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 στ 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 CAα is a subvariety

• f 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 Lσ in the signature σ (with the underlying set of variables Varα= { vi : i <α } is generated by the standard grammar for first-order languages. Atomic formulas of the form P(v0i,…, vn-1i) Restricted first-order languages contain only atoms of the form P(v0,…,vn-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, LR

σ ⊆ Lσ . Moreover, if we have Varα where α ≥ ω then

LR

σ and Lσare guaranteed to be equally expressive.

This result does not hold in general for a finite variable fragment. Example P(v0,v1) and P(v1,v0) and just two variables It is possible to regard substituted atomic formulas as somehow complex. (1) ℑ |= P(v0,…,vn-1) [α] iff < ℑv0[α],…,ℑ vn-1 [α] > ∈ Iℑ (Pn) (2) ℑ |= P(vτ(0),…,v τ (n-1)) [α] iff P(v0,…,vn-1) [ α ο τ ] Complexity is a semantic complexity. The simultaneous substitution

• peration is not a part of the syntax.

Reverse Correspondence Theory Ideally, we would like to define a translation from ordinary first-

• rder language with Varω = { vi : 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 Varn= { vi : i < n } The notion of a first-order structure for this languages is the usual

• ne.

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 Dℑ

Modal Logic of Relations

• Def. A modal logic of relations MLRn is generated by the following

grammar in BNF: φ = p | χ∧ψ | χ∨ψ | ∼ψ | δij | ◆i φ | <τ>φ

• Def. The n - cube over D is the tuple

CD = < nD, ≡i , ≈τ , δij >, τ ∈ nn 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) idij = { s ∈ nD : s(i) = s(j) }. The cube model over D is ℑ = < CD , V > where V is a valuation function

Modal Logic of Relations Semantics. (1) ℑ, s |= δij iff s ∈ δij (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) (vi = vj)V = δij (2) P(v0 ,...vn-1)V = p (3) P(vτ(0) ,…,vτ(n-1) ) V = <τ> p (4) (∃vi φ) V = ♦i (φ) V

Modal Logic of Relations

• Prop. For each ℑ, assignment β and formula φ ∈ LF

σ ,

ℑ |= φ [β] iff ℑ V+ , β |= φV

• 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) Cn , W = n U (2) Rn , W ⊆ nU (3)Dn , if s ∈ W then s ο [i\j] ∈ W ; for each i,j < n (4) LCn if s ∈ W then s ο τ ∈ W ; for each τ ∈ nn

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 Cω propositional variables are interpreted as the set of

• mega 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
• f 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 ωU which are finitary. R ⊆ ω U is finitary if and only if there exists n ∈ ω and Rn ⊆ nU s.t R = Rn x ω U Given ℑ = <C(U)ω, V > ℑD = <C, VD > where VD 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 vi in a structure ℑ is ℑvi : Assℑ → Dℑ s.t

ℑvi (α) = α (vi) ; the satisfaction by an assignment

The meaning of a variable vi in a structure ℑ is i-th projection function ℑvi : αDℑ → Dℑ 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 → Dℑ s.t ℑci (w) = Iℑ (ti) 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.

VFML – Its Syntax and Semantics A first-order multi-modal signature (with no function symbols) is a tuple σ : = < Prα , Conα , Oprα , Bool, τ > where: (1) Prα := {Pi : i < α } (2) Conα : = { ci : i < α } where α ≠ 0 , α ∈ Ord (3) Oprα : = {♦i, ■i : i < α } where α ≠ 0, α ∈ Ord (4) τ : Prα ∪ Oprα→ ω Restriction: unary modalities, recursively enumerable languages Given a modal first-order signature σ the language QFMLσ is generated by the following grammar in BNF φ = ci = cj |P(ci0,...,cin-1) | χ∧ψ | χ ∨ψ | ∼φ | ♦i φ | ■i φ

Syntax and Semantics Given a signature σ a first-order modal structure is a tuple ℑ = < W, D, < ∼i : i < α > , δ, Iℑ > (a) W is a non-empty set (b) D is a non-empty set (c) ∼i W x W ⊆ (d) δ : W → ℘ ( D ) ; each δ(w) ≠∅ . (e) Iℑ : Pn

i → ( W → Dn )

Iℑ : ci → ( W → D ) For brevity, I

ℑ (Pn i ) (w) = Iℑ w(Pn i ); similarly for the constant

A constant ci denotes rigidly iff for each w,w' , it holds that I ℑ (ci ) (w) = I ℑ (ci ) (w) A constant ci denotes non-rigidly iff it does not denote rigidly

VFML – Syntax and Semantics Observe that we do not require the existence property. In other words it might hold that Iw(Pn

i ) ⊄ δ(w) n

A frame F = < W, D, < ∼i : i < α > , δ > A frame F is called globally constant domain frame if and only if for each if for each w,w' it is the case that δ(w) = δ (w') . A frame F is called locally constant domain if and only if for each w,w' and each ∼i , if w ∼i w' then δ(w) = δ (w'). A frame F is called varying domain if and only it is not a constant domain frame In the sequel we shall work primarily with globally constant frames.

VFML – Syntax and Semantics

• For every φ and each pointed Kripke structure we define

inductively the satisfaction relation as follows: (1) ℑ ,w |= ci = cj iff Iℑ

w(c I) = Iℑ w(c j)

(2) P(c i0 ,...,cin-1) iff <Iℑ

w(c i0),…, Iℑ w(c i0) > ∈ Iℑ w (Pn i )

(3) ℑ ,w |= ♦i φ iff for some w' ∼i w, ℑ ,w' |= φ (4) ℑ ,w |= ■i φ iff for each w' ∼i w, ℑ ,w' |= φ The truth relation in a Kripke structure is defined (non-inductively) as follows ℑ |= T φ iff for each w ∈ W, ℑ ,w |= φ A formula is valid in a frame F if and only if it is true in each ℑ based on F.

Bisimulation and the expressive power

• As one can guess the expressive power of VFMLσ is quite limited.
• Definitely, it is strictly weaker than the expressive power of its

quantified extension.

• In order to formally establish that we need a bisimulation relation

appropriate for ours language.

• Bisimulation. Given ℑ , ℑ' ∈ Xσ a non-empty binary relation

Z ⊆ W x W’ is called a bisimulation iff (1) ℑ, w and ℑ , w’ satisfy the same atomic formula (2) For each ∼i and every w', if w ∼i w' then there exists w such that (3) vice versa

Bisimulation and the Expressiveness By induction on the complexity of a formula one shows that if two worlds w and w’ are bisimilar then they are modally equivalent Our relation of bisimulation looks pretty much as a propositional bisimulation. It does not collapse to the relation of modal equivalence between holding between worlds. The standard example is applicable in our case. With this at hand it is easy to provide examples of neither increasing and decreasing structures and worlds which bisimilate. Consequently, these classes are not definable in VFML On the other hand, Barcan formula ♦ ∃ vi φ → ∃ vi♦ φ defines the class of decreasing frames and its converse the class of increasing frames.

Bisimulation and Expressiveness In general, there is no unique bisimulation relation as an appropriate invariance relation for first-order modal languages The another is world object bisimulation. Designed for full first-order modal languages with the standard semantics. Roughly speaking the world object bisimulation merges back and forth conditions of a system of partial isomorphisms of the length ω with back and forth clauses of plain bisimulation. By Fraisse’s Theorem two worlds matched by this bisimulation are elementarily equivalent Below we state the definition of world-object bisimulation (without cross existence property) We will need it in the sequel

Bisimulation and Expressiveness

• Definition. Given ℑ, ℑ + a world – object bisimulation is a binary

relation Z ⊆ (W x Dℑ

∗ ) x (W + x Dℑ+ ∗ ) s.t

(1) if <w, <ai0,…,ain-1> > Z <w +, <a+

i0,…,a+ in-1> > then f: (ai) = a+ i is

a partial isomorphism (2) for each w’ and each ∼i such that w ∼i w’ there exists w’+ such that <w’, <ai0,…,ain-1> > Z <w’+, <a’+

i0,…,a’+ in-1> >

(3) vice versa (4) for each a ∈ Dw there exists a+ ∈ Dw+ s.t <w,<ai0,…,ain-1a> > Z <w +, <a+

i0,…,a+ in-1 a+> >

(5) for every a+ ∈ Dw+ there exists a ∈ Dw s.t <w,<ai0,…,ain-1a> > Z <w +, <a+

i0,…,a+ in-1 a+> >

Translating Ordinary First-Order Languages into VFMLσ

• As usual translation is going to have two components a structure

transformation function and syntactic translation

• We modify the semantics for first-order languages slightly
• We borrow an idea from Wolfgang Rautenberg and distinguish

between a structure and a model structure

• A structure is ℑ and a model structure is a pair ℑM = < ℑ, α >

where α is an assignment. Observe that this is a complete theory.

• The space of model structures is S = {ℑ} x Assℑ
• The modification of first-order semantics to the space of model of

structures is a cosmetic one.

• We wish to generate the corresponding constant domain frame with

rigid interpretation of predicate symbols from S

Translating Ordinary First-Order Logic into VFML Given an ordinary first-order language L with Varα we define the corresponding modal signature σm(σ) = <Pr, Conα ,Oprα , τ > in a straightforward way. This obviously fixes the corresponding language Trσm(σ), VFMLσm(σ) and class of structures.

• Def. For each Varα and Conα define the obvious bijection ∗ such

that vi

∗ = ci

• Def. For each Lσ and VFMLσm(σ) we define inductively translation

t: Lσ → VFMLσm(σ) as follows: (1) (vi = vj ) t = vi

∗ = vj ∗

(2) P(vi0,…,vin-1) t = P(vi0

∗ ,…, vin-1 ∗ )

(2) (∃vi φ ) t = ♦i (φ) t

Translating Ordinary First-Order Logic into VFML Given a space of model structures Sℑ define the corresponding constant domain Kripke structure ℑT as follows: (1) W t = { wβ : β ∈ Assℑ } (2) D t = Dℑ (3) δ(wβ ) = δ(wβ

’) for each wβ , wβ ’ ∈ W t

(4) Iℑ

wβ ( Pn) = Iℑ wβ’ ( Pn) for each wβ , wβ ’ ∈ W t and each Pn

(5) Iℑ

wβ (ci ) := β(vi ) for each i < α

(6) wβ ∼ i wβ

’ iff for all j ≠ i, Iℑ wβ (cj ) = Iℑ wβ ’(cj)

Observe that the frame is not primitively defined. It is parasitic on the interpretation function.

Translating Ordinary First-Order Logic into VFML

• Prop. For each Sℑ and each φ ∈ Lσ it holds that

< ℑ, β > | = φ if and only if ℑ t , wβ | = φ t

• Proof. A straightforward induction on the complexity of a formula

Consequently, Λ |= φ if and only if Λ t |= φ t In a moment we shall define the converse relation. However, some care is needed in order to arrive at it.

Converse Translation The following subclasses of Xσ The class of globally constant domain structures CG R := {ℑ ∈ Xσ : for each w, w’ ∈ ℑ and each Pi

n , Iw ℑ (Pn i ) = Iw ℑ (Pn i)}

A : = { ℑ ∈ Xσ : | W | = Dℑ

Con and for each ci , w , Iw ℑ (ci) ∈ δ(w) }

Vi : = { ℑ ∈ Xσ: for each w, w’ ∈ W , ci ∈ Cons , w ∼i w’ iff Iw

ℑ (cj) = Iw ℑ (cj) , j ≠ i }

Q = R ∩ A ∩ Vi ∩ CG

Converse Translation It is easy to see that each member of Q has the unique corresponding space

• f

model structures. Moreover the correspondence μ+ is injective . The syntactic translation is completely analogous to t. Namely we define μ: (1) μ(P(c0i,...cn-10) = P(c0i

• ,…, c0i
• )

(2) μ(♦i φ) = ∃vi (φ)μ The semantic correctness of this translation follows immediately. Hence over Q these languages can be seen as syntactic variants of each other. They are equally expressive. Inheritance of metatheoretic properties such as Compactness, Undecidabilty and cardinality transfer theorems is immediate.

Substitution Principles for VFML Our next goal is to define a Hilbert Style Axiom System for Q . To this end we must discuss substitution operation for VFMLσ Typically, variables are substituted for. Nonetheless, non-rigidly designating constants are ‘variable objects’ . Consequently, this type

• f constants are good objects to be substituted for. See also

Goldblatt Quantifiers Propositions and Identity Before we proceed we stress that we can only inductively define the set of constants occurring in a formula (a) Const P(c0i,...,cn-1i) = { c0i,...,cn-1i } (b) Const (♦i φ) = Const (φ)

Substitution Principles for VFML The clause (b) stands in contrast with the analogous definition in the case of first-order logic. The role of an index attached to a modality is to point to the matching accessibility relation The notions of bound and free occurrences of constants simply nonsensical Therefore we believe that the right notion of substitution is just the

• peration of (simultaneous) rewriting of a constant by a constant.

Substitution Principles for VFML

• Def. Given a function i: Consi → Consti s.t Si = {ci ∈ Ci : i(ci) ≠ ci }

is finite denote the corresponding substitution operation [i]S inductively defined as follows: (1) [i]S P(c0i,…,cn-1i) = P(i(c0i),…,i(c0n-1)) (2) [i]S (♦i φ) = ♦i [i]S φ It is possible to prove the substitution lemma for this operation. For each ℑ ∈ Xσ define ℑi such that ℑi is alike as ℑ except for the fact that for each ci ∈ Si and each wi ∈ Wi , Iwi

ℑi (ci) = (Iw ℑ ο i) (ci)

Substitution Principles for VFML

• Prop. Given [i]s , for each formula φ ∈ VFMLσ and each pointed

Kripke structure , ℑ, w |= [i]s φ if and only if ℑi , wi |= φ

• Proof. By the induction on the complexity of a formula

Axiomatization of Q A normal modal logic : Λ ⊆ VFMLσ s.t (1) Λ contains all instantiations of distributivity schemas (2) Λ is closed under MP (3) Λ is closed under N (4) Λ is closed under substitution operation A quasi normal modal logic: Λ ⊆ VFMLσ s.t (1) , (2), (3) We define ΛQ semantically. Namely, ΛQ = { φ ∈ VFMLσ : ℑ |= φ for each ℑ ∈ Q }.

Axiomatization of Q Our goal is to find a strongly complete axiomatization of ΛQ . A natural approach is to axiomatize ΛQ by a Hilbert – Style Axiom System which mirrors a Hilbert – Style Axiom System for FOL Nonetheless, a straightforward choice (such Enderton’s Axiom System) encounters complications: (1) Tautologies schemas (2) ∀vi (φ → ψ ) → (∀vi φ → ∀vi ψ ) (3) ∀vi φ → [vi := vj ] φ ; if vj is free for vi in φ (4) φ → ∀vi φ ; if vi ∉ Fr(φ) (5) vi = vi (6) vi = vj → (φ → [vi = vj ] φ) ; φ is an atomic formula Rules: if |-H φ → ψ and |-H φ then |-H ψ

Axiomatization of Q The predicament is that we can’t mirror the side condition in the axiom schema (3) and the notion of a free occurrence of a variable in a formula in the case of axiom schema (4) Hence the task is to find suitable restrictions We propose a solution which employs ours translation functions Namely, for each φ ∈ VFMLσ define: C ∗ : = { ci ∈ Cons(φ): ci = μ (vi) : vi ∈ Fr(μ(φ) } Furthermore for each first-order simultaneous substitution operation [i]s define its modal variant [i]s

m s.t for each φ ∈ VFMLσ

[i]s

m φ = ([i]s (φμ) t .

Of course, this rewrite operation can be defined inductively

Axiomatization of Q The problematic axiom schemas can be now captured as follows (3’) φ → ■i φ ; if ci ∉ Cφ

and (4’) ■i φ → [i]s m φ ; if ci ∉ Cφ

Well, this gives the desired procedure. Nonetheless, the resulting Hilbert- Style Axiom System is (perhaps) not that transparent. There exists an alternative. In Simplified Formalization of Predicate Logic with Identity Tarski defined a Hilbert – Style Axiom System which dispenses with notion of a substitution of a free occurrence

• f a variable by a free occurrence of a variable and the notion of a

free occurrence of a variable in a formula. This axiom system employs only the notion of an occurrence of a variable in a formula and replacement operation defined for atomic formulas.

Hilbert Style Axiom Systems for Q Hilbert – Style Axiom System T (1) (φ→ψ) → (( ψ→ χ ) → (φ→ χ)) (2) (∼φ → φ) → φ (3) φ → (∼φ→ψ) (4) ∀vi (φ→ψ) → (∀vi φ → ∀vi ψ) (5) ∀vi φ → φ (6) φ → ∀vi φ , vi ∉ Oc (φ) (7) ∼ ∀vi ∼ ( vi = vj ) where i≠j (8) vi = vj → (φ→[vi := v] φ); φ is atomic if |-T φ→ψ and |-T φ then |-T ψ if |-T φ then |-T ∀vi φ

Hilbert Style Axiom Systems for Q Furthermore, Tarski shown that T is equivalent to the standard systems (such as Enderton’s ). Consequently T is a strongly complete Hilbert – Style Axiom System. Now we can either choose either the modal variant of Enderton’s Axiom System EM or the modal variant of Tarski’s Axiom System TM . We stick to the latter. In both cases it easy to show by the induction on the length of proofs that: (1) Λ |-E φ iff Λt |-E φ t and (1’) Λ |-EM φ iff Λ μ |-E φ μ (2) Λ |-T φ iff Λt |-TM φ t and (1 ) Λ ’

|- TM φ iff Λ μ |- T φ μ

Completeness of Q Is this axiomatization strongly complete with reference to Q ? Equivalently, we ask whether every TM consistent set of formulas is satisfiable in some ℑ ∈ Q It is natural to approach this question is by the canonical structure construction. We choose to proceed in the manner of Henkin

Completeness of Q Given a maximally consistent set of formulas ΛM define the corresponding wΛM : (1) DwΛM = { [ci]⁄ΛM : {cj ∈ Const : ci = cj ∈ ΛM }}. (2) I wΛM

ℑ (ci) = [ci]⁄ΛM

(3) I wΛM

ℑ (Pi n) = {<[c0i]⁄ΛM,…, [cn-1i]⁄ΛM > ∈ nDwΛM :

P(c0i,…,cn-1i) ∈ ΛM } (4) For each wΛM, w’Λ’M , wΛM ∼iw’Λ’M iff for each φ, if φ ∈ Λ‘M then ♦i φ ∈ ΛM This fixes the canonical structure ℑC in an obvious way. The domain of ℑC is the set of all equivalence classes of all equivalence relations over Constσ

Completeness of Q Truth Lemma. For each φ ∈ VFMLσ and each wc ∈ Wc ℑC , wΛ MC |= φ if and only if φ ∈ Λ MC

• Proof. By the induction on the complexity of a formula

However, we do not obtain the completeness wrt to Q,

Completeness of Q Firstly observe that the truth lemma is not proved relatively to the intended semantics. Namely, the canonical accessibility relation does not relate the worlds which are necessarily i-variants. Hence, (1) ℑC V ∉

i

Secondly, (2) R , ℑ ∉ (3) ℑC C ∉ As we have seen (1) and (2) are undefinable in VFML and even in its quantified extension. Consequently, there is no hope to strengthen axiomatization in

• rder to obtain an another canonical structure to behave in the

desired way Nonetheless, the strong completeness holds (as expected) indirectly via the translation functions

Completeness of Q

• Suppose that not Λ |-T(m) φ . Hence, Λ ∪ ∼ φ is T(m) consistent.

Consequently Λμ ∪ ∼ φ μ is consistent. As FOL is strongly complete it follows that ℑ, β |= Λμ ∪ ∼ φ μ . But now, ℑT , wβ |= (Λμ)T ∪(∼φμ)T