Download The initial question: “What is the meaning of a first

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)).