carlos maldonado memorias vec

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NCLs and ComplexityCarlos Eduardo MaldonadoFull ProfessorUniversidad del Rosario



Preliminary

Abstract thought:

Philosophy

Mathematics

Logic



Logic and Science

Logic was traditionally conceived as the organon for science.

It has been often claimed that society, the world,and reality are logical or should be logical.

Cognitiion, languag, psychology, and logic



Logic and the World

Le coeur a ses raisons que la raison ne connaît

point (Pascal)



Formal Classical Logic

1847: G. Boole publishes The mathematical

analysis of logic , and simultaneuously, A. DeMorgan publishes Formal Logic

1879: G. Frege publishes the Begriffschrift

1899: D. Hilbert introduces the concept ofmetamathematics in his Foundations of

Geometry 1933-1936: Tarski, The concept of Truth in

Formalized Languages, and On the Concept of

Logical Consequence



LOGIC AS SCIENCE1847: G. Boole publishes The mathematical analysis oflogic , and simultaneuously, A. De Morgan publishesFormal Logic

1879: G. Frege publishes the Begriffschrift

1899: D. Hilbert introduces the concept ofmetamathematics in his Foundations of Geometry

1933-1936: Tarski, The concept of Truth in Formalized

Languages, and On the Concept of Logical

Consequence



WHAT IS LOGIC?

Valid inference(entailment)(1930s)

Definability :Language andits expressive

powerComputation

Proof Theory

Model Theory

RecursionTheory



GÖDEL

1931: Incompleteness Theorems

“There are things that are true and

we do not know how the are [true]”

If a system is consistent it is incomplete, and ifit is complete it is inconsistent

Any given theory can be translated as a settheory!



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 isthe non-Euclidean)

• Recognizsing that logic is thus quantumsolves the Schrödingers cat problem



(IN)FALLIBILISM IN

LOGICThe critique to absolute certaintyin knowledge

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

I. Berlin, B. Williams (in ethics)



LOGICAL PLURALISM

It is namely, the pluralism about logical consequence:more than one relationship regarding logicalconsequence exist

There is no one true logic

There is no a formal canon for logic

Numerous forms of notation, quantification are

possibleVarious worlds are possible – logically speaking



Logical Pluralism



NCLs – PhilosophicalLogics



NON MONOTONICITY

Classical logic is monotonic

Given a A set of assumptions and a formula „x‟such that A then x. If we add more assumptionsto A so as to get A* we will still have A* then x.„More information‟ cannot invalidate inferencesdrawn on the basis of „less information‟

Learning and Adaptation



THE SEMANTICS OFNCLs

The semantics of NCLs is the

notion of the semantic ofpossible worlds (whereas thesemantics of formal classical

logic is the semantics of thereal world, or also, of the worldin general (überhaupt ))



When solving a problem

We can:

1. Carry out some calculi

2. Build a reasoning

3. Solve an equation



Reasoning, calculus,research

Calculus entails the importance ofalgorithms

Reasoning is more free

Time as limit for research

Thinking throughout signs, andimagination



The Rationale for NCLs



Thinking logically and logicalthought

Logic as a canon

Rigid formalized

quantificationValidity

First order logic

One single form

Intuition

Alternative

formal systemsTruth(s)

Higher orderlogics

Plurality



About Complexity

Complexity is about:

Unpredictable, unexpected, irreversible,sudden changes

The importance of time, hence learning andadaptation



SCIENCES OFCOMPLEXITY

Non-Equilibrium Thermodynamics (I. Prigogine)

Chaos (E. Lorenz)

Catastrophe theory (R. Thom)Fractals (B. Mandelbrot)

Complex Networks (D. Watts, L. Barabasi, S.Strogatz)

Artificial Life (Ch. Langton)

NCLS (*)



THEORIES, MODELS,EXPLANATIONS

Turbulence

Fluctuations

Cascade effects

Attractors

Percolation

Swarm intelligence

Hypercomputation

Principle of uncertainty

Principle of quantumdecoherence

P versus NP problems

Entropy

Adaptation

Self-Organization

Unpredictability



Claim 1: We can explaincomplexity thanks to the NCLs



Cascade failures

Emergence

Self-organization

Non-linearity TurbulenceFluctuations

Instabilities

Percolation

Bursts

Synchronicity

Free-scale networks

Phase transitions

Underdetermination

Power laws

Synergy

Positive and/or negative feedbacks

Uncertainty

Free (of existence assumptions)

Multideductive systems

Paraconsistency

Non-monotonicity

Probabilities

Many-value

Fuzziness

Dynamics

Time-dependency

Knower

Modality and multimodalities

Counterfactuals

Intuitionism

Set theory

Labelling

Fibring and Combination

Deviant

The sciences of Complexity Non-classical logics

Claim 2:



Claim 2:NCLs as a Science of

Complexity



2011



Lógicas No Clásicas (Lógicas filosóficas)

Lógicas Probabilísticas Lógicas No Monotónicas

Lógica libre

Lógica defabrica

Lógicaparaconsistente

Lógica de larelevancia

Lógica deltiempo

Lógica cuántica

Lógica formalclásica

Lógica difusaLógicas

polivalentes

Lógicaintuicionista

Lógica decontrafácticos

Lógica modal

Lógicadeóntica

Extensiones a lalógica clásica

Alternativas a lalógica clásica

Lógicadinámica

Lógicamultimodal

Lógicaabductiva

Lógicaepistémica



Cascade failures

Emergence

Self-organization

Non-linearity TurbulenceFluctuations

Instabilities

Percolation

Bursts

Synchronicity

Free-scale networks

Phase transitions

Underdetermination

Power laws

Synergy

Positive and/or negative feedbacks

Uncertainty



Paraconsistency

Non-monotonicity

Probabilities

Many-value

Fuzziness

Dynamics

Time-dependency

Knower


Counterfactuals

Intuitionism

Set theory

Labelling


Deviant

The sciences of Complexity Non-classical logics





Paraconsistency

Non-monotonicity

Probabilities

Many-value

Fuzziness

Dynamics

Time-dependency

Knower


Counterfactuals

Intuitionism

Set theory

Labelling


Deviant

Non-classical logics Agents

Multi-deduction

Non-linerarity

The arrow oftime

Diversity

A D A P T A T I O N





Paraconsistency

Non-monotonicity

Probabilities

Many-value

Fuzziness

Dynamics

Time-dependency

Knower


Counterfactuals

Intuitionism

Set theory

Labelling


Deviant

Non-classical logics Agents

Multi-deduction

Non-linerarity

The arrow oftime

Diversity

POS

SIBILI

TIES

L E A R N I N G

A D A P T A T I O N



NCLS and DiscreteSystems

Several NCLs are part of discrete mathematics

Discrete systems versus continuous systems

Complexity is about the discussion between

continuous and discrete systems



NCLs

Complex systems are essentially discrete

All the best current maths are (about) discretephenomena

Reality is intrinsically discrete (for the hypothesisof the continuum has been proved to beundecidable)



SUMMARY

Logical gut, i.e.emptiness

Indeterminacy

Open systems

Gaps of information

Many agents

Plurality of truths

Variance

Time, i.e. temporality

Changes in/ofinformation

Lack of a certaindeparture

Change of contexts,i.e. frameworks

Contradictions andinconsistencies

COMPLEXITY

D ti l i



Non-classical

logics

Deontic logic

Modal logic

Multi-modal logics

Counterfactual logic

Many-valued logic

Fuz

Epistemic logic

Intuitionistic logic

Free logic

Time (temporal)

logic

Quantum logic

Relevant

(relevance) logic

Abductive logic

Paraconsistent logic

Dynamic logic

Fibring logic

Hybrid logic

Provability logic Interpre

Proba

ext

is a

sha

du

we

str

Doxastic logic

Alethic logic

Conditional logic

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