Nondeviant logics of relative identity

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Non-deviant logics of relative identity
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Non-deviant logics of relative identity
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Non-deviant logics of relative identity
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Non-deviant logics of relative identity
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Non-deviant logics of relative identity
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Non-deviant logics of relative identity
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Non-deviant logics of relative identity
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Non-deviant logics of relative identity
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Non-deviant logics of relative identity

• (F3) (i) ?C2 ?x1(A(x1) ? B(x1)) ? (x1A x2 ? x1B
  x2).
• (ii) ?C3 ?x1(A(x1)?B(x1)) ? (x1A x2 ? x2B x3
  ? x1B x3).
• (iii) ?C4 ?x1(A(x1)?B(x1)) ? (x1A x2 ? x2B x3
  ? x1A x3).
• (iv) ?C5 ?x1(A(x1)?B(x1)) ? (x1A x2 ? x2B x3
  ?
• x1A x3 ? x1B x3 ).
• (v) ?C6 x1A x2 ? x2B x3 ? x1Ax3 ? x1B x3.
• (vi) ?C7 x1A x2 ? B(x1) ? B(x2).
• (vii) ?C8 x1A x2 ? B(x1) ? B(x2) ? x1B x2 .
• (viii) ?C9 x1A x2 ? B(x1) ? x1B x2 .

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Deviant logics of relative identity

• (C10) If X?F(?), then ?u??F(?) Xu.
• (C11) ?F(?)?F(?).

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Monadic logics of relative identity

• C1
• C2
• C3 C4
• C5 C6 C7 C8
• C9
• C11 C10

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Sequent calculi of relative identity

• (D8) A sequent ? ? is derivable in a sequent
  calculus SQR1, R2, , Rn iff there exists a
  finite sequence of sequents ?1 ?1, ?2 ?2, , ?n
  ?n such that
• (i) ?n? and ?n?,
• (ii) every element ?i ?i in that sequence
  either is the conclusion of some non-premise
  rule from SQ or there is a rule in SQ such that
  ?i ?i is its conclusion and its premises are
  among precedent sequents in the sequence.

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Sequent calculi of relative identity

• (D9) A formula ? is derivable in SQ from a set ?
  of formulas, ??SQ?, iff there is a finite
  number of formulas ?1, ?2, , ?n in ? such that
  ?SQ ?1?2 ?n ?.

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Calculi of relative identity

• (R11A) ? ?1?2
• ? ?(?1)
• ---------------------------
• ? ?1??2
• (R11B) ? ?1??2
• ---------------------------
• ? ?(?1)

• (R11C) ? ?1??2
• ---------------------------
• ? ?2??1
• (R11D) ? ?1??2
• ? ?2??3
• ---------------------------
• ? ?1??3