Laws of classical logic

Laws of classical logic

The laws of classical logic are a small collection of fundamental sentences of propositional logic and Boolean algebra, from which may be derived all true sentences in both of these elementary formal systems.

The syntax of the laws of classical logic includes the:
* Truth values T, logically true, and F, logically false;
* Sentential variables "p", "q", and "r", each ranging over {T,F};
*Unary logical operator ¬, "not";
* Binary connectives ∧ "and," ∨, "or," and ≡, "logically equivalent." Where '≡' appears below, Boolean algebra would write '='.

Truth tables embody the principle 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 using truth tables. 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 of Boolean algebra took identity, commutativity, excluded middle, and distributivity 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 is self-dual. This organization into dual pairs embodies the principle of duality, fundamental to Boolean algebra and classical logic. By duality and the functional 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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