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Derivation Schemes for Topological Logics

Derivation Schemes for Topological Logics. Derived Logics. What Are They? Why Do We Need Them? How Can We Use Them? Colleague: Michael Westmoreland. History. 1936 Von Neumann and Birkhoff a lattice of propositions based on the closed subspaces of Hilbert space

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Derivation Schemes for Topological Logics

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  1. Derivation Schemes for Topological Logics

  2. Derived Logics • What Are They? • Why Do We Need Them? • How Can We Use Them? • Colleague: Michael Westmoreland

  3. History • 1936 Von Neumann and Birkhoff • a lattice of propositions based on the closed subspaces of Hilbert space • now known as “quantum logic” • based on measurement • non-Boolean (fails to meet distributive properties) • No satisfying way to do implication

  4. A Topological Logic • A proposition is an equivalence class of sets • S  S’ iff int(S) = int(S’) • [S] = [ (int S)c ] • [S]  [S’] = [ (int S)  (int S’) ] • [S]  [S’] = [ (int S)  (int S’) ] • Most Boolean properties hold • Law of noncontradiction: [S]  [S] = [int S]  [ (int S)c ]= [S  (int Sc) ] choosing canonical representation= [] • But not all:  [S] =? [S] No! •  [S] =  [ (int S)c ] So [S]  [int( (int Sc )c ] =  [ int Sc ] =  [S]

  5. Logic Properties • No tertium non datur: [S]  [S]  [U] where U is the universal set. • What about truth assignment? A measurement (open set) m verifies a proposition P iff mPiPiP. • Example: the real line with the standard topology. P = [ (-3, 5) ]. m = (0, 4) verifies P since (0,4)  (-3, 5), [-3, 5), [-3,5], (-3,5] • We speak of “verification” rather than truth. • Rationale: Let S be a classical system and P a proposition about S with P0 as the canonical representative of [P]. Then P0 = int Pj • Pj [P]. A measurement m that contains points of P0 but does not lie entirely in P0 would not verify P.

  6. More Properties • P = (-3, 5). m = (0,6) does not verify P. • Should we conclude P is false? The state of S could lie in P0 and still be consistent with the result of the measurement m. In fact, there is a more precise measurement, say m’ that lies entirely in P0 and the result of m. Hence, we cannot conclude that P is false. • New concept for assigning truth values: associated with a given measurement (set) , three possibilities: verifiability set, falsifiability set, indeterminate. • Twin Open Set Phase Space Logic (TOSPS) • A measurement m verifies P if m  P0 where P0 is the canonical rep of P. • A measurement falsifies a proposition if m  Cl(P0)c.

  7. Twin Open Set Phase Space Logic • Definition: P is a proposition in TOSPS logic if P = ( [V0], [F0] ), where V0 and F0 are disjoint open sets. • Definition: Let P = ( [V0], [F0] ) be a proposition in TOSPS logic and m be a measurement. P will be assigned the truth value true if mV0; false if mF0; indeterminate otherwise. • Logical Operators P = ( [PV], [PF] ) Q = ( [QV], [QF] ) PQ = ([int PV int QV], [int PF int QF] ) PQ = ( [int PV int QV], [int PF int QF] ) P = ( [int PF], [int PV] )

  8. Properties • P = ([PF], [QF]) = ( [PV], [PF] ) = P • P P = ( [PV], [PF] )  ( [PF], [PV] ) = ( [int PV int PF], [int PF int PV] )= ([], [U]) • P P = ([U], []) • DeMorgan’s laws • Ditributivity • All Boolean properties, but tertium non datur.

  9. Note:  fails to be truth functional • P = [(-1,2), (5,9)] Q = [(1,3), (8,11)] • P  Q = [(-1,3), (8,9)] The measurement m = (0, 2.5) assigns I to P, I to Q, and T to P  Q, since m  PV  QV • m = (0,4) assigns I to P  Q and I to P and I to Q, since m ⊈PV  QV

  10. Twin Open Set Logic Based on Exact Measurement (Discrete)

  11. Truth Tables for TOPSL

  12. Example • Example to illustrate lack of truth functionality for disjunction P = [(-2, 2), (5,9) ] Q = [(1, 3), (8, 11) ] P  Q = [(-2, 3), (8,9) ] Suppose m = (0, 2.5) m assigns “I” to each of P and Q, “T” to P  Q since m  PV QV Now suppose m = ( 0, 4) m assigns “I” to P  Q as well as to P and to Q since m  (PV QV)

  13. Applications to Billiard Ball Model of Computation

  14. OR-Gate

  15. AND-Gate

  16. NOT-Gate

  17. Derivation Gate Input the value of P and the value of P → Q

  18. Derivation • For a Boolean lattice, define P ≤ Q when P  Q is valid where ≤ is the lattice ordering • Modus Ponens

  19. Three Questions to Consider • What is a proper ordering for the propositions in twin open set logic? • What is a proper implication operator in twin open set logic? • What derivation method can be implemented given the answers to 1 and 2?

  20. Characterization Theorem Let (A, , , ) be a DeMorgan algebra. If we define an ordering ≼ on the algebra by P ≼ Q def P  Q = Q, then P  (P  Q) ≼ Q iff (A, , , ) is a boolean algebra . Reminder: TOPSL is a DeMorgan algebra.

  21. Proof Need: (A, , , ) satisfies the law of non contradiction. In any DeMorgan algebra satisfying our hypothesis, 0 ≼ P  P. Substituting Q = 0 in the modus ponens scheme, P  (P  0) ≼0

  22. Using distributivity, P  (P  0) ≼ 0 (P  P)  (P  0) ≼ 0 Since P  0 = 0 and Q  0 = Q for any Q, P  P ≼ 0 By antisymmetry of ≼, P  P = 0 and so (A, , , ) is boolean.

  23. Implications of the theorem: Any DeMorgan algebra in which • Entailment is given by  (), • The implication operator is given by P  Q, and • Modus ponens is satisfied must be a Boolean algebra.

  24. Non Standard Derivation • TOSPL is a DeMorgan algebra, but not a boolean algebra. • At least one of the three properties above must fail.

  25. Modus Ponens Fails • Ordering for TOSL (suggested by  or ) P  Q  PV  QV and QF  PF motivated by either P  Q = P  PV  QV and QF  PF P  Q = Q  PV  QV and QF  PF Q is more readily verified and less easily falsified than P.

  26. Implication • P→Q def P  Q • So P→Q = P  Q = [(PF  QV),(PV  QF)] • Previous Theorem tells us that modus ponens fails. Why does it?

  27. Theorem: With the ordering given by , it is not the case that P  (P→Q )  Q Proof: P  (P→Q ) = P  (P  Q ) = (P  P)  (P  Q) = [(PV PF), (PVPF)]  [(PV  QV), (PF  QF)] = [, (PVPF)]  [(PV  QV), (PF  QF)] = [(PV  QV),, ((PVPF)  (PF  QF))] = [(PV  QV), ((PV  QF)  PF)]

  28. For P  (P→Q ) ≼ Q, QF  (PV  QF)  PF But whenever PV  PF  X (the whole space), the containment fails. In any nondiscrete topology we have disjoint open sets PV and PF such that PV  PF  X and the claim is established.

  29. Need: a proposition that contains [(PV  QV), ((PV  QF)  PF)] One possibility [QV, ] Given P and P→Q [QV, ]

  30. Good Point: It works. Not so good: So does any proposition of the form [QV, Y] where Y is any open subset of int(QVC) Cannot falsify

  31. Modus Tollens P→Q = P  Q = (Q) P  P = Q → P Consider Q  (P→Q ) = [(PF  QF), ((PV  QF)  QV)] Analog to modus ponens: [PF, ]

  32. Another Possibility • For Modus Ponens: Given P and P→Q, Conclude [QV, PV  QF] =def QP • For Modus Tollens: Given Q  (P→Q ), Conclude [PF, PV  QF] =def PQ Now P  (P→Q ) ≼ QP

  33. Moreover, PV  QV  QV and PV  QF  (PV  QF)  PF thereby respecting entailment

  34. Non Standard Entailment P  Q def Pv Qv • Not antisymmetric, but is reflexive and transitive (a quasi ordering relation) Theorem:  satisfies: P  (P→Q )  Q

  35. What about falsifiability? P  Q def QF PF Does not give a valid modus ponens!

  36. Both Verifiability and Falsifiability Quasi ordering: Reminder: PS = PV PF P ≤ Q def

  37. theorem The quasi ordering defined gives P  (P→Q ) ≤ Q

  38. Non Standard Implication Instead of P→Q = P  Q P ↪ Q =def [PV  QV, QF\ ]

  39. Motivation sup(X | P  X ≤ Q) well defined for any orthonormal lattice. Propositions in TOSL make a lattice, but not orthonormal sup(X | P  X ≲ Q) where X = [XV, XF] and XV = sup(Y | PV Y  QV) and XF = inf(Y | QF PF Y)

  40. To get existence need: PF QV XV This blocks inf(Y | QF PF Y) Leading to XF = (QF \ )

  41. Theorem: P  (P ↪ Q) ≴ Q

  42. Why ? • We get the usual implication operator when considering the discrete twin logic. • Natural interpretation of implication when measurement P verifies P and P ↪ Q, whatever form ↪ may take.

  43. Discoveries • In any derivation scheme that is given by the lattice theoretic entailment, an implication P  Q that is equivalent to P  Q must be Boolean. • Define P  Q = P  Q = [(PF QV), (PV QF)] • m will assign a value of true to P  Q iff m assigns a value of true to either P or Q. i.e., m  (PF QV). • Alternately, m will assign a value of false to P  Q iff m  (PV QF). • m assigns indeterminate to P  Q iff m  (PV QF) and m  (PF QV)

  44. Derivation in Collision Models Replace modus ponens by P and P → Q yield [QV, PV QF] Replace modus tollens by Q and P → Q yield [QV, PV QF]

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