Negation introduction is a
rule of inference
In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of in ...
, or
transformation rule
In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of in ...
, in the field of
propositional calculus
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...
.
Negation introduction states that if a given antecedent implies both the consequent and its complement, then the antecedent is a contradiction.
[
]
Formal notation
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.
Proof
References
{{DEFAULTSORT:Negation introduction
Propositional calculus
Rules of inference