# 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.

### Jim

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.

2. Een mooi spel ja.Nu Z 4R biedt, kan N misschien alsnog 4H bieden. Na zijn passen op 3H kan dat vermoedelijk geen 4kaart zijn? Geen simpel 15 En een beetje achteraf-commentaar?Met een 4kaart H had W misschien wel 3S geboden..Maar misschien is het beter om na 3S nogmaals te doubleren. Noord zou kunnen passen en de 500 incasseren. Als 5R er in zit zal 3S vast wel (genoeg?) down gaan.Maar doublet met een singleton klaver?……Vandaar: een mooi spel!

3. How do I win my ex back who is now dating his ex?Okay, my boyfriend and I have been dating for 2 1/2 months. We’ve had problems with his ex girlfriend(which was his first love) for awhile. Anyway, last week he broke up with me because I supposedly “changed”. Which now I know was not why. He and his ex had been talking. And now they are dating. All I want is for him to be happy. But I also want to be with him. How do I win him back? I love him with all my heart.

4. Quisiera saber si el Partido Politico ESPERANZA ROSARINA sigue vigente.Desde ya agradezco su atenciÃ³n.Y respondiÃ³ el 13 de julio de 2010 a las 16:25:Fernanda: No figura haber participado ningÃºn partido de ese nombre ni en las elecciones de 2009 ni de 2007, lo que hace presumir que no existe

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