Lessons in Logic: Conjunction & Addition

Today we’ll cover two pretty simple rules of inference, addition and conjunction. They sound the same, but they’re distinct in some pretty essential ways. 

Conjunction

Conjunction works exactly like the operator of the same name, and arguments using it take this form:

  1. Cats are furry (C)
  2. Snow is white (S)
  3. Therefore, C ∧ S

In other words, if it’s true that C, and it’s true that S, then joining C and S also has to be true. This is pretty straightforward, and can apply to any two premises, regardless of their content. Remember, what matters in this respect is whether they’re true or not, not what they actually say. If one of the premises is false, then the conclusion is also necessarily false. It’s a reasonably common rhetorical tactic to conjoin something false with something true in order to pass it off as true. A rather topical example of this can be found in the Kony 2012 campaign:

  1. Joseph Kony is a terrible man (True).
  2. Joseph Kony is a threat to Uganda (False).
  3. Therefore, Joseph Kony is a terrible man and a threat to Uganda (False).

Since one of the premises is false, the conclusion is also false.

Addition

Addition works by adding another proposition to create a disjunction. The most important thing to know is that you can add any proposition that you want, because of how “Or” statements work in logic. As long as at least one half of the disjunction is true, the conclusion is true. For example,

  1. Cats are furry. (C)
  2. Therefore, cats are furry or I am the leader of the nation of North Dakota (C ∨ D)

This is true, because it’s true that cats are furry. Addition can be done an infinite amount of times to create incredibly long premises, complex disjunctions which express an array of options. However, a popular rhetorical tactic is to try to limit it to the options one prefers. To pick on Vic Toews again (Remember him from Modus Ponens?), here’s an example:

  1. You support Bill C-30. (C)
  2. Therefore, you support Bill C-30 or you support child pornography (C ∨ P)

Now, you might not fit into either of those categories, but have no fear, we can apply more addition, and get this:

3.  Therefore, you support Bill C-30 or you support child pornography or you value your privacy. ((C ∨ P) ∨ V)

If one of those is true, then it’s true. And that’s addition.

So today we’ve covered addition and conjunction, which leaves only four more rules of inference to get through. Thanks for reading, and I’ll see you next week for something equally exciting.

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