Metalogic: The non-algorithmic side of the mind

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Metalogic: The non-algorithmic side of the mind

Paola Zizzi
paola.zizzi@unipv.it

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Definition of metalogic
Metalogic is the study of the metatheory of logic.

A metatheory is a theory whose subject matter is some other theory. In other
words it is a theory about a theory.
Statements made in the metatheory about the theory are called metatheorems

While logic is the study of the manner in which logical systems can be used to
decide the correctness of arguments, metalogic studies the properties of the
logical systems themselves.

A formal system (also called a logical calculus, or a logical system) consists of a
formal language together with a deductive system.
The latter may consist of a set of transformation rules (inference rules) or a set
of axioms, or have both.

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An interpretation of a formal system is the assignment of meanings, to the
symbols, and truth-values to the sentences of the formal system.

The study of interpretations is called formal semantics. Giving an interpretation
is 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 a
metalanguage (ML).

This distinction is a key difference between logic and metalogic.
While logic deals with proofs in a formal system, expressed in some formal
language (OL), metalogic deals with proofs about a formal system which are
expressed in a metalanguage (ML) about some object language.

In metalogic, 'syntax' has to do with formal languages or formal systems without
regard to any interpretation of them, whereas, 'semantics' has to do with
interpretations of formal languages.

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Classical metalanguage
A metalanguage (ML) is a language which talks about another language, called
object-language (OL).

A (classical) formal ML consists of (classical) assertions, and meta-linguistic
links among them. (By classical assertions, we mean assertions which are stated
with certitude). It consists of:

i) Atomic assertions: − A (A declared, or asserted), where A is a proposition of
the OL.
ii) Meta-linguistic links: − (“yelds”, or “entails”), and (metalinguistic
“and”).
iii) Compound assertions. Example: − A and − B .

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Let us consider the introduction of the (classical) logical connective & in Basic
Logic (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 − B
Definitional equation of the connective & in BL.

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Quantum Metalanguage (QML)
It consists of:
i) Quantum atomic assertions: −λ p
p is a proposition of the quantum object-language (QOL)
λ is a complex number, called the assertion degree, which indicates the degree
of 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]
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( partial truth-value as in Fuzzy Logic).
ii) Meta-linguistic links: − (“yelds”, or “entails”), and (metalinguistic “and”),
as in the classical case.
λ0 λ1
iii) Compound assertions. Example: − p0 and − p1

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n −1

iv) 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 the
quantum connective λ0 & λ1 (connective of quantum superposition).

Definitional equation of λ0 & λ1 :

λ0 λ1
− p 0 λ0 & λ1 p1 iff − p0 and − p1

with the constraint:

λ0 + λ1 =1
2 2

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Tarski “Convention T”

By Tarski Convention T , every sentence p of the object-language (OL) must
satisfy:

(T): ‘p’ is true iff p

where ‘p’ stands for the name of the proposition p, which is the translation in the
metalanguage 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 a
sentence is true if it denotes the existing state of affairs (or, if it is conform to
reality).

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By the point of view of a physicist this would mean that a sentence is true if it
states something that is observable, measurable, computable.
In the classical context of Tarski, a true sentence has truth value 1, which
corresponds 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 of
compound 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 of
assertions and metalinguistic links to recover the definitional equation of the
reflection principle.

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We do so for a precise scope, that is, to show that the mathematician asserting
the truth of the Gödel sentence G in his (non-algorithmic) metalanguage is
operating in Tarski semantic theory of truth, where the material adequacy
condition 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 B

The T-Schema gives:
‘A & B’ true iff ‘A’ true and ‘B’ true

In terms of assertions, we have:
− ' A ' iff A

− ' B ' iff B

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From A and B in the OL, we can form the compound proposition A & B , to
which we apply again Convention T:

(T): − ' ( A & B)' iff A & B

The T-Schema gives:

− A & B iff − A and − B

which is the (classical) definitional equation for the (classical) logical connective
&.

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Convention PT
The quantum case is based on a different kind of Convention T, namely the
Convention “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 a
new formula P (ϕ ) , read “probably ϕ ” and define the truth value of P(ϕ ) to be
the 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 )

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Let 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
i

Then, 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 )) = 1
i =1
i

We can then reformulate Tarski Convention T for any sentences P ( p i ) as
convention PT.
(PT): ‘ p i ’ is probably true iff P( pi ) .

Example: The proposition ‘Snow is white” is probably true if and only if
probably snow is white.

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The expression “is probably true” means that I am asserting the truth of a
sentence with a certain degree of assertion, not with complete certitude.
In terms of assertions, convention PT reads:
λ
− ' pi ' iff P ( pi )
i

which means that proposition ' pi ' is asserted with assertion degree λi if and
only if probably p i , with probability λi ∈ [0,1] .
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From above, it follows that by assigning a probability to a sentence p i of a
classical OL, the corresponding assertion belongs to a QML, that is, the fuzzy
probabilistic proposition P ( p i ) does not belong anymore to the classical OL, but
to a QOL.
The truth-value of P ( p i ) is just the probability of p i
v( P ( pi )) = p ( pi ) = λi
2

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Let 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 ' iff
Let us now form, in the QOL, the new conjunction & λ1 taking into account
λ0

the weights λ0 , λ1 by which the two propositions p 0 , p1 contribute to the
conjunction itself.

We define then:

p 0 λ0 & λ1 p1 ≡ P ( p 0 ) & P ( p1 ) .

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Apply 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 p1

Apply the T-Schema:
λ0
− p 0 λ0 & λ1 p1 iff − p0 and −
λ1
p1

which is the definitional equation for the quantum connective λ0 & λ1 .

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The non-computational mode of the mind

Since Hilbert’s program, all true mathematical statements were assumed to be
provable within the formal axiomatic system.
This assumption was shown to be wrong by Gödel’s First Incompleteness
Theorem for which there exist true statements which are not provable within the
formal 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 program
which can enumerate all the axioms of the system
“Consistent” means that there is no statement of the system, such that both the
statement and its negation are provable from the axioms
“Complete” means that for any statement of the system, either the statement or
its negation are provable from the axioms.

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For any effectively generated, consistent formal system F that includes
arithmetic, 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 recognize
the truth of the Gödel sentence G(F) although the latter is not demonstrable
within 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), while
we 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 be
provable in F, which is absurd, because G(F) states that it cannot be proved in F.

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In the task of recognizing the truth of G(F) the human mind can develop
mathematical insight, or intuition, a property which is not shared by any
algorithmically 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 of
language.

We humans have both the ML by which we give instruction to the machine, and
the 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 the
level of QML, where assertions (with a degree of assertion) live.

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The 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

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Goedel’s Second Incompleteness Theorem.
For any consistent formal system F within which a
certain amount of elementary arithmetic can be
carried out, the consistency of F cannot be proved
in F itself.

The “certain amount of arithmetic” in the second
theorem is not the same we ask for in the first one.
“F is consistent” can be expressed in said systems
thanks to a technique called arithmetization of syntax
that uses a way of representing syntactical objects
such as sentences and proofs as numbers called
Gödel numbering

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The proof of the 1st Incompleteness Theorem establishes in the metalanguage
that 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 the
metalanguage of F, can be formalized within the object language, that is, within
the F system itself. For Peano arithmetic, or any familiar explicitly axiomatized
theory F, it is possible to canonically define a formula C(F) expressing the
consistency of F; this formula expresses the property that "there does not exist a
natural number coding a sequence of formulas, such that each formula is either
of the axioms of F, a logical axiom, or an immediate consequence of preceding
formulas according to the rules of inference of first-order logic, and such that the
last formula is a contradiction".
When formalized, some well formed formula (wff) of F that expresses the
following becomes a theorem:
If the arithmetical theory F is consistent, then G.

C (F ) → G(F )

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But suppose, for reduction ad absurd, that the consistency of F were provable in
the object language for F. Then by modus ponens:

[C ( F ) → G ( F ) ∧ C ( F )] − G ( F )

we'd have a proof of G in F.
Yet we know from the 1st Incompleteness Theorem there is no proof of G if F is
consistent. So, we've arrived at a contradiction, and we must reject our
assumption that the consistency of F is provable in F.
In summary:
Goedel originally stated his second incompleteness theorem: a sufficiently
strong theory F can formalise the argument just given and prove C( F ) → G .

Since F ⊬ G , Gödel concludes that F ⊬ C( F ) .

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In the quantum case, the assertion: − λ G means that G is asserted with an
assertion degree λ ≠ 1.
Or, in other words, G is probably an axiom with probability λ :
2

λ
2

G− G

And the formal system F can prove G with probability 1 − λ :
2

1− λ
2

F− G

For λ = 1 , it reduces to the classical case:

F −0 G That is: F −G
/

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In 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) is
probably provable in F.
This is not anymore a contradiction, because both the truth of G and the
consistency of F are probabilistic.

It follows that C(F) can be probably proven in F, with probability p = 1 − λ .
2

The system F is then probabilistic consistent and probabilistic incomplete.

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The non-algorithmic side of the mind, described by a quantum metalanguage,
where probabilistic assertions operate, is then reflected into a:
Quantum formal system FQ
namely the quantum computational logic of the unconscious, which is
“probably consistent” (sentences like yes&not are allowed with a certain
probability) and
“probably incomplete”, that is, some “probably true” G-sentences can be
probably proven in the system.

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Conclusions
The original conjecture of Penrose about the existence of non-algorithmic
aspects of the mind regarded mainly consciousness.
However, we believe that conscious, rational human thought consists of a very
rapid sequence of decoherence processes from the quantum computational mode
to the classical one.
More specifically, in the Penrose-Hameroff Orch-Or theory, superposed
tubulins/qubits decohere to classical bits at a fast rate.
Accordingly to this theory, it looks like consciousness is made of “flashes” of
classical computation.
Consciousness is not in a persisting classical mode of the mind, because that
would lead to an absurd conclusion: A classical Turing machine, which persists
in the classical mode, would be more “conscious” than a human mind.
The problem is that the static conscious state of a classical computer is totally
useless for any kind of aware reasoning, which is dynamical by definition. It is
the never-ending supply of new data coming from decoherence, which makes
the difference. What is really non-algorithmic, is the origin of consciousness in
the quantum metalanguage, not consciousness itself.

Metalogic: The non-algorithmic side of the mind

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