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-REPRESENTATION OF S4-LOGICS
Alexei Muravitsky
Northwestern State University
Oxford August 2007
Alexei Muravitsky 2
Logics in focus: The consistent extensions of S4
• ExtInt be the lattice of the intermediate logics;• ExtS4 be the lattice of normal extensions of S4;
(w.r.t. as join and as meet)
• Grz=S4+▢(▢(p▢p)p)p (Grzegorczyk logic),
• M0 = Grz⋂S5,
• [S0, S1] : S5=S0…S2S1=S4+p!▢p
[Scroggs 1951],
• [M0, Grz] : M0…M2M1=Grz.
[Muravitsky 2006]
Alexei Muravitsky 3
Preliminaries
Definitions:
• Let AAt denote McKinsey-Tarski translation and t={At : A}, for any set of assertoric formulas.
(L)=S4Lt, for any LExtInt. (M)={A : AtM}, for any MExtS4.
(L)= (L)Grz, for any LExtInt.
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Starting Point
Well-known facts: : ExtInt ExtS4 (monomorphism) : ExtS4 ExtInt (epimorphism) : ExtInt ExtGrz (isomorphism)[L.L. Maksimova and V.V. Rybakov 1974]
Theorem (Blok-Esakia inequality) For any MExtS4,
(M)M(M).[W. Blok 1976, L. Esakia 1976]
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Focusing on (M)M(M):
Let M=M*(M) for some logic M*.
Where can M* lie?
Main Definition (1st variant): The equality
M=M*(L), where M*[S4,Grz] and LExtInt, is
called a -representation of the logic M. In this
equality, M* is called a modal component of M
and L an assertoric component of M.
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Let M=M*(L) be a -representation of M.
Observations:• An assertoric component, L, is uniquely
determined by M so that L=(M) , since (M)=(M*)(L)=IntL=L.
• A modal component, M*, may vary.
Examples:
1. Let M=S5. Then all M*[S4,M0] are modal components of M. In fact, this interval comprises all the modal components of M.
2. Let M= S1=S4+p!▢p. Then the interval [S4.1,Grz] consists of all the modal components of M.
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Examples (continued):3. If M[S4,Grz] then M*=M is the only
modal component of M.4. If M=(L) for some LExtInt then M*=S4 is
the only modal component of M.
Observation:Given logic M, all its modal componentsform a dense sublattice of [S4,Grz].
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Definition (well-known): L={(L):LExtInt}(Reminder: Lattices L and ExtInt are isomorphic.)
Main Definition (2nd variant): The equalityM=M*, where M*[S4,Grz] and L, iscalled a -representation of the logic M. In thisequality, M* is called a modal component of Mand a -component of M.
Remark: If M=M*, where M*[S4,Grz], then=(M); that is, is determined uniquely by M.
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Observation:• Given logic M, the logic M*=MGrz is a
greatest modal component of M w.r.t. .
Examples:
1. For all logics in [M0,S1], their greatest modal components are M0, M1,…, which form the interval [M0,Grz]. (It will be illustrated on the next slide.)
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Lattice ExtS4
Figure 1
S1=S4+p▢p
S2
S3
S0=S5
S4
L+?
Grz=M1
M3
M0
M2
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Examples (continued):2. There are greatest modal components
different from M0, M1,… (Cf. Example 3 (above): M=MS4, if M[S4,Grz]; and Example 4 (above): M=S4M, if M=L.)
Observations:• However, every modal component either is
included in each logic of an initial segment of M1, M2,…or is included in M0. (This will be explained in Theorem on slide 16.)
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Observations (continued):
• Some logics have a least modal component w.r.t. .
Examples:
1. All logics in the interval [Grz,S1] have logic S4.1 as their least modal component.
2. All logics in L have logic S4 as their least modal component.
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A least modal component
S1=S4+p▢p
S2
S3
S0=S5
S4
L+?K1=S4.1
M3
M0
M2
Figure 2
Grz=M1
M
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Definition: A -representation M=M* is
called fine if M* is the least modal
component of M.
Main Question: Does every logic MExtS4
have a fine -representation?
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Slicing ExtS4Definition: The logics {Sn}n0, which form
the interval [S5,S1], we call S-series. The
logics {Mn}n0, which form the interval [M0,Grz], wecall M-series.Definition of an nth S-slice of ExtS4:• the 0th S-slice is {M:n1(MSn)}=[S4,S5];• the nth S-slice, n1, is {M:MSn and M⊈Sn+1}.Definition of an nth M-slice of [S4,Grz]:• the 0th M-slice is {M:n1(MMn)}=[S4,M0];• the nth M-slice, n1, is {M:MMn and M⊈Mn+1}.
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Observations:• S-slices form a partition of ExtS4.• M-slices form a partition of [S4,Grz].• Each nth M-slice is properly included in the nth S-
slice, respectively.• The nth M-slice is an interval [Kn,Mn] for some
logic Kn and the logic Mn from the M-series.
Theorem Given logic M, M lies in the nth S-slice ifand only if all its modal components are in the nth
M-slice.
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Lattice ExtS4
Figure 3
S1=S4+p▢p
S2
S3
S0=S5
S4
E1
E2E3
E0
L+?
Grz=M1
M3
M0K1=S4.1
K2
K3
M2
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Definition of a partial binary operationd(X,Y) on ExtS4: Given logics X,YExtS4,we define d(X,Y) to be such a logic C that
Z(CZX=ZY).Remark: Operation d(X,Y) is a slightmodification of the pseudo-difference introduced in[C. Rauszer 1974].
Proposition Let M=M* be a -representation. If d(M,) isdefined, then d(M,)M* and d(M,) is a modal component of M.In other words, d(M,) is the least modal component of M.
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Operation d(X,Y):
d(M,)
M
Grz M
●
●●
●
Figure 4
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Definition (well-known): Let ≃ be the equalityrelation in Kleene’s 3-valued weak logic. That is,given f(x,y) and g(x,y), f(x,y)≃g(x,y) if and only ifeither both f(x,y) and g(x,y) are defined andf(x,y)=g(x,y), or both f(x,y) are g(x,y) are
undefined
Theorem Let be the -component of a logic Mand all modal components of M lie the nth M-slice,that is, in [Kn,Mn]. Then
d(M,)≃d(MGrz,(Grz)Kn).
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Commenting on d(M,)≃d(MGrz,(Grz)Kn)
We note that if M belongs in the nth S-slice (of
ExtS4), then the arguments of the right side,
that is, MGrz and (Grz)Kn, belong in the
nth M-slice (of [S4,Grz]).
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● Sn
●
●MGrz ●
● (Grz)Kn
Mn
Kn
Figure 5
M●
●
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Comments (continued)
Thus, in order to answer the main question,
whether a logic M has a fine -representation,
we can only focus on the M-slice corresponding
to M to examine for which elements of this slice
the function d(x,y) is defined.
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Final view
Kn ●
d(M,) ≃ d(MGrz,(Grz)Kn)●
MGrz ●
Mn ●
● (Grz)Kn
M ●
●
Figure 6
The modal components of M
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Thank you
