NationMaster - Encyclopedia: Laws of logic

These laws of classical logic are valid in propositional logic and any boolean algebra. Some are axioms and others derived with truth tables. The logical operators ¬ 'not', ∧ 'and', ∨ 'or', the values T 'logically true', F 'logically false', and the relation ≡ 'logically equivalent to' are applied to propositions p, q, r. Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. ... The propositional calculus is a formal deduction system whose atomic formulas are propositional variables. ... In formal logic, mathematics and computer science, Boolean algebras, or Boolean lattices, are algebraic structures which capture the essence of the logical operations AND, OR and NOT as well as the corresponding set-theoretic operations intersection, union and complement. ... Truth tables are a type of mathematical table used in logic to determine whether an expression is true or whether an argument is valid. ... In logical calculus, logical operators or logical connectors serve to connect statements into more complicated compound statements. ... Negation, in its most basic sense, changes the truth value of a statement to its opposite. ... AND Logic Gate Logical conjunction (usual symbol and) is a logical operator that results in true if both of the operands are true. ... OR Logic Gate Logical disjunction (usual symbol or) is a logical operator that results in true if either of the operands is true. ... In logic, a truth value, or truth-value, is a value indicating to what extent a statement is true. ... In logic, a truth value, or truth-value, is a value indicating to what extent a statement is true. ... In logic, statements p and q are logically equivalent if they have the same logical content. ... In modern philosophy, logic and linguistics, a proposition is what is asserted as the result of uttering a declarative sentence. ...

Basic Principles of Classical, Propositional and Boolean Logic

* Bivalency	¬ T	≡	F
* Bivalency	¬ F	≡	T
* Involution	¬ ¬ p	≡	p
* Idempotency	p ∧ p	≡	p
* Idempotency	p ∨ p	≡	p
Identity	p ∧ T	≡	p
Identity	p ∨ F	≡	p
(Non-)Contradiction	p ∧ ¬ p	≡	F
(Non-)Contradiction	¬ ( p ∧ ¬ p )	≡	T
Excluded Middle	p ∨ ¬ p	≡	T
Excluded Middle	¬ ( p ∨ ¬ p )	≡	F
* Contraction	p ∧ ( p ∨ q )	≡	p
* Contraction	p ∨ ( p ∧ q )	≡	p
Commutativity	p ∧ q	≡	q ∧ p
Commutativity	p ∨ q	≡	q ∨ p
* DeMorgan's	¬ ( p ∧ q )	≡	¬ p ∨ ¬ q
* DeMorgan's	¬ ( p ∨ q )	≡	¬ p ∧ ¬ q
Associativity	p ∧ ( q ∧ r )	≡	( p ∧ q ) ∧ r
Associativity	p ∨ ( q ∨ r )	≡	( p ∨ q ) ∨ r
Distributivity	p ∧ ( q ∨ r )	≡	( p ∧ q ) ∨ ( p ∧ r )
Distributivity	p ∨ ( q ∧ r )	≡	( p ∨ q ) ∧ ( p ∨ r )

Categories: Logic In logic, the principle of bivalence states that for any proposition P, either P is true or P is false. ... In mathematics, an involution is a function that is its own inverse, so that f(f(x)) = x for all x in the domain of f. ... In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ... An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ... In logic, the law of noncontradiction judges as false any proposition P asserting that both proposition Q and its denial, proposition not-Q, are true at the same time and in the same respect. In the words of Aristotle, One cannot say of something that it is and that it... The law of excluded middle (tertium non datur in Latin) states that for any proposition P, it is true that (P or ~P). ... Idempotency of entailment is a property of logical systems that states that one may derive the same consequences from many instances of a hypothesis as from just one. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In logic, De Morgans laws (or De Morgans theorem) are the two rules of propositional logic, boolean algebra and set theory not (P and Q) = (not P) or (not Q) not (P or Q) = (not P) and (not Q) which allow us to move a negation over a... In mathematics, associativity is a property that a binary operation can have. ... In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ...