QUANTUM LOGIC

And the Complexification of the

World

CARLOS EDUARDO MALDONADO

Full Professor

Universidad del Rosario

The Copenhaguen Debate

Bohr

• Heisenberg

• Dirac

Einstein

• Born

• Pauli

De BroglieSchrödinger

The Copenhaguen Debate

Indeterminism

• Complementarity

• Uncertainty

Determinism

• Quantum Mechanics

• Realism

Schrödinger´s

Paradox

Hidden

Variables

THE ORIGINS OF QL

BIRKHOFF

• Father of universal algebra

• (Birkhoff Theorem)

• Lattice theory

• Hidrodynamics

VON NEUMANN

• Game Theory

• Arquitecture von VN

• Sets theory

• Economics

• Cibernetics

• Manhattan Project

• …

QUANTUM LOGIC

• Main concern: Quantum logic is the logic ofempirical world, very much in the same way as quantum mechanics states

• Antecedent: Quantum mechanics, the theory ofrelativity, the principle of uncertainty

• Origin and development: Birkhoff and vonNeumann (1936) ; semiinterpreted language (Van Fraasen, 1970); Kripke´s semantics ; ortho-logic(which is a translation of a weak quantum logics) by Goldblatt (1977)

THE GATE TO QL • “The Logic of QM”, in: Annals of Mathematics, 37: 823-843, 1936

• “The object of the presentpaper is to discover whatlogical structure we mayhope to find in physicaltheories that, like QM, do not conform to classicallogic”.

The most

fundamental

problems that

QM raises are

conceptual in

nature

THE CONCERNS FOR QUANTUM LOGIC

• Establish a conceptual coherence

between different things “quantum”:

- (Quantum) Logic

- (Quantum) Probability

- (Quantum) Mechanics

CLASSICAL LOGIC

• 1847: G. Boole publishes The Mathematical Analysis of

Logic, and simultaneously, A. De Morgan publishes

Formal Logic

• 1879: G. Frege publishes el Begriffschrift (Notación

conceptual)

• 1899: D. Hilbert introduces in his Grundlegung der

Geometrie the concept of metamathematics

• 1933-1936: Tarski, The Concept of Truth in Formalized

Languages, and On te Concept of Logical Consequence

AND YET, WHAT IS LOGIC?

• Valid inference

(entailment)

(1930s)

• Definability:

Language and its

expressive power

• Computation

• Proof Theory

• Model Theory

• Recursion Theory

LOGICS AS SCIENCE

• 1847: G. Boole: The mathematical analysis of logic,

and contemporarily A. De Morgan: Formal Logic

• 1879: G. Frege: Begriffschrift

• 1899: D. Hilbert introduces the concept of

metamathematics in his The Foundations of

Geometry

• 1933-1936: Tarski, The Concept of Truth in

Formalized Logics, and Concerning the Concept of

Logical Consequence

IS LOGIC EMPIRICAL?

• Putnam (1968):

• Revise our classical notions in favour of “quantum

logical” ones;

• The revision of logic is not merely local, but it is truly

global. Quantum logic is the ‘true’ one (just as the

‘true’ geometry is the non-Euclidean)

• Recognizsing that logic is thus quantum solves the

Schrödingers cat problem

(IN)FALLIBILISM IN LOGIC

• The critique to absolute certainty in

knowledge

(Ch. S. Peirce, Dewey, [Popper], Quine) �

Gödel

• I. Berlin, B. Williams (in ethics)

(Brakets)

1) A � B ∧ C

(Brakets)

2) A � B ∨C

• (A � B) ∨C

• A � (B ∨C)

AND WHAT ABOUT CLASSICAL LOGIC?

• Classical logic is sometimes described as

the logic of an omniscient mind in a

deterministic universe

• Any problem is semantic decided

• Meanings behave in a compositional way

• Meanings are sharp and unambiguous

CL

• It is to be understood as classical

two-valued propositional logic,

• Or, predicate logic of first order with

identity

QUANTUM LOGIC

• Birkhoff and Von Neumann conceive of QL as

a weak (weaker) logic (rather) than classical

formal logic

• It can be formulated as a modified version of

propositional logic, or also as al

noncommutative and non-associative many-

valued logic

Lógicas No Clásicas(Lógicas filosóficas)

Lógicas ProbabilísticasLógicas ProbabilísticasLógicas No MonotónicasLógicas No Monotónicas

Lógica libre

Lógica de

fabrica

Lógica

paraconsistente

Lógica de la

relevancia

Lógica del

tiempo

Lógica cuántica

LÓGICAS NO CLÁSICAS

Lógica formal

clásica

Lógica difusaLógicas

polivalentes

Lógica

epistémica

Lógica de

contrafácticos

Lógica modal

Lógica

deóntica

Extensiones a la lógica clásica

Alternativas a la lógica clásica

Lógicas No Clásicas(Lógicas filosóficas)

Lógicas ProbabilísticasLógicas ProbabilísticasLógicas No MonotónicasLógicas No Monotónicas

Lógica libre

Lógica de

fabrica

Lógica

paraconsistente

Lógica de la

relevancia

Lógica del

tiempo

Lógica cuántica

Lógica formal

clásica

Lógica difusaLógicas

polivalentes

Lógica

intuicionista

Lógica de

contrafácticos

Lógica modal

Lógica

deóntica

Extensiones a la lógica clásica

Alternativas a la lógica clásica

Lógica

dinámica

Lógica

multimodal

Lógica

abductiva

Lógica

epistémica

(Brakets 2)

• Problem: Popper:

• Weaker logics � Weaker testability strategies

• � A change in the validity concept

THE SEMANTICS OF NCLs

• The semantics of NCLs is the notion

of the semantic of possible worlds

(whereas the semantics of formal

classical logic is the semantics of the

real world, or also, of the world in

general (überhaupt))

QL LOGIC AND LOGICAL PLURALISM

• Logic is about anything, and it can be applied

to a number of fields

• QL means weakening clasical logic, i.e. first-

order logic

• The whole framework is the call for alternative

logics

THE PROBLEM!

• The interpretation of Qphenomena

• Discrete versus Continuum

• Foundational studies on QL and QP. For, in

complexity there is no room any longer to

foundations, as key cornerstones that once

shape the evolution of a system

A DISCRETE WORLD

• QPhysics � Energy (non continuous)

E = m c2

• Information theory � 1 / 0

• Qinformation theory 1 and 0

QUANTUM LOGIC AND QP

• It is the most general, universal logic of

physical propositions

• And yet, what is physics? Thanks to QP, physics

is not anymore about what is reality, but

about what do we know about reality

• Wave collapse

• QM is formulated in terms of classical logic,

but can it give rise to a new nonclassical logic?

• If logic is empirical, have we to think that QL is

more fundamental than classical logic in view

of the fact that QM provides a more

fundamental description of natural

phenomena than classical mechanics does?

WHAT IS QUANTUM LOGIC ABOUT?

• Two grand domains of reality:

The macroscopic universe

The microscopic universe

• And yet, reality is ultimately quantum

mechanic

REALITY

MACROSCOPIC

second = 1/60 m

minute = 1/60 h

day = 24 hs

year = 365 ds

century = 100 ys

millions of years = 106

billions of years = 1012

MICROSCOPIC

mili = 10-3

micro = 10-6

nano = 10-9

pico = 10-12

femto = 10-15

atto = 10-18

zepto = 10-21

yocto = 10-24

UNIVERSE

MACROSCOPIC

• Kilo = 103

• Mega = 106

• Giga = 109

• Tera = 1012

• Peta = 1015

• Exa = 1018

• Zetta = 1021

• Yocta = 1024

MICROSCOPIC

• Mili = 10-3

• Micro = 10-6

• Nano = 10-9

• Pico = 10-12

• Femto = 10-15

• Atto = 10-18

• Zepto = 10-21

• Yocto = 10-24

The Planck time/scale = 10-43 secs

OUR UNIVERSE, i. e. REALITY

• Kolmogorov and Heisenberg

• Probability and Uncertainty

• Orthomodular nondistributive lattices are

models of QL

• very much as Boolean algebras are models of

classical logic

QL AND NON-MONTONIC LOGIC

• Classical logic is monotonic

� Given a A set of assumptions and a formula

‘x’ such that A then x. If we add more

assumptions to A so as to get A* we will still

have A* then x. ‘More information’ cannot

invalidate inferences drawn on the basis of

‘less information’

QUANTUM LOGIC IS NON-

MONOTONIC

• In non-monotonic logics ‘old inferences’ may be invalidated by ‘new information’

• If so, then it from bit, and bit from qu-bit!

• [A qubit is any unitary vector in the Hilbertspace (of dimension 2) on the set of allordered pairs of complex numbers]

ALTERNATIVES TO THE FIRST-ORDER SEMANTICS

• Substitutive semantics

• Truth-value semantics

• Probabilistic semantics

TWO PROGRAMS IN QL

• The weak program(Birkhoff and Von Neumann)

• It has a purealgebraicconstruction

• It aims at clarifyingsuperpositions and the structure of QP

• The strong program(Putnam)

• It has or aims at a logical construction

• It supplements butdoes not depose standard logic

STANDARD QL

• Abstract (also

called

orthomodular) QL

• Concrete (also

called Hilbert) QL

BIRKHOFF AND VON NEUMANN

• The propositional calculus of QM has the

same structure as an abstract projective

geometry

HOWEVER…

• Many authors understand Ql as simply a

lattice or a poset

ONE MORE TIME

• Is there a sound and universal language fordescribing physical events?

• Dispute between experimentalists and theoreticians

• Because of the random character of Qevents, one history is not enough for an account of allpossibilities

QL

• Follows “non-classical” rules of disjunction

and implication

• Conspicously, superposition and possibility

structure

• Classical principles: Uncertainty,

Qdecoherence, Superposition, Non-locality,

Entanglement, Complementarity, …

• In the domain of Qphysical reality, the strong

ontological preconditions of the classical

language are no longer fulfilled

• According to our present knowledge QP is

universally valid in all domains of the physical

reality

• The application of CL requires a justification in

every individual case

• Parallel reasoning! (� Qcomputation)

DISCRETE MATHS AND COMPLEXITY

• Partially ordered sets

• Extreme sets

• Discrete and combinatorial geometry

• Theory of discrete probabilities

• Comnatorial problems (combinatorial complexity)

• Game theory and rational choice theory

• Topology

• Some NCLs

• Maths of computational systems

PATTERNS AND THE DISCRETE

• Tiles

• Extreme sets

• Posets

• Numbering

• Networks theory

• Graphs and hypergraphs

• Code/coding theory

COMPLEXITY

• From QP to complexity means

logizicing Qphenomena

MODULAR LATTICE

ORTHOMODULAR LATTICE

• A logicization of certain non-Boolean

lattices is possible, although the

procedure of logicization raises a

number of interpretational and

philosophical issues that remain

controversial

• The two natural candidates for algebraic

structures that represent waeker structures

than Boolean algebras: orthomodular and

modular lattices

• The heart of the conceptual problem is how to

relate the non-Boolean logic to

noncommutative probability theory

• Ql is the ‘true’ logic just as the ‘true’ geometry

of space-time is non-Euclidean

• Empirical considerations alone cannot force us

to revise our logic; a distinctly philosophical

component will be needed to justify a revision

of our logic

QUANTUM

LOGIC

QUANTUM

LOGIC

MANY-VALUED LOGIC

FUZZY LOGIC

PARACONSISTENT LOGIC

NON-MONOTONICITY