Formal Logic 101 – Part 6: Rules of Inference (Part 1)

We’ll be covering inference rules today. We’ve previously discussed Modus Ponens and Modus Tollens. We’ll very briefly touch on them as a refresher, and then move on to the other inference rules that we haven’t covered yet.

A Recap of the Modus-es

Modus Ponens: Given the premises “If A, then B” and “A”, conclude “B”.
1. A→B    Premise
2. A          Premise
3. B          1,2 MP

Modus Tollens: Given the premises “If A, then B” and “Not B”, conclude “Not A”.
1. A→B     Premise
2. ~B         Premise
3. ~A         1,2 MT

Modus Ponens and Modus Tollens are both examples of creating a “Syllogism” (basically, a syllogism is roughly the same as a logical proof. It’s a line of reasoning where certain things are assumed, and it’s determined that something else must necessarily be the case as a result). Using the example of Modus Ponens given above, neither of the premises by themselves state “B is true”, but the premises allow us to reason our way to that unavoidable conclusion.

Hypothetical Syllogism (HS)

Our next rule is a “Hypothetical Syllogism”. The Hypothetical Syllogism takes the form:
1. A→B        Premise
2. B→C        Premise
3. A→C        1,2 HS

This rule should seem somewhat obviously true at face value. After all from the premises
1) If corn is a plant, then it has seeds
and
2) if corn has seeds, then it can be farmed
the conclusion
3) If corn is a plant, then it can be farmed.
seems obviously true (and later on, when we cover the conditional and indirect proofs we’ll be able to prove that the inference rule is valid).
 
Note that this is a “hypothetical syllogism” because we’re not actually claiming that A is true (note that “A” is not a premise in this proof) and we’re not proving that "C" is true. We’re only proving that "C" would be true if, hypothetically speaking, "A" were true.

Conjunction (Conj.)

Conjunction is the first of our “And” rules, and states simply: Given the premises “A” and “B”, conclude “A^B”.
1. A          Premise
2. B          Premise
3. A^B    1,2 Conj.
Again, this should be intuitively correct. After all, if we grant that “Bill Gates founded Microsoft” and “Jeff Bezos founded Amazon” then the statement “Bill Gates founded Microsoft, and Jeff Bezos founded Amazon” must also be true.

Simplification

Simplification is also an “And” rule and is effectively Conjunction’s opposite number. Given the premise A^B, conclude either of the conjuncts.

1. A^B        Premise
2. A           1 Simp. (If you don’t make a joke in the comments, this abbreviation will have been for nothing.)
Or:
1. A^B        Premise
2. B           1 Simp.

Continuing our example from above, if we accept the statement “Bill Gates founded Microsoft, and Jeff Bezos founded Amazon” as a premise, then certainly we can conclude that the statement “Bill Gates founded Microsoft” must be correct as a standalone.

*****

There are a total of ten inference rules that we’ll be covering in this series, and we’ve touched on five of them so far, so I think this is a good place to end for right now. We’ll come back next time and cover three more inference rules, and a couple of restrictions on using them. 

*****

Practice
Using the rules we’ve covered so far, attempt to prove the following symbolized arguments.

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

Note: If you find yourself overwhelmed, try finding some phrases to help you translate the phrases back to regular English. You can generally pick any phrases you like. For instance, I might say something like:
1.      If I like Apple pie, I will also like Banana bread.
2.      If I like Banana Bread, I will not like Cheesecake.
3.      I both like Cheesecake, and dislike Danish pastries.

And then attempt to prove that I don’t like Apple pie and don’t like Banana bread.

Argument B
1.      ~P^~S                                  P
2.      (P→~Q) ^ (~Q→P)             P – Prove Q

Argument C
1.      (AvB)→(CvD)                     P
2.      C→E                                    P
3.      Cv~F                                    P
4.      A^~E                                    P
5.      Fv(D→Z)                             P – Prove Z



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

Argument A
1.      A→B                                   Premise (P)
2.      B→~C                                 P
3.      C^~D                                   P – Prove ~A^~B
4.      C                                          3 simp
5.      ~B                                        2,4, MT
6.      ~A                                        1,5 MT
7.      ~A^~B                                 5,6 Conj

Argument B
1.      ~P^~S                                   P
2.      (P→~Q) ^ (~Q→P)             P – Prove Q
3.      ~P                                        1 simp
4.      ~Q→P                                  2 simp
5.      Q                                           3,4 MP

Argument C
1.      A→(~D→C)                        P
2.      C→E                                     P
3.      ~C→~F                                P
4.      A^~E                                     P
5.      ~F→(D→Z)                          P – Prove Z
6.      ~E                                          4 Simp
7.      ~C                                         2,6 MT
8.      ~C→(D→Z)                         3,5 HS
9.      D→Z                                     7,8 MP
10.   A                                             4 simp
11.   ~D→C                                   1,10 MP
12.   D                                            7,11 MT
13.   Z                                             9,10 MP

Comments

Post a Comment

Popular posts from this blog

Formal Logic 101 – Part 3: Intro to Symbolization

So far, we’ve seen fairly short three-line proofs. For those proofs, writing them out longhand is fine. However, proofs can get much longer. Consider: Argument 1 1. If Clara will come to the party then we’ll have a good time. premise 2. Clara will come to the party. premise 3. If we have a good time then we’ll come home after midnight. premise 4. If we come home after midnight then we’ll sleep in late. premise 5. We’ll have a good time. from 1 and 2 6. We’ll come home after midnight. from 3 and 5 7. We’ll sleep in late. from 4 and 6 That’s only 7 lines with fairly simple sentences, and 3 applications of modus ponens, and we’re rapidly approaching the point where I think a lot of people – if they had to do i...
Read more

Formal Logic 101 – Part 19: Proofs in Predicate Logic (Part 2)

Last time we covered two of our final 4 inference rules (Universal Instantiation and Existential Generalization). Today we’ll be wrapping up the last of our new rules. Beyond this I have a couple posts planned for less-common situations, and then the series will be done. So let’s get to it. Existential Instantiation (EI) If you remember our last past, you might have a guess about how this next rule is going to work. EG let us go from Dm to (∃x)Dx, so EI is probably going to let us go from (∃x)Dx to Dm. If that was your guess, you’re basically right, however, EI is going to have some additional restrictions on how it can be used. To understand the need for these restrictions, I will present the following example where I will INCORRECTLY use EI by treating it exactly the same way as UI. Consider the following premises: Some numbers are odd. Some numbers are even. Let us say that “Nx” is “x is a Number”, “Ox” is “x is Odd”, and “Ex” is “x is Even”. Observe the following argument: 1. (∃x)(...
Read more

Formal Logic 101 – Part 11: Replacement Rules (Part 2)

Welcome back. Today we’ll be picking up where we left off with more replacement rules. Commutation (Comm.) Commutation is a handy step for dealing with “And” or “Or” statements, where the statement you have is close to what you would need to complete another step (such Modus Ponens, for example) but is not quite exactly right. I think the easiest way to explain this would be by example. Consider the premises: If the Sun is very Large but looks very Small, then it must be Far away. The Sun looks very Small, but it is very Large. Our symbolism would be as follows: 1. (L^S)→ F                Premise (P) 2. S^L                            P Now, instinctively, I think we understand that “L^S” and “S^L” mean the same thing, but as was stated in the introduction, formal logic requires you to show your work. Every step must be documented. It’s not enough to assume that th...
Read more