Proof theory is a branch of mathematical logic Mathematical logic is a subfield of mathematics with close connections to computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the that represents proofs In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single exception. An unproved proposition that is believed to as formal mathematical objects In mathematics and its philosophy, a mathematical object is an abstract object arising in mathematics, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures In computer science, a data structure is a particular way of storing and organizing data in a computer so that it can be used efficiently such as plain lists, boxed lists, or trees, which are constructed according to the axioms In traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject to necessary decision. Therefore, its truth is taken for granted, and serves as a starting point for deducing and inferring other truths and rules of inference In logic, a transformation rule is a syntactic rule used in a formal system which may be interpreted as a valid rule of inference for constructing true propositions. Rules of inference, along with any axioms or axiom schemata it uses to derive valid formulas, comprise the deductive system of the formal system of the logical system. As such, proof theory is syntactic In logic, syntax is anything having to do with formal languages or formal systems without regard to any interpretation or meaning given to them. Syntax is concerned with the rules used for constructing, or transforming the symbols and words of a language, as contrasted with the semantics of a language which is concerned with its meaning in nature, in contrast to model theory In mathematics, model theory is the study of mathematical structures such as groups, fields, graphs, or even universes of set theory, using tools from mathematical logic. A structure that gives meaning to the sentences of a formal language is called a model for the language. If a model for a language moreover satisfies a particular sentence or, which is semantic Formal semantics is the study of the semantics, or interpretations, of formal and also natural languages. A formal language can be defined apart from any interpretation of it. This is done by designating a set of symbols and a set of formation rules (also called a formal grammar) which determine which strings of symbols are well-formed formulas in nature. Together with model theory In mathematics, model theory is the study of mathematical structures such as groups, fields, graphs, or even universes of set theory, using tools from mathematical logic. A structure that gives meaning to the sentences of a formal language is called a model for the language. If a model for a language moreover satisfies a particular sentence or, axiomatic set theory Set theory is the branch of mathematics that studies sets, which 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, and recursion theory Computability theory, also called recursion theory, is a branch of mathematical logic that originated in the 1930s with the study of computable functions and Turing degrees. The field has grown to include the study of generalized computability and definability. In these areas, recursion theory overlaps with proof theory and effective descriptive, proof theory is one of the so-called four pillars of the foundations of mathematics Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory. The search for foundations of mathematics is also a central question of the philosophy of mathematics: On what ultimate basis can mathematical.[1]
Proof theory is important in philosophical logic Philosophical logic is the study of the more specifically philosophical aspects of logic. The term contrasts with philosophy of logic, metalogic, and mathematical logic; and since the development of mathematical logic in the late nineteenth century, it has come to include most of those topics traditionally treated by logic in general.[citation, where the primary interest is in the idea of a proof-theoretic semantics Proof-theoretic semantics is an approach to the semantics of logic that attempts to locate the meaning of propositions and logical connectives not in terms of interpretations, as in Tarskian approaches to semantics, but in the role that the proposition or logical connective plays within the system of inference, an idea which depends upon technical ideas in structural proof theory In mathematical logic, structural proof theory is the subdiscipline of proof theory that studies proof calculi that support a notion of analytic proof to be feasible.
Contents |
History
Although the formalisation of logic was much advanced by the work of such figures as Gottlob Frege Friedrich Ludwig Gottlob Frege was a German mathematician who became a logician and philosopher. He was one of the founders of modern logic, and made major contributions to the foundations of mathematics. As a philosopher, he is generally considered to be the father of analytic philosophy, for his writings on the philosophy of language and, Giuseppe Peano Giuseppe Peano was an Italian mathematician, whose work was of exceptional philosophical value. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The standard axiomatization of the natural numbers is named in his honor. As part of this axiomatization effort, he, Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, socialist, pacifist, and social critic. He spent most of his life in England; he was born in Wales where he also died, aged 97, and Richard Dedekind Julius Wilhelm Richard Dedekind was a German mathematician who did important work in abstract algebra (particularly ring theory), algebraic number theory and the foundations of the real numbers, the story of modern proof theory is often seen as being established by David Hilbert David Hilbert /ˈdaːfɪt ˈhɪlbʌt/ was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. He discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of, who initiated what is called Hilbert's program In mathematics, Hilbert's program, formulated by German mathematician David Hilbert in the 1920s, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies. As a solution, Hilbert proposed to ground all existing theories in the foundations of mathematics Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory. The search for foundations of mathematics is also a central question of the philosophy of mathematics: On what ultimate basis can mathematical. Kurt Gödel Kurt Gödel (German pronunciation: [kʊʁt ˈɡøːdl̩] ; April 28, 1906, Brno, Moravia – January 14, 1978, Princeton, New Jersey, USA) was an Austrian-American logician, mathematician and philosopher. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century,'s seminal work on proof theory first advanced, then refuted this program: his completeness theorem Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. It was first proved by Kurt Gödel in 1929 initially seemed to bode well for Hilbert's aim of reducing all mathematics to a finitist formal system; then his incompleteness theorems Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems for mathematics. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The two results are widely interpreted as showing that showed that this is unattainable. All of this work was carried out with the proof calculi called the Hilbert systems.
In parallel, the foundations of structural proof theory In mathematical logic, structural proof theory is the subdiscipline of proof theory that studies proof calculi that support a notion of analytic proof were being founded. Jan Łukasiewicz Jan Łukasiewicz (21 December 1878 – 13 February 1956) was a Polish logician and philosopher born in Lwów (Lemberg in German), Galicia, Austria–Hungary (now Lviv, Ukraine). His work centred on analytical philosophy and mathematical logic. He thought innovatively about traditional propositional logic, the principle of non-contradiction and the suggested in 1926 that one could improve on Hilbert systems as a basis for the axiomatic presentation of logic if one allowed the drawing of conclusions from assumptions in the inference rules of the logic. In response to this Stanisław Jaśkowski (1929) and Gerhard Gentzen Gerhard Karl Erich Gentzen was a German mathematician and logician. He had his major contributions in the foundation 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 (1934) independently provided such systems, called calculi of natural deduction 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 the axiomatic systems which instead use axioms as much as possible to express the logical laws of deductive reasoning, with Gentzen's approach introducing the idea of symmetry between the grounds for asserting propositions, expressed in introduction rules, and the consequences of accepting propositions in the elimination rules, an idea that has proved very important in proof theory[2]. Gentzen (1934) further introduced the idea of the sequent calculus In proof theory and mathematical logic, sequent calculus is a widely known family of formal systems sharing a certain style of inference and certain formal properties. The first sequent calculi, systems LK and LJ, were introduced by Gerhard Gentzen in 1934, as a tool for studying natural deduction in first-order logic. Gentzen's so-called ", a calculus advanced in a similar spirit that better expressed the duality of the logical connectives[3], and went on to make fundamental advances in the formalisation of intuitionistic logic, and provide the first combinatorial proof of the consistency of Peano arithmetic In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental. Together, the presentation of natural deduction and the sequent calculus introduced the fundamental idea of analytic proof In mathematical analysis, an analytical proof is a proof of a theorem in analysis that only makes use of methods from analysis, and which does not make use of results from geometry. The term was first used by Bernard Bolzano, who first provided a non-analytic proof of his intermediate value theorem and then, several years later provided proof of to proof theory,
Formal and informal proof
Main article: Formal proof A formal proof or derivation is a finite sequence of sentences each of which is an axiom or follows from the preceding sentences in the sequence by a rule of inference. The last sentence in the sequence is a theorem of a formal system. The notion of theorem is not in general effective, therefore there may be no method by which we can always find aThe informal proofs of everyday mathematical practice are unlike the formal proofs of proof theory. They are rather like high-level sketches that would allow an expert to reconstruct a formal proof at least in principle, given enough time and patience. For most mathematicians, writing a fully formal proof is too pedantic and long-winded to be in common use.
Formal proofs are constructed with the help of computers in interactive theorem proving Interactive theorem proving is the field of computer science and mathematical logic concerned with tools to develop formal proofs by man-machine collaboration. This involves some sort of proof assistant: an interactive proof editor, or other interface, with which a human can guide the search for proofs, the details of which are stored in, and some. Significantly, these proofs can be checked automatically, also by computer. (Checking formal proofs is usually simple, whereas finding proofs (automated theorem proving Automated theorem proving or automated deduction, currently the most well-developed subfield of automated reasoning (AR), is the proving of mathematical theorems by a computer program) is generally hard.) An informal proof in the mathematics literature, by contrast, requires weeks of peer review Peer review is a generic term that is used to describe a process of self-regulation by a profession or a process of evaluation involving qualified individuals with the related field. Peer review methods are employed to maintain standards, improve performance, and provide credibility to be checked, and may still contain errors.
Kinds of proof calculi
The three most well-known styles of proof calculi In mathematical logic, a proof calculus corresponds to a family of formal systems that use a common style of formal inference for its inference rules. The specific inference rules of a member of such a family characterize the theory of a logic are:
- The Hilbert calculi
- The natural deduction calculi
- The sequent calculi In proof theory and mathematical logic, sequent calculus is a widely known family of formal systems sharing a certain style of inference and certain formal properties. The first sequent calculi, systems LK and LJ, were introduced by Gerhard Gentzen in 1934, as a tool for studying natural deduction in first-order logic. Gentzen's so-called "
Each of these can give a complete and axiomatic formalization of propositional In mathematical logic, a propositional calculus or logic is a formal system in which formulas of a formal language may be interpreted as representing propositions. A system of inference rules and axioms allows certain formulas to be derived, called theorems; which may be interpreted as true propositions. The series of formulas which is constructed or predicate logic In mathematical logic, predicate logic is the generic term for symbolic formal systems like first-order logic, second-order logic, many-sorted logic or infinitary logic. This formal system is distinguished from other systems in that its formulas contain variables which can be quantified. Two common quantifiers are the existential ∃ and universal of either the classical Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. They are characterised by a number of properties: or intuitionistic Intuitionistic logic, or constructive logic, is the symbolic logic system originally developed by Arend Heyting to provide a formal basis for Brouwer's programme of intuitionism. The system preserves justification, rather than truth, across transformations yielding derived propositions. From a practical point of view, there is also a strong flavour, almost any modal logic Modal logic is a type of formal logic that extends the standards of formal logic to include the elements of modality . Modals qualify the truth of a judgment. For example, if it is true that "John is happy," we might qualify this statement by saying that "John is very happy," in which case the term "very" would be a, and many substructural logics In mathematical logic, in particular in connection with proof theory, a number of substructural logics have been introduced, as systems of propositional calculus that are weaker than the conventional one. They differ in having fewer structural rules available: the concept of structural rule is based on the sequent presentation, rather than the, such as relevance logic Relevance logic, also called relevant logic, is a kind of non-classical logic requiring the antecedent and consequent of implications be relevantly related. They may be viewed as a family of substructural or modal logics or linear logic Linear logic is a substructural logic proposed by Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the dualities of the former with many of the constructive properties of the latter. Although the logic has also been studied for its own sake, more broadly, ideas from linear logic have been influential in fields such. Indeed it is unusual to find a logic that resists being represented in one of these calculi.
Consistency proofs
Main article: Consistency proof In logic, a consistent theory is one that does not contain a contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if it has a model; this is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the termAs previously mentioned, the spur for the mathematical investigation of proofs in formal theories was Hilbert's program. The central idea of this program was that if we could give finitary proofs of consistency for all the sophisticated formal theories needed by mathematicians, then we could ground these theories by means of a metamathematical argument, which shows that all of their purely universal assertions (more technically their provable sentences In mathematics, Hilbert's program, formulated by German mathematician David Hilbert in the 1920s, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies. As a solution, Hilbert proposed to ground all existing theories) are finitarily true; once so grounded we do not care about the non-finitary meaning of their existential theorems, regarding these as pseudo-meaningful stipulations of the existence of ideal entities.
The failure of the program was induced by Kurt Gödel Kurt Gödel (German pronunciation: [kʊʁt ˈɡøːdl̩] ; April 28, 1906, Brno, Moravia – January 14, 1978, Princeton, New Jersey, USA) was an Austrian-American logician, mathematician and philosopher. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century,'s incompleteness theorems, which showed that any ω-consistent theory that is sufficiently strong to express certain simple arithmetic truths, cannot prove its own consistency, which on Gödel's formulation is a sentence.
Much investigation has been carried out on this topic since, which has in particular led to:
- Refinement of Gödel's result, particularly J. Barkley Rosser's refinement, weakening the above requirement of ω-consistency to simple consistency;
- Axiomatisation of the core of Gödel's result in terms of a modal language, provability logic;
- Transfinite iteration of theories, due to Alan Turing and Solomon Feferman;
- The recent discovery of self-verifying theories, systems strong enough to talk about themselves, but too weak to carry out the diagonal argument that is the key to Gödel's unprovability argument.
See also Mathematical logic
Structural proof theory
Main article: Structural proof theoryStructural proof theory is the subdiscipline of proof theory that studies proof calculi that support a notion of analytic proof. The notion of analytic proof was introduced by Gentzen for the sequent calculus; there the analytic proofs are those that are cut-free. His natural deduction calculus also supports a notion of analytic proof, as shown by Dag Prawitz. The definition is slightly more complex: we say the analytic proofs are the normal forms, which are related to the notion of normal form in term rewriting. More exotic proof calculi such as Jean-Yves Girard's proof nets also support a notion of analytic proof.
Structural proof theory is connected to type theory by means of the Curry-Howard correspondence, which observes a structural analogy between the process of normalisation in the natural deduction calculus and beta reduction in the typed lambda calculus. This provides the foundation for the intuitionistic type theory developed by Per Martin-Löf, and is often extended to a three way correspondence, the third leg of which are the cartesian closed categories.
Proof-theoretic semantics
Main articles: proof-theoretic semantics and logical harmonyIn linguistics, type-logical grammar, categorial grammar and Montague grammar apply formalisms based on structural proof theory to give a formal natural language semantics.
Tableau systems
Main article: Method of analytic tableauxAnalytic tableaux apply the central idea of analytic proof from structural proof theory to provide decision procedures and semi-decision procedures for a wide range of logics.
Ordinal analysis
Main article: Ordinal analysisOrdinal analysis is a powerful technique for providing combinatorial consistency proofs for theories formalising arithmetic and analysis.
Logics from proof analysis
Main article: substructural logicSeveral important logics have come from insights into logical structure arising in structural proof theory.
See also
Notes
| Logic portal |
- ^ E.g., Wang (1981), pp. 3–4, and Barwise (1978).
- ^ Prawitz (1965).
- ^ Girard, Lafont, and Taylor (1988).
|
Thu, 05 Aug 2010 05:49:33 GMT+00:00
National Post A scientist who touts his own private research as proof of his theory doesn't just raise red flags, it sets off an AWOOGA-AWOOGA bomb-shelter klaxon. ...
317px x 541px | 76.50kB
[source page]
Two Three Four There are two ideas supporting the universe yes and no There are three ideas yes no and maybe There are four ideas yes no maybe maybe not All of these are true Simultaneoulsy
Casey Kazan Daily Galaxy Editorial Staff
Fri, 03 Sep 2010 07:48:00 GM
For those who did not get what Hawking meant by "spontaneous creation", I recomend you look up quantum fluctuations, false vacua, and the anthropic string . theory. land scape to shed light on the issue. ... Dark Energy is . proof. of anti- gravity and a dumb Hawking statement for more fame and money. Likely dark energy expansion cools the universe to absolute zero, where Bose-Einstein condensates by condensation forms larger black holes. other big-bangs are fractal size- scaled ...
Q. hasnt it been announced on every major cable network. Note: if you say why hasnt God been announced, answer: because we will never see him in our physical form, because he dwells outside of our physical universe, somewhere we wont discover if we had billions of years to do so. and if the Atheist King Richard Dawkins says that you are working on it, what is the timeline for the proof?
Asked by Atheists Nightmare - Tue Sep 29 00:37:17 2009 - - 23 Answers - 0 Comments
A. There is a voluminous amount of evidence for evolution. This evidence has been growing for more than 100 years. The evidence did not all come about all at once. As they are discovered, the bits and pieces of the evidence are occasionally reported if the networks consider it interesting enough. The theory of evolution says absolutely nothing about the existence or non-existence of a (emphasizing "a") god. On the other hand, the theory of evolution is contrary to the myths that certain primitives living thousands of years ago attributed to their particular god.
Answered by Gilgamesh - Tue Sep 29 19:00:40 2009
