A Lesson in Logic

On to our last two rules of inference today, and then we can really start on the interesting things. The dilemma uses the material conditional and a disjunction of the antecedents to show the disjunction of the consequents. It sounds more complicated than it is, really. No guest argument this week, and instead the last rule of inference, and the easiest, simplification. Now, onto the logic!

Here’s our example:

  1. If snow is white, then the cat is on the mat. (W → M)
  2. If sparrows fly, then the dog is in the yard. (F → Y)
  3. Snow is white or sparrows fly. (W ∨ F)
  4. Therefore, The cat is on the mat or the dog is in the yard. (M ∨ Y)

Remember, the content doesn’t have to be connected in any way, it just has to be able to be judged true or false. The way this works is as follows. If premises 1 and 2 are true, then M and Y follow from W and F respectively. If premise 3 is true, then at least one of W and F are true, which means that at least one of M and Y are true. Here’s a quick example of how a dilemma might play out in real life. Note how the truth of the conclusion follows necessarily from the premises. This is what makes it a valid argument.

  1. If we get what I want for dinner, then we’re having pizza. (I → P)
  2. If we get what you want for dinner, then we’re having chicken. (Y → C)
  3. We’re getting what I want or what you want. (I ∨ Y)
  4. Therefore, we’re getting pizza or chicken. (P ∨ C)

Bonus Rule: Simplification

The final rule of inference is Simplification, which is really straightforward. It looks like this:

  1. Snow is white and it is raining (S ∧ R)
  2. Therefore, Snow is white. (S)

Alternately, you could say therefore, it is raining (R). Simplification is a rule that lets you break up conjunctions into their parts, provided those conjunctions are the only element in the premise. If it’s true that snow is white and it’s true that it’s raining, then it follows that those things are as true separately as they are together.

And that’s it. That’s all the rules of inference, from Modus Ponens to Simplification. I hope you’ve enjoyed the lessons in logic so far, and if you’ve just come into them, I recommend going to my post index and checking out the rest. Each post builds on previous ones in some way. Next week, I’ll show you how these rules of inference can help you evaluate arguments. That’s right, we’re going to do a couple of proofs. Thanks for reading!


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