Substitution Rules

A Lesson in Logic

Now that we’ve finished all of the rules of inference, it’s time to get into substitution rules. See, you can’t change a set of symbols without applying some kind of rule to do so, in order to make sure that the change actually follows from. Substitution rules are a different set of rules with a different function, and should be pretty straightforward. 

Rules of inference, you’ll remember, let you draw conclusions from premises and, as long as the rules are followed and the premises are true, the conclusion has to be true. Substitution rules show that some logical statements can be exchanged for other ones without losing any of the information involved. This means that every time a premise is true, its substitution is true, every time it’s false, its substitution is false. They’re equivalent, alternate constructions of the same thing. We use them because sometimes we need one construction, and sometimes we need another. This argument will show why.

1. ~P → Q. If the pencil case isn’t red, then there is a quick brown fox on the mat
2. ~Q. There is not a quick brown fox on the mat.
3. Therefore ~~P. The pencil case isn’t not red.

This is a pretty straightforward  use of Modus Tollens, where the negation of the antecedent (P) follows from the negation of the consequent (Q). But P was already negated, so the rule gives us ~~P. But we wouldn’t say “The pencil case isn’t not red”. It has the right meaning, but it’s awkward. So we apply a substitution rule, the rule of Double Negation. ~~P is equivalent to P, so we add a fourth line.

4. Therefore, P. Double Negation from 3.

Every time ~~P is true, P is true. And now, instead of saying “The pencil case isn’t not red”, we can show that it follows that “The pencil case is red.” You can have triple negations and quadruple negations as well, but the same rule applies. Removing two negations leaves you would the same meaning as if they were there. This is probably the most intuitive of the substitution rules, but there aren’t a lot, and we’ll get through them pretty quickly. Check out the whole series for more lessons in logic, and how we can use formal logic in our every day lives.


  1. I see my calculus stnetuds make these sorts of mistakes all the time. I also plug in numbers to show them that it doesn’t work, but they continually make the same mistake. The (a+b)^2 versus a^2+b^2 thing comes up at least once a month. I find that going into the theory helps a lot more, but there are still just some things that look so right. And so the kids (and the teachers sometimes!) just keep making silly mistakes.

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