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In propositional logic, conjunction elimination (also called and elimination, ∧ elimination, [1] or simplification) [2][3][4] is a valid immediate inference, argument form and rule of inference which makes the inference that, if the conjunction A and B is true, then A is true, and B is true. The rule makes it possible to shorten longer proofs ...
Transformation rules. In propositional logic, disjunction elimination[1][2] (sometimes named proof by cases, case analysis, or or elimination) is the valid argument form and rule of inference that allows one to eliminate a disjunctive statement from a logical proof. It is the inference that if a statement implies a statement and a statement ...
The rules allow the expression of conjunctions and disjunctions purely in terms of each other via 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
from line 1 by rule (B) 3. Bp: from line 2 by conjunction elimination 4. BBp: from line 3 by rule (E) 5. B¬Bp: from line 2 by conjunction elimination 6. BBp & B¬Bp: from lines 4 and 5 by conjunction introduction: 7. ¬(BBp & B¬Bp) by rule (F) 8. ¬B(p & ¬Bp) from lines 6 and 7 by reductio ad absurdum, discharging assumption 1
Rules of inference. Implication introduction / elimination (modus ponens) Biconditional introduction / elimination. Conjunction introduction / elimination. Disjunction introduction / elimination. Disjunctive / hypothetical syllogism. Constructive / destructive dilemma. Absorption / modus tollens / modus ponendo tollens.
Associative property. In mathematics, the associative property[1] is a property of some binary operations that means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs.
First-order logic —also called predicate logic, predicate calculus, quantificational logic —is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables.
In propositional logic, tautology is either of two commonly used rules of replacement. [1][2][3] The rules are used to eliminate redundancy in disjunctions and conjunctions when they occur in logical proofs. They are: The principle of idempotency of disjunction: and the principle of idempotency of conjunction: Where " " is a metalogical symbol ...