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With this premise, we also conclude that q=T, p∨q=T, etc. as shown by columns 9–15. The column-11 operator (IF/THEN), shows Modus ponens rule: when p→q=T and p=T only one line of the truth table (the first) satisfies these two conditions. On this line, q is also true. Therefore, whenever p → q is true and p is true, q must also be true.
In propositional logic, tautology is either of two commonly used rules of replacement. [ 1 ] [ 2 ] [ 3 ] The rules are used to eliminate redundancy in disjunctions and conjunctions when they occur in logical proofs .
Formulas and are logically equivalent if and only if the statement of their material equivalence is a tautology. [ 2 ] The material equivalence of p {\displaystyle p} and q {\displaystyle q} (often written as p ↔ q {\displaystyle p\leftrightarrow q} ) is itself another statement in the same object language as p {\displaystyle p} and q ...
Some of these connectives may be defined in terms of others: for instance, implication, p → q, may be defined in terms of disjunction and negation, as ¬p ∨ q; [74] and disjunction may be defined in terms of negation and conjunction, as ¬(¬p ∧ ¬q). [51]
Many logicians in the early 20th century used the term 'tautology' for any formula that is universally valid, whether a formula of propositional logic or of predicate logic. In this broad sense, a tautology is a formula that is true under all interpretations, or that is logically equivalent to the negation of a contradiction.
Irving Anellis's research shows that C.S. Peirce appears to be the earliest logician (in 1883) to devise a truth table matrix. [4]From the summary of Anellis's paper: [4] In 1997, John Shosky discovered, on the verso of a page of the typed transcript of Bertrand Russell's 1912 lecture on "The Philosophy of Logical Atomism" truth table matrices.
Logical equality is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both operands are false or both operands are true. The truth table of p EQ q (also written as p = q, p ↔ q, Epq, p ≡ q, or p == q) is as follows: The Venn diagram of A EQ B (red part is true)
Tautological consequence can also be defined as ∧ ∧ ... ∧ → is a substitution instance of a tautology, with the same effect. [2]It follows from the definition that if a proposition p is a contradiction then p tautologically implies every proposition, because there is no truth valuation that causes p to be true and so the definition of tautological implication is trivially satisfied.