Question 1190255: Topics In Contemporary Math
Arguments
Use truth tables to determine if each of the following arguments are valid or invalid.
4) If you back up your hard drive, then you are protected.
Either you are protected or you are daring.
Therefore, if you are daring, then you won’t back up your hard drive.
Answer by math_tutor2020(3816)   (Show Source):
You can put this solution on YOUR website!
I'll define the following
B = you back up your hard drive
P = you are protected
D = you are daring
We have 3 variables that can either take on values of true (T) or false (F). So there will be 2^3 = 8 different combos of those T's and F's.
Here's one way we can set up the first part of the table
| B | P | D | | T | T | T | | T | T | F | | T | F | T | | T | F | F | | F | T | T | | F | T | F | | F | F | T | | F | F | F |
Notice that the first column has four T's followed by four F's
The next column has two blocks of TTFF
The third columnn has four copies of TF
Let's add in a column denoting ~B, which means "not B". This negates everything in the B column
B = you back up your hard drive
~B = you do not back up your hard drive
This will be useful later to set up the conclusion.
| B | P | D | ~B | | T | T | T | F | | T | T | F | F | | T | F | T | F | | T | F | F | F | | F | T | T | T | | F | T | F | T | | F | F | T | T | | F | F | F | T |
Premise #1 is "If you back up your hard drive, then you are protected."
That translates to B -> P
The arrow notation is used in conditional statements of the form "if this, then that".
Writing B -> P means "If B, then P".
Sometimes a sideways horseshoe symbol is used in place of the arrow.
The conditional statement B -> P will be false if B = T and P = F
Otherwise, B -> P is true.
Let's add that to our table
| | | | Premise 1 | | B | P | D | ~B | B -> P | | T | T | T | F | T | | T | T | F | F | T | | T | F | T | F | F | | T | F | F | F | F | | F | T | T | T | T | | F | T | F | T | T | | F | F | T | T | T | | F | F | F | T | T |
The second premise is "Either you are protected or you are daring" which translates to P v D
The "v" symbol means "or", used for disjunctions.
Disjunctions are only false when both pieces are false. Otherwise, the entire thing is true.
So P v D is only false when P = F and D = F
Otherwise, P v D is true.
Let's add that to the truth table
| | | | Premise 1 | Premise 2 | | B | P | D | ~B | B -> P | P v D | | T | T | T | F | T | T | | T | T | F | F | T | T | | T | F | T | F | F | T | | T | F | F | F | F | F | | F | T | T | T | T | T | | F | T | F | T | T | T | | F | F | T | T | T | T | | F | F | F | T | T | F |
Finally, the conclusion is "If you are daring, then you won't back up your hard drive"
This translates to the symbolic form of D -> ~B
Here's the full truth table
| | | | Premise 1 | Premise 2 | Conclusion | | B | P | D | ~B | B -> P | P v D | D -> ~B | | T | T | T | F | T | T | F | | T | T | F | F | T | T | T | | T | F | T | F | F | T | F | | T | F | F | F | F | F | T | | F | T | T | T | T | T | T | | F | T | F | T | T | T | T | | F | F | T | T | T | T | T | | F | F | F | T | T | F | T |
Now to answer the question if this argument is valid or not.
Notice that in row 1, which I've marked in red above, we have- Premise B -> P is true
- Premise P v D is true
- Conclusion D -> ~B is false
Invalid arguments are where all true premises lead to a false conclusion.
So the argument
B -> P
P v D
:. D -> ~B
is invalid
The :. means "therefore"
It could be written as .:
It's supposed to represent an equilateral triangle of dots
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Here's a shortcut
Let's assume that the argument is invalid.
If so, then conclusion D -> ~B must be false and the goal is to try to get all premises to be true.
If D -> ~B is false, then that leads to D = T and ~B = F
~B = F flips to B = T
If B = T, then B -> P is true if and only if P = T as well
Put another way: if B = T and P = F, then B -> P would be false.
Since we found P = T and D = T, premise 2 P v D is true
Though we technically didn't need the truth value of P to figure out that P v D is true because we found that D = T earlier.
P v D = T v T = T
P v D = F v T = T
With these values
B = T
P = T
D = T
we have shown that all the premises are true but they lead to a false conclusion.
Therefore, the entire argument posed by your teacher/textbook is invalid.
While this shortcut is nice to use (perhaps in an exam environment), it's still helpful practice to construct the full truth table when it comes to homework assignments.
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