Formal Logic 101 – Part 8: Rules of Inference (Part 3)

Today we’ll start covering the two more-complex inference rules. We will only get to one today, but I’d rather spread this out over two posts and have it be understandable than try to crunch it into one and have folks not get it.

Conditional proof

From time to time, it is useful to prove an “If/Then” statement. In order to prove an If/Then statement, we sometimes can use hypothetical syllogism. For instance, Consider:
1. If a nuclear Missile is launched, it will start a Nuclear war.
2. If there’s a Nuclear war, then nuclear Winter will follow.

We can use Hypothetical Syllogism to prove “if a nuclear Missile is launched, then nuclear Winter will follow”, as follows:
1. M→N            Premise
2. N→W            P
3. M→W            1,2 HS

However some situations are more complex, and simple Hypothetical Syllogism will not do. Consider this more complex argument:
1. If a nuclear Missile is launched, it will start a Nuclear war.
2. If there’s a Nuclear war, then nuclear Winter will follow.
3. If there’s nuclear Winter, then unless we can grow plants Indoors, there will be no Plants
4. If there are no Plants, then there will also be no Animals.
5. If there are no Plants and no Animals, then humanity will Starve and go Extinct.
6. Therefore, if a nuclear Missile is launched, then unless we can grow plants Indoors, humanity will go Extinct.

1. M→N                     Premise
2. N→W                     P
3. W→(~I→~P)        P
4. ~P→~A                 P
5. (~P^~A)→(S^E)   P – Prove M→(~I→E)

(Note: in premise 3, “unless we can grow plants indoors” is symbolized “~I”, this is because that portion of our sentence means the same thing as “If we cannot grow plants indoors”.)

Now we could use Hypothetical Syllogism to prove “M→W”, and even “M→(~I→~P)”, but we really don’t have anywhere we can go from there. Yet just looking at the proof in plain English, it certainly appears correct, so there should be some way to prove it. That is where Conditional Proof comes in.

To begin a Conditional Proof, we need to make an Assumption. An Assumption is functionally very similar to a Premise, but with a Premise we are saying “this is true” while, with an Assumption, we are essentially saying “Let’s pretend that this is true for a minute, just to see where it would lead”. 

Also, in order to ensure that our logic remains truth preserving, by the time we’re done, our argument should be based only on our actual premises (which we are asserting the truth of) rather than assumptions (which we are merely stating might be true). As such we will need to discharge our assumptions at the end. We will cover that in more detail later, but for now just keep it in the back of your mind that it will be a thing that needs to happen.

So what should our assumption be? It’s generally best to assume the antecedent (the front part) of our intended conclusion and see if we can prove the consequent (the back part) as a result. For right now, our proof should look something like this:

1. M→N                        Premise
2. N→W                         P
3. W→(~I→~P)            P
4. ~P→~A                     P
5. (~P^~A)→(S^E)       P – Prove M→(~I→E)
6. M                               Assumption – Prove ~I→E
7. N                                1,6 MP
8. W                               2,7 MP
9. ~I→~P                     3,8 MP

From here we are stuck again. You’ll note that there is no line that gives us ~I, so line 9 seems to be a dead end. That said, you might notice that we are trying to use our Assumption “M” to prove “~I→E”. We are still trying to prove an “If/Then” statement, and there is no rule saying we can only have one assumption So we can just make one more assumption and see if it clears matters up, as such:

10. ~I                        Assumption – Prove E
11. ~P                       9,10 MP
12. ~A                      4,11 MP
13. ~P^~A               11,12 Conj
14. S^E                     5,13 MP
15. E                         14 Simp

So, we have made it to E. We established that if we assume M (that a nuclear Missile is launched) then, if we assume ~I (we cannot grow plants Indoors), then E (humanity goes Extinct) results, which is exactly what we set out to prove. All that’s left is to discharge the assumptions, which basically means that we stop speaking of them as though we know them to be true. For starters, let’s look at just lines 10-15 (when discharging assumptions, you’ll always want the last one introduced to be the first one discharged). What lines 10-15 actually do is say “If ~I were true, then by these steps, we can prove that E would have to be true”, so to “discharge the assumption, we rephrase that whole section into an “If/Then” statement as such:

16. ~I→E                10-15 CP
(Note, not “10,15”, but “10-15”. Every step along the way is needed to prove that E necessarily follows from ~I, so every step should be cited.)

So instead of line 10 essentially treating ~I as though it is true, we’ve walked our claim back to “Well… if ~I were true…” so the proof no longer depends on ~I actually being true.

To wrap up our final conclusion, we just use Conditional Proof one more time (and walk back our claim just a bit further so we’re no longer speaking of “M” as though it is true) as such:

17. M→(~I→E)             6-16 CP
So we've now stopped treating M like it's true (as we did on line 6) and walked it back to "If M were true... this would result". Since we're no longer asserting that M is true, our proof no longer depends on the assumption. The whole this is based purely on our premises.

*****

And that’s conditional proof! I know that that was easily the longest proof that we've dealt with so far, so if you feel like your head is swimming, don't worry, that's natural. Give yourself a few minutes to process, and maybe reread it and see if it feels less foreign. If you hung on through that, that should be probably the most difficult rule, so it should all be gravy from here. For our next post we’ll learn how to prove a negative!

Practice


Using the previously discussed inference rules and Conditional Proof (CP) prove the following arguments.

Our Rules so far:



Argument A
1. (D^E)→~F        Premise (P)
2. Fv(G^W)           P
3. D→E                 P – Prove D→G

Argument B
1. F→(~Z^~Y)     P
2. ~(GvZ)→~H    P
3. ~(F^H)vY         P – Prove F→(H→G)

Argument C
1. Q                            P
2. (R^Q)→(S→T)      P - Prove (R^S)→T

Argument D
1. S→(A→B)         P
2. A^C                    P
3. C→D                  P – Prove S→(B^D)





Answers

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


Argument A
1. (D^E)→~F          Premise (P)
2. Fv(G^W)             P
3. D→E                   P – Prove D→G
4. D                          Assumption – Prove G
5. E                          3,4 MP
6. D^E                      4,5, conj
7. ~F                        1,6, MP
8. G^W                     2,7 DS
9. G                          8 simp
10. D→G                 4-10 CP

Argument B
1. F→(~Z^~Y)        P
2. ~(GvZ)→~H       P
3. ~(F^H)vY            P – Prove F→(H→G)
4. F                          Assumption – Prove H→G
5. H                         A – Prove G
6. F^H                     4,5 Conj
7. Y                         3,6 DS
8. GvZ                    2,5 MT
9. ~Z^~Y                1,4 MT
10. ~Z                     9 simp
11. G                       8,10 DS
12. H→G                5-11 CP
13. F→(H→G)       4-12 CP

Argument C
1. Q                              P
2. (R^Q)→(S→T)       P - Prove (R^S)→T
3. R^S                          A – Prove T
4. R                              3 simp
5. S                              3 simp
6. R^Q                         1,4 conj
7. S→T                        2,6 MP
8. T                              5,7 MP
9. (R^S)→T                 3-8 CP

Argument D
1. S→(A→B)              P
2. A^C                         P
3. C→D                       P – Prove S→(B^D)
4. S                              A – Prove B^D
5. A→B                       1, 4 MP
6. A                              2 simp
7. B                              5,6 MP
8. C                              2 simp
9. D                              3,8 MP
10. B^D                       7,9 Conj
11. S→(B^D)               4-10 CP

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