Transcript What is logic?

Jean-Yves Béziau
Federal University
of Rio de Janeiro
Brazilian Research
Council
What is logic?
We are logical (rational) animals
Relation
Logic and logic
 Logic : reasoning
 logic : the theory of reasoning
 History : the series of events
 history : 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)
 Before 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 Aristotle’s theory of
reasoning
Pythagoras
The Paradox of Descartes
 Descartes was against logic
 But he was very logical
DESCARTES 4 PRECEPTS
Clarity
Division
Ascension
Exhaustivity
Never 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.
To divide each of the difficulties under examination into as
many parts as possible, and as might be necessary for its adequate
solution.
To 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.
To make enumerations so complete, and reviews so general, that
I might be assured that nothing was omitted.
PASCAL 8 RULES
Rules
for
Definitions
Not 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.
Rules
for
Axioms
Rules
for
Proofs
Not 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.
Not 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 Method
36 Fundamental constituents of a deductive
theory—primitive and defined terms, axioms and
theorems (Sur la méthode déductive‘, in Travaux du IXe
Congrès 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 thought
KANT
BOOLE
 Logic and logic
 Logic is eternal,
 are eternal
 logic is changing
Modern Logic
Different names for modern logic
 Formal logic
 Symbolic logic
 Algebra of logic
 Logistic
 Metamathematics
 Methodology of deductive sciences
 Mathematical logic
 Logic
Different systems
 Classical logic
 Intuitionistic logic
 Many-valued logic
 Modal logic
 Non monotonic logic
 Fuzzy logic
 Substructural logic
 Linear logic
 Paraconsistent logic …
Universal Logic
 A general theory of logics,
of the different theories of reasoning,
of the different logical structures
 Not a universal system of Logic
 Not a Logic, not a system that is the description of the
right way of reaoning
Languages and Linguistics
 There 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 languages
Ferdinand de Saussure
 The structure of language
 The originator of
structuralism
Universal Algebra
 J.J.Sylvester
 A.N.Whitehead
 Garrett Birkhoff
But Universal Algebra is different from
Universal Logic
 Different structures, differents objects, differents
tools
 Logics are structures but not necessarily algebraic
structures
Structure = Lattice
 To be is to be an element of a structure
(a class of structures)
 4 does not exist by itself
Algebra
 Muḥammad ibn
Mūsā al-Khwārizmī
(780 – 850 Persia)
 Algorithm
 Algarismo = digit
Two reasons to reject axioms
 Theoretical reasons
 Practical reasons
Practical reasons
Anti-classical logic
 A simple example of a logic not obeying any standard
axioms
 Non-reflexive, non-monotonic, non-transitive, non-
structural
 But proof theory and semantics
Special Issue of
Logica Universalis
Vol4 n2 2010
1. 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?
2. 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 ?
3. Do we reason in different ways depending on the situation?
Do we use the same logic for everyday life, physics,
economy?
4. Do the different systems of logic reflect the diversity of
reasonings?
5. 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)
 Leçons de logique
algébrique 1952
 Translated and presented by
Jonathan Seldin
Leon Henkin
 La structure algébrique
des théories
mathématiques
1956
DEVIATION/EXPANSION
Deviations
Intuitionistic logic
Relevant logic
Expansions Modal logic
Causal logic
GRADES
Subsystems
Positive classical propositional
logic
Full classical propositional logic
Supersystems Many-sorted classical first-order
logic
Second order classical logic
TECHNIQUES
Proof
Hilbert systems
Sequents systems
Semantics
Logical matrices
Kripke structures
http://www.uni-log.org/