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The following statement is a contradiction:

"We are happy, and it is not the case that Cole having ordered the pizza implies that we are happy."

Express this statement in symbolic logic, clearly specifying what English statements any variables represent. Then, use logical equivalence rules to show that this is a contradiction.

Matti Ovitt
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  • What have you tried? Have you expressed it in logical symbols? Where are you stuck? – Ross Millikan Nov 05 '20 at 21:21
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    Welcome to MSE. You'll get a lot more help, and fewer votes to close, if you show that you have made a real effort to solve the problem yourself. What are your thoughts? What have you tried? How far did you get? Where are you stuck? This question is likely to be closed if you don't add more context. Please respond by editing the question body. Many people browsing questions will vote to close without reading the comments. – saulspatz Nov 05 '20 at 21:21
  • I have: P = We are happy Q=It is not the case that Cole ordered the pizza (P ^ ~Q) -> P Does this look right so far? If so, how do I continue? – Matti Ovitt Nov 05 '20 at 21:26
  • I assume that by ^ you mean $\land$? And by ~Q I assume that you mean negation, i.e. $\lnot Q$? – Stokolos Ilya Nov 05 '20 at 21:54
  • Yes, that is correct – Matti Ovitt Nov 05 '20 at 21:58
  • No, $(P \land \lnot Q) \implies P$ is not a contradiction. By definition, contradiction is a statement which is always false (i.e. regardless of how you assign truth values to the variables, the statement will always be false). However in your case, if for example, you suppose that $P$ is false, then $(P \land \lnot Q) \implies P$ will be true. In fact, you can check that the statement you came up with a is tautology (i.e it is always true), which is the opposite of what you wanted. – Stokolos Ilya Nov 05 '20 at 22:06
  • If you want a contradiction, you can define $P := \text{"We are happy"}$ and $Q := \text{"Cole have ordered a pizza"}$. Then the desired statement will be: $P \land \lnot (Q \implies P)$. You can use truth table to check that it is a contradiction. – Stokolos Ilya Nov 05 '20 at 22:15
  • This was a problem, that my professor gave me. – Matti Ovitt Nov 05 '20 at 22:15

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