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# 6.3 Truth Tables For Propositions

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Course lecture I developed over section 6.3 of Patrick Hurley\'s &quot;A Concise Introduction to Logic&quot;.

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### 6.3 Truth Tables For Propositions

1. 1. 6.3 Truth Tables for Propositions
2. 2. Truth tables <ul><li>These are ways of labeling every possible truth value for each symbol and variable in a proposition. </li></ul><ul><li>To figure out the number of lines, use the following equation: </li></ul><ul><ul><li>L = 2^n </li></ul></ul><ul><ul><li>Or more simply put, start with 2 lines and double that number for every “unique” variable in a proposition. </li></ul></ul><ul><ul><ul><li>One variable (A) = 2 </li></ul></ul></ul><ul><ul><ul><li>Two variables (A, B) = 4 </li></ul></ul></ul><ul><ul><ul><li>Three variables (A, B, C) = 8 </li></ul></ul></ul><ul><li>Note: There are two ways of doing truth tables so I’ll be showing you each one. </li></ul>
3. 3. Making truth tables <ul><li>Now let’s apply these rules in practice. </li></ul><ul><ul><li>Example: </li></ul></ul><ul><ul><ul><li>There are two unique variables, A and B, so this table is four lines long. </li></ul></ul></ul><ul><ul><ul><li>For each unique variable you encounter after the first, divide the number of lines in half and use that as a measurement for how often you should alternate T’s and F’s. We have two lines so half of that is one, so there will be one T followed by one F for each spot under B. </li></ul></ul></ul>F F F T T F T T T F F F F F F T T T T T T F T T B > B) ~ v (A
4. 4. Alternate method <ul><li>There is a different method for doing truth tables; it uses the same rules but in a different way. </li></ul><ul><li>Write out the proposition but make two columns to the left: one for each variable you have. Follow the same rules for figuring out the number of lines and which value goes on each one, then just plug them into the columns and use the rules for the symbols to decide which ones are true and which are false. </li></ul>T T F F F F T F F T T F F T T T F T T T T T T T B > A)] > (B · B) V [(A B A
5. 5. Ways to classify propositions <ul><li>There are a number of different qualities a proposition can have that we can use to classify them. </li></ul><ul><li>Tautologous (logically true) </li></ul><ul><ul><li>The truth values in the column under the main operator are all true. </li></ul></ul><ul><li>Self-contradictory (logically false) </li></ul><ul><ul><li>Truth values under the main operator are all false. </li></ul></ul><ul><li>Contingent </li></ul><ul><ul><li>Truth values under the main operator have at least one truth value that is true and at least one that is false. </li></ul></ul>
6. 6. Ways to compare propositions <ul><li>These definitions are used in comparing two separate propositions and showing the relationship between them. </li></ul><ul><li>Logically equivalent </li></ul><ul><ul><li>The truth values under the main operator are the same in each proposition. </li></ul></ul><ul><li>Contradictory </li></ul><ul><ul><li>The truth values under the main operator are opposites in each proposition. </li></ul></ul><ul><li>Consistent </li></ul><ul><ul><li>There’s at least one line in each proposition under the main operator where both truth values are true. </li></ul></ul>
7. 7. Comparing propositions, continued <ul><li>Inconsistent </li></ul><ul><ul><li>There’s no line in each proposition under the main operator where both truth values are true. </li></ul></ul>