 Hello and welcome back to fill 320 deductive logic. I'm Professor Matthew J Brown and today we're going to be talking again about proofs in SL and we're going to be learning about the derived rules and rules of replacement that we use in SL. Now having seen the basic rules of SL both direct and indirect proof all the introduction and elimination rules for our different connectives you might ask yourself do we need more rules. The rules we've seen so far for our natural deduction system for SL are systematic and they could let us prove anything that we need to prove right but there are some other rules that are some of which are very intuitive others are just convenient and many you might say many different proof systems include such rules in their systems and so we're going to look at some of those rules now we call these rules derived rules because they don't add new content to our proof system and they themselves can be can be established or shown to be valid based on the existing rules. One example is the dilemma rule right the dilemma rule shows that if we have three lines the disjunction a or b and the conditional if a then c and if b then c then we can derive the conclusion c right a b and c here are our script letter meta variables right not just sentence letters but this the intuitive idea here is that you don't know if a or b is true but you know that one or the other or both are true and if you have these two conditionals from a to c or b to c then either way c is going to end up being true that's kind of the intuitive idea PD Magnus actually shows the derivation for this rule in chapter six of for all x so if you want to see how a derived rule is derived you can go look at that hypothetical syllogism is another fairly intuitive derived rule this tells us that if we have the two lines mnm if a then b if b then c then we can conclude if a then c it seems like a fairly intuitive rule it's derivation is also shown in the book and actually we we performed a version of this derivation for a specific example in one of our previous problems that we worked out in a previous lecture the last derived rule which is in some proof systems treated as a basic rule is called modus tollens right and it is kind of the reverse of a conditional elimination rule instead of starting with the conditional and the antecedent we start with the conditional and the negation of the consequent and then we derive the negation of the antecedent you'll actually have the opportunity to prove this one in one of the practice exercises in chapter six now let me say again you don't strictly need any of the derived rules they only make proofs more convenient another kind of rule that makes proofs more convenient we call rules of replacement and so far all of the rules we have seen only apply to whole expressions right so if we look at this argument and we want to prove if a then b and c therefore if a then c we might be tempted to use like the conditional elimination rule here we've got b and c let's just use conditional elimination to pull out that c and and do that directly but you can't do that that won't work the conjunction elimination rule only applies to whole sentences where the conjunction is the main connective right in general all of the introduction elimination rules and all of the previous rules we've looked at only apply to the entire sentence and are focused on the main connective of that sentence right so to prove this we'd have to go a different direction we'd have to start with our premise we in this case would probably want to do a conditional proof since we're proving a conditional right we start with a we know we want to get c we can use conditional elimination to get b and c we can use conjunction elimination to get c then we can discharge our sub proof and get if a then c and we did end up using conjunction elimination but we couldn't apply it directly we had to go through this sub proof process let's look at another example this seems like a fairly intuitive thing to prove if a then b and c therefore if a then c and b seems like we ought to be able to do that pretty easily and we could but notice that these two sub expressions these two components of the expression are logically equivalent sentences we know that the conjunction the order doesn't matter b and c c and b they mean the same thing right and there is a rule that allows us to replace that part of the expression we call these rules of replacement they involve substituting completely logically equivalent expressions or sub expressions for one another and let's go through the examples from the book we have first the commutivity rules right they allow us to take any uh any conjunction disjunction or biconditional and substitute it for the reverse version the order doesn't matter for these connectives and the commutivity rule reflects that right and again you can apply this to sub expressions right to parts of expressions that fit that mold demorgan's laws are another kind of replacement rule having to do with the interaction between negation disjunction and conjunction right so if we have the negation of the of a or b that's the same as not a and not b right if we have the the negation of the conjunction of a and b that's the same as not a or not b right we've seen that previously we can show that with truth tables that the examples of that are the case and here we the rule allows us to make that substitution double negation also a very helpful replacement rule whenever you've got two negations you can you can eliminate the double negation so not not a it's the same as a also occasionally you might want to start from a and substitute in not not a and that's allowed as well we also have replacement rules based on the definition of the material conditional and the biconditional right we know our material conditional if a then b is equivalent to not a or b right and so we can use that fact to do this substitution of we call this material conditional the biconditional exchange also allows us to use the fact that a biconditional if a if and only if b is equivalent to if a then b and if b then a right so we can we can go in either direction here as well and replace one expression with the other just to give you a sense of how we might justify some of these replacement rules there's many different approaches you can do it with a with a kind of meta language derivation you can do it with truth tables I can show you with some Venn diagrams how we would do it for de Morgan's law here right so let's take this version the negation of a or b is equivalent to not a and not b right so let's consider this circle a and consider this circle b we know a or b is whatever is in either or both circles right that whole area in red is a or b and the negation of that right is everything outside of that area right so the whole left hand expression there is highlighted in green now let's look at the right side right so the red here is what is a right that's that's all of the places where a applies right and not a is going to be everything outside of that circle right similar deal with b right everything that is not b is what's outside that circle right and then in the conjunction of not b and not a is going to be every place where you have both which is highlighted in both this figure and this figure and again that's just everything that is outside of both circles and that is the same as what we had here right in the one side so that is perhaps the least formal way to show it but it's interesting you could also show it using truth tables and I'd suggest you give it a try on your own and try it for the other example of de Morgan's law and any of the other replacement rules as well so that's a quick tour of the derived rules and rules of replacement of course the best way to learn how these rules can be used is to give it a try so I suggest you go ahead and give the next set of practice problems ago and see if these any of these rules help you out in those cases right next time we're going to talk about strategies for doing proofs and we'll work some additional examples together okay bye