- Laws of classical logic
The laws of
classical logic are a small collection of fundamental sentences ofpropositional logic and Boolean algebra, from which may be derived all truesentence s in both of these elementaryformal system s.The syntax of the laws of classical logic includes the:
*Truth value s T, logically true, and F, logically false;
*Sentential variable s "p", "q", and "r", each ranging over {T,F};
*Unary logical operator ¬, "not";
* Binaryconnectives ∧ "and," ∨, "or," and ≡, "logically equivalent ." Where '≡' appears below, Boolean algebra would write '='.Truth table s embody theprinciple of bivalence (included in the table below) and the "arithmetical" laws T∧T≡T, F∨F≡F, T∧F≡F, and T∨F≡T. All other laws of classical logic can be verified usingtruth table s. This is the sense in which no law of logic is truly an axiom.
The Laws of Classical Sentential/Boolean Logic.* Bivalence ¬ T ≡ F ¬ F ≡ T * Involution ¬ ¬ p ≡ p * Idempotence p ∧ p ≡ p p ∨ p ≡ p Identity p ∧ T ≡ p p ∨ F ≡ p (Non-)Contradiction ¬ ( p ∧ ¬ p ) ≡ T Excluded Middle p ∨ ¬ p ≡ T * Absorption p ∧ ( p ∨ q ) ≡ p p ∨ ( p ∧ q ) ≡ p Commutativity p ∧ q ≡ q ∧ p p ∨ q ≡ q ∨ p * DeMorgan's ¬ ( p ∧ q ) ≡ ¬ p ∨ ¬ q ¬ ( p ∨ q ) ≡ ¬ p ∧ ¬ q Associativity p ∧ ( q ∧ r ) ≡ ( p ∧ q ) ∧ r p ∨ ( q ∨ r ) ≡ ( p ∨ q ) ∨ r Distributivity p ∧ ( q ∨ r ) ≡ ( p ∧ q ) ∨ ( p ∧ r ) p ∨ ( q ∧ r ) ≡ ( p ∨ q ) ∧ ( p ∨ r ) '*' The starred principles (bivalency, involution, idempotency, contraction, DeMorgan, and others not shown above, such as the pair p ∨ T ≡ T, p ∧ F ≡ F, called annihilation) are traditionally derived from the remaining six principles, which are deemed axioms. Many axiom sets for sentential logic and
Boolean algebra are known. Hence there are many possible partitions of the above table into axioms and consequences. For example, Huntington's classic axiomatization ofBoolean algebra took identity,commutativity ,excluded middle , anddistributivity as axioms.With three exceptions, each row of the table above consists of a pair of sentences, called a dual pair. Each member of a dual pair differs from the other in two ways. Where one sentence has:
*∧, the other sentence of the pair has ∨;
*¬, the other sentence applies ¬ to each variable and to the whole side of the equation. Then apply involution to all nested ¬.Excluded middle and Noncontradiction together make up a dual pair. Involution is not a dual pair because it isself-dual . This organization into dual pairs embodies the principle ofduality , fundamental toBoolean algebra andclassical logic . Byduality and thefunctional completeness of ¬ and one of ∧ or ∨, the laws of classical logic can be formulated using only one equation from each of the pairs in the table above.Intuitionistic logic accepts all the laws in the above table except, famously,excluded middle .ee also
*
Boolean algebra
*Propositional calculus
*Laws of thought
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