1. ~A ⊃ ~B
2. A ⊃ C
3. Z ⊃ W
4. ~C • ~W /∴ ~B v W
---------------------------------------------------------
5. ~C 4 Simplification
6. ~A 2,5 Modus Tollens.
7. ~B 1,6 Modus Ponens.
8. ~B v W 7 Addition.
QED.
I want to also show the difference in rules set in some LOGIC texts. The rule set you see above is known as Copi Rules. This set of rules use English words to describe the justification. Some LOGIC systems do NOT use such rules. You may find other LOGIC systems that use a set of rules called NATIRAL DEDUCTION RULES. The two most common sets of rules are COPI RULES and NATURAL DEDUCTION RULES.
Natural Deduction Rules do NOT use Words like the Copi method above. Natural Deduction has Introduction Rules and Elimination Rules.
In Natural Deduction, you MAY NOT USE the following frequently used rules: Modus Tollens, Disjunctive Syllogism, Material Implication, etc. NONE of those justifications are in the rule set, BUT you can DERRIVE those rules with a bit of work.
Without the rules Modus Tollens, Disjunctive Syllogism, and Material Implication the proofs are more difficult and as a result be more frustrating to you. You can do it though. You just need a wider creative side!
I will demonstrate what a Natural Deduction Proof would look like using the same problem above with the different rules.
[I am doing this so you can see they are NOT identical sets of rules].
Note you will justify a line by the connective followed by and either an I for introduction or an E for elimination. For example, VI, VE, &I, &E, -->I or -->E and so on.
Here we go:
1. ~A ⊃ ~B
2. A ⊃ C
3. Z ⊃ W
4. ~C • ~W /∴ ~B v W
------------------------------------------------------------
5. |A Assumption
6. |C 2,5 -->E
7. |~C 4 &E
8. |C & ~C 6,7 &I
9. ~A 5-8 ~I
10. ~B 1,9 -->E (another name for Modus Ponens)
11. ~B v W 10 VI
QED.
Which rules set would you prefer?
Also I want you to be aware that some of the rules just go by different names but the justification is identical. For instance, -->E is another name for Modus Ponens, &E is another name for Simplification, -->I is another name for Conditional proof, VI is another name for Addition, and so on. BUT do not think all the rules are identical! Natural Deduction has a VE RULE that is also called proof by cases that the Copi Rule set does NOT have. There is also other rules like Reiteration Rule and a Falsum Rule not found in Copi.
All of that to illustrate that just saying the word LOGIC is ambiguous. There are different kinds of systems out there that may have different rules. The rules set can add difficulty because you are limited to those strict rules.
You need to be more creative than ever before to solve the exact same problem given less rules. After the hard work of solving many problems you will smile more often. You will smile more because of all you have been through up to now trying to make this work for you and finally getting the hang of it over time. :)
