57 lines
1.6 KiB
Markdown
57 lines
1.6 KiB
Markdown
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---
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title: Tautologies
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---
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## Tautologies
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### Definition
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In logic, a tautology is a statement that is true in every possible case. The opposite of a tautology is a contradiction, a statement being false in every possible cases.
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### Example
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<table>
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<tr>
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<th>p</th>
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<th>q</th>
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<th>p OR q</th>
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<th>p → p OR q</th>
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</tr>
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<tr>
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<td>T</td>
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<td>T</td>
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<td>T</tq>
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<td>T</td>
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</tr>
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<tr>
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<td>T</td>
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<td>F</td>
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<td>T</td>
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<td>T</td>
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</tr>
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<tr>
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<td>F</td>
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<td>T</td>
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<td>T</td>
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<td>T</td>
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</tr>
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<tr>
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<td>F</td>
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<td>F</td>
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<td>F</td>
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<td>T</td>
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</tr>
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</table>
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As we can see in the truth table, the statement "p → p OR q" is always true (see last column).
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An example in terms of Boolean logic is `B || !B`. It is always true that B is true or B is not true.
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The opposite of a tautology is a contradiction, a formula which is "always false". In other words, a contradiction is false for every assignment of truth values to its simple components.
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An example of a contradiction with Boolean logic is `B && !B`. It is impossible for B to be both true and false at the same time.
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#### Note
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The arrow simply means "implies". p implies p OR q, it can also mean <i>if...then...</i>
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#### More Information:
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<!-- Please add any articles you think might be helpful to read before writing the article -->
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[Wikipedia Tautology (Logic)](https://en.wikipedia.org/wiki/Tautology_(logic))
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[Youtube Truth Tables](https://www.youtube.com/watch?v=O0KbymjE7xU)
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[Wikipedia Logic Symbols](https://en.wikipedia.org/wiki/List_of_logic_symbols)
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