Formal system

Last updated

A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A formal system is essentially an "axiomatic system". [1]

Contents

In 1921, David Hilbert proposed to use such a system as the foundation for the knowledge in mathematics. [2] A formal system may represent a well-defined system of abstract thought.

The term formalism is sometimes a rough synonym for formal system, but it also refers to a given style of notation, for example, Paul Dirac's bra–ket notation.

Background

Each formal system uses primitive symbols (which collectively form an alphabet) to finitely construct a formal language from a set of axioms through inferential rules of formation.

The system thus consists of valid formulas built up through finite combinations of the primitive symbols—combinations that are formed from the axioms in accordance with the stated rules. [3]

More formally, this can be expressed as the following:

  1. A finite set of symbols, known as the alphabet, which concatenate formulas, so that a formula is just a finite string of symbols taken from the alphabet.
  2. A grammar consisting of rules to form formulas from simpler formulas. A formula is said to be well-formed if it can be formed using the rules of the formal grammar. It is often required that there be a decision procedure for deciding whether a formula is well-formed.
  3. A set of axioms, or axiom schemata, consisting of well-formed formulas.
  4. A set of inference rules. A well-formed formula that can be inferred from the axioms is known as a theorem of the formal system.

Recursive

A formal system is said to be recursive (i.e. effective) or recursively enumerable if the set of axioms and the set of inference rules are decidable sets or semidecidable sets, respectively.

Inference and entailment

The entailment of the system by its logical foundation is what distinguishes a formal system from others which may have some basis in an abstract model. Often the formal system will be the basis for or even identified with a larger theory or field (e.g. Euclidean geometry) consistent with the usage in modern mathematics such as model theory.[ clarification needed ]

Formal language

A formal language is a language that is defined by a formal system. Like languages in linguistics, formal languages generally have two aspects:

  • the syntax of a language is what the language looks like (more formally: the set of possible expressions that are valid utterances in the language) studied in formal language theory
  • the semantics of a language are what the utterances of the language mean (which is formalized in various ways, depending on the type of language in question)

In computer science and linguistics usually only the syntax of a formal language is considered via the notion of a formal grammar. A formal grammar is a precise description of the syntax of a formal language: a set of strings. The two main categories of formal grammar are that of generative grammars, which are sets of rules for how strings in a language can be generated, and that of analytic grammars (or reductive grammar, [4] [5] ) which are sets of rules for how a string can be analyzed to determine whether it is a member of the language. In short, an analytic grammar describes how to recognize when strings are members in the set, whereas a generative grammar describes how to write only those strings in the set.

In mathematics, a formal language is usually not described by a formal grammar but by (a) natural language, such as English. Logical systems are defined by both a deductive system and natural language. Deductive systems in turn are only defined by natural language (see below).

Deductive system

A deductive system, also called a deductive apparatus or a logic, consists of the axioms (or axiom schemata) and rules of inference that can be used to derive theorems of the system. [6]

Such deductive systems preserve deductive qualities in the formulas that are expressed in the system. Usually the quality we are concerned with is truth as opposed to falsehood. However, other modalities, such as justification or belief may be preserved instead.

In order to sustain its deductive integrity, a deductive apparatus must be definable without reference to any intended interpretation of the language. The aim is to ensure that each line of a derivation is merely a syntactic consequence of the lines that precede it. There should be no element of any interpretation of the language that gets involved with the deductive nature of the system.

An example of deductive system is first order predicate logic.

Logical system

A logical system or language (not be confused with the kind of "formal language" discussed above which is described by a formal grammar), is a deductive system (see section above; most commonly first order predicate logic) together with additional (non-logical) axioms and a semantics [ disputed ]. According to model-theoretic interpretation, the semantics of a logical system describe whether a well-formed formula is satisfied by a given structure. A structure that satisfies all the axioms of the formal system is known as a model of the logical system. A logical system is sound if each well-formed formula that can be inferred from the axioms is satisfied by every model of the logical system. Conversely, a logic system is complete if each well-formed formula that is satisfied by every model of the logical system can be inferred from the axioms.

An example of a logical system is Peano arithmetic.

History

Early logic systems includes Indian logic of Pāṇini, syllogistic logic of Aristotle, propositional logic of Stoicism, and Chinese logic of Gongsun Long (c. 325–250 BCE) . In more recent times, contributors include George Boole, Augustus De Morgan, and Gottlob Frege. Mathematical logic was developed in 19th century Europe.

Formalism

Hilbert's program

David Hilbert instigated a formalist movement that was eventually tempered by Gödel's incompleteness theorems.

QED manifesto

The QED manifesto represented a subsequent, as yet unsuccessful, effort at formalization of known mathematics.

Examples

Examples of formal systems include:

Variants

The following systems are variations of formal systems[ clarification needed ].

Proof system

Formal proofs are sequences of well-formed formulas (or wff for short). For a wff to qualify as part of a proof, it might either be an axiom or be the product of applying an inference rule on previous wffs in the proof sequence. The last wff in the sequence is recognized as a theorem.

The point of view that generating formal proofs is all there is to mathematics is often called formalism . David Hilbert founded metamathematics as a discipline for discussing formal systems. Any language that one uses to talk about a formal system is called a metalanguage . The metalanguage may be a natural language, or it may be partially formalized itself, but it is generally less completely formalized than the formal language component of the formal system under examination, which is then called the object language , that is, the object of the discussion in question.

Once a formal system is given, one can define the set of theorems which can be proved inside the formal system. This set consists of all wffs for which there is a proof. Thus all axioms are considered theorems. Unlike the grammar for wffs, there is no guarantee that there will be a decision procedure for deciding whether a given wff is a theorem or not. The notion of theorem just defined should not be confused with theorems about the formal system, which, in order to avoid confusion, are usually called metatheorems.

See also

Related Research Articles

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 Greek axíōma (ἀξίωμα) 'that which is thought worthy or fit' or 'that which commends itself as evident.'

Formal language Words whose letters are taken from an alphabet and are well-formed according to a specific set of rules

In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules.

First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—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, while x is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic.

Gödels completeness theorem Fundamental theorem in mathematical logic

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.

Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives. Propositions that contain no logical connectives are called atomic propositions.

Mathematical logic, also called formal logic, is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, philosophy, 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.

Theorem In mathematics, a statement that has been proved

In mathematics and logic, a theorem is a non-self-evident statement that has been proven to be true, either on the basis of generally accepted statements such as axioms or on the basis of previously established statements such as other theorems. A theorem is hence a logical consequence of the axioms, with a proof of the theorem being a logical argument which establishes its truth through the inference rules of a deductive system. As a result, 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.

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.

In mathematical logic, sequent calculus is, in essence, a style of formal logical argumentation where every line of a proof is a conditional tautology 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 natural style of 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.

Metalogic is the study of the metatheory of logic. Whereas logic studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems. Logic concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derived about the languages and systems that are used to express truths.

In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language. A formal language can be identified with the set of formulas in the language.

Syntax (logic) Rules used for constructing, or transforming the symbols and words of a language

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 logic and mathematics, a formal proof or derivation is a finite sequence of sentences, each of which is an axiom, an assumption, or follows from the preceding sentences in the sequence by a rule of inference. It differs from a natural language argument in that it is rigorous, unambiguous and mechanically verifiable. If the set of assumptions is empty, then the last sentence in a formal proof is called a theorem of the formal system. The notion of theorem is not in general effective, therefore there may be no method by which we can always find a proof of a given sentence or determine that none exists. The concepts of Fitch-style proof, sequent calculus and natural deduction are generalizations of the concept of proof.

Object theory is a theory in mathematical logic concerning objects and the statements that can be made about objects.

In mathematical logic, a theory is a set of sentences in a formal language. In most scenarios, a deductive system is first understood from context, after which an element of a theory is then called a theorem of the theory. In many deductive systems there is usually a subset that is called "the set of axioms" of the theory , in which case the deductive system is also called an "axiomatic system". By definition, every axiom is automatically a theorem. A first-order theory is a set of first-order sentences (theorems) recursively obtained by the inference rules of the system applied to the set of axioms.

Metamath is a formal language and an associated computer program for archiving, verifying, and studying mathematical proofs. Several databases of proved theorems have been developed using Metamath covering standard results in logic, set theory, number theory, algebra, topology and analysis, among others.

Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics and to a lesser extent computer science. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct and incorrect inferences. Logicians study the criteria for the evaluation of arguments.

In logic, especially mathematical logic, a Hilbert system, sometimes called Hilbert calculus, Hilbert-style deductive system or Hilbert–Ackermann system, is a type of system of formal deduction attributed to Gottlob Frege and David Hilbert. These deductive systems are most often studied for first-order logic, but are of interest for other logics as well.

In mathematical logic, a judgment or assertion is a statement or enunciation in the metalanguage. For example, typical judgments in first-order logic would be that a string is a well-formed formula, or that a proposition is true. Similarly, a judgment may assert the occurrence of a free variable in an expression of the object language, or the provability of a proposition. In general, a judgment may be any inductively definable assertion in the metatheory.

In mathematical logic, formation rules are rules for describing which strings of symbols formed from the alphabet of a formal language are syntactically valid within the language. These rules only address the location and manipulation of the strings of the language. It does not describe anything else about a language, such as its semantics. .

References

  1. "Formal system, ENCYCLOPÆDIA BRITANNICA".
  2. "Hilbert's Program, Stanford Encyclopedia of Philosophy".
  3. Encyclopædia Britannica, Formal system definition, 2007.
  4. Reductive grammar: (computer science) A set of syntactic rules for the analysis of strings to determine whether the strings exist in a language. "Sci-Tech Dictionary McGraw-Hill Dictionary of Scientific and Technical Terms" (6th ed.). McGraw-Hill.[ unreliable source? ] About the Author Compiled by The Editors of the McGraw-Hill Encyclopedia of Science & Technology (New York, NY) an in-house staff who represents the cutting-edge of skill, knowledge, and innovation in science publishing.
  5. "There are two classes of formal-language definition compiler-writing schemes. The productive grammar approach is the most common. A productive grammar consists primarrly of a set of rules that describe a method of generating all possible strings of the language. The reductive or analytical grammar technique states a set of rules that describe a method of analyzing any string of characters and deciding whether that string is in the language." "The TREE-META Compiler-Compiler System: A Meta Compiler System for the Univac 1108 and General Electric 645, University of Utah Technical Report RADC-TR-69-83. C. Stephen Carr, David A. Luther, Sherian Erdmann" (PDF). Retrieved 5 January 2015.
  6. Hunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California Pres, 1971

Further reading