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# Logic question about strange implying?

Topic: R case statement
June 20, 2019 / By Shea
Question: how can I go from the statement "It is not the case that it is raining and it is cold outside." to: "If it is raining then I will not wear a jacket" ?? This doesnt seem to make any sense to me pls help im so confused I mean to go from the first to: "If it is raining then it is not cold outside." sorry

Patti | 6 days ago
"It is not the case that it is raining and it is cold outside." Let R = "It is raining." Let C = "It is cold outside." Therefore, the above sentence would be, "It is not the case that (R and C)" ---------------------------- Now, using the same symbols as above, let's translate "If it is raining then it is not cold outside." If R then not C. These are logically equivalent. Here is a proof: 1. ~(R and C) (Original premise) _________________ 2. ~R or ~C (De Morgan's Law) 3. If R then ~C (Material Implication) http://en.wikipedia.org/wiki/Material_im... http://en.wikipedia.org/wiki/De_Morgan's... --------- Now to explain how this makes sense from an informal perspective. It is not both raining and cold outside at the same time. This must mean that IF it IS raining, then it must not be cold outside, because they can't be true at the same time. It also entails the contrapositive (modus tollens) of the conditional. If it is cold outside, then it is not raining. This is exactly what the original statement also says. It can't be both raining and cold outside at once. Let me know if you have any more questions in additional details. Alternatively, you can email me at: [email protected] Have a great day. ^^
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Margot
We've got two premises here: A: It is raining B: It is cold outside Statement one can mean two things: either it means that right now, i's not raining and it's not cold outside (it's not the case that A and B are true) In this case, it would be a statement: Both A and B are not true. However, it can also mean that it's not true that A and B can be true at the same time: it's not the case that it's raining AND it's cold outside. In this case, it would be: A and B cannot be true. Either A is true or B is. Statement two is: If A then not B (if A is true, B is not true) Statement two is a logical conclusion based on the second version of statement one. Since we observe that A and B cannot be true at the same time (statement), we can conclude that if A is true, B is untrue (conclusion).
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