A Lesson in Logic

Now you’ve learned all of the rules of inference, and seen some of their applications independently, but arguments in real life are often far more intricate than any single rule. Often things work in conjunction, and it takes a lot more work to sort out what follows from what. You’re right, and we can apply the rules in sequence in a proof, a series of propositions that shows the conclusion follows from the premises. 

Here’s an example of a simple proof:

1. ((A ∧ J) → ~W) If Alice and Jane go to the party, then Wilhelm will not go to the party.
2. (A) Alice is going to the party.
3. (J) Jane is going to the party.
4. (~W) Therefore, Wilhelm will not go to the party.  (~W)
Now, we start the proof from here, applying our rules and referring to the lines where where drawing the information from to show that the conclusion follows.
5. (A ∧ J)                   Conjunction of 2 and 3.
We can join the propositions of lines 2 and 3 together with Conjunction.
6. (~W)                           Modus Ponens from 1 and 5.
The conjunction from 5 is the antecedent of line 1’s material conditional, so we can reach ~W through modus ponens.

In this way, you can follow every step of an argument. As long as the rules are applied correctly, the conclusion follows. This doesn’t speak to whether or not the conclusion is true, merely that it follows. If “Jane is going to the party” is false, then the conclusion will also be false. Here’s one that’s a little tougher for you to try. Leave your proofs in the comments, I look forward to reading them. Thanks for reading!

1. (C → (R ∨ F)) If the car is broken, it’s either the radiator or the fan belt.
2. (C) The car is broken.
3. (~F) It’s not the fan belt.
4. (R) Therefore, it’s the radiator.

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