?-REPRESENTATION OF S4-LOGICS

1
?-REPRESENTATION OF S4-LOGICS

• Alexei Muravitsky
• Northwestern State University
• Oxford August 2007

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)
• GrzS4?(?(p??p)?p)?p (Grzegorczyk logic),
• M0 Grz?S5,
• S0, S1 S5S0??S2?S1S4p!?p
• Scroggs 1951,
• M0, Grz M0??M2?M1Grz.
• Muravitsky 2006

3
Preliminaries

• Definitions
• Let A?At denote McKinsey-Tarski translation and
  ?tAt A??, for any set of assertoric
  formulas.
• ?(L)S4?Lt, for any L?ExtInt.
• ?(M)A At?M, for any M?ExtS4.
• ?(L) ?(L)?Grz, for any L?ExtInt.

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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 M?ExtS4,
• ??(M)?M???(M).
• W. Blok 1976, L. Esakia 1976

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Focusing on ??(M)?M???(M)

• Let MM???(M) for some logic M.
• Where can M lie?
• Main Definition (1st variant) The equality
• MM??(L), where M?S4,Grz and L?ExtInt, 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 MM??(L) be a ?-representation of M.
• Observations
• An assertoric component, L, is uniquely
  determined by M so that L?(M) , since
  ?(M)?(M)???(L)Int?LL.
• A modal component, M, may vary.
• Examples
• Let MS5. Then all M?S4,M0 are modal
  components of M. In fact, this interval comprises
  all the modal components of M.
• Let M S1S4p!?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 MM is the only modal
  component of M.
• 4. If M?(L) for some L?ExtInt then MS4 is the
  only modal component of M.
• Observation
• Given logic M, all its modal components
• form a dense sublattice of S4,Grz.

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• Definition (well-known) L?(L)L?ExtInt
• (Reminder Lattices L and ExtInt are isomorphic.)
• Main Definition (2nd variant) The equality
• MM??, where M?S4,Grz and ??L, is
• called a ?-representation of the logic M. In this
• equality, M is called a modal component of M
• and ? a ?-component of M.
• Remark If MM??, where M?S4,Grz, then
• ???(M) that is, ? is determined uniquely by M.

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• Observation
• Given logic M, the logic MM?Grz 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
S1S4p??p
S2
S3
GrzM1
S0S5
M2
M3
M0
L?


S4
Figure 1
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• Examples (continued)
• 2. There are greatest modal components different
  from M0, M1, (Cf. Example 3 (above) MM?S4, if
  M?S4,Grz and Example 4 (above) MS4?M, 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
S1S4p??p
S2
M
S3
GrzM1
S0S5
M2
M3
M0
L?
K1S4.1


S4
Figure 2
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• Definition A ?-representation MM?? is
• called fine if M is the least modal
• component of M.
• Main Question Does every logic M?ExtS4
• have a fine ?-representation?

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Slicing ExtS4

• Definition The logics Snn?0, which form
• the interval S5,S1, we call S-series. The
• logics Mnn?0, which form the interval M0,Grz,
  we
• call M-series.
• Definition of an nth S-slice of ExtS4
• the 0th S-slice is M?n?1(M?Sn)S4,S5
• the nth S-slice, n?1, is MM?Sn and M?Sn1.
• Definition of an nth M-slice of S4,Grz
• the 0th M-slice is M?n?1(M?Mn)S4,M0
• the nth M-slice, n?1, is MM?Mn and M?Mn1.

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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
  if
• and only if all its modal components are in the
  nth
• M-slice.

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Lattice ExtS4

Figure 3
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• Definition of a partial binary operation
• d(X,Y) on ExtS4 Given logics X,Y?ExtS4,
• we define d(X,Y) to be such a logic C that
• ?Z(C?Z?XZ?Y).
• Remark Operation d(X,Y) is a slight
• modification of the pseudo-difference introduced
  in
• C. Rauszer 1974.
• Proposition Let MM?? be a ?-representation. If
  d(M,?) is
• defined, 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)
M
?
Grz ? M
?
?
?
d(M,?)
?
Figure 4
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• Definition (well-known) Let ? be the equality
• relation in Kleenes 3-valued weak logic. That
  is,
• given f(x,y) and g(x,y), f(x,y)?g(x,y) if and
  only if
• either both f(x,y) and g(x,y) are defined and
• f(x,y)g(x,y), or both f(x,y) are g(x,y) are
  undefined
• Theorem Let ? be the ?-component of a logic M
• and all modal components of M lie the nth
  M-slice,
• that is, in Kn,Mn. Then
• d(M,?)?d(M?Grz,(??Grz)?Kn).

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Commenting on d(M,?)?d(M?Grz,(??Grz)?Kn)

• We note that if M belongs in the nth S-slice (of
• ExtS4), then the arguments of the right side,
• that is, M?Grz and (??Grz)?Kn, belong in the
• nth M-slice (of S4,Grz).

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? Sn
M
?
Mn
?
M?Grz ?
? ?
? (??Grz)?Kn
?
Kn
Figure 5
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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
M ?
Mn ?
? ?
M?Grz ?
? (??Grz)?Kn
d(M,?) ? d(M?Grz,(??Grz)?Kn)?
The modal components of M
Kn ?
Figure 6
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• Thank you