The principle of explosion is a logical rule of inference. According to the rule, from a set of premises in which a sentence A and its negation ¬A are both true (i.e., a contradiction is true), any sentence B may be inferred.
https://rationalwiki.org/wiki/Principle_of_explosion
Truth table for P ∧ ¬PIn symbolic logic, the principle of explosion can be expressed schematically in the following way:
P, ¬P ⊢ Q For any statements P and Q, if P and not-P are both true, then it logically follows that Q is true.
Step --- Proposition --- Derivation
1 -------- P ------------ Assumption
2 ------- ¬P ------------ Assumption
3 -------- P ∨ Q ------- Disjunction introduction (1)
4 -------- Q ------------ Disjunctive syllogism (3,2)
https://en.wikipedia.org/wiki/Principle ... sion#Proof
P----¬P----P ∧ ¬P
T-----F------F
F-----T------F
With arithmetic we know that we must perform the operations in a
specific order or we get the wrong answer. (2 * 3) + (4 * 5) = (6 + 20)
We resolve the inner operations before proceeding. If we don't do this
same thing in logic we get absurd results.
When the text says P, ¬P ⊢ Q it failed to resolve P ∧ ¬P ⊢ False
(according to the above truth table) before moving to the next step.
This is a key aspect of the absurd results. It is not a logic error. It is
the divergence of logic from correct reasoning. When two premises
contradict each other (in correct reasoning) they resolve to false.
When we resolve P ∨ ¬P to False then it becomes clear that Disjunction
introduction cannot be performed (False ∨ Q) ⊢ Q transforms the unknown
truth value of Q to true. Thus Disjunction introduction is not a truth
preserving operation in this case.