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negation: not propositional logic, Boolean algebra: The statement is true if and only if A is false. A slash placed through another operator is the same as placed in front. The prime symbol is placed after the negated thing, e.g. ′ [2]
Propositional logic deals with statements, which are defined as declarative sentences having truth value. [29] [1] Examples of statements might include: Wikipedia is a free online encyclopedia that anyone can edit. London is the capital of England. All Wikipedia editors speak at least three languages.
An example: we are given the conditional fact that if it is a bear, then it can swim. Then, all 4 possibilities in the truth table are compared to that fact. If it is a bear, then it can swim — T; If it is a bear, then it can not swim — F; If it is not a bear, then it can swim — T because it doesn’t contradict our initial fact.
" In this case, unlike the last example, the inverse of the statement is true. The converse is "If a polygon has four sides, then it is a quadrilateral." Again, in this case, unlike the last example, the converse of the statement is true. The negation is "There is at least one quadrilateral that does not have four sides.
De Morgan's laws represented with Venn diagrams.In each case, the resultant set is the set of all points in any shade of blue. In propositional logic and Boolean algebra, De Morgan's laws, [1] [2] [3] also known as De Morgan's theorem, [4] are a pair of transformation rules that are both valid rules of inference.
Examples: The column-14 operator (OR), shows Addition rule : when p =T (the hypothesis selects the first two lines of the table), we see (at column-14) that p ∨ q =T. We can see also that, with the same premise, another conclusions are valid: columns 12, 14 and 15 are T.
As a further example, negation can be defined in terms of NAND and can also be defined in terms of NOR. Algebraically, classical negation corresponds to complementation in a Boolean algebra, and intuitionistic negation to pseudocomplementation in a Heyting algebra. These algebras provide a semantics for classical and intuitionistic logic.
Equivalently, if P is true or Q is true and P is false, then Q is true. The name "disjunctive syllogism" derives from its being a syllogism, a three-step argument, and the use of a logical disjunction (any "or" statement.) For example, "P or Q" is a disjunction, where P and Q are called the statement's disjuncts.