meanings of all the N -variable, first-order formulas expressible in the language of A, where the meaning of a formula is defined by…”