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Duality Principle (Boolean Algebra)
In propositional logic and Boolean algebra, there is a duality between conjunction and disjunction, also called the duality principle. It is the most widely known example of duality in logic. The duality consists in these metalogical theorems: * In classical propositional logic, the connectives for conjunction and disjunction can be defined in terms of each other, and consequently, only one of them needs to be taken as primitive. * If \varphi^ is used as notation to designate the result of replacing every instance of conjunction with disjunction, and every instance of disjunction with conjunction (e.g. p \land q with q \lor p, or vice-versa), in a given formula \varphi, and if \overline is used as notation for replacing every sentence-letter in \varphi with its negation (e.g., p with \neg p), and if the symbol \models is used for semantic consequence and ⟚ for semantical equivalence between logical formulas, then it is demonstrable that \varphi^ ⟚ \neg \overline, and also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Propositional Logic
The propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called ''first-order'' propositional logic to contrast it with System F, but it should not be confused with first-order logic. It deals with propositions (which can be true or false) and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives representing the truth functions of conjunction, disjunction, implication, biconditional, and negation. Some sources include other connectives, as in the table below. Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or quantifiers. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Completeness
In Mathematical logic, logic, a functionally complete set of logical connectives or Boolean function, Boolean operators is one that can be used to express all possible truth tables by combining members of the Set (mathematics), set into a Boolean expression.. ("Complete set of logical connectives").. ("[F]unctional completeness of [a] set of logical operators"). A well-known complete set of connectives is . Each of the singleton (mathematics), singleton sets and is functionally complete. However, the set is incomplete, due to its inability to express NOT. A gate (or set of gates) that is functionally complete can also be called a universal gate (or a universal set of gates). In a context of propositional logic, functionally complete sets of connectives are also called (''expressively'') ''adequate''.. (Defines "expressively adequate", shortened to "adequate set of connectives" in a section heading.) From the point of view of digital electronics, functional completeness means t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjunctive Normal Form
In Boolean algebra, a formula is in conjunctive normal form (CNF) or clausal normal form if it is a conjunction of one or more clauses, where a clause is a disjunction of literals; otherwise put, it is a product of sums or an AND of ORs. In automated theorem proving, the notion "''clausal normal form''" is often used in a narrower sense, meaning a particular representation of a CNF formula as a set of sets of literals. Definition A logical formula is considered to be in CNF if it is a conjunction of one or more disjunctions of one or more literals. As in disjunctive normal form (DNF), the only propositional operators in CNF are or (\vee), and (\and), and not (\neg). The ''not'' operator can only be used as part of a literal, which means that it can only precede a propositional variable. The following is a context-free grammar for CNF: : ''CNF'' \, \to \, ''Disjunct'' \, \mid \, ''Disjunct'' \, \land \, ''CNF'' : ''Disjunct'' \, \to \, ''Literal'' \, \mid\, ''Literal'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disjunctive Normal Form
In boolean logic, a disjunctive normal form (DNF) is a canonical normal form of a logical formula consisting of a disjunction of conjunctions; it can also be described as an OR of ANDs, a sum of products, or in philosophical logic a ''cluster concept''. As a normal form, it is useful in automated theorem proving. Definition A logical formula is considered to be in DNF if it is a disjunction of one or more conjunctions of one or more literals. A DNF formula is in full disjunctive normal form if each of its variables appears exactly once in every conjunction and each conjunction appears at most once (up to the order of variables). As in conjunctive normal form (CNF), the only propositional operators in DNF are and (\wedge), or (\vee), and not (\neg). The ''not'' operator can only be used as part of a literal, which means that it can only precede a propositional variable. The following is a context-free grammar for DNF: : ''DNF'' \, \to \, ''Conjunct'' \, \mid \, ''Conjunc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof By Contrapositive
Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a construct in proof theory * Mathematical proof, a convincing demonstration that some mathematical statement is necessarily true * Proof complexity, computational resources required to prove statements * Proof procedure, method for producing proofs in proof theory * Proof theory, a branch of mathematical logic that represents proofs as formal mathematical objects * Statistical proof, demonstration of degree of certainty for a hypothesis Law and philosophy * Evidence, information which tends to determine or demonstrate the truth of a proposition * Evidence (law), tested evidence or a legal proof * Legal burden of proof, duty to establish the truth of facts in a trial * Philosophic burden of proof, obligation on a party in a dispute to provide ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Negation
In logic, negation, also called the logical not or logical complement, is an operation (mathematics), operation that takes a Proposition (mathematics), proposition P to another proposition "not P", written \neg P, \mathord P, P^\prime or \overline. It is interpreted intuitively as being true when P is false, and false when P is true. For example, if P is "Spot runs", then "not P" is "Spot does not run". An operand of a negation is called a ''negand'' or ''negatum''. Negation is a unary operation, unary logical connective. It may furthermore be applied not only to propositions, but also to notion (philosophy), notions, truth values, or interpretation (logic), semantic values more generally. In classical logic, negation is normally identified with 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 proofs are the re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Biconditional
In logic and mathematics, the logical biconditional, also known as material biconditional or equivalence or bidirectional implication or biimplication or bientailment, is the logical connective used to conjoin two statements P and Q to form the statement "P if and only if Q" (often abbreviated as "P iff Q"), where P is known as the ''antecedent (logic), antecedent'', and Q the ''consequent''. Nowadays, notations to represent equivalence include \leftrightarrow,\Leftrightarrow,\equiv. P\leftrightarrow Q is logically equivalent to both (P \rightarrow Q) \land (Q \rightarrow P) and (P \land Q) \lor (\neg P \land \neg Q) , and the XNOR gate, XNOR (exclusive NOR) Logical connective, Boolean operator, which means "both or neither". Semantically, the only case where a logical biconditional is different from a material conditional is the case where the hypothesis (antecedent) is false but the conclusion (consequent) is true. In this case, the result is true for the conditional, but ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Morgan's Laws
In propositional calculus, propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both Validity (logic), valid rule of inference, rules of inference. They are named after Augustus De Morgan, a 19th-century British mathematician. The rules allow the expression of Logical conjunction, conjunctions and Logical disjunction, disjunctions purely in terms of each other via logical negation, negation. The rules can be expressed in English as: * The negation of "A and B" is the same as "not A or not B". * The negation of "A or B" is the same as "not A and not B". or * The Complement (set theory), complement of the union of two sets is the same as the intersection of their complements * The complement of the intersection of two sets is the same as the union of their complements or * not (A or B) = (not A) and (not B) * not (A and B) = (not A) or (not B) where "A or B" is an "inclusive or" meaning ''at least' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disjunctive Normal Form
In boolean logic, a disjunctive normal form (DNF) is a canonical normal form of a logical formula consisting of a disjunction of conjunctions; it can also be described as an OR of ANDs, a sum of products, or in philosophical logic a ''cluster concept''. As a normal form, it is useful in automated theorem proving. Definition A logical formula is considered to be in DNF if it is a disjunction of one or more conjunctions of one or more literals. A DNF formula is in full disjunctive normal form if each of its variables appears exactly once in every conjunction and each conjunction appears at most once (up to the order of variables). As in conjunctive normal form (CNF), the only propositional operators in DNF are and (\wedge), or (\vee), and not (\neg). The ''not'' operator can only be used as part of a literal, which means that it can only precede a propositional variable. The following is a context-free grammar for DNF: : ''DNF'' \, \to \, ''Conjunct'' \, \mid \, ''Conjunc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denoted by 1 and 0, whereas in elementary algebra the values of the variables are numbers. Second, Boolean algebra uses logical operators such as Logical conjunction, conjunction (''and'') denoted as , disjunction (''or'') denoted as , and negation (''not'') denoted as . Elementary algebra, on the other hand, uses arithmetic operators such as addition, multiplication, subtraction, and division. Boolean algebra is therefore a formal way of describing logical operations in the same way that elementary algebra describes numerical operations. Boolean algebra was introduced by George Boole in his first book ''The Mathematical Analysis of Logic'' (1847), and set forth more fully in his ''An Investigation of the Laws of Thought'' (1854). According to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sentence Letter
In mathematical logic, a propositional variable (also called a sentence letter, sentential variable, or sentential letter) is an input variable (that can either be true or false) of a truth function. Propositional variables are the basic building-blocks of propositional formulas, used in propositional logic and higher-order logics. Uses Formulas in logic are typically built up recursively from some propositional variables, some number of logical connectives, and some logical quantifiers. Propositional variables are the atomic formulas of propositional logic, and are often denoted using capital roman letters such as P, Q and R. ;Example In a given propositional logic, a formula can be defined as follows: * Every propositional variable is a formula. * Given a formula ''X'', the negation ''¬X'' is a formula. * Given two formulas ''X'' and ''Y'', and a binary connective ''b'' (such as the logical conjunction ∧), the expression ''(X b Y)'' is a formula. (Note the parentheses.) T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjunction (logic)
In logic, mathematics and linguistics, ''and'' (\wedge) is the truth-functional operator of conjunction or logical conjunction. The logical connective of this operator is typically represented as \wedge or \& or K (prefix) or \times or \cdot in which \wedge is the most modern and widely used. The ''and'' of a set of operands is true if and only if ''all'' of its operands are true, i.e., A \land B is true if and only if A is true and B is true. An operand of a conjunction is a conjunct. Beyond logic, the term "conjunction" also refers to similar concepts in other fields: * In natural language, the denotation of expressions such as English " and"; * In programming languages, the short-circuit and control structure; * In set theory, intersection. * In lattice theory, logical conjunction ( greatest lower bound). Notation And is usually denoted by an infix operator: in mathematics and logic, it is denoted by a "wedge" \wedge (Unicode ), \& or \times; in electronics, \cdot; ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
