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Question 975438: I'm having real difficulty solving this, can you please help?
INSTRUCTIONS: Use indirect truth tables to answer the following problems.
Given the argument:
Premises: E ⊃ J / B ⊃ Q / D ⊃ (J • ∼Q) Conclusion: (E • B) ≡ D
This argument is:
Uncogent.
Sound.
Valid.
Invalid.
Cogent.
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! Symbols Note:
* I'm going to use "->" in place of the horseshoe
* I'm going to use & in place of the dot
* I'm going to use = in place of the triple equals sign
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First assume that the argument is invalid. Recall that an argument is invalid if all premises are true with a false conclusion
So in this form
| premise1 | / | premise2 | / | premise3 | // | conclusion |
|---|
| T | | T | | T | | F |
So for this particular argument, I'm going to place F under the outermost connector "=" in the conclusion. Also I'm going to place T's under the outermost connectors of each premise (see below)
| E | -> | J | / | B | -> | Q | / | D | -> | (J | & | ~ | Q) | // | (E | & | B) | = | D |
|---|
| T | | | | T | | | | T | | | | | | | | | F | |
Note: The table form is a bit ugly, but it's the only way I could separate each term enough (to give the proper amount of alignment and spacing).
Then we try to find the truth values of all the symbols to force it to be invalid
Now let's assume D is false. If that's the case, then (E&B) would have to be true. Place a T under the & in the conclusion. That forces E and B to both be true. So I'm going to place T's under the E and B in the conclusion
| E | -> | J | / | B | -> | Q | / | D | -> | (J | & | ~ | Q) | // | (E | & | B) | = | D |
|---|
| T | T | | | T | T | | | | T | | | | | | T | T | T | F | F |
If E is true, then J has to be true for E -> J to be true. Place a T under the J in premise1
If B is true, then Q has to be true for B -> Q to be true. Place a T under the Q in premise2
Place T's under the J and Q in premise3
Q is true, so ~Q is false. Place a F under the ~ next to the Q in premise3
overall, J&~Q is false, so place a F under the & in premise3
premise3 is forced to be true, (J&~Q) is false, so D must also be false (leading us back full circle to the initial assumption)
So here is what the full updated indirect truth table looks like
| E | -> | J | / | B | -> | Q | / | D | -> | (J | & | ~ | Q) | // | (E | & | B) | = | D |
|---|
| T | T | T | | T | T | T | | F | T | T | F | F | T | | T | T | T | F | F |
You'll notice that there are no contradictions. So assuming D is false leads to no contradictions.
Let's assume D is true
Following the same sort of logic, we will have this updated table (the second row of Ts and Fs corresponds to when D is true). I'm skipping showing the detailed step by step picture to save time/space. Hopefully my steps above will be enough to help you see how I'm getting this new row
| E | -> | J | / | B | -> | Q | / | D | -> | (J | & | ~ | Q) | // | (E | & | B) | = | D |
|---|
| T | T | T | | T | T | T | | F | T | T | F | F | T | | T | T | T | F | F | | T or F | T | T | | F | T | F | | T | T | T | T | T | F | | T or F | F | F | F | T |
Note: The E in premise1 and E in the conclusion could either be True or False. You'll find it doesn't matter. So that's why I put "T or F" under those "E"s
Again I couldn't find any contradictions for either row. Since there are no contradictions, this means that the claim "the argument is invalid" initially made goes unchallenged. The claim is true.
So the argument is indeed invalid. Those two rows prove that there is a way to set up the truth values to where the premises are all true which lead to a false conclusion.
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