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 mid-nineteenth 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However, except for the elementary part, the system of Principia is no longer much used, having been largely superseded by set theory.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 (post-Classic) 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 Proto-Greek. 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 Greek-like speech from the common Indo-European 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 Attic-Ionic 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 archaeological-linguistic 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
• Thessalian Boeotian
• Ionic-Attic Group
  • Attica Euboea and colonies in Italy Cyclades Asiatic Ionia
  Arcado-Cypriot 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, non-technical 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 half-truths 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 three-part 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, quasi-empiricism in mathematics, and so-called 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 1500BC-500 BC (ancient India), 1300-1200 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 man-made (designed) systems. Man-made 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 system-based 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.