Logic, Reasoning, and Computation: A Comprehensive Overview
6. Reasoning: Reasoning can be understood as a mental process based on certain claims to draw additional conclusions. To reason is to develop a discursive process, establishing some conclusions from other assumptions. To refer to linguistic reasoning, we have used terms such as deduction and inference.
7. Truth and Corrections: When an argument is correct, it is usually true. However, it is important to keep in mind the meaning of truth or falsehood from a logical point of view.
True or False Propositions:
There are two types of propositions: empirical and formal.
A) Empirical: The truth or falsity depends on the correspondence between what is stated in the proposition and reality, and can only be determined by empirical contrast.
B) Formal: The truth or falsity depends on the consistency or non-contradiction with the rest of the propositions of the system to which it belongs.
The Formal Truth of a Proposition:
The formal truth of a proposition is its deductibility from other propositions assumed to be accepted without real proof. This is important in both logic and mathematics, which are interested in the formal structure of arguments. For example, if all A is B and all B is C, then all A is C. Logic merely provides that deductions are correct when premises and conclusions are properly structured. It is better to discuss the validity of reasoning from a logical point of view than the truth of the premises.
8. Notion of Computation: A calculation is a relationship between uninterpreted symbols. It consists of a set of primitive elements (elementary symbols), formation rules, and transformation rules. A calculation rule is purely formal and only consists of syntactic and structural symbols. Symbols can be interpreted, and so the calculation becomes an artificial and formalized language.
9. Logic of Statements: The base of computation is the logic of statements, also called propositional calculus. It deals with the relations of inference between statements. In general, inference schemes are the most basic modes of reasoning, but they only take into account the relationship between propositions. At this level of analysis, only two types of signs are considered: sentences as propositions and particles. Therefore, two types of symbols are required: logical propositions and conjunctions.
10. Predicate Logic: There are arguments whose validity is not manifested through external connections with the propositions, but it is necessary to know their internal structure. Predicate Logic: An analysis of statements is given to the predicate. The relationship between subject and logical system differs between individuals and properties of individuals. It uses symbols that represent individuals and predicates, taking into account the properties or the number of individuals of a given class attributed to a certain property.
11. Class Logic: Class logic is an interpretation of extensional monadic predicates. In class logic, a class is a set of individuals or things that have a common property. The system analyzes the logical relationships between classes and between individuals and classes.
12. Axiomatic Calculus and Natural Deduction: Logic can take two different forms: axiomatic and natural deduction. The classic mode is axiomatic formalization, although it is not the only one and sometimes not the most recommendable. The fundamental difference between both forms of logical calculation lies in the starting set, the rules, and how they can be used.
Axiomatic Calculus:
To prove a statement is to show that it follows validly from other true sentences. Axiomatizing a theory is to organize a set of true statements, and by applying transformation rules, derive other statements (theorems). Axioms and theorems are expressions written in the language of calculus. The rules belong to the metalanguage. An axiomatic system must meet the following:
A) Consistency (not lead to internal contradictions).
B) complete (have the means
sufficient to derive all valid statements). C) deductibility (possibility of
determine whether cualkier formula is valid or not within the same). D) independence
of axioms: q None of the axioms of the system can be deduced from the others.
12.2.calculo of natural deduction: the calculations are systems of natural deduction
deductive language used to analyze them formal.en one or several
initial premises, which are considered hypothetically q formulas given from the
principio.a them from other formulas are obtained up to the conclusión.las
premises and conclusion are not stated formally true, but true
empirically, and they do not belong to logic formal.las q if they belong are the rules of
q deduction applied to obtain the conclusion from the rules allow premisas.estas
compose and decompose formulas through the introduction and elimination of conjunctions
and logical quantifiers.
BRIEF HISTORY OF LOGIC:
1.Aristoteles: 384-322aC.importante Greek philosopher, may be considered the first formal logic
history, was 1 study and encode q argument forms correcta.en his earlier work
developing analytical theory of the syllogism, the type of argument q, put certain
assumptions, it follows from them a conclusion d under the terms of relations between component q
the premisas.presento is an axiomatic theory as a set of logical laws demonstrated
from the first figure of syllogism, Barbara and Celarent, q axiomas.entendio use as logic
as analysis of language, but only analyze the structure of natural language, and by not distinguishing
between use and mention it is unclear whether abla cosas.en on words or on their works are
some notes for propositional logic, the modal and d relations.
2.The Stoics: the Stoics (DSD philosophy school 300aC the s.IIdC ast) developed the logic of
enunciados.tenian the distinction between use and mention, and possessed a distinguished semantic theory q
between the sign, its meaning and denotacion.un argument is a system composed d premises and a
conclusion, and is defined as correct if the conditional is interested in problems verdadero.se
as the paradoxes and contradictions.
3.What is your age Ages and Renaissance: medieval logic does not add new systems axiomaticos.el
Renaissance was a period of relative inactivity in the history of logic and reaction
against medieval scholasticism.
4.Leibniz: 1646-1716.admite logic symbolization aristotelica.intenta to get into
logic matematica.la the rigor of mathematics has progressed a lot for the construction
a manageable and symbolism penso q seguro.por it suited the logic-linking
analogous to the content in semantic proposiciones.insiste q d the logical deduction
s pure calculo.pero not get to build the system of symbols and their contribution is reduced
to a simple program.
5.Siglo XIX: the middle of the nineteenth century when they start to become real logic systems
MARK. George Boole (1815-1864): in his mathematical analysis of the test logic as
for a calculus of deductive reasoning, construct a purely algebraic calculation using
symbols and operations defined from the well mismos.transforma traditional theory
syllogistic terms and especially in logic equations d, q considers the calculation is somewhat artificial,
purely formal. Charles Sanders Peirce (1839-1914): development and introduced propositional logic
the idea of axiomatization and the method of truth tables. Gottlob Frege (1848-1925): formula d
clearly the difference between variable and constant. Distinguish between law and rule and started the theory of
descriptions. Giuseppe Peano (1858-1912): q logic and considered a powerful tool for
matematico.realiza carefully systematize knowledge of arithmetic axiomatized using
logic. Bertrand Russell (1872-1970): seeing mathematics as a branch of logic and
tries its logical foundation in the BOUT Principia Mathematica (1910-1913).
6.Nuevos approaches: in early ls dl sXX, one d ls most worrying problems for the
Logic has been the fundamentals d d matematicas.Peano had made the systematization
d arithmetic concepts and an axiomatization of arithmetic. George Cantor (1845-1918):
and other specialists attempted to reduce arithmetic to a deeper base and so all arithmetic
can be integrated into the algebra class or theories of conjuntos.hubo a problem appeared
logical paradoxes. Burali-Forti (1861-1931): the first paradox discovered in 1897 and elaborated Russell
1902.intentando other in solving this problem, Ernerst Zermelo (1871-1953) in 1908 axiomatizing
set theory. David Hilbert (1862-1943): Conducts research on these issues in their
Foundations of Geometry, consider q classical mathematics can be formalized in three systems
axiomatic arithmetic, analysis and theory d sets. Kurt Gödel (1906-1978) Demonstrations in
Q 1931 proof of the consistency of arithmetic can not Obetener with instruments
belonging to the same formal system is expressed aritmetica.Gödel q show the complete
of elementary logic and establishes the theorem of incompleteness of the higher order logics.
THE Fixisme: s.IV in BC, Aristotle established a classification of the
No living especies.partia the idea of these remained unchanged q, l
ijos you are equal to their parents, and q abia other beings, such as worms
, q arose from inorganic clay in a matter od decomposition,
called spontaneous generation.
INORGANIC ORIGIN OF EVOLUTION VIDA.LA: in sXIX,
Q Pasteur showed no living organisms were formed were not descendants q
similar organisms, Darwin was up on the idea of changing species q
cn dl the passing slowly we started to believe and tiemp.se q ls probably planets
abian emerged from inorganic powder d d q smples gases floating in space.
in the twentieth century, A. Oparin suggested d q Earth’s atmosphere at one time could be very
actual.en different from 1953, S. Miller’s hypothesis and showed experiemntalment dsd
entoncs known molecular biology to an extraordinary development.
THE ORIGIN OF MAN: Linnaeus established the call q Systema Naturae
in q A classification d ls d the plants animals, their species according
q d similarity degree of effect appeared ellas.acabo stablciendo existence d kinship
among primates especies.Somos very similar to living species with q is
are the family what they pungidos.con and other species extingudas, proceed d
1 line common ancestral: the Hominoidea, q d during evolution was dividing
in different branches.
Lamarck’s theory:Lamarck was in his Philosophy zoology kien made a presentation
rigorous theory d the geological and paleontological studies evolucion.sus joined
review d q abia relations between subgroups grups and organics, led to his conviction
Nature of q part d was a continuum in the q ls plants and animals
into 2 evolutionary lines in the q ls d sucesivamnte beings are producing according to
1 trend toward perfeccionismo.en this evolution, the species do not follow a line
continua.por Thus, the evolution would be the result of the need to have the species q d
adapt to the environment they live in the q d ls by developing appropriate organs: the law of
use and disuse organs d ls d always in accordance with the principle of q is the q function creates the organ.
Natural Selection: Darwin published The Origin of Species x Half Day
assuming natural selection and evolutionary theory by providing evidence in favor
q abia pickup in his travels, taking in the fact TRMalthus dq d populations beings
living increase in greater proportion q disponible.y food considering the selection
q farmers succeeded by crossing breeds cn d to some d d improve the
qualities d ls animals in his book Darwin formulated the fundamental principle of his theory:
evol natural.la selection. Biological process is explained by natural selection and not d
through a process d q and adaptation to the environment stems from the same living being and not
the medium in q d is the population growth desarrolla.el requires individuals d ls species
a struggle for existence in the q ls only fittest survive.
CONTRIBUTIONS FROM GENETICS: to explain the mechanisms of evolution was
scientifically necessary distiguir hereditary variations or mutations were not the q d
ereditarias or modifications adquiridas.En 1866 G. Mendel discovered the laws of heredity
biological hereditary factors parti d ls q stan d in germ cells, but later q
called genes, responsible for transmitting d variaciones.H.de Vries was able to interpret and
properly hereditary variations and call locates mutaciones.LMorgan
genes on chromosomes and mutations explained by chromosomal rearrangement.
Synthetic theory of evolution: evolution or t.sintetica d Darwinism
date has incorporated these new discoveries to expand and refine
the theory of characters evolucion.las mutations arise that produce variation
especies.dichas progressive mutations occur randomly, the effect of migration,
climate change or biological phenomena would occur between groups of isolates
especie.La same nature as dynamic, through natural selection, is responsible
d d direct the course of evolution by regulating the genetic variability of populations and producing d
the best adaptation of all living things.
The biped: progressive development of bones configuration d ls position posiblitar
vertical.el hominid needed to stay upright as much as possible in order to exploit d
and monitor the trend would entorno.esta already outlined in the animal when he lived in the tree,
straightening and strive and refine the progressive adaptation to the ground, causing the characters
convenintes acquired behavior and selccionando individuals with the best capabilities
congenital producirlos.resulta difficult to understand the value d q survival will confirm with
relatively quickly to the vertical position and q powers were developed to be congenital
upright, with no opposable thumb, which phase out its support over the entire length of the plant,
d narrowing the pelvis and the configuration of the hole between the long bones of the leg: tibia
and fibula.
THE LIBERATION OF HANDS: When the tendency to stay upright was modeled
enough of the hominid body to be moved as q always standing, there is another feature
Anatomic in its evolution towards ombre.se is the permanent release of the hands,
growing familiarity with this useful and use d to their effectiveness in educating management Irian hands
in developing their hands inervacion.las retain all five fingers extended, but develop in the
development of organic movement.
Brain development: the liberation d bipedalism and hands together with a behavior
increasingly complex were developing central nervous system and producing hominid
Cephalization and cerebration crecientes.es one to say the head-body ratio was progressively
increasing d for the head and Orebro evolved both in volume and neurological complexity.
in the vertical position needed the ominous yano defend and attack with the boot, but now q
hands.The aria with increased brain mass would begin with the development of capacity
d retain associate species-temporal perceptions would be exacerbated by the simultaneous cultivation of
visual areas, auditory, olfactory, tactile and motor cortex and d with the culture of cortical areas
related to the perception and the language manual.
Physiological traits: an immediate consequence of hand-brain relationship, technique-theory was that
, to provide the useful hominid ever more perfect, not only sought to better defend
but the farm tb d new food sources and even their processing in order to make them
more suitable for device ominous digestivo.de thus advancing in the acquisition of a new
physiological traits, as is the radical transformation of its food abito .. The change in
Supply: animal species adapt his body to food, while the ombre q adapting feeds
cuerpoun util his most essential to achieve this transformation in their food abito would be the
discovery of fire. neoteny: the need to protect the fire led subtle d prepare l aq
The hominid was not accustomed to camping in sheltered locations or easily once there vigilar.una
with a pattern of cooperation, with a very relieved and digestion with a large poblcion, not
q dificl understand the dependency relationships are becoming more Feran to lead
in another d ls q characterize physiological traits over hominid to man, or neoteny juvenilization
of the species.Natural curiosity: this echo epercute significantly in one comportamiento.Asi
youth important feature is the need to browse. Consequences of slow maturing: the
dl increase in the process d ombre nerve and brain maturation allows greater coexistence with
sociocultural processes of education and socialization.
Numeracy: the language is a city compared to a q antigua.de the old way
q is a lake is Eredar d mxas generaciones.de thus represent the old language
q natural while the new quarters would represent the artificial language, so the comparison is to
kimica symbolism of a suburb of our lenguje xq is to built in a short time
and its objective is different is the language natural.un q l.natural comprises a lexicon finite d d
a number d q rules allows us to combine these components d mxas ways ast reaching almost the
qa ERMISSION infinito.un mxo author on this subject was L. Wittgenstein considered one q sta d
ls dl s.XX.el more philosophical degrees Decia q language is like a life form but qa d been created
slowly from artificial language generaciones.el d mxas precision q is usually used in the
science to increase artificial language exactitud.los builders dl q acen only thing is to guide and
extension of a number q and d have the capabilities l.natural. q sta aimed at a specific meaning, so
ls q in artificial languages are being used tdo q expansion capabilities language bears within d if
same .* Components of a calculation: 1.Elementos primitives: these elements is vital to be q
q rigurosamente.se has to be defined clearly to decide whether or not an object primitivo.2.Reglas
training: determine which are the right combinations of primitives, the
appropriate combinations, formulas calculo.hay good care of a difference between natural language
and artificial.3.Reglas transformation: they are the d q allows us to pass it throws a formula to another, are
q rules give a computation operation, without which the calculation would be static.
Axiomatic system: an axiom is a statement not shown porq q assumes q is evident,
po both its truth is taken for supuesta.el axiom is used as a first principle to deduce other truths
. sule in mathematics and distinguish between non-logical axioms logicos.los-Logic are not defining properties
the field of q is a tratando.ekivale postulado.se this would be a clear statement q but not q
would be a formal expression or formula q izo logica.la first person axiomatic system was Euclidean
in his book the elements of geometry: 5 postulates (1.dados points can be drawn and only one straight
the une.2.cualkier segment may extend continuously in cualkier sentido.3.se can draw a
cualkier circle with center point and at right angles radio.4.todos cualkier are one iguales.5.si
cut straight to two other minor internal angles q are two right angles these lines extended
indefinitely cut on the side where q stan ls smallest angle.) 5 common notions: 1.cosas equal to
the same thing are equal among equals si.2.si added to equal the totals are equal iguales.3.si subtracted
the remains are equal match iguales.4.cosas q are q whole is greater iguales.5.el part.
Mathematical logic Euclid d also occurs with special interes.aki can distinguish between
logicos.1.los logical axioms and logical axioms are not universally valid and is normally used
tautology, from which we derived d tautologias.2.los other logical axioms are postulates no specific
of a particular theory, so they try a particular dqs peculiar d d estructura.no is tautologies
q q are nominated but are used to axiomatize teoria.en arithmetic is commonly used axioms
Peano arithmetic for the first Order. geometry have been used Euclides.analisis axioms of
relaes numbers used need logical axioms q d 2 º order to achieve them.