Question 1190300
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Premise 1: P v Q
Premise 2: P
Conclusion: ~Q


One way to form the truth table<table border = "1" cellpadding = "5"><tr><td></td><td></td><td>Premise 1</td><td>Premise 2</td><td>Conclusion</td></tr><tr><td>P</td><td>Q</td><td>P v Q</td><td>P</td><td>~Q</td></tr><tr><td>T</td><td>T</td><td><font color=red>T</font></td><td><font color=red>T</font></td><td><font color=red>F</font></td></tr><tr><td>T</td><td>F</td><td>T</td><td>T</td><td>T</td></tr><tr><td>F</td><td>T</td><td>T</td><td>F</td><td>F</td></tr><tr><td>F</td><td>F</td><td>F</td><td>F</td><td>T</td></tr></table>Row 1, marked in red, shows all true premises lead to a false conclusion. This is sufficient to prove the argument is invalid.


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Here's an alternative way to form the truth table
What we do is conjunct the list of premises to form the antecedent, and this will lead to the conclusion.
(P v Q) & P is the antecedent while ~Q is the conclusion
This forms the conditional [ (P v Q) & P ] -> ~Q
If that is ever false, for any row, then we have proven the argument is invalid.
This is because we have true premises point to a false conclusion. 


This is what the truth table looks like using this alternative method<table border = "1" cellpadding = "5"><tr><td>P</td><td>Q</td><td>P v Q</td><td>(P v Q) & P</td><td>~Q</td><td>[ (P v Q) & P ] -> ~Q</td></tr><tr><td>T</td><td>T</td><td>T</td><td>T</td><td>F</td><td><font color=red>F</font></td></tr><tr><td>T</td><td>F</td><td>T</td><td>T</td><td>T</td><td>T</td></tr><tr><td>F</td><td>T</td><td>T</td><td>F</td><td>F</td><td>T</td></tr><tr><td>F</td><td>F</td><td>F</td><td>F</td><td>T</td><td>T</td></tr></table>We have "F" at the very end of the first row (in red) to show that [ (P v Q) & P ] -> ~Q is false when P = T and Q = T
This confirms what the other table is showing (also in row 1).


Some side notes:<ul><li>P v Q is false if both P and Q are false, otherwise it's true.</li><li>P & Q is true if both P and Q are true, otherwise it's false.</li><li>P -> Q is false if P = T and Q = F, otherwise it's true.</li></ul>
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