An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word ἀξίωμα (axíōma), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'.
In logic, the law of non-contradiction (LNC) states that contradictory propositions cannot both be true in the same sense at the same time, e. g. the two propositions "the house is white" and "the house is not white" are mutually exclusive. Formally, this is expressed as the tautology ¬(p ∧ ¬p). For example it is tautologous to say "not both: the house is white and the house is not white" since this results from putting "the house is white" in that formula. The law is not to be confused with the law of excluded middle which states that at least one of two propositions like "the house is white" and "the house is not white" holds.
In logic, the law of excluded middle or the principle of excluded middle states that for every proposition, either this proposition or its negation is true. It is one of the three laws of thought, along with the law of noncontradiction, and the law of identity; however, no system of logic is built on just these laws, and none of these laws provides inference rules, such as modus ponens or De Morgan's laws. The law is also known as the law / principleof the excluded third, in Latin principium tertii exclusi. Another Latin designation for this law is tertium non datur or "no third [possibility] is given". In classical logic, the law is a tautology.
A syllogism is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true.
Friedrich Ludwig Gottlob Frege was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philosophy, concentrating on the philosophy of language, logic, and mathematics. Though he was largely ignored during his lifetime, Giuseppe Peano (1858–1932), Bertrand Russell (1872–1970), and, to some extent, Ludwig Wittgenstein (1889–1951) introduced his work to later generations of philosophers. Frege is widely considered to be the greatest logician since Aristotle, and one of the most profound philosophers of mathematics ever.
Classical logic or Frege–Russell logic is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy.
The history of logic deals with the study of the development of the science of valid inference (logic). Formal logics developed in ancient times in India, China, and Greece. Greek methods, particularly Aristotelian logic as found in the Organon, found wide application and acceptance in Western science and mathematics for millennia. The Stoics, especially Chrysippus, began the development of predicate logic.
In traditional logic, a contradiction occurs when a proposition conflicts either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle's law of noncontradiction states that "It is impossible that the same thing can at the same time both belong and not belong to the same object and in the same respect."
A proposition is a central concept in the philosophy of language, semantics, logic, and related fields, often characterized as the primary bearer of truth or falsity. Propositions are also often characterized as being the kind of thing that declarative sentences denote. For instance the sentence "The sky is blue" denotes the proposition that the sky is blue. However, crucially, propositions are not themselves linguistic expressions. For instance, the English sentence "Snow is white" denotes the same proposition as the German sentence "Schnee ist weiß" even though the two sentences are not the same. Similarly, propositions can also be characterized as the objects of belief and other propositional attitudes. For instance if one believes that the sky is blue, what one believes is the proposition that the sky is blue. A proposition can also be thought of as a kind of idea: Collins Dictionary has a definition for proposition as "a statement or an idea that people can consider or discuss whether it is true."
In logic and formal semantics, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to formal logic that began with Aristotle and was developed further in ancient history mostly by his followers, the Peripatetics. It was revived after the third century CE by Porphyry's Isagoge.
In the philosophy of mathematics, logicism is a programme comprising one or more of the theses that – for some coherent meaning of 'logic' – mathematics is an extension of logic, some or all of mathematics is reducible to logic, or some or all of mathematics may be modelled in logic. Bertrand Russell and Alfred North Whitehead championed this programme, initiated by Gottlob Frege and subsequently developed by Richard Dedekind and Giuseppe Peano.
The identity of indiscernibles is an ontological principle that states that there cannot be separate objects or entities that have all their properties in common. That is, entities x and y are identical if every predicate possessed by x is also possessed by y and vice versa. It states that no two distinct things can be exactly alike, but this is intended as a metaphysical principle rather than one of natural science. A related principle is the indiscernibility of identicals, discussed below.
Begriffsschrift is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book.
The principle of sufficient reason states that everything must have a reason or a cause. The principle was articulated and made prominent by Gottfried Wilhelm Leibniz, with many antecedents, and was further used and developed by Arthur Schopenhauer and William Hamilton.
The laws of thought are fundamental axiomatic rules upon which rational discourse itself is often considered to be based. The formulation and clarification of such rules have a long tradition in the history of philosophy and logic. Generally they are taken as laws that guide and underlie everyone's thinking, thoughts, expressions, discussions, etc. However, such classical ideas are often questioned or rejected in more recent developments, such as intuitionistic logic, dialetheism and fuzzy logic.
In propositional logic, a propositional formula is a type of syntactic formula which is well formed. If the values of all variables in a propositional formula are given, it determines a unique truth value. A propositional formula may also be called a propositional expression, a sentence, or a sentential formula.
In many philosophies of logic, statements are categorized into different logical qualities based on how they go about saying what they say. Doctrines of logical quality are an attempt to answer the question: "How many qualitatively different ways are there of saying something?" Aristotle answers, two: you can affirm something of something or deny something of something. Since Frege, the normal answer in the West, is only one, assertion, but what is said, the content of the claim, can vary. For Frege asserting the negation of a claim serves roughly the same role as denying a claim does in Aristotle. Other Western logicians such as Kant and Hegel answer, ultimately three; you can affirm, deny or make merely limiting affirmations, which transcend both affirmation and denial. In Indian logic, four logical qualities have been the norm, and Nāgārjuna is sometimes interpreted as arguing for five.
The Foundations of Arithmetic is a book by Gottlob Frege, published in 1884, which investigates the philosophical foundations of arithmetic. Frege refutes other idealist and materialist theories of number and develops his own platonist theory of numbers. The Grundlagen also helped to motivate Frege's later works in logicism.
The mathematical concept of a function dates from the 17th century in connection with the development of calculus; for example, the slope of a graph at a point was regarded as a function of the x-coordinate of the point. Functions were not explicitly considered in antiquity, but some precursors of the concept can perhaps be seen in the work of medieval philosophers and mathematicians such as Oresme.
