Now that we’ve covered all of the operators, we can move forward to rules of inference. This is where things start to get awesome. These are full argument constructions whose conclusions necessarily follow from their premises, assuming their premises are true. modus ponens (not to be confused with my one day famous philosophy rock band Modus Pwnens) is the simplest of these. Read on for more about rules of inference, and modus ponens specifically.

The classical modus ponens argument is this:

- If Socrates is a man, Socrates is mortal.
- Socrates is a man.
- Therefore, Socrates is mortal.

And we can symbolize that as the following.

- S → M
- S
- Therefore M

This is essentially an expression of how a conclusion can follow from the material conditional. In order to get M, it has to be true that S → M, and it has to be true that S. If S → M is false, then it doesn’t matter if S is true. S can have no bearing on whether M is true. If S → M is true and S is false, then we can’t derive M. It can be true that if Socrates is a man then he’s mortal, and false that Socrates is a man (maybe he’s a god, or an idea), so we can’t conclude that Socrates is mortal, though we know that it’s true that if he *were* a man, he would be mortal.

Rules of inference like this are the cornerstone of logic, because they can show clearly how the conclusion follow from the premises, and guarantee a true conclusion provided that the premises are true. An argument that follows the rules of inference is referred to as a *valid argument*, because it has a valid form. If the premises are true and the argument is valid, then it’s a *sound argument*, which means that the conclusion necessarily follows.

How is this useful? Let’s use an example from the news. Canadian Minister of Public Safety Vic Toews is championing a bill which would let police spy on people’s data traffic without warrants, ostensibly in an effort to combat child pornography. He said, in Parliament that Canadians are either for the bill or for child pornography (link). Well, let’s try Vic’s logic in modus ponens.

- If you are against the bill, then you are with the child pornographers.
- You are against the bill.
- Therefore, you are with the child pornographers.

Over the next few weeks I’ll take a look at the rules of inference and the substitution rules, and dig around for some more concrete ways to apply them. I hope this won’t involve my government doing crazy things, but I’ll take my arguments where I can find them.