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1 INCOHERENT QUANTUM META-LANGUAGE AND SCHIZOPHRENIA Paola Zizzi “Paolo Sotgiu” Institute for research in Quantitative & Quantum Psychiatry & Cardiology L.U.de.S. of Lugano, Switzerland paola.zizzi@uniludes.ch A Long Shadow over the Soul:Molecular and Quantum Approaches to Psychopathology An Interdisciplinary Dialog with Psychiatrists FANO - March 2012
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2CONTENTS:1. Three computational modes of the mind2. The logic of human reasoning3. Quantum Meta-Language4. The probabilistic identity axiom and disintegration of the Self5. Quantum coherent states of the mind Conclusions References
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31. Three computational modes of the mind(Zizzi, 2012)We make the formal distinction between ordinary thought and meta-thought:Ordinary thought: i) conscious – classical computation, classical formal language ii) unconscious -quantum computation, quantum formal languageMeta-thought: unconscious, non-algorithmic, Quantum Meta-Language (QML)(Zizzi 2010).We call the MIND items i) and ii). MIND ii) is the Quantum Mind a) The quantum mode b) The classical mode c) The non-algorithmic mode
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4a) The quantum modeThe unconscious ordinary thought: driven by mental processes which are extremely fast,much more than those concerning the conscious thought.This already suggests that the above processes are quantum-computational(a quantum computer is exponentially faster than its classical counterpart).The sudden decision-makings or understandings, creativity, imagination and discoveriesArising from an unconscious state of the mind, are just the results of aquantum mental process, whose intermediate steps, however, remain unknowable.In quantum computing: one can get the result of a computationwith a given probability, but the intermediate steps are not available.Then, these two features seem to indicate that the unconscious mind is indeed quantum-computational. The Quantum Mind.b) The classical mode
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5The unconscious mind computes in the quantum mode, and it “prepares”,at highest speed, what we then recognize as a conscious thought.The conscious thought derives from a choice (a projective measurement)made on the quantum computational state, and thereafter uses a classical mode..We don’t have much time to re-elaborate the outputs of the unconscious mind(half a second) then, our conscious thought looks more like a succession of flashes ofconsciousness rather than a proper classical computation.We use the partial information obtained from quantum measurements.But in effect, we do not compute anything new.In fact, most of the time, humans compute in a quantum mode..c) The non-algorithmic mode
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6Meta-thought is the process of thinking about our own thought.It has no computational mode, neither classical, nor quantum.Quantum meta-thought, which thinks about the quantum, unconscious thought,can be viewed as the roots of the unconscious mind (or Quantum Mind)It is the aspect of thought most closely related to matter(physical processes in the brain).The latter are supposed to be described by a Dissipative Quantum Field Theories(DQFT) of the brain (Vitiello 2001).Quantum Meta-thought co-ordinates intuition, intentions, and (quantum) control.Meta-thought processes could be interpreted as aiming to keep some sort of coherence inordinary thought (coherent states in DQFT).2. The logic of human reasoning
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7The study of the “natural” logic of mental processes is very important in thecontext of a constructivist approach to logic.Constructivism: logic is not pre-existent in the Platonic “world of ideas”,but instead is a by-product of the mind.In this regard, one can adopt either the microscopic or the macroscopic point of view.Macroscopic point of view: Phenomenology of “thought processes” (Cognitive Science) Philosophy of Mind Dynamical constructivism (Sambin, 2002), cognitive/social interpretation of mental processes.Microscopic point of view: focuses on the quantum processesoccurring in the brain, and can be formalized by Quantum Theory.Classical laws of thought :Law of identity: A→ A ( states that an object is the same as itself)
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8Law of excluded middle: ( A ∨ ¬ A) = 1 (A or non A is true)Law of non-contradiction: ( A ∧¬ A) = 0 ( A and non A is false)The law of identity partitions the Universe into two parts: the Self and the Other Universe The Other The Self It is a dichotomy: a partition of the whole into two parts that are: Mutually exhaustive S ∪O =U → (excluded middle) Mutually exclusive S IO = Ø → (non-contradiction) The Other is the complement of the Self in U.However, we claim that human logic is not Aristotelian classical logic.The latter is in fact: i) too abstract (the resources-the premises-can be used as many times as one likes). ii) too much structured
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9 (has structural rules-contraction-weakening-exchange)In sequent calculus notation: − turnstile (or yelds) , Γ,∆... contexts A,B...propositions Γ , AA − ∆Contraction: Γ , A − ∆ (data can be copied) Γ−∆Weakening: Γ, A − ∆ (data can be deleted) Γ , A, B , Γ − ∆Exchange: Γ , B , A, Γ − ∆Instead the mind operates in a very simple way at the fundamental level.Then the mind must be described by a weaker (that is, with fewer structural rules) and less abstract logic than classical logic, which should, nevertheless, have extensions(among which, classical logic).Macroscopic point of view: Basic Logic (Sambin et al., 2000) is the best choice for the logic of the conscious, classical MIND.Basic Logic: Substructural (lacks the structural rules of weakening and contraction)
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10 Symmetry: (every connective has its dual) The law of identity hold: A −A Γ − A A−∆ The cut rule holds: Γ −∆ (this is a meta-rule) The principle of the excluded middle does not hold(As in Intuitionist logic, proving the excluded middle would require producing proof forthe truth or falsity of all possible statements, which is impossible) By symmetry, the principle of non-contradiction does not hold as well Visibility: all active formula are isolated from the context Reflection: metalinguistic links between assertions of the metalanguage are reflected into logical connectives between propositions of the object-language through the definitional equationThe BL Metalanguage consists of:Metalinguistic links: − (yelds or entails), andAtomic assertions: −A (proposition A “is”),
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11Compound assertions. Example: − A and − BObject-language consists of:Propositions A, B,….Logical connectives &, ,→ ←⊗,℘ ∨ , , (and, or, implies, counter-implies, times, par).The connectives ⊗,℘ , borrowed from Linear logic, are the multiplicative conjunction and disjunctionrespectively.The negation ¬ A is an abbreviation for: A →⊥ (A implies the false)The double negation can be introduced: A → ¬¬ ABut the elimination ¬¬ A → A does not hold.Definitional equation for the connective &:Γ − A& B iff Γ −A and Γ −BMicroscopic point of view: Logic Lq (Zizzi, 2010) Symmetry, as in BL Visibility, as in BL Reflection principle, as in BL
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12 The law of the excluded middle does not hold The law of non-contradiction does not hold Is many-valued (fuzzy) and probabilistic The law of identity holds probabilistically None of the structural rules holds The (quantum) cut rule holdsTwo new connectives: Quantum superposition λ0 &λ1 Quantum entanglement @ = f (℘,&)Fuzzy probabilistic propositions in LqThere is a relation between the probability p and the fuzzy notion probably(H’ajek et al, 1995).In fact, the latter can be axiomatized as a fuzzy modality. Having a probability p on Boolean formulas, define for each such formula p i
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13a new formula P( pi ) , read “probably p i ”, and define the truth value ofP ( p i ) to be the probability of p i :v( P( pi )) = p( pi ) ∈ [0,1] .Then, it holds: n ∑ v( P( p )) = 1 i =1 iwhere n is the number of atomic propositions p i3. Quantum MetalanguageMetalinguistic sequents with no antecedent −A
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14are assertions in the (classical) metalanguage ML.There is a close relation between assertions, the truth values of propositions in the OL and the truthpredicate of Tarski, T, which is formulated in ML (Tarski (1944)).Tarski Convention T:By Tarski Convention T, every sentence p of OL must satisfy:(T): ‘p’ is true iff p‘p’ stands for the name of the proposition p, which is the translation in the metalanguage ML, of the corresponding proposition in the OLor:(T): −p iff pT-schema (inductive definition of truth): a sentence of the form "A and B" is true if and only if A is true and B is trueIn the formalism of sequent calculus: − A & B iff − A and − Bwhich is nothing else than the definitional eq. of & in BL
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15Convention PTWhen the certitude in the assertion is not full, also the truth values of the propositions arepartial, and Convention T must be modified as Convention PT:(PT): ‘p’ is probably true iff P( p)In sequent notation: λ(PT): − p iff P( p)The proposition p is asserted with assertion degree λ if and only if“probably” p, with probability λ ∈ [0,1] , and the partial truth value of P ( p ) 2is just the probability of p :v ( P ( p )) = p ( p ) = λ 2Given p 0 , p1 of the QOL: λ0(PT): − p0 iff P( p 0 ) λ1(PT): − p1 iff P( p1 )with: v ( P ( p 0 )) = p ( p 0 ) = λ0 , v ( P ( p1 )) = p ( p1 ) = λ1 2 2
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16Apply T-schema:P( p 0 ) & P( p1 ) is true iff P( p 0 ) is true and P( p1 ) is true.Define: P( p 0 ) & P( p1 ) ≡ p0 λ &λ p1 0 1In the sequent formalism: − p0 λ0 & λ1 p1 iff − λ0 p0 and − λ1 p1Definitional equation for the connective λ0 & λ1 “quantum superposition”The QML consists of the same metalinguistic links of the classical ML, but quantum assertions areasserted with an assertion degree, which is a complex number, λ interpreted as a probabilityamplitude. λWe physically interpret the atomic quantum assertions − pi i as quantum field states of theDQFT of the brain.
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174. The probabilistic identity axiom and the disintegration of the SelfIn the classical case: −A proposition A is true( − A) ⊥ = A − proposition A is false ( ⊥ Sambin-Girard duality ).The identity axiom:A − A stays for: ( − A ) − ( − A )In the quantum case: λ − A A is (probably) true for λ = 1 it reduce sto the classical case − A(− A ) λ ⊥ = A− λ* A is (probably) false ( λ is the complex coniugate of * λ)gluing operator o:
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18 λ* λA− o − A gives: λ 2 A− A Probabilistic identity axiomStates that an object is probabilistically the same of itself.In particular: The Self is probabilistically the same of itself. (Disintegration of the Self in schizophrenia).Truth-value of the “Probably” Self:vS = 1 − vO where vO is the truth of the OtherIn the classical case, it was:A in S, ¬ A in Onow we have A ∧ ¬ A true in S ∩ O.The non validity of the law of non-contradiction here
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19is not just a philosophical choice as in Intuitionist logic and in BL,but follows from the probabilistic nature of the identity axiom.As the dichotomy is lost, A and its negation are no more mutually exclusive.This is a feature of para-consistent logic.In the logic of the quantum mind, a para-consistent logicallows for quantum superpositions of bits 0 and 1, and then, for quantum computation.5. Quantum coherent states of the mindConsider:A set S of N atomic Boolean propositions:ψi ( i =1, 2 ,...... N )and
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20 pi ( i =1, 2....... n ) with n < N , propositions of a subset S⊂ S , nto which it is possible to assign a probability p such that ∑ p( p ) = 1 . i i =1Then, it holds: n∑ v( P( p )) = 1i =1 iCall: ϕ i ( i = n +1,........ n + r = N ) the remaining r = N − n Boolean propositionsto which it was not assigned a probability.Assuming the propositions ϕi true, with full truth value1, it holds:N =n+r ∑ v(ϕ ) = r i = n +1 iIt follows: N∑ v(ψ i =1 i ) = 1+ rIn the limit case where all the propositions had the same truth valuev ⋅ N = 1+ rUncertainty relation for quantum logical propositions: (Zizzi, 2012)
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21“It is impossible to fully determine both the truth values of the propositionsbelonging to a set S, and the power of S”:∆v ⋅ ∆N ≥ 1For r = 0, N = n :The uncertainty relation is saturated:∆v ⋅ ∆n = 1 v= λ 2As it is: , the uncertainty of the truth value ∆v can be expressedin terms of the uncertainty of the assertion degree ∆λ as: ∆v = 2 λ ∆λ ,The uncertainty relation: k∆λ ⋅ ∆N ≥ 2
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22 k 1 ∆λ ⋅ ∆ n = with: k= 2 λ .reminds of the uncertainty relation phase-number of quantum optics,which is saturated by coherent Glauber states (Glauber, 1963).Definition:“Quantum-coherent atomic propositions are those fuzzy- probabilistic atomic propositions whose partial truth values are all equal and sum up to1”.Physical interpretationQuantum coherence in QFT is a property of some particular quantum field states, the coherentstates α , which are eigenvectors ofthe annihilation operator a , with eigenvalues α:aα =α α .As the operator a is non-hermitian, the eigenvalue α is in general a complex number.A coherent state α is itself a superposition of states, in the Fock basis { n }:
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23Coherent states are the “most classical” among quantum states, and are very robust againstdecoherence.In quantum optics, Glauber coherent states minimize the phase-number uncertainty 1relation: ∆ϕ ⋅ ∆n = 2The metalogical equivalent of a coherent state is the ensemble of sequents: α− pi with: α ∈C , i = 1,………nNotice:
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24The superposition of two coherent states α α +β β is not, in general a coherent state, 1unless α = β = . 2The resulting coherent state: 1 ( α + −α ) 2is called the “cat” coherent state in quantum optics.Cat states are very fragile against decoherence.Compound quantum-coherent propositionsThe identification p 0 λ0 & λ1 p1 = P( p 0 & p1 ) 1can be made only in the particular case with λ0 = λ1 = 2 :P( p 0 & p1 ) ≡ p0 1 & 1 p1 2 2
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25In this particular case, we can apply Convention PT : 1 λ=− 2 ( p0 & p1 ) iff P ( p0 & p1 )A compound quantum-coherent proposition is a “cat state” proposition.A general, incoherent state is, in the Fock basis:ψ = ∑ λn n nThis corresponds to the n-ple of incoherent atomic assertions: λi − pi i = 1,….n Two kinds of quantum metalanguagesCoherent QML Incoherent QML α λi− pi − pi
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26 1 1 λ0 λ1− p0 2 and − p1 2 − p0 and − p1ConclusionsIt is a fact that an healthy mind is capable of a rational abstract thought,which is coherent and logical.Then, such a mind process should come from a coherent metalanguage,which should also provide a logic.Such a metalanguage has coherent “cat” assertions.In the physical model, such coherent cat states are very fragile against decoherence,which means that the passage to the classical mode of consciousness is very fast,
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27as it should be.Otherwise, the mind would remain for too long trapped in the quantum mode(the inconscious).On the other side, an healthy mind should also be capable of a creative,unpredictable, fuzzy thought.This kind of thought would be incoherent, perhaps disorganized,but nevertheless would be ruled by a particular kind of logic.It correspond to an incoherent quantum metalanguage.This metalanguage provides the logic of quantum information Lq which isthe logic of the quantum mode of the mind, the unconscious.For some reasons, in schizophrenia, the incoherent QML is dominanton the coherent QML (work in progress).
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28 Schizophrenia The healthy mind The Double IQML= The Double IQML CQML CQML HallucinationsReferences:
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29P. Zizzi, “When Humans Do Compute Quantum”,in: A Computable Universe, Hector Zenil (Ed), Word Scientific Publishing (2012).P. Zizzi, “From Quantum Metalanguage to the Logic of Qubits”.PhD Thesis, arXiv:1003.5976 (2010).G. Sambin , G. Battilotti, C. Faggian, “Basic logic: reflection, symmetry, visibility”.The Journal of Symbolic Logic, 65, 979-1013 (2000).G. Sambin, “Steps towards a dynamical constructivism”. In the scope of Logic, Methodology and Philosophy of Science, Vol.1, 263-289. Kluwer, Dordrecht (2002).G. Vitiello. My double unveiled. Amsterdam: Benjamins (2001).H´ajek P., Godo L., Esteva F., Probability and Fuzzy Logic.In Proc. Of Uncertainty in Artificial Intelligence UAI’95, (Besnard and Hanks, Eds.)Morgan Kaufmann. San Francisco, 237–244 (1995).A. Tarski, “The semantic conception of truth”.Philosophy and Phenomenological Research, 4, 13-47 (1944).R. J. Glauber, “Coherent and incoherent states of radiation field”,Phys. Rev. 131, 2766-2788 (1963).P. Zizzi, “The Uncertainty Relation for quantum Propositions”,arXiv:1112.2923 (2012), submitted to IJTP.
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