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Variables, Constants and The Modal Logic of Quantification Maciej Kleczek The Univeristy of Bielefeld. The initial question: “What is the meaning of a firstorder variable ?”. Frege answered that the question is devoid of substance. A variable is a syncategorematic constituent of a language. It is merely a placeholder for a closed term. This calls for the substitutional semantics, which is markedly different than the Tarskian Semantics. Subsequently, this question was undertook by Alonzo Church. Church's semantics of a firstorder variable The opening chapter of Church's “Introduction to Mathematical Logic” conveys important insights related to the semantics of a first-order variable: (1) A variable acquires a meaning relatively to a range of values. (2) Nevertheless, the meaning of a variable must not be identified with the range of values. This follows from the assumption according to which substitution of synonymous expressions preserves the synonymy relation induced by a semantics. Church Cont. (3) Obviously, Church implicitly assumes that a synonymy relation is a congruence relation on the set of constituents of a first-order language. (4) One can easily generalize Church view and claim that synonymy relation restricted to the set of variables is not the universal relation. (5) A structural semantic feature of a variable is ambiguity and denotational behaviour. Church Cont. The following quotation serves as an illustration: “A variable is a symbol whose meaning is like that of a proper name or constant except that the single denotation of the constant is replaced by the possibility of various values of the variable”. Moreover, Church writes in a footnote: “It is indeed possible, as we shall see later by particular examples to construct languages of so restricted vocabulary as to contain no constants but only variables and forms. Church Cont. But it would seem that the most natural way to arrive at the meaning of forms which occur in these languages is by contemplating languages which are extensions of them and which do contain constants – or else, what is nearly the same thing, by allowing a temporary change in the meaning of the variables (“fixing the values of the variables”) so that they become constants” . Church Cont. My aim is to rigorously capture Church's insights. We shall provide a translation from a first-order language of a signature σ (no constants) into a variable free firstorder modal language of a signature σ. Hopefully, this translation turns out to be illuminating to the same extent as the translation of IPC into S4. Nevertheless, before we proceed we shall discuss one more proposal dealing with the problem of a meaning of the variable. Kit Fine Lately, this issue was approached by Kit Fine in “Semantic Relationism”. Fine distinguishes between two problems of the meaning of a variable: (1) the meaning of a variable in isolation. (2) the meaning of a variable in a context. Where by context is understood a set of formulae or possibly text. It is not entirely clear which semantics should receive priority. Kit Fine Cont. (1) Accordingly, Fine identifies the meaning of a variable in isolation with its range of values. This is subject to Church's objections. (2) Moreover, an additional argument against Fine's proposal appears to be plausible. Recall the definition of type <1> generalized quantifier Q:= D → ℘(D) where ∀ = {D}. Of course it holds that a domain (the range of values) D is not identical to {D}. One might argue from the conceptual angle. Implicitly universally quantified. This semantics is naturally associated with the truth relation not with the satisfaction relation. The principle of alphabetic innocence Given a function s: Var → Terms we define in the standard way define operations of capture avoiding (1) single substitution (2) simultaneous substitution on L (terms for free occurrences of a variable). Of course, one can put additional constraints on s. Though not explicitly stated Fine requires that the meaning of a formula should be invariant under a bijective substitution operation induced by s where the range of s is restricted to the set of variables. σ . Fine Cont. It is worthwhile to note that under bijectivity a single substitution operation and simultaneous substitution operations coincide. Clearly, one can generalize the principle of the alphabetic innocence and state that a bijective substitution operation of free and bound occurrences of variables by variables preserves the meaning of a formula. Bijectivity guarantees that such operation is capture avoiding. Fine Cont. It is plausible to interpret alphabetic innocence as the extension of alpha equivalence to the entire first-order language under consideration. Occasionally, Fine's text tend to be admittedly vague. The following can be perceived as shortcomings. There is neither (1) mature semantics nor (2) the proof theory for FOL under the principle of the alphabetic innocence (what about the rule of UG?). The concept of the meaning of a formula is highly intuitive and it stands in need of detailed investigation. Defining variable free first order modal logic Languages have at most a countable signature and the identity symbol is in an alphabet. If |Con|:=κ then there are κ distinct modal operators. Given a signature σ the language Lσ is generated by the following grammar in BNF: P(t , ..., t ) | φ ∧ ψ | φ ∨ ψ | ∼ φ | <i> φ | [i]φ | <i>φ 1 n The translation Fix the natural bijection s: Var → Con s(x )=c and define the translation recursively as follows: *: L → LMσ such that: (1)(P(t1,…,tn)*:=P([x1/c1,…xn/cn,...xn+1/cn+1...]t1 [x1/c1,xn/cn,...xn+1/cn+1 ...]tn) (2) ( φ @ ψ ) * = ( φ )* @ ( ψ )* for @ ∈ { ∧ , ∨} (3) ( ∼φ ) * = ∼ ( φ ) * (4) ( ∃ xi φ )* = <i> ( φ )* (5) ( ∀ xi φ )* = [i] ( φ )* i σ i Semantic structures We introduce the intended class of structures in which a modal language of this kind is interpretable. Here think This particular class of structures There are might be many really. They do not have to be necessarily constant domain. However, in order to stress a connection we generate the unique constant domain frame out of what can be dubbed the space of σ – models in the narrow sense. Semantic Structures Cont. Given an σ - structure ℑ and Ass we take the space of models in the narrow sense to be Φσ:={ℑ} x Ass. By DSL theorem we can consider a countable elementary submodel of . Each element of is called a model structure whereas is called a structure. Th( ) is a prime theory. Since the underlying logic is classical it follows that Th( ) is a complete theory. Of course, these properties do not hold for Th ( ) Semantic Structures Cont. (1) Dw:= D (2) For all Pn , Iw(Pn):=I(Pn) (3) I(s(x )) := ( x ) for all i, 0 i This determines a set W. Moreover, for every i, 0 i we define an equivalence relation ∼ :={<w,w' > W x W: Iw(c ) = Iw'(c ) for all j i but possibly Iw(ci) Iw'(ci)}. A world can be identified with a sequence of elements from D. i i i j j Semantic Structure It is worthwhile to observe that we do not have to augment a frame with an assignment in order to obtain a model. A frame contains all necessary information in order evaluate a formula at a point w. It occupies an intermediate status between a propositional and firstorder frame. Semantic Structures Semantics: (1) ℑ*, w |= <i>φ iff for some w' ∼i w ℑ*, w |= φ (2) ℑ*, w |= <i>φ iff for every w' ∼i w ℑ*, w |= φ With this at hand we are ready to prove the correctness of the translation * with respect to the satisfaction relation and the single conclusion local semantic consequence relation. Obviously, by compactness this consequence relation is finitary. Correctness of The Translation (1) A local semantic consequence relation for Lσ is a binary relation |=l :={<Δ,φ> ∈ ℘(Lσ) x Lσ : for every σ –structure ℑ and every assignment α, if ℑ ,α Δ then ℑ ,α |=φ}. σ (2) A local semantic consequence relation for L is a binary relation |=*l :={<Δ,φ> ∈ ℘(Lσ) x Lσ : for every constant domain frame F and every w, if ℑ ,w|=Δ then ℑ , α |=φ}. Correctness of The Translation Proposition 1. For every formula φ ∈ Lσ and every σ structure and assignment, ℑ,α |=φ iff ℑ*,w |= φ* where w corresponds to α. Proof. By a routine induction on the complexity of a formula Proposition 2. <Δ, φ >∈ |=l iff <Δ,φ > ∈ |=*l Proof. By Proposition 1. Correctness of The Translation It follows that Lσ ≥ LMσ. There is also the converse translation and converse semantic construction for which analogous results can be proved. Therefore we conclude that Lσ ≡ LMσ Moreover, it is well known that the decidability/undecidability is preserved from the source language into the target language. Therefore LMσ is undecidable. Axiomatization The logic is recursively axiomatized by All substitution instance of CPL (?) [i] (φ → ψ) → ([i] φ → [i]ψ) [i] φ → φ [i]φ → [i][i] φ φ → [i]<i>φ <i> <j>φ → <j> <i>φ – Left commutativity <j> <i>φ → <i> <j>φ – Right commutativity <i>[j]φ →[j]<i>φ – Church Rosser Property Axiomatization Rules of inference: (1)If |- φ and |-φ → ψ then |- ψ (2)If |- φ then |- [i] φ (possibly subject to some restrictions) (3)The rule of substitution if |- φ then |- φ' Completeness We can take an advantage of Sahlqvist's Completeness Theorem. Every formula appearing in axiomatization is Sahlqvist. Therefore it is canonical. Therefore the logic is strongly Kripke complete. Bisimulation and Isomorphism Structural invariance relations for modal logics are bisimulations. It is well known that if two pointed structures are bismilar then they are modally equivalent. Of course, it possible to define a bisimulation in our case for each equivalence relation ∼i . Bisimulation and Isomorphism Given two frames ℑ and ℑ*, the bisimulation corresponding to an equivalence relation ∼i is a binary relation Ri ⊆ W x W* such that <w, w* > ∈ Ri if and only if (1) w and w* satisfy the same atomic formulae (2) for every w' such that w ∼i w' there exists w*' such that w* ∼i w*' and and <w', w*' > ∈ Ri (3) Vice versa. Bisimulation and Isomorphism Proposition 3. For any σ frames ℑ and ℑ* , if w ≅ w' then <w, w' > ∈ Ri for all i, 0≤i≤n. Proof. Suppose that w ≅ w'. The condition (1) is satisfied by the assumption. Suppose that w ∼i w* . One can easily find a matching w*'. Bisimulation and Isomorphism This fact is hardly surprising. However, it illustrates a conceptual point. Frames mirror faithfully the space of models in the narrow sense. It is possible to strengthen the standard model-theoretic concept of isomorphism to the isomorphism of model structures. Bisimulation and Isomorphism Given <ℑ,α> and <ℑ*,α'> we say that <ℑ, α > ≅' <ℑ*, α' > are isomorphic: (1) ℑ ≅ ℑ* (2) for every xi, h(α(xi)) := α'(xi). The relation of model structure isomorphism does not coincide with the relation of structure isomorphism. Proposition 4. If <ℑ,α>≅'<ℑ*,α' > then <ℑ,α >≡' <ℑ*,α' >. Proof. The routine induction. Interpreted Languages It is well known that the algebraic structure of FOL with identity is modelled by cylindric algebras of dimension κ. Whereas the algebraic structure of modal logics are boolean algebras with operators. These algebras are total. Therefore concepts of strong and a weak homomorphism coincide and reduce to the standard homomorphism. They are equationally definable and consequently form a variety. Cont. A diagonal free cylindric algebra of dimension κ is a tuple Cκ := < A, 0, 1, ∧, ∨, ∼, cλ , dλζ > for all λ ,ζ <κ (1) < A, 0, 1, ∧, ∨, ∼ > is a boolean algebra (2) cλ 0 : = 0 (3) x ≤ cλ x (4) cλ (x ∧ cλ y ) := (cλ x ∧ cλ y) (5)cλ cζ x:=cζ cλ x (6) dκκ := 1 Cont. In the present context, we are interested in the class of cylindric algebras of assignments. These are a special case of the cylindric set algebras whose carriers is the power set of a set of sequences. Cont. The cylindric algebra of assignments ( over ℑ ) of dimension κ is a tuple Cκ : = < ℘(Ass), ∅, Ass, ∩, ∪, ∼, cλ, d > where: (1) the reduct < ℘(Ass), ∅, Ass, ∩, ∪ > is the field of sets over Ass (2)cλ : ℘(Ass) → ℘(Ass) such that cλ (X): = {α∈Ass : α∼λ α' some α' ∈A}. In the context of first-order logic we identify a dimension (an ordinal) with the cardinality of the set Var. Cont. Typically, the algebraization of a first-order language in the signature σ is done relatively to the Lindenbaum's algebra modulo a theory Γ (possibly empty). We have not introduced the consequence relation defined by means of proof and derivability from a theory Γ. Therefore we will approach algebraization directly. Cont Fix an σ-structure ℑ. With each formula φ we correlate a subset of Assℑ. This is the local meaning of φ in ℑ. It is a philosophically contentious issue whether this is an adequate explication of the concep of the meaning of a formula. Nevertheless, we postpone this question. A grammar for first-order logic can be viewed as an algebra whose signature is similar to the signature of the class of cylindric algebras. Cont More formally, the overall picture looks as follows. Suppose that μ*ℑ: Atσ→ ℘(Assℑ) where: (a)℘(Assℑ) is the carrier of the cylindric algebras of assignments (b) μ* ℑ(φ) := {α:ℑ |= φ[α]}. Cont. A κ-modal algebra is tuple ℜ := <A,0,1,∧,∨,∼,<♦λ: for all λ< κ, > where: <A,0,1,∧,∨,∼ > is a Boolean algebra (1) ♦λ 0:= 0 (2) ♦λ ( x1∨ x2 ) := ♦λ x1 ∨ ♦λ x2 (3) ♦λx:=♦λ♦λx (4) ♦ς ♦λx = ♦λ♦ς x Cont. Given a frame F it is known that one can build the unique modal algebra F+ := ℜF called the dual of a frame follows. Suppose that F:= <W, Ri > where is a frame. A boolean algebra with operators over F is a tuple ℑF:= <℘(W), ∅, W, ∩, ∪, ∼, ♦i> where: (1) <℘(W), ∅, 1, ∩, ∪, ∼, > a field of sets (2) ♦i: ℘(W) → ℘(W) such that ♦i(A):={w∈W: there exists w' such that w ∼i ' and w' ∈A}. Cont. Given a frame F we wish to define a Boolean algebra with operators which could serve as the meaning algebra. We shall not use dual construction since it forces us to consider the entire field of sets as the carrier of algebra. We will introduce a subtler method of constructing a modal algebra, which finds applications when it comes to defining a general frame. Cont. A general frame is a tuple FG := <F, Ω > where F is a frame and Ω ⊆ F+ In PML there is a natural way to define a generalized frame out of a particular Kripke's structure. Even though our models are not propositional models and the valuation function is not present there is a natural way to convert a frame into a general frame. Cont. Given a polymodal constant domain frame of our type F := <W, ∼i> we define the general frame FG = <F, Ω> where Ω: = < A,0,1,∩,∪,∼, ♦i> such that (1) A: = {X: for every φ ∈ L:w |= φ} Here how the carrier set is defined. (2) 0 = V(⊥) (3) 1= V(T) (4) operations are defined in the standard way. Cont. σ Suppose that V* : AtM → A where: (a) A is the carrier of Ω of the polymodal algebra generated by ℑ*. V* (φ) := {w ∈W: ℑ, |= φ } where φ is an atom. Then there exists the unique extension V of V* to the entire language LMσ Cont. Proposition 4. Fix σ– model ℑ let ℜ be a cylindric algebra of assignments defined by Lσ to ℑ. Let ℑ* the corresponding constant domain frame and let FℑG a general frame generated by Fℑ. Then it is the case that ℜ ≅ Ω. Cont. Proof: A bijective homomorphism h is defined as ℑ follows: h(φ ): = V(φ*). It proceeds by cases. We show only the case for cylindrification. Cont. Suppose that w ∈ h(ci(φℑ)). By definition ci(μℑ ℑ (φ)) := μ (∃xiφ) . Therefore w ∈ V(<i>φ*). Once again by definition it follows that w ∈ ♦i V(φ*). From the definition of h we infer that w ∈ ♦i h(φ*). n the reverse direction suppose that w ∈ ♦i h(μℑ(φ)). It follows that w ∈ ♦i V(φ*). ℑ ℑ Hence w ∈ V( <i>φ*) = h(μ (∃xiφ)) =h(ci(μ (φ)).