Answer the following question: Information: - A formal system or logical calculus is any well-defined system of abstract thought based on the model of mathematics. A formal system need not be mathematical as such; for example, Spinoza's "Ethics" imitates the form of Euclid's "Elements".. Spinoza employed Euclidiean elements such as "axioms" or "primitive truths", rules of inferences etc. so that a calculus can be build using these. For nature of such primitive truths, one can consult Tarski's "Concept of truth for a formalized language". - Sequent calculus is, in essence, a style of formal logical argumentation where every line of a proof is a conditional tautology (called a sequent by Gerhard Gentzen) instead of an unconditional tautology. Each conditional tautology is inferred from other conditional tautologies on earlier lines in a formal argument according to rules and procedures of inference, giving a better approximation to the style of natural deduction used by mathematicians than David Hilbert's earlier style of formal logic where every line was an unconditional tautology. There may be more subtle distinctions to be made; for example, there may be non-logical axioms upon which all propositions are implicitly dependent. Then sequents signify conditional theorems in a first-order language rather than conditional tautologies. - Inferences are steps in reasoning, moving from premises to conclusions. Charles Sanders Peirce divided inference into three kinds: deduction, induction, and abduction. Deduction is inference deriving logical conclusions from premises known or assumed to be true, with the laws of valid inference being studied in logic. Induction is inference from particular premises to a universal conclusion. Abduction is inference to the best explanation. - Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems. Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory. These areas share basic results on logic, particularly first-order logic, and definability. In computer science (particularly in the ACM Classification) mathematical logic encompasses additional topics not detailed in this article; see Logic in computer science for those. - Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects. - Mathematics (from Greek "mathma", knowledge, study, learning) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics. - Proof theory is a major branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of the logical system. As such, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature. - Gerhard Karl Erich Gentzen (November 24, 1909 August 4, 1945) was a German mathematician and logician. He made major contributions to the foundations of mathematics, proof theory, especially on natural deduction and sequent calculus. He died in 1945 after the Second World War, because he was deprived of food after being arrested in Prague. - In mathematics, model theory is the study of classes of mathematical structures (e.g. groups, fields, graphs, universes of set theory) from the perspective of mathematical logic. The objects of study are models of theories in a formal language. A set of sentences in a formal language is called a theory; a model of a theory is a structure (e.g. an interpretation) that satisfies the sentences of that theory. - First-order logic also known as first-order predicate calculus and predicate logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a man" one can have expressions in the form "there exists X such that X is Socrates and X is a man" where "there exists" is a quantifier and "X" is a variable. This distinguishes it from propositional logic, which does not use quantifiers. - In logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of inference called "modus ponens" takes two premises, one in the form "If p then q" and another in the form "p", and returns the conclusion "q". The rule is valid with respect to the semantics of classical logic (as well as the semantics of many other non-classical logics), in the sense that if the premises are true (under an interpretation), then so is the conclusion. - Latin (Latin: ) is a classical language belonging to the Italic branch of the Indo-European languages. The Latin alphabet is derived from the Etruscan and Greek alphabets. - Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories. Emphasis on metamathematics (and perhaps the creation of the term itself) owes itself to David Hilbert's attempt to secure the foundations of mathematics in the early part of the 20th Century. Metamathematics provides "a rigorous mathematical technique for investigating a great variety of foundation problems for mathematics and logic" (Kleene 1952, p.59). An important feature of metamathematics is its emphasis on differentiating between reasoning from inside a system and from outside a system. An informal illustration of this is categorizing the proposition "2+2=4" as belonging to mathematics while categorizing the proposition "'2+2=4' is valid" as belonging to metamathematics. - Logic in computer science covers the overlap between the field of logic and that of computer science. The topic can essentially be divided into three main areas: - In propositional logic, modus ponendo ponens (Latin for "the way that affirms by affirming"; generally abbreviated to MP or modus ponens) or implication elimination is a rule of inference. It can be summarized as ""P" implies "Q" and "P" is asserted to be true, so therefore "Q" must be true." The history of "modus ponens" goes back to antiquity. - In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning. This contrasts with Hilbert-style systems, which instead use axioms as much as possible to express the logical laws of deductive reasoning. - Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics. In this latter sense, the distinction between foundations of mathematics and philosophy of mathematics turns out to be quite vague. Foundations of mathematics can be conceived as the study of the basic mathematical concepts (number, geometrical figure, set, function, etc.) and how they form hierarchies of more complex structures and concepts, especially the fundamentally important structures that form the language of mathematics (formulas, theories and their models giving a meaning to formulas, definitions, proofs, algorithms, etc.) also called metamathematical concepts, with an eye to the philosophical aspects and the unity of mathematics. The search for foundations of mathematics is a central question of the philosophy of mathematics; the abstract nature of mathematical objects presents special philosophical challenges. - In mathematics, a theorem is a statement that has been proved on the basis of previously established statements, such as other theoremsand generally accepted statements, such as axioms. A theorem is a logical consequence of the axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally "deductive", in contrast to the notion of a scientific law, which is "experimental". - The term proposition has a broad use in contemporary philosophy. It is used to refer to some or all of the following: the primary bearers of truth-value, the objects of belief and other "propositional attitudes" (i.e., what is believed, doubted, etc.), the referents of that-clauses, and the meanings of declarative sentences. Propositions are the sharable objects of attitudes and the primary bearers of truth and falsity. This stipulation rules out certain candidates for propositions, including thought- and utterance-tokens which are not sharable, and concrete events or facts, which cannot be false. - Theoretical computer science, or TCS, is a division or subset of general computer science and mathematics that focuses on more abstract or mathematical aspects of computing and includes the theory of computation. - In philosophy and logic, an argument is a series of statements typically used to persuade someone of something or to present reasons for accepting a conclusion. The general form of an argument in a natural language is that of premises (typically in the form of propositions, statements or sentences) in support of a claim: the conclusion. The structure of some arguments can also be set out in a formal language, and formally defined "arguments" can be made independently of natural language arguments, as in math, logic, and computer science. - In mathematical logic , the cut rule is an inference rule of sequent calculus . It is a generalisation of the classical modus ponens inference rule . Its meaning is that , if a formula A appears as a conclusion in one proof and an hypothesis in another , then another proof in which the formula A does not appear can be deduced . In the particular case of the modus ponens , for example occurrences of man are eliminated of Every man is mortal , Socrates is a man to deduce Socrates is mortal . - In mathematical logic, a sequent is a very general kind of conditional assertion. After reading the paragraphs above, choose the best answer for the entity that related to 'cut rule' with the relationship of 'instance of'. Choices: - alphabet - approximation - article - assertion - august - branch - century - change - complex - computer - concept - division - feature - field - formal language - formal system - foundation - function - history - inference - language - latin alphabet - line - logic - mathematics - model - natural language - november - order - part - philosophy - predicate - quantity - reasoning - rule - rule of inference - science - sense - series - set - set theory - space - statement - structure - study - style - subset - term - theorem - theory - three - truth - utterance - variable - variety - war - world war
Answer:
rule of inference