If any explanation of a formula is true, it is called tautology (eternal truth).
Mathematical logic aims to derive as many tautologies as possible by using limited axioms. In addition, tautology also has important applications in the field of computer lexical analysis.
1. Definition
Given a propositional formula, if the corresponding truth value is always T (true) no matter how the components are assigned, the propositional formula is called tautology or eternal truth formula.
Let a be an arbitrary propositional formula, and if a is true under its various assignments, it is called tautology.
Logical tautology is a statement that is always true regardless of the truth of its components. For example, the statement that "all crows are black or not all crows are black" is tautological, because it holds true no matter what color crows are. Formal expression is a proposition that "crows are all black". It is also true if X: X is used or not, because no matter whether X is true or not, there is a disjunctive term that makes the whole proposition true.
Regardless of the true value of its components, statements that are always false are called contradictions.
Eternal truth and eternal fallacy are mutually negative.
2. Correlation Theorem
Theorem 1: The conjunction or disjunction of any two tautology is still tautology.
Theorem 2: A tautology in which the same component is replaced by an arbitrary formula is still a tautology.
Theorem 3: Let A and B be two propositional formulas. A and B are logically equivalent if and only if the double conditional proposition "A is and only if B" holds.
Theorem 4: Let A, B and C be compound formulas. If A contains B, A is tautology, then B is tautology.
Theorem 5: If A contains B and B contains C, A contains C, that is, the implication relation is transitive.
Step 3 look for tautology
The simplest way to find tautology in Boolean algebra is to use truth tables. However, with the increase of the number of variables involved, the size of the truth table increases by a power of 2, which is not conducive to the tautology of four or more variables, so simplification and algebra become more useful.
4. Computer fields
As a logical word, tautology is widely used in the computer field. In the field of lexical analysis of natural language processing, tautology is often used as the standard of logical judgment. Tautology is often generalized in computer field, and it is often combined with fuzzy implication operator of binary logic in discrete mathematics. In recent years, it has been successfully applied to many fields, such as fuzzy control, approximate reasoning, word calculation, fuzzy image processing and so on, which has attracted extensive attention of scholars. Among them, the generalized tautology related to implication operators has become the focus of current research. At present, people gradually realize the important role of generalized tautology in the theory and application of fuzzy logic.