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, a logical connective (also called a logical operator) is a symbol 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 or word A word is the smallest free form in a language, in contrast to a morpheme, which is the smallest unit of meaning. A word may consist of only one morpheme (e.g. wolf), but a single morpheme may not be able to exist as a free form (e.g. the English plural morpheme -s) used to connect two or more sentences 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 (of either a formal 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 a natural In the philosophy of language, a natural language is any language which arises in an unpremeditated fashion as the result of the innate facility for language possessed by the human intellect. A natural language is typically used for communication, and may be spoken, signed, or written. Natural language is distinguished from constructed languages language) in a grammatically valid 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 way, such that the compound sentence produced has a truth value In logic and mathematics, a logical value, also called a truth value, is a value indicating the relation of a proposition to truth dependent on the respective truth values of the original sentences.
Each logical connective can be expressed as a function The mathematical concept of a function expresses the intuitive idea that one quantity completely determines another quantity (the value, or the output). A function assigns a unique value to each input of a specified type. The argument and the value may be real numbers, but they can also be elements from any given sets: the domain and the codomain, called a truth function In mathematical logic, a truth function is a function from a set of truth values to truth-values. Classically the domain and range of a truth function are {truth,falsehood}, but they may have any number of truth-values, including an infinity of them. For this reason, logical connectives are sometimes called truth-functional connectives. The most common logical connectives are binary connectives (also called dyadic connectives) which join two sentences whose truth values can be thought of as the function's operands 1) In computing an operand is the part of a computer instruction which specifies what data is to be manipulated or operated on, whilst at the same time represeting the data itself. A computer instruction describes an operation such as add or multiply X, while the operand specify on which X to operate as well as the value of X. Also commonly, negation In logic and mathematics, negation is an operation on propositions. For example, in classical logic negation is normally interpreted by the truth function that takes truth to falsity and vice versa. In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov interpretation, the negation of a proposition P is the proposition whose is considered to be a unary connective.
Logical connectives along with quantifiers Quantification has several distinct senses. In mathematics and empirical science, it is the act of counting and measuring that maps human sense observations and experiences into members of some set of numbers. Quantification in this sense is fundamental to the scientific method are the two main types of 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 used in 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 such as propositional logic 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 and 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.
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In language
Natural language
In the grammar of natural languages two sentences may be joined by a grammatical conjunction In grammar, a conjunction is a part of speech that connects two words, phrases or clauses together. This definition may overlap with that of other parts of speech, so what constitutes a "conjunction" should be defined for each language. In general, a conjunction is an invariable grammatical particle, and it may or may not stand between to form a grammatically compound sentence In the English language, a compound sentence is composed of at least two independent clauses. It does not require a dependent clause. The clauses are joined by a coordinating conjunction , a correlative conjunction (with or without a comma), or a semicolon that functions as a conjunction. A conjunction can be used to make a compound sentence. The. Some but not all such grammatical conjunctions are truth functions. For example, consider the following sentences:
- A: Jack went up the hill.
- B: Jill went up the hill.
- C: Jack went up the hill and Jill went up the hill.
- D: Jack went up the hill so Jill went up the hill.
The words and and so are grammatical conjunctions joining the sentences (A) and (B) to form the compound sentences (C) and (D). The and in (C) is a logical connective, since the truth of (C) is completely determined by (A) and (B): it would make no sense to affirm (A) and (B) but deny (C). However so in (D) is not a logical connective, since it would be quite reasonable to affirm (A) and (B) but deny (D): perhaps, after all, Jill went up the hill to fetch a pail of water, not because Jack had gone up the Hill at all.
Various English words and word pairs express truth functions, and some of them are synonymous. Examples (with the name of the relationship in parentheses) are:
- "and" (conjunction)
- "or" (inclusive or exclusive disjunction)
- "implies" (implication)
- "if...then" (implication)
- "if and only if" (equivalence)
- "only if" (implication)
- "just in case" (equivalence)
- "but" (conjunction)
- "however" (conjunction)
- "not both" (NAND)
- "neither...nor" (NOR)
The word "not" (negation) and the phrases "it is false that" (negation) and "it is not the case that" (negation) also express a logical connective – even though they are applied to a single statement, and do not connect two statements.
Formal languages
In formal languages, truth functions are represented by unambiguous symbols; these can be exactly defined by means of truth tables A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—to compute the functional values of logical expressions on each of their functional arguments, that is, on each combination of values taken by their logical variables . In particular, truth tables. There are 16 binary truth tables, and so 16 different logical connectives which connect exactly two statements, that can be defined. Not all of them are in common use. These symbols are called "truth-functional connectives", "logical connectives", "logical operators" or "propositional operators". See well-formed formula 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 for the rules which allow new well-formed formulas to be constructed by joining other well-formed formulas using truth-functional connectives.
Venn diagrams Venn diagrams or set diagrams are diagrams that show all hypothetically possible logical relations between a finite collection of sets . Venn diagrams were conceived around 1880 by John Venn. They are used in many fields, including set theory, probability, logic, statistics, and computer science illustrate the logical connective limitation of all quantifiers Quantification has several distinct senses. In mathematics and empirical science, it is the act of counting and measuring that maps human sense observations and experiences into members of some set of numbers. Quantification in this sense is fundamental to the scientific method to a fixed domain of discourse The domain of discourse, sometimes called the universe of discourse, logical discourse, or simply discourse, is an analytic tool used in deductive logic, especially predicate logic. It indicates the relevant set of entities that are being dealt with by quantifiers in a formal language.
Logical connectives can be used to link more than two statements. A more technical definition is that an "n-ary logical connective" is a function The mathematical concept of a function expresses the intuitive idea that one quantity completely determines another quantity (the value, or the output). A function assigns a unique value to each input of a specified type. The argument and the value may be real numbers, but they can also be elements from any given sets: the domain and the codomain which assigns truth values "true" or "false" to n-tuples of truth values.
Common logical connectives
Commonly used logical connectives include:
- Negation (not) In logic and mathematics, negation is an operation on propositions. For example, in classical logic negation is normally interpreted by the truth function that takes truth to falsity and vice versa. In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov interpretation, the negation of a proposition P is the proposition whose (¬ or ~)
- Conjunction (and) In logic and mathematics, logical conjunction or and is a two-place logical connective that has the value true if both of its operands are true, otherwise a value of false (, &, or · )
- Disjunction (or) In logic and mathematics, or, also known as logical disjunction or inclusive disjunction, is a logical operator that results in true whenever one or more of its operands are true. E.g. in this context, "A or B" is true if A is true, or if B is true, or if both A and B are true. In grammar, or is a coordinating conjunction. In ordinary ( or ∨ )
- Material implication (if...then) The material conditional, also known as the material implication or truth functional conditional, expresses a property of certain conditionals in logic. In propositional logic, it expresses a binary truth function from truth-values to truth-values. In predicate logic, it can be viewed as a subset relation between the extension of predicates (, or )
- Biconditional (if and only if) (iff) (xnor) In logic and mathematics, the logical biconditional is a logical operator connecting two statements to assert "p if and only if q", where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). The operator is denoted using a doubleheaded arrow (↔), an equality sign (=), an equivalence sign (≡), or EQV. It is (, , or = )
For example, the meaning of the statements it is raining and I am indoors is transformed when the two are combined with logical connectives:
- It is raining and I am indoors (P Q)
- If it is raining, then I am indoors (P Q)
- It is raining if I am indoors (Q P)
- It is raining if and only if I am indoors (P Q)
- It is not raining (¬P)
For statement P = It is raining and Q = I am indoors.
Fri, 13 Aug 2010 07:55:10 GMT+00:00
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