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In 1968 Shirley Chisholm became the first Black woman elected to the United States Congress. In 1972 she launched a campaign for the Democratic presidential nomination, breaking another political barrier.
MORE →Reflects the personal views, recollections, and perspective of the author, Mike Davis.
This is a personal recollection on the Move fire on May 13, 1985
| Negation introduction | |
|---|---|
| Type | Rule of inference |
| Field | Propositional calculus |
| Statement | If a given antecedent implies both the consequent and its complement, then the antecedent is a contradiction. |
| Symbolic statement | |
Negation introduction is a rule of inference, or transformation rule, in the field of propositional calculus.
Negation introduction states that if a given antecedent implies both the consequent and its complement, then this implies the negated antecedent.[1][2]
This can be written as:
An example of its use would be an attempt to prove two contradictory statements from a single fact. For example, if a person were to state "Whenever I hear the phone ringing I am happy" and then state "Whenever I hear the phone ringing I am not happy", one can infer that the person never hears the phone ringing.
Many proofs by contradiction use negation introduction as reasoning scheme: to prove ¬P, assume for contradiction P, then derive from it two contradictory inferences Q and ¬Q. Since the latter contradiction renders P impossible, ¬P must hold.
With identified as , the principle is as a special case of Frege's theorem, already in minimal logic.
Another derivation makes use of as the curried, equivalent form of . Using this twice, the principle is seen equivalent to the negation of which, via modus ponens and rules for conjunctions, is itself equivalent to the valid noncontradiction principle for .
A classical derivation passing through the introduction of a disjunction may be given as follows:
| Step | Proposition | Derivation |
|---|---|---|
| 1 | Given | |
| 2 | Classical equivalence of the material implication | |
| 3 | Distributivity | |
| 4 | Law of noncontradiction for | |
| 5 | Disjunctive syllogism (3,4) |
Source: Wikipedia. Article content is retrieved live through the MediaWiki API.
Negation introduction is a rule of inference, or transformation rule, in the field of propositional calculus. Negation introduction states that if a given antecedent implies both the consequent and its complement, then this implies the negated antecedent.
In logic, negation, also called the logical not or logical complement, is an operation that takes a proposition P {\displaystyle P} to another proposition "not P {\displaystyle P} ", written ¬ P {\displaystyle \neg P} , ∼ P {\displaystyle {\mathord {\sim }}P} , P ′ {\displaystyle P^{\prime }} or P ¯ {\displaystyle {\overline {P}}} . It is interpreted intuitively as being true when P {\displaystyle P} is false, and false when P {\displaystyle P} is true. For example, if P {\displaystyle P} is "The dog runs", then "not P {\displaystyle P} " is "The dog does not run". An operand of a negation is called a negand or negatum. Negation is a unary logical connective. It may furthermore be applied not only to propositions, but also to notions, truth values, or semantic values more generally. In classical logic, negation is normally identified with the truth function that takes truth to falsity (and vice versa). In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov interpretation, the negation of a proposition P {\displaystyle P} is the proposition whose proofs are the refutations of P {\displaystyle P} .
In propositional logic, biconditional introduction is a valid rule of inference. It allows for one to infer a biconditional from two conditional statements. The rule makes it possible to introduce a biconditional statement into a logical proof. If P → Q {\displaystyle P\to Q} is true, and if Q → P {\displaystyle Q\to P} is true, then one may infer that P ↔ Q {\displaystyle P\leftrightarrow Q} is true. For example, from the statements "if I'm breathing, then I'm alive" and "if I'm alive, then I'm breathing", it can be inferred that "I'm breathing if and only if I'm alive". Biconditional introduction is the converse of biconditional elimination. The rule can be stated formally as: P → Q , Q → P ∴ P ↔ Q {\displaystyle {\frac {P\to Q,Q\to P}{\therefore P\leftrightarrow Q}}} where the rule is that wherever instances of " P → Q {\displaystyle P\to Q} " and " Q → P {\displaystyle Q\to P} " appear on lines of a proof, " P ↔ Q {\displaystyle P\leftrightarrow Q} " can validly be placed on a subsequent line.
Conjunction introduction (often abbreviated simply as conjunction and also called and introduction or adjunction) is a valid rule of inference of propositional logic. The rule makes it possible to introduce a conjunction into a logical proof. It is the inference that if the proposition P {\displaystyle P} is true, and the proposition Q {\displaystyle Q} is true, then the logical conjunction of the two propositions P {\displaystyle P} and Q {\displaystyle Q} is true. For example, if it is true that "it is raining", and it is true that "the cat is inside", then it is true that "it is raining and the cat is inside". The rule can be stated: P , Q ∴ P ∧ Q {\displaystyle {\frac {P,Q}{\therefore P\land Q}}} where the rule is that wherever an instance of " P {\displaystyle P} " and " Q {\displaystyle Q} " appear on lines of a proof, a " P ∧ Q {\displaystyle P\land Q} " can be placed on a subsequent line.
Before the 1921 destruction of Tulsa’s Greenwood District, Black residents had created a remarkable center of business and community life. The district included stores, professional offices, entertainment venues and homes owned by Black citizens. Understanding Greenwood means learning what was built—not only what was burned.
MORE →Madam C.J. Walker
