Is AvB Equivalent to Av~~B in Logical Proofs?

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I am trying to prove that AvB (which reads "A or B") is equivalent to Av~~B (which reads "A or not not B"). My steps are wrong... I checked them out on Fitch (the program we use in class to check validity of proofs). I can't write them out in here... I don't have the right symbols ... so maybe somebody can suggest a starting step that will help?

 

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NM! I got it all on my own! Wow, I am sooo super proud right now!

 

[wais]

First, let's define the logical symbols we will use in this proof:
- A: statement A
- B: statement B
- ~: negation (not)
- v: disjunction (or)
- ~~: double negation (not not)

To prove that AvB is equivalent to Av~~B, we need to show that they have the same truth values in all possible scenarios. This can be done by showing that each statement implies the other.

Starting with AvB, we can use the law of excluded middle to break it down into two cases: either A is true or B is true.

Case 1: A is true
In this case, AvB is automatically true since one of the disjuncts (A) is true.

Case 2: B is true
Similarly, AvB is true since one of the disjuncts (B) is true.

Therefore, we have shown that AvB implies Av~~B in all possible scenarios.

To show the reverse implication, we can use a proof by contradiction. Assume that AvB is false. This means that both A and B are false. Using the double negation law, we can rewrite this as ~~A and ~~B.

Since ~~A is equivalent to A, we can substitute and get A and ~~B. From this, we can apply the law of excluded middle again to get two cases:

Case 1: A is true
This leads to a contradiction since we assumed A to be false.

Case 2: B is true
This also leads to a contradiction since we assumed B to be false.

Therefore, our initial assumption that AvB is false must be incorrect, and thus AvB must be true. This shows that Av~~B implies AvB in all possible scenarios.

Hence, we have shown that AvB and Av~~B are equivalent, and our proof is complete.