Logic in the context of Logical proof

Ancient Greek philosophy arose in the 6th century BC. Philosophy was used to make sense of the world using reason. It dealt with a wide variety of subjects, including astronomy, epistemology, mathematics, political philosophy, ethics, metaphysics, ontology, logic, biology, rhetoric and aesthetics. Greek philosophy continued throughout the Hellenistic period and later evolved into Roman philosophy.

Greek philosophy has influenced much of Western culture since its inception, and can be found in many aspects of public education. Alfred North Whitehead once claimed: "The safest general characterization of the European philosophical tradition is that it consists of a series of footnotes to Plato". Clear, unbroken lines of influence lead from ancient Greek and Hellenistic philosophers to Roman philosophy, early Islamic philosophy, medieval scholasticism, the European Renaissance and the Age of Enlightenment.

Reason is the capacity of consciously applying logic by drawing valid conclusions from new or existing information, with the aim of seeking truth. It is associated with such characteristically human activities as philosophy, religion, science, language, and mathematics, and is normally considered to be a distinguishing ability possessed by humans. Reason is sometimes referred to as rationality, although the latter is more about its application.

Reasoning involves using more-or-less rational processes of thinking and cognition to extrapolate from one's existing knowledge to generate new knowledge, and involves the use of one's intellect. The field of logic studies the ways in which humans can use formal reasoning to produce logically valid arguments and true conclusions. Reasoning may be subdivided into forms of logical reasoning, such as deductive reasoning, inductive reasoning, and abductive reasoning.

Rhetoric is the art of persuasion. It is one of the three ancient arts of discourse (trivium) along with grammar and logic/dialectic. As an academic discipline within the humanities, rhetoric aims to study the techniques that speakers or writers use to inform, persuade, and motivate their audiences. Rhetoric also provides heuristics for understanding, discovering, and developing arguments for particular situations.

Aristotle defined rhetoric as "the faculty of observing in any given case the available means of persuasion", and since mastery of the art was necessary for victory in a case at law, for passage of proposals in the assembly, or for fame as a speaker in civic ceremonies, he called it "a combination of the science of logic and of the ethical branch of politics". Aristotle also identified three persuasive audience appeals: logos, pathos, and ethos. The five canons of rhetoric, or phases of developing a persuasive speech, were first codified in classical Rome: invention, arrangement, style, memory, and delivery.

Alfred North Whitehead OM FRS FBA (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He created the philosophical school known as process philosophy, which has been applied in a wide variety of disciplines, including ecology, theology, education, physics, biology, economics, and psychology.

In his early career Whitehead wrote primarily on mathematics, logic, and physics. He wrote the three-volume Principia Mathematica (1910–1913), with his former student Bertrand Russell. Principia Mathematica is considered one of the twentieth century's most important works in mathematical logic, and placed 23rd in a list of the top 100 English-language nonfiction books of the twentieth century by Modern Library.

Aelius Galenus or Claudius Galenus (Greek: Κλαύδιος Γαληνός; September 129 – c. 216 AD), often anglicized as Galen (/ˈɡeɪlən/) or Galen of Pergamon, was a Roman and Greek physician, surgeon, and philosopher. Considered to be one of the most accomplished of all medical researchers of antiquity, Galen influenced the development of various scientific disciplines, including anatomy, physiology, pathology, pharmacology, and neurology, as well as philosophy and logic.

The son of Aelius Nicon, a wealthy Greek architect with scholarly interests, Galen received a comprehensive education that prepared him for a successful career as a physician and philosopher. Born in the ancient city of Pergamon (present-day Bergama, Turkey), Galen traveled extensively, exposing himself to a wide variety of medical theories and discoveries before settling in Rome, where he served prominent members of Roman society and eventually was given the position of personal physician to several emperors.

Deductive reasoning is the process of drawing valid inferences. An inference is valid if its conclusion follows logically from its premises, meaning that it is impossible for the premises to be true and the conclusion to be false. For example, the inference from the premises "all men are mortal" and "Socrates is a man" to the conclusion "Socrates is mortal" is deductively valid. An argument is sound if it is valid and all its premises are true. One approach defines deduction in terms of the intentions of the author: they have to intend for the premises to offer deductive support to the conclusion. With the help of this modification, it is possible to distinguish valid from invalid deductive reasoning: it is invalid if the author's belief about the deductive support is false, but even invalid deductive reasoning is a form of deductive reasoning.

Deductive logic studies under what conditions an argument is valid. According to the semantic approach, an argument is valid if there is no possible interpretation of the argument whereby its premises are true and its conclusion is false. The syntactic approach, by contrast, focuses on rules of inference, that is, schemas of drawing a conclusion from a set of premises based only on their logical form. There are various rules of inference, such as modus ponens and modus tollens. Invalid deductive arguments, which do not follow a rule of inference, are called formal fallacies. Rules of inference are definitory rules and contrast with strategic rules, which specify what inferences one needs to draw in order to arrive at an intended conclusion.

A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning that establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning that establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work.

Proofs employ logic expressed in mathematical symbols, along with natural language that usually admits some ambiguity. In most mathematical literature, proofs are written in terms of rigorous informal logic. Purely formal proofs, written fully in symbolic language without the involvement of natural language, 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, oral traditions in the mainstream mathematical community or in other cultures. The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.

Positivism is a philosophical school that holds that all genuine knowledge is either true by definition or positive – meaning a posteriori facts derived by reason and logic from sensory experience. Other ways of knowing, such as intuition, introspection, or religious faith, are rejected or considered meaningless.

Although the positivist approach has been a recurrent theme in the history of Western thought, modern positivism was first articulated in the early 19th century by Auguste Comte. His school of sociological positivism holds that society, like the physical world, operates according to scientific laws. After Comte, positivist schools arose in logic, psychology, economics, historiography, and other fields of thought. Generally, positivists attempted to introduce scientific methods to their respective fields. Since the turn of the 20th century, positivism, although still popular, has declined under criticism within the social sciences by antipositivists and critical theorists, among others, for its alleged scientism, reductionism, overgeneralizations, and methodological limitations. Positivism also exerted an unusual influence on Kardecism.

Science is a systematic discipline that builds and organises knowledge in the form of testable hypotheses and predictions about the universe. Modern science is typically divided into two – or three – major branches: the natural sciences, which study the physical world, and the social sciences, which study individuals and societies. While referred to as the formal sciences, the study of logic, mathematics, and theoretical computer science are typically regarded as separate because they rely on deductive reasoning instead of the scientific method as their main methodology. Meanwhile, applied sciences are disciplines that use scientific knowledge for practical purposes, such as engineering and medicine.

The history of science spans the majority of the historical record, with the earliest identifiable predecessors to modern science dating to the Bronze Age in Egypt and Mesopotamia (c. 3000–1200 BCE). Their contributions to mathematics, astronomy, and medicine entered and shaped the Greek natural philosophy of classical antiquity and later medieval scholarship, whereby formal attempts were made to provide explanations of events in the physical world based on natural causes; while further advancements, including the introduction of the Hindu–Arabic numeral system, were made during the Golden Age of India and Islamic Golden Age. The recovery and assimilation of Greek works and Islamic inquiries into Western Europe during the Renaissance revived natural philosophy, which was later transformed by the Scientific Revolution that began in the 16th century as new ideas and discoveries departed from previous Greek conceptions and traditions. The scientific method soon played a greater role in the acquisition of knowledge, and in the 19th century, many of the institutional and professional features of science began to take shape, along with the changing of "natural philosophy" to "natural science".

A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true or apparently true premises, leads to a seemingly self-contradictory or a logically unacceptable conclusion. A paradox usually involves contradictory-yet-interrelated elements that exist simultaneously and persist over time. They result in "persistent contradiction between interdependent elements" leading to a lasting "unity of opposites".

In logic, many paradoxes exist that are invalid arguments, yet are nevertheless valuable in promoting critical thinking, while other paradoxes have revealed errors in definitions that were assumed to be rigorous, and have caused axioms of mathematics and logic to be re-examined. One example is Russell's paradox, which questions whether a "list of all lists that do not contain themselves" would include itself and showed that attempts to found set theory on the identification of sets with properties or predicates were flawed. Others, such as Curry's paradox, cannot be easily resolved by making foundational changes in a logical system.

Existence is the state of having being or reality in contrast to nonexistence and nonbeing. Existence is often contrasted with essence: the essence of an entity is its essential features or qualities, which can be understood even if one does not know whether the entity exists.

Ontology is the philosophical discipline studying the nature and types of existence. Singular existence is the existence of individual entities while general existence refers to the existence of concepts or universals. Entities present in space and time have concrete existence in contrast to abstract entities, like numbers and sets. Other distinctions are between possible, contingent, and necessary existence and between physical and mental existence. The common view is that an entity either exists or not with nothing in between, but some philosophers say that there are degrees of existence, meaning that some entities exist to a higher degree than others.

Islamic philosophy is philosophy that emerges from the Islamic tradition. Two terms traditionally used in the Islamic world are sometimes translated as philosophy—falsafa (lit. 'philosophy'), which refers to philosophy as well as logic, mathematics, and physics; and kalam (lit. 'speech'), which refers to a rationalist form of Scholastic Islamic theology which includes the schools of Ash'arism, Maturidism and Mu'tazilism.

Early Islamic philosophy began with al-Kindi in the 2nd century of the Islamic calendar (early 9th century CE) and declined with Ibn Rushd (Averroes) in the 6th century AH (late 12th century CE), broadly coinciding with the period known as the Islamic Golden Age. The death of Ibn Rushd effectively marked the end of a specific discipline of Islamic philosophy usually called the Islamic peripatetic school, and philosophical activity declined significantly in the west of the Islamic world, including al-Andalus and the Maghreb.

The problem of universals is an ancient question from metaphysics that has inspired a range of philosophical topics and disputes: "Should the properties an object has in common with other objects, such as color and shape, be considered to exist beyond those objects? And if a property exists separately from objects, what is the nature of that existence?"

The problem of universals relates to various inquiries closely related to metaphysics, logic, and epistemology, as far back as Plato and Aristotle, in efforts to define the mental connections humans make when understanding a property such as shape or color to be the same in nonidentical objects.

Natural science or empirical science is a branch of science concerned with the description, understanding, and prediction of natural phenomena, based on empirical evidence from observation and experimentation. Mechanisms such as peer review and reproducibility of findings are used to try to ensure the validity of scientific advances.

Natural science can be divided into two main branches: life science and physical science. Life science is alternatively known as biology. Physical science is subdivided into physics, astronomy, Earth science, and chemistry. These branches of natural science may be further divided into more specialized branches, also known as fields. As empirical sciences, natural sciences use tools from the formal sciences, such as mathematics and logic, converting information about nature into measurements that can be explained as clear statements of the "laws of nature".

Aristotelianism (/ˌærɪstəˈtiːliənɪzəm/ ARR-i-stə-TEE-lee-ə-niz-əm) is a philosophical tradition inspired by the work of Aristotle, usually characterized by deductive logic and an analytic inductive method in the study of natural philosophy and metaphysics. It covers the treatment of the social sciences under a system of natural law. It answers why-questions by a scheme of four causes, including purpose or teleology, and emphasizes virtue ethics. Aristotle and his school wrote tractates on physics, biology, metaphysics, logic, ethics, aesthetics, poetry, theatre, music, rhetoric, psychology, linguistics, economics, politics, and government. Any school of thought that takes one of Aristotle's distinctive positions as its starting point can be considered "Aristotelian" in the widest sense. This means that different Aristotelian theories (e.g. in ethics or in ontology) may not have much in common as far as their actual content is concerned besides their shared reference to Aristotle.

In Aristotle's time, philosophy included natural philosophy, which preceded the advent of modern science during the Scientific Revolution. The works of Aristotle were initially defended by the members of the Peripatetic school and later on by the Neoplatonists, who produced many commentaries on Aristotle's writings. In the Islamic Golden Age, Avicenna and Averroes translated the works of Aristotle into Arabic and under them, along with philosophers such as Al-Kindi and Al-Farabi, Aristotelianism became a major part of early Islamic philosophy.

Deliberation is a process of thoughtfully weighing options, for example prior to voting. Deliberation emphasizes the use of logic and reason as opposed to power-struggle, creativity, or dialogue. Group decisions are generally made after deliberation through a vote or consensus of those involved.

In legal settings a jury famously uses deliberation because it is given specific options, like guilty or not guilty, along with information and arguments to evaluate. In "deliberative democracy", the aim is for both elected officials and the general public to use deliberation rather than power-struggle as the basis for their vote.