Difference between revisions of "Logical Implication"
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| − | 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 | + | A logical implication is a relationship between two statements. If a statement <math> Q </math> is always true when another statement <math> P </math> is true, then we say that <math> P </math> implies <math> Q </math>, which is denoted symbolically as <math> P \implies Q </math>. Note that if <math> P </math> is false, <math> Q </math> does not necessarily have to be false. For example, if <math> x > 10 </math>, then <math> x > 0 </math>, so we can say that "<math> x > 10 \implies x > 0 </math>". However, if x is less than 10, it doesn't necessarily mean that x isn't greater than 0. That is, <math> x > 10 \implies x > 0 </math> does NOT mean that <math> x \le 10 \implies x \le 0 </math>. The truth table for logical implication is as follows: |
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| − | Note that while the inverse of <math> P \implies Q </math> (that is, <math> \neg P \implies \neg Q </math>) does not necessarily have the same truth value as <math> P \implies Q </math>, the contrapositive (<math> \neg Q \implies \neg P </math>) does. For example, <math> x > 10 \implies x > 0 </math> and its contrapositive, <math> x \leq 0 \implies x \leq 10 </math>, are logically equivalent, and always have the same truth value for any number x. | + | Note that while the inverse of <math> P \implies Q </math> (that is, <math> \neg P \implies \neg Q </math>) does not necessarily have the same truth value as <math> P \implies Q </math>, the contrapositive (<math> \neg Q \implies \neg P </math>) does. For example, <math> x > 10 \implies x > 0 </math> and its contrapositive, <math> x \leq 0 \implies x \leq 10 </math>, are logically equivalent, and always have the same truth value for any number <math> x </math>. |
==Resources== | ==Resources== | ||
* [https://sites.millersville.edu/bikenaga/math-proof/truth-tables/truth-tables.html Truth Tables, Tautologies, and Logical Equivalences], Millersville University | * [https://sites.millersville.edu/bikenaga/math-proof/truth-tables/truth-tables.html Truth Tables, Tautologies, and Logical Equivalences], Millersville University | ||
Revision as of 13:13, 27 September 2021
A logical implication is a relationship between two statements. If a statement 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 Q } is always true when another statement is true, then we say that 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 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 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 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 } is false, 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 Q } does not necessarily have to be false. For example, if 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 } , then 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 > 0 } , so we can say that "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 } ". However, if x is less than 10, it doesn't necessarily mean that x isn't greater than 0. That is, 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 } does NOT mean that 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 \le 10 \implies x \le 0 } . The truth table for logical implication is as follows:
| 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} | 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 Q} | 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 \Rightarrow Q} |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Note that while the inverse of 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 } (that is, 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 P \implies \neg Q } ) does not necessarily have the same truth value 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 } , 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 logically equivalent, and always have the same truth value for any number 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 } .
Resources
- Truth Tables, Tautologies, and Logical Equivalences, Millersville University
