Maciej Kleczek The University of Bielefeld Faculty of Linguistics - PowerPoint PPT Presentation

Maciej Kleczek The University of Bielefeld Faculty of Linguistics Variable as a Non-Rigidly Designating Modal Constant

What is Reverse Correspondence Theory ? Modal Correspondence Theory Modal Languages in n-ary modal similarity type are n+1 - variable bounded fragments of first-order logic This enables use powerful tools of first-order modal theory in modal domain. For example one obtains van Benthem's – Rosser Characterization theorem A first-order formula φ with one free variable in the signature σ τ is equivalent to the standard translation of a modal formula if and only if it is invariant for bisimulation. Proved via proved via the existence of omega saturated first-order structures .

What is Reverse Correspondence Theory ? A similar question can be asked whether there is a converse translation ? Namely, whether there exists a semantically correct translation from a first-order language to a modal language ? Why one should be interested in defining such a translation ? It is believed that the semantics of first-order languages has modal features. Tarskian satisfaction relation involves a binary i-variance relation between assignments or sequences Furthermore, axiomatizations of first-order logic resemble modal axiomatization. Take a look at S5. The class of cylindric algebras of dimension α CA α is a subvariety of the variety of boolean algebras with operators.

What is Reverse Correspondence Theory ? Even though these initial insights are plausible it is a non-trivial question how to provided a desired translation Admittedly, we used central concepts in a loose manner. The notion of a first-order language is ambiguous We distinguish between: (1) ordinary first-order languages (2) restricted first-order languages (3) full first-order languages

First-Order Languages and Their Semantics An ordinary first-order language L σ in the signature σ ( with the underlying set of variables Var α = { v i : i < α } is generated by the standard grammar for first-order languages. Atomic formulas of the form P(v 0i ,…, v n-1i ) Restricted first-order languages contain only atoms of the form P(v 0 ,…,v n-1 ) where the string of variables is the proper initial segment of an ordinal α Full n-ary first-order languages. Each predicate symbol has n-ary arity

Reverse Correspondence Theory Of course, L R σ ⊆ L σ . Moreover, if we have Var α where α ≥ ω then L R σ and L σ are guaranteed to be equally expressive. This result does not hold in general for a finite variable fragment. Example P(v 0 ,v 1 ) and P(v 1 ,v 0 ) and just two variables It is possible to regard substituted atomic formulas as somehow complex. (1) ℑ |= P(v 0 ,…,v n-1 ) [ α ] iff < ℑ v 0 [α],…, ℑ v n-1 [ α ] > ∈ I ℑ (P n ) (2) ℑ |= P(v τ (0) ,…,v τ (n-1) ) [ α ] iff P(v 0 ,…,v n-1 ) [ α ο τ ] Complexity is a semantic complexity. The simultaneous substitution operation is not a part of the syntax.

Reverse Correspondence Theory Ideally, we would like to define a translation from ordinary first- order language with Var ω = { v i : i < ω } with the usual Tarskian semantics (no extra clause for substituted atomic formulas ) However, it is not a straightforward task to define such a translation To date there exists two frameworks which grapple with this problem Many-sorted modal logic of Steven Kuhn ( Quantifiers as Modal Operators ). Modal Logic of Relations (or Cylindric Modal Logic) developed around 20 years ago in the series of papers by Yde Venema and co- authors.

Modal Logic of Relations The starting point of modal logic of relations is limited. Namely, source languages of translation are n-ary full languages with Var n = { v i : i < n } The notion of a first-order structure for this languages is the usual one. The semantics has the extra clause for substituted atomic formulas. This semantics is chosen in order to avoid complications in defining syntactic translation Venema’s translation has as usual two components: (a) the syntactic translation V (b) the structure transformation function V + The latter associates with each ℑ for σ , the n-ary cube over D ℑ

Modal Logic of Relations Def. A modal logic of relations MLR n is generated by the following grammar in BNF: φ = p | χ∧ψ | χ∨ψ | ∼ψ | δ ij | ◆ i φ | <τ>φ Def. The n - cube over D is the tuple C D = < n D, ≡ i , ≈ τ , δ ij >, τ ∈ n n and i,j < n (1) s ≡ i s’ iff for all j ≠ i, s(j) = s(j) (2) s ≈ τ s’ iff s’ = s ο τ (3) id ij = { s ∈ n D : s(i) = s(j) }. The cube model over D is ℑ = < C D , V > where V is a valuation function

Modal Logic of Relations Semantics. (1) ℑ , s |= δ ij iff s ∈ δ ij (2) ℑ , s |= <τ>φ iff for some s’ s.t s ≈ τ s’ , ℑ , s’ |= φ (3) ℑ , s |= ♦ i φ iff for some s’ s.t s ≡ i s’ , ℑ , s’ |= φ Venema’s Translation. (1) (v i = v j ) V = δ ij (2) P(v 0 ,...v n-1 ) V = p (3) P(v τ( 0) ,…,v τ (n-1) ) V = <τ> p (4) ( ∃ v i φ) V = ♦ i ( φ) V

Modal Logic of Relations Prop. For each ℑ, assignment β and formula φ ∈ L F σ , ℑ |= φ [ β ] iff ℑ V+ , β |= φ V Proof. A straightforward induction on the complexity of a formula The following classes of structures are distinguished as suitable for modal logics of relations. (1) C n , W = n U (2) R n , W ⊆ n U (3)D n , if s ∈ W then s ο [i\j] ∈ W ; for each i,j < n (4) LC n if s ∈ W then s ο τ ∈ W ; for each τ ∈ n n

Modal Logic of Relations Even though it all sounds pleasing the framework of cylindric modal logic has not quite succeeded when it comes to fulfilling the basic task. So far we have seen no translation for (1) ordinary first-order languages in at least omega many variables with (2) the standard semantics What are the proposals to deal with these issues ? (or at least with (1)).

Modal Logic of Relations Firstly one would like to have recursively enumerable languages. Restrict the set of substitution diamonds to those with finite support <τ> where s( τ ) = { i ∈ ω : τ( i) ≠ i } is finite Define a language in this signature.

Modal Logic of Relations There are many issues involved when it comes to this extension. We single out the following. The choice of intended structures. We restrict attention to cubes (or its mild variants). This are the closer to the standard notion of a first-order structure. Relatively to C ω propositional variables are interpreted as the set of omega sequences (any set of omega sequences). Finite dimension of a formula interpreted by a set of omega sequences. This does not meshes well with the usual interpretation of predicate symbols. Change the syntax of first-order logic to an infinitary one. Hmm …

Modal Logic of Relations The second proposal is to introduce the modification of a structure based on ω -cube. We want to restrict valuations only to subsets of ω U which are finitary. R ⊆ ω U is finitary if and only if there exists n ∈ ω and R n ⊆ n U s.t R = R n x ω U Given ℑ = <C(U) ω , V > ℑ D = <C, V D > where V D is a dummy valuation. Prop. For every formula φ , φ is satisfiable in a cube model iff φ is satisfiable in a dummy cube model .

Modal Logic of Relations We loose the compactness over the class of dummy models { p ∧ ∼ ♦ i p : i < ω } is satisfiable over the class of cube models but it is not satisfiable over the class of dummy cube models.

Towards a New Beginning The question whether there exists a semantically correct modal translation from ordinary first-order languages (with infinitely many variables) with the standard Tarskian semantics remains open. It happens that such a translation exists. Moreover, this translation is grammatical. We drop the assumption that first-order languages must be translated into propositional modal languages The crux of our approach is to take the semantics of a first-order variable semantics seriously. But what is the meaning of a first-order variable ? Unfortunately, there are no clear answers

Towards a New Beginning Kit Fine claims that the meaning of a first-order variable is the range of its values … The meaning of a variable v i in a structure ℑ is ℑ v i : Ass ℑ → D ℑ s.t ℑ v i ( α ) = α (v i ) ; the satisfaction by an assignment The meaning of a variable v i in a structure ℑ is i-th projection function ℑ v i : α D ℑ → D ℑ In the literature it is often stated (see for example Church’s “Introduction to Mathematical Logic) that the semantics of a variable is ‘the same’ as the semantics of an individual constant This is partly true: the similarity holds just for free occurrences of constants

Towards a New Beginning Furthermore, the analogy is more accurate when instead of an individual constant we consider a non-rigidly designating modal constant. Its interpretation is not fixed. Moreover, observe that the meaning of a closed modal term in a modal first-order structure ℑ is an individual concept. Namely, ℑ t i : W → D ℑ s.t ℑ c i (w) = I ℑ (t i ) Notice the analogy with the meaning of a variable when assignments ‘play’ the role of worlds (not necessarily are the worlds themselves). We believe that we can combine these ideas in a fruitful way.