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The R-diagrams for p and q are shown below, respectively: For more than two statements, the four spaces formed by the intersection of lines p and q must be subdivided into more lines. In the case of r, a single upward sloping line (/) is added in each of the four spaces. The R-diagram for r is shown below:
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 .
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.
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]
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.
Inspection of the tabular derivations for NAND and NOR, under each assignment of logical values to the functional arguments p and q, produces the identical patterns of functional values for ¬(p ∧ q) as for (¬p) ∨ (¬q), and for ¬(p ∨ q) as for (¬p) ∧ (¬q). Thus the first and second expressions in each pair are logically equivalent ...
Of its five connectives, {∧, ∨, →, ¬, ⊥}, only negation "¬" can be reduced to other connectives (see False (logic) § False, negation and contradiction for more). Neither conjunction, disjunction, nor material conditional has an equivalent form constructed from the other four logical connectives.