Double negation
Encyclopedia
In the theory of logic, double negation is expressed by saying that a proposition A is identical to (equivalent to) not (not-A), or by the formula A = ~~A. Like the Law of Excluded Middle
, this principle when extended to an infinite collection of individuals is disallowed by Intuitionistic logic
. Some writers grant this "law" the status of a Law of thought
.
Law of excluded middle
In logic, the law of excluded middle is the third of the so-called three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is....
, this principle when extended to an infinite collection of individuals is disallowed by Intuitionistic logic
Intuitionism
In the philosophy of mathematics, intuitionism, or neointuitionism , is an approach to mathematics as the constructive mental activity of humans. That is, mathematics does not consist of analytic activities wherein deep properties of existence are revealed and applied...
. Some writers grant this "law" the status of a Law of thought
Law of thought
The laws of thought are fundamental axiomatic rules upon which rational discourse itself is based. The rules have a long tradition in the history of philosophy. They are laws that guide and underlie everyone's thinking, thoughts, expressions, discussions, etc.The three classic laws of thought are...
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- The principium contradictiones of modern logicians (particularly Leibnitz and Kant) in the formula A is not not-A, differs entirely in meaning and application from the Aristotelian proposition [ i.e. Law of Contradiction: not (A and not-A) i.e. ~(A & ~A), or not (( B is A) and (B is not-A))]. This latter refers to the relation between an affirmative and a negative judgment. According to Aristotle, one judgment [B is judged to be an A] contradicts another [B is judged to be a not-A]. The later proposition [ A is not not-A ] refers to the relation between subject and predicate in a single judgment; the predicate contradicts the subject. Aristotle states that one judgment is false when another is true; the later writers [Leibniz and Kant] state that a judgment is in itself and absolutely false, because the predicate contradicts the subject. What the later writers desire is a principle from which it can be known whether certain propositions are in themselves true. From the Aristotelian proposition we cannot immediately infer the truth or falsehood of any particular proposition, but only the impossibility of believing both affirmation and negation at the same time.