Formal Logic 101 – Part 9: Rules of Inference (Part 4) - Yes Virginia, you CAN prove a negative

At one point or other, you have probably seen some theist say "Well prove that God DOESN'T exist" and seem some atheist respond “You can’t prove a negative”. Don’t worry, I’ve said it too. There’s no shame in being wrong from time to time, the point is to grow, and learn, and do better. I have no idea how that idea first entered circulation, but the thing is IT IS FALSE. And today, we’re going to use an Indirect Proof to prove a negative.

An Indirect Proof is somewhat similar to a Conditional Proof, insofar as we will be using (and later discharging) Assumptions. However, in this case, instead of assuming the antecedent and hoping to prove the consequent, we will assume the OPPOSITE of what we intend to prove, and see if it leads to absurdity (specifically, if we can create a contradiction such as A^~A).

Contradictions are a big deal for us because, in logic, every proposition has exactly one truth value. Schrodinger’s Cat is either alive or dead, but the cat can’t be both alive and not alive at the same time! (In fact, the entire Schrodinger’s Cat thought experiment was intended to prove that certain ideas in quantum mechanics cannot be true, because they would lead to a contradiction.) If you allow a contradiction, where a given proposition is both true and false, then logic explodes.

Consider the following, as an example:
1) S^~S        Premise ("Schrodinger's Cat is alive and Schrodinger's Cat is not alive.")
2) S               1 simp.
3) SvG           2 add. (G="Garden gnomes are plotting to conquer Mars")
4) ~S            1 simp
5) G               3,4 D.S.

Recall that “Addition” allows you to take a single proposition and make it into an “Or” statement, and the other half of that “Or” can be anything you like. My usage of inference rules is perfectly correct and yet I have concluded something absolutely bonkers. And remember, I could make “G” any statement I like and the proof would be exactly the same. From a single contradiction and three simple inference rules, every single proposition can be proven true AND false at the same time. That is what I mean when I say that logic explodes if there’s a contradiction. It becomes useless, the distinction between “true” and “false” is rendered meaningless. And really we may as well pack it up and go home.

To preserve the idea of “true” and “false” as being meaningful concepts, we adopt the stance that – if you adopt an assumption, and from that assumption you are able to reach a contradiction – then you may conclude that your assumption is false. This is what allows us to use Indirect Proof to prove a negative.

For our example argument, let’s consider:
If the Bible is literally true, then both God and the Devil exist, and the Story of Adam and Eve is correct. If the Story of Adam and Eve is correct, then God is Wrathful and not Kind. If God exists, then he is Omniscient and Kind. Therefore, the Bible is not literally true.

Let us symbolize this argument as such:
1. B →[(G^D)^S]        P
2. S→(W^~K)            P
3. G→(O^K)                P – Prove ~B

Now for our assumption we assume the opposite of “~B”, which would be just “B”

4. B                            A – Prove Contradiction.
5. (G^D)^S                1,4 MP
6. G^D                       5 simp
7. S                            5 simp
8. G                            6 simp
9. W^~K                    2,7 MP
10. O^K                     3,8 MP
11. ~K                       9 simp
12. K                         10 simp
13. K^~K                   11,12 conj.
So we have reached our contradiction. We have reached a point where we have established God as being simultaneously kind and unkind. All that remains is to discharge our assumption. With an Indirect Proof, we discharge an Assumption by saying “obviously our assumption was false”, yeeting it out the window, and concluding the opposite of our assumption. So the final line of our proof will be:

14. ~B                        4-13 IP (again, as with Conditional Proof, it is “4-13”, not “4,13”, because our conclusion depends on that entire sub-proof.)

Do note that, in some cases, you’ll need Conditional Proof or Indirect proof to get you to your desired conclusion, but you’ll have more to do after you’ve used them. (For instance, maybe you’ve used IP to prove ~J, but you need to prove “(~J^~K)^(C→R), so you still have some work ahead of you.) With both Conditional Proof and Indirect Proof, once you have discharged an assumption you cannot use that assumption (or any line that depends on that assumption) ever again. So, if you have a line that looks like:
9. ~J                        3-8 IP
Lines 3-8 are now off limits for the rest of the proof. We reached all of those lines by using an assumption rather than a premise, so we cannot be sure of their truth (especially considering that, with Indirect Proof, we actually prove the falsity of the assumption, so using conclusions that we reached via a false assumption is a sure-fire way to reach bad conclusions on down the line).


Before we end, I know that some folks reading this may want to say “Ok, I see what you did there with proving a negative, but that’s not the same as proving that something doesn’t exist”, to which I respond: It is exactly identical. If we agree to a series of premises and can show that adding the assumption “at least one deity exists” leads to a contradiction, then we can conclude that no deities exist. The problem with disproving deities (the Christian deity being a prime example) is that their followers often make the deity unfalsifiable, and indistinguishable from chance. If everything good that happens if proof your deity is real and loves you, and everything bad that happens is proof that your deity is real and is testing you… if every “answered” prayer is proof of your deity, and every unanswered one just shows that your deity knows that that’s not what’s best for you… then the universe where the deity exists and the one where it doesn’t are indistinguishable. There’s no null hypothesis to test against. There’s no statement where a believer would grant “If [x] were true it would prove that God doesn’t exist”. So the problem does not rest with the ability of logic to prove a negative, the problem rests with believers carefully crafting their deity to try to circumvent logic’s ability to disprove things.

Practice


Using the rules we’ve learned so far, including Conditional Proof and Indirect Proof, prove the following arguments.

Our Rules so far:


Argument A

1. A→(B→C)            Premise (P)
2. (C^D)→E              P
3. F→(D^~E)            P – Prove A→(B→~F)

Argument B
1. (AvB)→(~Fv~D)     P
2. ~AvD                        P
3. ~F→~(C^D)             P – Prove ~(A^C)

Argument C
1. ~(~A^~D)→W        P
2. ~B^~C                     P
3. R→(C^T)                 P
4. ~R→A                     P – Prove ~(W→C)

Argument D
1. (X^Y)v~(ZvW)         P
2. (Z^X)→(Y→~B)      P
3. C→~C                      P
4. ~(BvC)→~Y             P – Prove ~Z





Answers

Note that solutions provided are not necessarily the only possible solutions.

Argument A
1. A→(B→C)             Premise (P)
2. (C^D)→E                P
3. F→(D^~E)             P – Prove A→(B→~F)
4. A                             Assumption (A) – Prove B→~F
5. B                             A – Prove ~F
6. F                             A – Prove contradiction
7. D^~E                      3,6 MP
8. B→C                      1,4 MP
9. C                             5,8 MP
10. D                           7 simp
11. C^D                      9,10 conj
12. E                           2,11 MP
13. ~E                        7 simp
14. E^~E                    12,13 conj
15. ~F                         6-14 IP
16. B→~F                  5-15 CP
17. A→(B→F~)         4-16 CP

Argument B
1. (AvB)→(~Fv~D)      P
2. ~AvD                         P
3. ~F→~(C^D)             P – Prove ~(A^C)
4. A^C                            A – Prove contradiction
5. A                                4 simp
6. C                                4 simp
7. AvB                            5 conj
8. ~Fv~D                       1,6 MP
9. D                                 2,5 DS
10. ~F                            8,9 DS
11. ~(C^D)                     3,10 DS
12. C^D                          6,9 Conj
13. (C^D)^~(C^D)        11,12 conj
14. ~(A^C)                    4-13 IP

Argument C
1. ~(~A^~D)→W         P
2. ~B^~C                      P
3. R→(C^T)                  P
4. ~R→A                      P – Prove ~(W→C)
5. W→C                        A – Prove contradiction
6. ~C                             2 simp
7. ~W                            5,6 MT
8. ~A^~D                      1,7 MT
9. ~A                             8 simp
10. R                             4,9 MT
11. C^T                         3,10 MP
12. C                             11 simp
13. C^~C                      6,12 Conj
14. ~(W→C)                5-13 IP

Argument D
1. (X^Y)v~(ZvW)          P
2. (Z^X)→(Y→~B)       P
3. C→~C                       P
4. ~(BvC)→~Y              P – Prove ~Z
5. Z                                 A – Prove contradiction
6. ZvW                            5 Add
7. X^Y                             1,6 DS
8. X                                 7 simp
9. Z^X                             5,8 conj
10. Y→~B                      2,9 MP
11. Y                               7 simp
12. ~B                            10,11 MP
13. BvC                           4,11 MT
14. C                               12,13 DS
15. ~C                             3,14 MP
16. C^~C                        14,15 Conj
17. ~Z                             5-16 IP

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