Universal Logic and Modern Logic

The Relativity and Universality of Logic

Jean-Yves Bziau

Federal University of Rio de Janeiro

Brazilian Research Council

What is logic?

We are logical (rational) animals

The Brain according to Aristotle

Relation

Logic is not logic

Logic and logic

Logic : reasoning logic : the theory of reasoning

History : the series of eventshistory : the science which studies History

Logics or logics?Are there different logics?

Are there different Logics?Is Aristotle the creator of logic?Aristotle was maybe the first to have a logic, a theory of reasoning

But he was not the first person to have a Logic, to reason (not the first logical animal)

PythagorasBefore Aristotle, the Greeks introduced a new way of reasoning, a new Logic, based on the reduction to the absurd - Irrationality

Some people consider that this was the birth of Mathematics

Mathematicians have never used Aristotles theory of reasoning

The Paradox of DescartesDescartes was against logic

But he was very logical

DESCARTES 4 PRECEPTSClarityNever to accept anything for true which I did not clearly know to be such; that is to say, carefully to avoid precipitancy and prejudice, and to comprise nothing more in my judgment than what was presented to my mind so clearly and distinctly as to exclude all ground of doubt.DivisionTo divide each of the difficulties under examination into as many parts as possible, and as might be necessary for its adequate solution.AscensionTo conduct my thoughts in such order that, by commencing with objects the simplest and easiest to know, I might ascend by little and little, and, as it were, step by step, to the knowledge of the more complex; assigning in thought a certain order even to those objects which in their own nature do not stand in a relation of antecedence and sequence.ExhaustivityTo make enumerations so complete, and reviews so general, that I might be assured that nothing was omitted.PASCAL 8 RULESRulesforDefinitionsNot to undertake to define any of the things so well known of themselves that clearer terms cannot be had to explain them.Not to leave any terms that are at all obscure or ambiguous without definition.Not to employ in the definition of terms any words but such as are perfectly known or already explained.RulesforAxiomsNot to omit any necessary principle without asking whether it is admitted, however clear and evident it may be.Not to demand, in axioms, any but things that are perfectly evident of themselves.RulesforProofsNot to undertake to demonstrate any thing that is so evident of itself that nothing can be given that is clearer to prove it.To prove all propositions at all obscure, and to employ in their proof only very evident maxims or propositions already admitted or demonstrated.To always mentally substitute definitions in the place of things defined, in order not to be misled by the ambiguity of terms which have been restricted by definitions.TARSKI: Introduction to logic and the methodology of deductive sciences - VI On the Deductive Method36 Fundamental constituents of a deductivetheoryprimitive and defined terms, axioms and theorems (Sur la mthode dductive, in Travaux du IXe Congrs International de Philosophie, VI, Paris: Hermann, pp.95-103)

Ideas which are closely related to those presented in this section can be found in earlier literature. See, for instance, the opusculum (posthumously published), De I'esprit geometrique et de I'art de persuader, of the great French philosopher and mathematician B. PASCAL (1623-1662).

Logic: the laws of thoughtKANTBOOLE Logic and logic are eternal Logic is eternal, logic is changing

Modern Logic

Different names for modern logicFormal logicSymbolic logicAlgebra of logicLogisticMetamathematicsMethodology of deductive sciencesMathematical logicLogicDifferent systemsClassical logicIntuitionistic logicMany-valued logicModal logicNon monotonic logicFuzzy logicSubstructural logicLinear logicParaconsistent logic Universal LogicA general theory of logics, of the different theories of reasoning, of the different logical structures

Not a universal system of LogicNot a Logic, not a system that is the description of the right way of reaoning

Languages and LinguisticsThere are many languages

They have something in common despite very strong differences, i.e. chinese, english, arabic

This thing in common is not a language itself, the essence of language is not a language, it is the object of linguistics

Linguistics is not a universal language but the study of the universal features of languagesFerdinand de SaussureThe structure of language

The originator of structuralism

Universal AlgebraJ.J.Sylvester

A.N.Whitehead

Garrett Birkhoff

But Universal Algebra is different from Universal LogicDifferent structures, differents objects, differents tools

Logics are structures but not necessarily algebraic structuresStructure = Lattice

To be is to be an element of a structure(a class of structures)

4 does not exist by itself

What is a logical structure?

AlgebraMuammad ibn Ms al-Khwrizm (780 850 Persia)

Algorithm

Algarismo = digit

Axiomatic emptiness

Two reasons to reject axiomsTheoretical reasons

Practical reasonsPractical reasons

Anti-classical logicA simple example of a logic not obeying any standard axioms

Non-reflexive, non-monotonic, non-transitive, non-structural

But proof theory and semanticsIs Logic Universal?Special Issue ofLogica UniversalisVol4 n2 2010

Do all human beings have the same capacity of reasoning? Do a man, a woman, a child, a papuan, a yuppie, reason in the same way?Does reasoning evolve?Did human beings reason in the same way two centuries ago?In the future will human beings reason in the same way?Did computers change our way to reason?Is a mathematical proof independent of time and culture ?Do we reason in different ways depending on the situation?Do we use the same logic for everyday life, physics, economy?Do the different systems of logic reflect the diversity of reasonings?Is there any absolute true way of reasoning Logic and logic are relative

Nevertheless logic as a science can be universal

(1) science is not a private business, it is objective, not subjective, not a question of taste

(2) science explains not the idiosyncrasies of a particular phenomenon, but some general patterns of phenomena

Science is concerned with a double ALL, ALL minds and ALL objects.

Chuaqui and Suppes (1995) have shown that classical physics can be described with a first-order logic theory with only universal quantifiers

logic as a science is universal(physics as a science is universal)

There is no universal system of logic(there is no universal theory of the universe)

Louis Rougier (1889- 1982) The relativity of logic 1941

With the discovery of the conventional and relative character of logic, human spirit has burned his last idol.

Haskell Curry (1900 - 1982) Leons de logique algbrique 1952

Translated and presented by Jonathan Seldin

Leon HenkinLa structure algbrique des thories mathmatiques 1956

The Universe of Logics(The world of possible logics)DEVIATION/EXPANSIONDeviationsIntuitionistic logicRelevant logicExpansionsModal logicCausal logicGRADESSubsystemsPositive classical propositional logicFull classical propositional logicSupersystemsMany-sorted classical first-order logicSecond order classical logicTECHNIQUESProofHilbert systemsSequents systemsSemanticsLogical matricesKripke structures

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