1Metalogic: The non-algorithmic side of the mind Paola Zizzi firstname.lastname@example.org
2Definition of metalogicMetalogic is the study of the metatheory of logic.A metatheory is a theory whose subject matter is some other theory. In otherwords it is a theory about a theory.Statements made in the metatheory about the theory are called metatheoremsWhile logic is the study of the manner in which logical systems can be used todecide the correctness of arguments, metalogic studies the properties of thelogical systems themselves.A formal system (also called a logical calculus, or a logical system) consists of aformal language together with a deductive system.The latter may consist of a set of transformation rules (inference rules) or a setof axioms, or have both.
3An interpretation of a formal system is the assignment of meanings, to thesymbols, and truth-values to the sentences of the formal system.The study of interpretations is called formal semantics. Giving an interpretationis synonymous with constructing a model.In metalogic, formal languages are sometimes called object languages (OL).The language used to make statements about an object language is called ametalanguage (ML).This distinction is a key difference between logic and metalogic.While logic deals with proofs in a formal system, expressed in some formallanguage (OL), metalogic deals with proofs about a formal system which areexpressed in a metalanguage (ML) about some object language.In metalogic, syntax has to do with formal languages or formal systems withoutregard to any interpretation of them, whereas, semantics has to do withinterpretations of formal languages.
4Classical metalanguageA metalanguage (ML) is a language which talks about another language, calledobject-language (OL).A (classical) formal ML consists of (classical) assertions, and meta-linguisticlinks among them. (By classical assertions, we mean assertions which are statedwith certitude). It consists of:i) Atomic assertions: − A (A declared, or asserted), where A is a proposition ofthe OL.ii) Meta-linguistic links: − (“yelds”, or “entails”), and (metalinguistic“and”).iii) Compound assertions. Example: − A and − B .
5Let us consider the introduction of the (classical) logical connective & in BasicLogic (BL)In the OL, let A, B be propositions.In the ML, I read: A decl. , B decl, that is: −A , −B respectively.Let us introduce a new proposition A&B in the OL.In the ML, we will read: A&B decl., that is: − A & B .The question is: From A &B decl., can we understand A decl. and B decl. ?More formally, from − A & B can we understand − A and − B ?To be able to understand A decl. and B decl. from A&B decl, we should solve: −A& B iff − A and − BDefinitional equation of the connective & in BL.
6Quantum Metalanguage (QML)It consists of:i) Quantum atomic assertions: −λ pp is a proposition of the quantum object-language (QOL) λ is a complex number, called the assertion degree, which indicates the degreeof certitude in stating the assertion.In the limit case λ = 1 , quantum assertions reduces to classical ones.The truth-value of the corresponding proposition p in the QOL, is given by:v ( p ) = λ ∈ [0 ,1] 2( partial truth-value as in Fuzzy Logic).ii) Meta-linguistic links: − (“yelds”, or “entails”), and (metalinguistic “and”),as in the classical case. λ0 λ1iii) Compound assertions. Example: − p0 and − p1
7 n −1iv) Meta-data: ∑ v( p ) = 1 i =0 i (n= number of propositions in the QOL)As in the classical case, one should solve the definitional equation of thequantum connective λ0 & λ1 (connective of quantum superposition).Definitional equation of λ0 & λ1 : λ0 λ1− p 0 λ0 & λ1 p1 iff − p0 and − p1with the constraint:λ0 + λ1 =1 2 2
8Tarski “Convention T”By Tarski Convention T , every sentence p of the object-language (OL) mustsatisfy:(T): ‘p’ is true iff pwhere ‘p’ stands for the name of the proposition p, which is the translation in themetalanguage ML of the corresponding proposition in the OL.The standard example is:‘Snow is white’ is true iff snow is white.Convention T is also called “material adequacy condition”, in the sense that asentence is true if it denotes the existing state of affairs (or, if it is conform toreality).
9By the point of view of a physicist this would mean that a sentence is true if itstates something that is observable, measurable, computable.In the classical context of Tarski, a true sentence has truth value 1, whichcorresponds to probability 1 in the measurement procedure.But this state of affairs changes when we deal with a quantum metalanguage.Tarski T-Schema (equivalence schema) allows to state inductively the truth ofcompound propositions.For example, for the conjunction A&B of two propositions A and B, the T-Schema gives:‘A & B’ is true iff A is true and B is true.There is a close relation between the concepts of assertion and truth.Then, we will “translate” Tarski Convention T and T-Schema in terms ofassertions and metalinguistic links to recover the definitional equation of thereflection principle.
10We do so for a precise scope, that is, to show that the mathematician assertingthe truth of the Gödel sentence G in his (non-algorithmic) metalanguage isoperating in Tarski semantic theory of truth, where the material adequacycondition hold.The classical case:Let us apply Convention T to the two sentences A and B of the object-language:(T): ‘A’ true iff A(T): ‘B’ true iff BThe T-Schema gives:‘A & B’ true iff ‘A’ true and ‘B’ trueIn terms of assertions, we have: − A iff A− B iff B
11From A and B in the OL, we can form the compound proposition A & B , towhich we apply again Convention T:(T): − ( A & B) iff A & BThe T-Schema gives:− A & B iff − A and − Bwhich is the (classical) definitional equation for the (classical) logical connective&.
12 Convention PTThe quantum case is based on a different kind of Convention T, namely theConvention “Probably” T.The fuzzy notion probably can be axiomatized as a fuzzy modality.Having a probability on Boolean formulas, define for each such formula ϕ anew formula P (ϕ ) , read “probably ϕ ” and define the truth value of P(ϕ ) to bethe probability of ϕ :v( Pϕ ) = p(ϕ ) ∈ [0,1].Let us consider a set S of N atomic Boolean propositions of OL:ψi ( i =1, 2 ,...... N )
13Let us call p i ( i =1, 2....... n ) with n < N , the propositions of a subset S⊂ S ,to which it is possible to assign a probability p such that: n ∑ p( p ) = 1 . i =1 iThen, we can define n new propositions P ( p i ) , for which it holds:v( P ( pi )) = p ( pi ) ∈ [0,1] .And it follows: n∑ v( P( p )) = 1i =1 iWe can then reformulate Tarski Convention T for any sentences P ( p i ) asconvention PT.(PT): ‘ p i ’ is probably true iff P( pi ) .Example: The proposition ‘Snow is white” is probably true if and only ifprobably snow is white.
14The expression “is probably true” means that I am asserting the truth of asentence with a certain degree of assertion, not with complete certitude.In terms of assertions, convention PT reads: λ − pi iff P ( pi ) iwhich means that proposition pi is asserted with assertion degree λi if andonly if probably p i , with probability λi ∈ [0,1] . 2From above, it follows that by assigning a probability to a sentence p i of aclassical OL, the corresponding assertion belongs to a QML, that is, the fuzzyprobabilistic proposition P ( p i ) does not belong anymore to the classical OL, butto a QOL.The truth-value of P ( p i ) is just the probability of p iv( P ( pi )) = p ( pi ) = λi 2
15Let us consider two probabilistic propositions p 0 , p1 of the OL.From Convention PT: P( p 0 ) with: v ( P ( p 0 )) = p ( p 0 ) = λ0 λ0− p0 iff 2 P( p1 ) with: v( P( p1 )) = p ( p1 ) = λ1 2 λ1− p1 iffLet us now form, in the QOL, the new conjunction & λ1 taking into account λ0the weights λ0 , λ1 by which the two propositions p 0 , p1 contribute to theconjunction itself.We define then:p 0 λ0 & λ1 p1 ≡ P ( p 0 ) & P ( p1 ) .
16Apply Convention T to the new formed proposition P( p0 ) & P( p1 ) :− P( p0 ) & P( p1 ) iff P( p 0 ) & P( p1 ) .That is:− p0 λ0 & λ1 p1 iff p 0 λ 0 & λ1 p1Apply the T-Schema: λ0− p 0 λ0 & λ1 p1 iff − p0 and − λ1 p1which is the definitional equation for the quantum connective λ0 & λ1 .
17The non-computational mode of the mindSince Hilbert’s program, all true mathematical statements were assumed to beprovable within the formal axiomatic system.This assumption was shown to be wrong by Gödel’s First IncompletenessTheorem for which there exist true statements which are not provable within theformal system.Gödel’s First Incompleteness Theorem states that: Any effectively generated formal system capable of expressing arithmetic,cannot be both consistent and complete. “Effectively generated” means that in principle there exist a computer programwhich can enumerate all the axioms of the system “Consistent” means that there is no statement of the system, such that both thestatement and its negation are provable from the axioms“Complete” means that for any statement of the system, either the statement orits negation are provable from the axioms.
18For any effectively generated, consistent formal system F that includesarithmetic, there is a statement which is true, but not provable within the theory.Such a statement is called the Gödel sentence G(F).The Penrose’s conjecture states that some aspects of the mind have a non-algorithmic nature in relation with Gödel’s First Incompleteness Theorem.Penrose bases his conjecture on the fact that the human mind is able to recognizethe truth of the Gödel sentence G(F) although the latter is not demonstrablewithin the axiomatic system.The Gödel sentence G(F) is:G(F)= “This sentence cannot be proved in F”.Penrose says that the First Incompleteness Theorem tells us that no computer,working within a consistent formal system F can prove the sentence G(F), whilewe humans can “see” the truth of G(F).In fact, we “see” that G(F) is true, because, if it were false, then it would beprovable in F, which is absurd, because G(F) states that it cannot be proved in F.
19In the task of recognizing the truth of G(F) the human mind can developmathematical insight, or intuition, a property which is not shared by anyalgorithmically based system of logic.We suggest: Mathematical intuition is described by a quantum metalanguage, which is non-algorithmic.As a metalanguage cannot be given to a machine, which uses only the object-language, it is obvious that humans and machines have different levels oflanguage.We humans have both the ML by which we give instruction to the machine, andthe OL, already contained in the ML., while machines can utilize only the OL.In particular, QML organizes and controls our own QOL. When the mathematician asserts the truth of G(F), in fact he is operating at thelevel of QML, where assertions (with a degree of assertion) live.
20The mathematician asserts: λ− G (F )This is equivalent to Convention PT:(PT): ‘G(F)’ is probably true iff P(G(F))with:v ( P (G ( F ))) = p (G ( F )) = λ 2
21Goedel’s Second Incompleteness Theorem.For any consistent formal system F within which acertain amount of elementary arithmetic can becarried out, the consistency of F cannot be provedin F itself.The “certain amount of arithmetic” in the secondtheorem is not the same we ask for in the first one.“F is consistent” can be expressed in said systemsthanks to a technique called arithmetization of syntaxthat uses a way of representing syntactical objectssuch as sentences and proofs as numbers calledGödel numbering
22The proof of the 1st Incompleteness Theorem establishes in the metalanguagethat if the arithmetical theory F is consistent, then G(F) is not provable.G(F) "says" it is not provable in F. So, if the theory F is consistent, then G.But the previous reasoning of the 1st Incompleteness theorem, which is all in themetalanguage of F, can be formalized within the object language, that is, withinthe F system itself. For Peano arithmetic, or any familiar explicitly axiomatizedtheory F, it is possible to canonically define a formula C(F) expressing theconsistency of F; this formula expresses the property that "there does not exist anatural number coding a sequence of formulas, such that each formula is eitherof the axioms of F, a logical axiom, or an immediate consequence of precedingformulas according to the rules of inference of first-order logic, and such that thelast formula is a contradiction".When formalized, some well formed formula (wff) of F that expresses thefollowing becomes a theorem:If the arithmetical theory F is consistent, then G.C (F ) → G(F )
23But suppose, for reduction ad absurd, that the consistency of F were provable inthe object language for F. Then by modus ponens:[C ( F ) → G ( F ) ∧ C ( F )] − G ( F ) wed have a proof of G in F.Yet we know from the 1st Incompleteness Theorem there is no proof of G if F isconsistent. So, weve arrived at a contradiction, and we must reject ourassumption that the consistency of F is provable in F.In summary:Goedel originally stated his second incompleteness theorem: a sufficientlystrong theory F can formalise the argument just given and prove C( F ) → G .Since F ⊬ G , Gödel concludes that F ⊬ C( F ) .
24In the quantum case, the assertion: − λ G means that G is asserted with anassertion degree λ ≠ 1.Or, in other words, G is probably an axiom with probability λ : 2 λ 2 G− GAnd the formal system F can prove G with probability 1 − λ : 2 1− λ 2 F− GFor λ = 1 , it reduces to the classical case: F −0 G That is: F −G /
25In the quantum case, then, F is probably consistent, and probably incomplete.In terms of the modality P = “probably”The sentence P (C ( F ) → P (G ( F ))Gives, by modus ponens and by the use of the T-schema:P[(C ( F )) → (G ( F )) ∧ (C ( F ))] − PG( F )If C(F) is probably true, and is probably true that C(F) implies G, then G(F) isprobably provable in F.This is not anymore a contradiction, because both the truth of G and theconsistency of F are probabilistic.It follows that C(F) can be probably proven in F, with probability p = 1 − λ . 2The system F is then probabilistic consistent and probabilistic incomplete.
26The non-algorithmic side of the mind, described by a quantum metalanguage,where probabilistic assertions operate, is then reflected into a:Quantum formal system FQnamely the quantum computational logic of the unconscious, which is“probably consistent” (sentences like yes¬ are allowed with a certainprobability) and“probably incomplete”, that is, some “probably true” G-sentences can beprobably proven in the system.
27ConclusionsThe original conjecture of Penrose about the existence of non-algorithmicaspects of the mind regarded mainly consciousness.However, we believe that conscious, rational human thought consists of a veryrapid sequence of decoherence processes from the quantum computational modeto the classical one.More specifically, in the Penrose-Hameroff Orch-Or theory, superposedtubulins/qubits decohere to classical bits at a fast rate.Accordingly to this theory, it looks like consciousness is made of “flashes” ofclassical computation.Consciousness is not in a persisting classical mode of the mind, because thatwould lead to an absurd conclusion: A classical Turing machine, which persistsin the classical mode, would be more “conscious” than a human mind.The problem is that the static conscious state of a classical computer is totallyuseless for any kind of aware reasoning, which is dynamical by definition. It isthe never-ending supply of new data coming from decoherence, which makesthe difference. What is really non-algorithmic, is the origin of consciousness inthe quantum metalanguage, not consciousness itself.