Upload
others
View
8
Download
1
Embed Size (px)
Citation preview
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