In logic Logic is the study of arguments. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, and computer science. Logic examines general forms which arguments may take, which forms are valid, and which are fallacies. It is one kind of critical thinking. In philosophy, the study of logic, syntax is anything having to do with formal languages A formal language is a set of words, i.e. finite strings of letters, symbols, or tokens. The set from which these letters are taken is called the alphabet over which the language is defined. A formal language is often defined by means of a formal grammar ; accordingly, words that belong to a formal language are sometimes called well-formed words ( or formal systems In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive (to conclude) one expression from one or more other expressions (premises) antecedently supposed (axioms) or derived (theorems). The axioms and rules may be called a deductive apparatus. A formal system may be formulated and studied for its without regard to any interpretation An interpretation is an assignment of meaning to the symbols of a language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation. The general study of interpretations of formal languages is called formal or meaning The field of semantics is often understood as a branch of linguistics, but non-idealized meaning as a type of semantics is more accurately a branch of psychology and ethics. Meaning in so far is it is objectified by not considering particular situations and the real intentions of speakers and writers examines the ways in which words, phrases, and 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 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 of a language which is concerned with its meaning.

The symbols A symbol is an idea, abstraction or concept, tokens of which may be marks or a configuration of marks which form a particular pattern. Although the term "symbol" in common use refers at some times to the idea being symbolized, and at other times to the marks on a piece of paper or chalkboard which are being used to express that idea; in, formulas In the formal languages used in mathematical logic and computer science, a well-formed formula or simply formula is an idea, abstraction or concept which is expressed using the symbols and formation rules (also called the formal grammar) of a particular formal language. To say that a string of symbols is a wff with respect to a given formal, systems In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive (to conclude) one expression from one or more other expressions (premises) antecedently supposed (axioms) or derived (theorems). The axioms and rules may be called a deductive apparatus. A formal system may be formulated and studied for its, theorems In mathematics, a theorem is a statement which has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms. The derivation of a theorem is often interpreted as a proof of the truth of the resulting expression, but different deductive systems can yield other, proofs 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 a, and interpretations An interpretation is an assignment of meaning to the symbols of a language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation. The general study of interpretations of formal languages is called formal expressed in formal languages are syntactic entities whose properties may be studied without regard to any meaning they may be given, and, in fact, need not be given any.

Syntax is usually associated with the rules (or grammar) governing the composition of texts in a formal language that constitute the well-formed formulas In the formal languages used in mathematical logic and computer science, a well-formed formula or simply formula is an idea, abstraction or concept which is expressed using the symbols and formation rules (also called the formal grammar) of a particular formal language. To say that a string of symbols is a wff with respect to a given formal of a formal system.

In computer science Computer science or computing science is the study of the theoretical foundations of information and computation, and of practical techniques for their implementation and application in computer systems. It is frequently described as the systematic study of algorithmic processes that create, describe, and transform information. Computer science, the term syntax In computer science, the syntax of a programming language is the set of rules that define the combinations of symbols that are considered to be correctly structured programs in that language. The syntax of a language defines its surface form. Text-based programming languages are based on sequences of characters, while visual programming languages refers to the rules governing the composition of meaningful texts in a formal language, such as a programming language A programming language is an artificial language designed to express computations that can be performed by a machine, particularly a computer. Programming languages can be used to create programs that control the behavior of a machine, to express algorithms precisely, or as a mode of human communication, that is, those texts for which it makes sense to define the semantics Semantics is the study of meaning. It typically focuses on the relation between signifiers, such as words, phrases, signs and symbols, and what they stand for or meaning, or otherwise provide an interpretation.

Syntactic entities

Symbols

A symbol is an idea In the most narrow sense, an idea is just whatever is before the mind when one thinks. Very often, ideas are construed as representational images; i.e. images of some object. In other contexts, ideas are taken to be concepts, although abstract concepts do not necessarily appear as images. Many philosophers consider ideas to be a fundamental, abstraction Abstraction is a conceptual process by which higher, more abstract concepts are derived from the usage and classification of literal concepts. An "abstraction" (noun) is a concept that acts as super-categorical noun for all subordinate concepts, and connects any related concepts as a group, field, or category or concept A concept is a cognitive unit of meaning—an abstract idea or a mental symbol sometimes defined as a "unit of knowledge," built from other units which act as a concept's characteristics. A concept is typically associated with a corresponding representation in a language or symbology[citation needed] such as a single meaning of a term, tokens In philosophy and knowledge representation, the type-token distinction is a distinction that separates an abstract concept from the objects which are particular instances of the concept. For example, the particular bicycle in your garage is a token of the type of thing known as "The bicycle." Whereas, the bicycle in your garage is in a of which may be marks or a configuration of marks which form a particular pattern. Symbols of a formal language need not be symbols of anything. For instance there are logical constants In mathematical logic, a logical constant of a language is a symbol that has the same semantic value under every interpretation of . Two important types of logical constants are logical connectives and quantifiers. The equality predicate is also treated as a logical constant in many systems of logic which do not refer to any idea, but rather serve as a form of punctuation in the language (e.g. parentheses). A symbol or string of symbols may comprise a well-formed formula if the formulation is consistent with the formation rules of the language. Symbols of a formal language must be capable of being specified without any reference to any interpretation of them.

Formal language

A formal language is a syntactic entity which consists of a set A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. Although it was invented at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In of finite strings In mathematical logic, more precisely in the theory of formal languages, and in computer science, a string is a sequence of symbols that are chosen from a set or alphabet.[citation needed] of symbols A symbol is an idea, abstraction or concept, tokens of which may be marks or a configuration of marks which form a particular pattern. Although the term "symbol" in common use refers at some times to the idea being symbolized, and at other times to the marks on a piece of paper or chalkboard which are being used to express that idea; in which are its words (usually called its well-formed formulas In the formal languages used in mathematical logic and computer science, a well-formed formula or simply formula is an idea, abstraction or concept which is expressed using the symbols and formation rules (also called the formal grammar) of a particular formal language. To say that a string of symbols is a wff with respect to a given formal). Which strings of symbols are words is determined by fiat by the creator of the language, usually by specifying a set of formation rules 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 . (See. Such a language can be defined without reference A reference, or a reference point, is the intensional use of one thing, a point of reference or reference state, to indicate something else[citation needed]. When reference is intended, what the reference points to is called the referent to any meanings The field of semantics is often understood as a branch of linguistics, but non-idealized meaning as a type of semantics is more accurately a branch of psychology and ethics. Meaning in so far is it is objectified by not considering particular situations and the real intentions of speakers and writers examines the ways in which words, phrases, and of any of its expressions; it can exist before any interpretation An interpretation is an assignment of meaning to the symbols of a language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation. The general study of interpretations of formal languages is called formal is assigned to it – that is, before it has any meaning.

Formation rules

Formation rules are a precise description of which strings In mathematical logic, more precisely in the theory of formal languages, and in computer science, a string is a sequence of symbols that are chosen from a set or alphabet.[citation needed] of symbols A symbol is an idea, abstraction or concept, tokens of which may be marks or a configuration of marks which form a particular pattern. Although the term "symbol" in common use refers at some times to the idea being symbolized, and at other times to the marks on a piece of paper or chalkboard which are being used to express that idea; in are the well-formed formulas In the formal languages used in mathematical logic and computer science, a well-formed formula or simply formula is an idea, abstraction or concept which is expressed using the symbols and formation rules (also called the formal grammar) of a particular formal language. To say that a string of symbols is a wff with respect to a given formal of a formal language. It is synonymous with the set of strings In mathematical logic, more precisely in the theory of formal languages, and in computer science, a string is a sequence of symbols that are chosen from a set or alphabet.[citation needed] over the alphabet An alphabet is a standardized set of letters — basic written symbols or graphemes — each of which roughly represents a phoneme in a spoken language, either as it exists now or as it was in the past. There are other systems, such as logographies, in which each character represents a word, morpheme, or semantic unit, and syllabaries, in which of the formal language which constitute well formed formulas. However, it does not describe their semantics Semantics is the study of meaning. It typically focuses on the relation between signifiers, such as words, phrases, signs and symbols, and what they stand for (i.e. what they mean).

Propositions

A proposition is a sentence In the field of linguistics, a sentence is an expression in natural language, often defined to indicate a grammatical and lexical unit consisting of one or more words that represent distinct concepts. A sentence can include words grouped meaningfully to express a statement, question, exclamation, request or command expressing something true Truth can have a variety of meanings, from the state of being the case, being in accord with a particular fact or reality, being in accord with the body of real things, events, actuality, or fidelity to an original or to a standard, truth "behind" everything, the ontological truth. In archaic usage it could be fidelity, constancy or or false Falsity or falsehood is a perversion of truth originating in the deceitfulness of one party, and culminating in the damage of another party. Falsity is also a measure of the quality or extent of the falseness of something, while a falsehood may also mean simply an incorrect (false) statement, independent of any intention to deceive. A proposition is identified ontologically Ontology (from the Greek ὄν, genitive ὄντος: "of being" and -λογία, -logia: science, study, theory) is the philosophical study of the nature of being, existence or reality in general, as well as the basic categories of being and their relations. Traditionally listed as a part of the major branch of philosophy known as as an idea In the most narrow sense, an idea is just whatever is before the mind when one thinks. Very often, ideas are construed as representational images; i.e. images of some object. In other contexts, ideas are taken to be concepts, although abstract concepts do not necessarily appear as images. Many philosophers consider ideas to be a fundamental, concept A concept is a cognitive unit of meaning—an abstract idea or a mental symbol sometimes defined as a "unit of knowledge," built from other units which act as a concept's characteristics. A concept is typically associated with a corresponding representation in a language or symbology[citation needed] such as a single meaning of a term or abstraction Abstraction is a conceptual process by which higher, more abstract concepts are derived from the usage and classification of literal concepts. An "abstraction" (noun) is a concept that acts as super-categorical noun for all subordinate concepts, and connects any related concepts as a group, field, or category whose token instances In philosophy and knowledge representation, the type-token distinction is a distinction that separates an abstract concept from the objects which are particular instances of the concept. For example, the particular bicycle in your garage is a token of the type of thing known as "The bicycle." Whereas, the bicycle in your garage is in a are patterns of symbols A symbol is an idea, abstraction or concept, tokens of which may be marks or a configuration of marks which form a particular pattern. Although the term "symbol" in common use refers at some times to the idea being symbolized, and at other times to the marks on a piece of paper or chalkboard which are being used to express that idea; in, marks, sounds, or strings In mathematical logic, more precisely in the theory of formal languages, and in computer science, a string is a sequence of symbols that are chosen from a set or alphabet.[citation needed] of words.^[1] Propositions are considered to be syntactic entities and also truthbearers.

Formal theories

A formal theory is a set of sentences in a formal language.

Formal systems

A formal system (also called a logical calculus, or a logical system) consists of a formal language together with a deductive apparatus (also called a deductive system). The deductive apparatus may consist of a set of transformation rules (also called inference rules) or a set of axioms, or have both. A formal system is used to derive one expression from one or more other expressions. Formal systems, like other syntactic entities may be defined without any interpretation given to it (as being, for instance, a system of arithmetic).

Syntactic consequence within a formal system

A formula A is a syntactic consequence^[2][3][4][5] within some formal system of a set Г of formulas if there is a formal proof in formal system of A from the set Г.

Syntactic consequence does not depend on any interpretation of the formal system.^[6]

Syntactic completeness of a formal system

A formal system is syntactically complete^{[7][8][9][10]} (also deductively complete, maximally complete, negation complete or simply complete) iff for each formula A of the language of the system either A or ¬A is a theorem of . In another sense, a formal system is syntactically complete iff no unprovable axiom can be added to it as an axiom without introducing an inconsistency. Truth-functional propositional logic and first-order predicate logic are semantically complete, but not syntactically complete (for example the propositional logic statement consisting of a single variable "a" is not a theorem, and neither is its negation, but these are not tautologies). Gödel's incompleteness theorem shows that no recursive system that is sufficiently powerful, such as the Peano axioms, can be both consistent and complete.

Interpretations

An interpretation of a formal system is the assignment of meanings to the symbols, and truth values to the sentences of a formal system. The study of interpretations is called formal semantics. Giving an interpretation is synonymous with constructing a model. An interpretation is expressed in a metalanguage, which may itself be a formal language, and as such itself is a syntactic entity.

Sources

1. ^ Metalogic, Geoffrey Hunter
2. ^ http://books.google.com/books?id=EYP7uCZIRQYC&pg=PA82&lpg=PA82&dq=syntactic+consequence&source=bl&ots=Ms58438B6w&sig=FE38FCaZpRpAr18gtG7INX4wieM&hl=en&ei=qOy7SoLlEI7KsQPgnYG7BA&sa=X&oi=book_result&ct=result&resnum=6#v=onepage&q=syntactic%20consequence&f=false
3. ^ http://books.google.com/books?id=lXI7AAAAIAAJ&pg=PA1&lpg=PA1&dq=syntactic+consequence&source=bl&ots=8IYWyFYTN-&sig=wrOg75cFxQwn1Uq-8LShBNXf9w0&hl=en&ei=I-y7SpHtLZLotgOsnLHcBQ&sa=X&oi=book_result&ct=result&resnum=10#v=onepage&q=syntactic%20consequence&f=false
4. ^ http://books.google.com/books?id=87BcFLgJmxMC&pg=PA189&lpg=PA189&dq=syntactic+consequence&source=bl&ots=Fn2zomcMZP&sig=8hnJWsJFysNhmWLskICo4IQDYAc&hl=en&ei=I-y7SpHtLZLotgOsnLHcBQ&sa=X&oi=book_result&ct=result&resnum=6#v=onepage&q=syntactic%20consequence&f=false
5. ^ http://www.swif.uniba.it/lei/foldop/foldoc.cgi?syntactic+consequence
6. ^ Hunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California Pres, 1971, p. 75.
7. ^ http://jigpal.oxfordjournals.org/cgi/reprint/11/5/513.pdf
8. ^ http://portal.acm.org/citation.cfm?id=504575
9. ^ http://books.google.com/books?id=CHfgmt4w5QEC&pg=PA236&lpg=PA236&dq=syntactic+completeness&source=bl&ots=WmLkbHF-uj&sig=reo9rQk1NWySJEmRY5JTBbIgkXY&hl=en&ei=TPG7SrjFFJLUsQOJ-encBQ&sa=X&oi=book_result&ct=result&resnum=4#v=onepage&q=syntactic%20completeness&f=false
10. ^ http://www.swif.uniba.it/lei/foldop/foldoc.cgi?syntactic+completeness

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