
What, When, Where, How, Who?
Logic
Introduction, Important Definitions and Related
Concepts:
Logic (from
Classical Greek λόγος
logos; meaning word, thought, idea, argument,
account, reason, or principle) is the study of the
principles of valid
inference and
demonstration. As a
formal science, logic investigates and classifies
the structure of statements and arguments, both through
the study of
formal systems of
inference and through the study of arguments in
natural language. The field of logic ranges from core
topics such as the study of
fallacies and
paradoxes, to specialized analysis of reasoning
using
probability and to arguments involving
causality. Logic is also commonly used today in
argumentation theory.
^{
[1] }
Traditionally, logic was considered a branch of
philosophy, a part of the classical
trivium of
grammar, logic, and
rhetoric. Since the midnineteenth century formal
logic has been studied in the context of
foundations of mathematics, where it was often
called
symbolic logic. In 1903
Alfred North Whitehead and
Bertrand Russell attempted to establish logic
formally as the cornerstone of mathematics with the
publication of
Principia Mathematica.^{[2]}
However, except for the elementary part, the system of
Principia is no longer much used, having been largely
superseded by
set theory. As the study of formal logic expanded,
research no longer focused solely on foundational
issues, and the study of several resulting areas of
mathematics came to be called
mathematical logic. The development of formal logic
and its implementation in computing machinery is
fundamental to
computer science. Form is central to logic. It
complicates exposition that 'formal' in "formal logic"
is commonly used in an ambiguous manner. Symbolic
language is just one kind of formal logic, and is
distinguished from another kind of formal logic,
traditional
Aristotelian syllogistic logic, which deals solely
with
categorical propositions.

Informal logic is the study of
natural language
arguments. The study of
fallacies is an especially important branch of
informal logic. The dialogues of
Plato ^{
[3]} are a
good example of informal logic. Formal logic
is the study of
inference with purely formal content, where that
content is made explicit. (An inference possesses a
purely formal content if it can be expressed
as a particular application of a wholly abstract
rule, that is, a rule that is not about any
particular thing or property. The first rules of
formal logic that have come down to us were written
by
Aristotle.
^{
[4]} In
many definitions of logic, logical inference and
inference with purely formal content are the same.
This does not render the notion of informal logic
vacuous, because no formal language captures all of
the nuance of natural language.
Symbolic logic is the study of symbolic
abstractions that capture the formal features of
logical inference.^{[2]}^{[5]}
Symbolic logic is often divided into two branches,
propositional logic and
predicate logic.
Mathematical logic is an extension of
symbolic logic into other areas, in particular to
the study of
model theory,
proof theory,
set theory, and
recursion theory.
The Ancient Greek
language is the historical stage of the
Greek language^{[1]}
as it existed during the
Archaic (9th–6th centuries
BC) and
Classical (5th–4th centuries BC) periods in
Ancient Greece. The
Hellenistic (postClassic) period of Ancient
Greece formally constitutes its own stage in the
Greek language known as
Koine Greek. Ancient Greek is subdivided into
various
dialects, including the
Homeric Greek of the
Homeric poems, and the
Attic Greek of great works of literature and
philosophy of the
Athenian Golden Age. For information on the
Hellenic language family prior to the creation of
the
Greek alphabet, see articles
Mycenaean Greek and
ProtoGreek. The origins, early forms, and early
development of the
Hellenic language family are not well
understood, owing to the lack of contemporaneous
evidence. There are several theories about what
Hellenic dialect groups may have existed between the
divergence of early Greeklike speech from the
common
IndoEuropean language (not later than 2000 BC),
and about 1200 BC. They have the same general
outline but differ in some of the detail. The only
attested dialect from this period^{[2]}
is
Mycenaean, but its relationship to the
historical dialects and the historical circumstances
of the times imply that the overall groups already
existed in some form. The major dialect groups of
the Ancient Greek period can be assumed to have
developed not later than 1100 BC, at the time of the
Dorian invasion(s), and their first appearances
as precise alphabetic writing began in the
8th Century BC. The invasion would not be
"Dorian" unless the invaders had some cultural
relationship to the historical Dorians; moreover,
the invasion is known to have displaced population
to the later AtticIonic regions, who regarded
themselves as descendants of the population
displaced by or contending with the Dorians. The
ancient Greeks themselves considered there to be
three major divisions of the Greek people, into
Dorians, Aeolians, and Ionians (including
Athenians), each with their own defining and
distinctive dialects. Allowing for their oversight
of Arcadian, an obscure mountain dialect, and
Cyprian, far from the center of Greek scholarship,
this division of people and language is quite
similar to the results of modern
archaeologicallinguistic investigation. This is
very important to realize because of the content and
the change that has occurred. One standard
formulation for the dialects is:^{[3]}
West Group Northwest Greek

Doric
Aeolic Group Aegean/Asiatic Aeolic

IonicAttic Group

Attica Euboea and colonies in Italy
Cyclades Asiatic Ionia
ArcadoCypriot Arcadian Cypriot
 Pamphylian.
Logos (Greek
λόγος)
is an important term in
philosophy,
analytical psychology,
rhetoric and
religion. It derives from the verb
λέγω legō: to count, tell, say,
or speak.^{[1]}
The primary meaning of logos is: something
said; by implication a subject, topic of
discourse or reasoning. Secondary meanings
such as logic, reasoning, etc. derive from
the fact that if one is capable of
λέγειν (infinitive) i.e. speech, then
intelligence and
reason are assumed. Its
semantic field extends beyond "word"
to notions such as "thought, speech,
account,
meaning,
reason,
proportion, principle, standard", or "logic".
In English, the word is the root of "logic,"
and of the "ology" suffix (e.g., geology).^{[2]
}
Heraclitus established the term in
Western philosophy as meaning both the
source and fundamental order of the cosmos.
The
sophists used the term to mean
discourse, and
Aristotle applied the term to argument
from reason. After Judaism came under
Hellenistic influence,
Philo adopted the term into Jewish
philosophy. The Gospel of John identifies
Jesus as the incarnation of the Logos,
through which all things are made. The
gospel further identifies the Logos as God (theos),
providing scriptural support for the
Trinity. It is this sense, the Logos as
Jesus Christ and God, that is most common in
popular culture. Psychologist
Carl Jung used the term for the
masculine principle of rationality. In
ordinary, nontechnical Greek, logos
had two overlapping meanings. One meaning
referred to an instance of speaking:
"sentence, saying, oration"; the other
meaning was the
antithesis of ergon ("action" or
"work"), which was commonplace. Despite the
conventional translation as "word", it is
not used for a
word in the grammatical sense; instead,
the term lexis is used. However, both
logos and lexis derive from
the same verb
λέγω. It also means the inward intention
underlying the speech act: "opinion,
thought, grounds for belief, common sense."
^{
[3]
}Inference is the act or process
of deriving a
conclusion based solely on what one
already knows. Inference is studied within
several different fields.
Human inference (i.e. how humans draw
conclusions) is traditionally studied within
the field of
cognitive psychology.

Logic studies the laws of valid
inference.
Statisticians have developed formal
rules for inference from quantitative
data.
Artificial intelligence researchers
develop automated inference systems.
The conclusion inferred from multiple
observations is made by the process of
inductive reasoning. The conclusion
may be
correct or incorrect, and may be
tested by additional observations. In
contrast, the conclusion of a
valid
deductive inference is true if the
premises are true. The conclusion is
inferred using the process of
deductive reasoning. A valid
deductive inference is never false. This
is because the validity of a deductive
inference is formal. The inferred
conclusion of a valid deductive
inference is necessarily true if the
premises it is based on are true. The
field of
halftruths as they relate to the
truth of observations, is another area
of concern impacting inference based on
observations. Inferences are either
valid or invalid, but not both.
Philosophical logic has attempted to
define the rules of proper inference,
i.e. the formal rules that, when
correctly applied to true premises, lead
to true conclusions.
Aristotle has given one of the most
famous statements of those rules in his
Organon. Modern [[mathematical
logic, beginning in the 19th century,
has built numerous
formal systems that embody
Aristotelian logic (or variants
thereof).
Greek philosophers defined a number
of
syllogisms, correct threepart
inferences, that can be used as building
blocks for more complex reasoning. Most
famous of them all: All men are mortal
Socrates is a man 
Therefore Socrates is mortal. In
mathematics, a proof is a
convincing demonstration that some
mathematical statement is
necessarily true, within the accepted
standards of the field. A proof is a
logical argument, not an
empirical one. That is, the proof
must demonstrate that a proposition is
true in all cases to which it applies,
without a single exception. An unproven
proposition believed or strongly
suspected to be true is known as a
conjecture. Proofs employ
logic but usually include some
amount of
natural language which usually
admits some ambiguity. In fact, the vast
majority of proofs in written
mathematics can be considered as
applications of
informal logic. Purely formal proofs
are considered in
proof theory. The distinction
between
formal and informal proofs has led
to much examination of current and
historical
mathematical practice,
quasiempiricism in mathematics, and
socalled
folk mathematics (in both senses of
that term). The
philosophy of mathematics is
concerned with the role of language and
logic in proofs, and
mathematics as a language.
Regardless of one's attitude to
formalism, the result that is proved to
be true is a
theorem; in a completely formal
proof it would be the final line, and
the complete proof shows how it follows
from the
axioms alone by the application of
the rules of inference. Once a theorem
is proved, it can be used as the basis
to prove further statements. A theorem
may also be referred to as a
lemma if it is used as a stepping
stone in the proof of a theorem. The
axioms are those statements one cannot,
or need not, prove. These were once the
primary study of philosophers of
mathematics. Today focus is more on
practice, i.e. acceptable
techniques. In direct proof, the
conclusion is established by logically
combining the axioms, definitions, and
earlier theorems. For example, direct
proof can be used to establish that the
sum of two
even
integers is always even:
For any two even integers
x and
y we
can write x
= 2a and
y = 2b
for some integers
a and
b, since both
x and
y are
multiples of 2. But the sum
x + y
= 2a + 2b = 2(a +
b) is also a multiple of
2, so it is therefore even by
definition. A formal science is a
theoretical study that is concerned with
theoretical
formal systems, for instance,
logic,
mathematics, and the theoretical
branches of
computer science,
information theory, and
statistics. The formal sciences are
built up of theoretical symbols and
rules. The formal sciences can sometimes
be applied to the reality, and, within
certain limitations, they can be useful.
People often make the mistake of
confusing theoretical systems with
reality, applying theoretical models as
if they represent reality perfectly, or
believing that the theoretical model is
in fact the reality. The difference
between formal sciences and natural
science is that formal sciences start
from theoretical ideas and it leads to
other theoretical ideas through thinking
processes, while natural science starts
from observation of the real world and
leads to more or less useful models for
a part of reality. You can never learn
anything about reality from studying
formal sciences alone. You can never
prove anything about reality through the
use of formal sciences.
Applied mathematics is to try to
apply some theoretical mathematical
model to reality. It is possible within
certain limits and with certain
restrictions and with a certain limit of
precision. If the map and the reality
does not fit it is the map which is
wrong, not the reality. A map is a
theoretical representation (model)
of reality. The study of applied science
began earlier than formal science and
the formulation of
scientific method, with the most
ancient mathematical texts available
dates back to 1500BC500 BC (ancient
India), 13001200 BC (ancient
Egypt), and 1800 BC (Mesopotamia).
From then on different cultures such as
the
Indian,
Greek,
Islamic made major contributions to
mathematics. Besides mathematics, logic
is another oldest subject in formal
science. Logic as an explicit analysis
of the methods of reasoning received
sustained development originally in
three places: India in the 6th century
BC,
China in the 5th century BC, and
Greece between the 4th century BC and
the 1st century BC. The formally
sophisticated treatment of modern logic
descends from the Greek tradition, being
informed from the transmission
Aristotelian logic while the
tradition from other cultures do not
survive into the modern era.
As other disciplines of formal
science rely heavily on mathematics,
they did not exist until mathematics had
developed into a relatively advanced
level.
Pierre de Fermat and
Blaise Pascal (1654), and
Christiaan Huygens (1657) started
the earliest study of
probability theory (statistics) in
the 17th century. Study on computer
science and information theory did not
begin until middle 20th century.
System (from
Latin systēma, in turn from
Greek
σύστημα
systēma) is a
set of interacting or interdependent
entities, real or abstract, forming
an integrated whole. The concept of an
'integrated whole' can also be stated in
terms of a system embodying a set of
relationships which are differentiated
from relationships of the set to other
elements, and from relationships between
an element of the set and elements not a
part of the relational regime. There are
natural and manmade (designed) systems.
Manmade systems normally have a certain
purpose, set of objectives. They are
“designed to work as a coherent entity”.
Natural systems may not have an apparent
objective but they are sustainable,
efficient and resilient. A system is a
fundamental concept of
systems theory, which views the
world as a complex system of
interconnected parts. We determine a
system by choosing the relevant
interactions we want to consider, plus
choosing the system boundary —–
or, equivalently, providing membership
criteria to determine which entities are
part of the system, and which entities
are outside of the system and are
therefore part of the environment
of the system. We then make simplified
representations (models)
of the system in order to understand it
and to predict or impact its future
behavior. An
open system usually interacts with
some entities in their environment. A
closed system is isolated from its
environment. A subsystem is a set
of elements, which is a system itself,
and a
part of a larger system. The
scientific research field which is
engaged in the transdisciplinary study
of universal systembased properties of
the world is general
system theory,
systems science and recently
systemics. They investigate the
abstract properties of the matter and
mind, their
organization, searching concepts and
principles which are independent of the
specific domain, independent of their
substance, type, or spatial or temporal
scales of existence. The term system
has multiple meanings
^{
[1]}:
 A collection of organized things;
as, a solar system. A way of organising
or planning. A whole composed of
relationships among the members.
Most systems share the same common
characteristics.

