Logical Implication
A logical implication is a relationship between two statements. If a statement Q is always true when another statement P is true, then we say that "P implies Q", which is denoted symbolically as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P \implies Q } . Note that if P is false, Q does not necessarily have to be false. For example, if x > 10, then x is also greater than 0, so we can say that "". However, if x is less than 10, it doesn't necessarily mean that x isn't greater than 0. That is, does NOT mean that . The truth table for logical implication is as follows:
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Note that while the inverse of (that is, ) does not necessarily have the same truth value as , the contrapositive (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \neg Q \implies \neg P } ) does. For example, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x > 10 \implies x > 0 } and its contrapositive, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \leq 0 \implies x \leq 10 } , are both true or both false at the same time for all values of x.
Resources
- Truth Tables, Tautologies, and Logical Equivalences, Millersville University
