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welcome back to logic 101 i'm william spaniel and today's topic is conditional proofs these are going to look very similar to what we covered last time proofs by contradiction in that we are going to start by assuming that something is true and see what falls out as a result of that rather than starting by just applying or rules of inference and replacement rules right off the bat so the idea behind the conditional proof is that we want to prove a conditional statement is true something like P implies Q where P and Q can be simple sentences or compound expressions doesn't matter we begin a conditional proof by assuming that the antecedent P is true and then using our replacement rules and our rules of inference we show that Q is true as well Q must follow given that we have assumed that P is true now the reason that this works is that conditional statements tell us what must follow given that the antecedent is true so we're sparing ourselves a little bit of hassle by going ahead and starting out by assuming that the antecedent is true and seeing where that takes us we are literally showing that an if-then statement holds if we assume that this is true then this other thing must be true as well this might help if we see an example so here we go right now we have three premises key or T implies you P implies Q and not s or not Q and we want to prove that s implies the implication T implies u so we have a conditional statement in the conclusion here actually we have two different conditional statements we have one inside the parenthesis and one outside of the parenthesis and so we need to be proving the overall statement here which means we're caring about the statement that's on the outside of the parenthesis not what's on the inside so we're going to begin our conditional proof by assuming that s is true so line for s and the justification for this is that we have an assumption for a conditional proof notice that I have indented this line just as I would in a proof by contradiction and the reason that I'm doing that is because anything that we show in this line as being true we can't show as being true overall all of the stuff that we show inside of the indentation is true as a result of us assuming that s is true or at least the way we're justifying it is based off of having s being true but if we've done that we can go through this invented line as we would normally using rules of inference and rules of replacement so if we have s as being true well us what else do we see hmm well I see something why do you suppose this video for a moment and see if you can figure out how to do the rest of this proof remembering that we want to ultimately conclude here that the consequent is true given that we have assumed that s is true so go ahead and pause the video for right now and if you're ready for the answer let's go ahead and see it if you have an answer please put it in the comment section but if not let's show what's what's actually going on here so we have s as being true and that means that through lines three and four using disjunctive syllogism we know that not Q is true that's because s is true and so if not S or not Q is true and we know that not s is not true then it must be that not Q is true if we have that not Q is true what else do we see well using lines 2 and also line 5 we can use modus tollens to back out that not P is true and if we have not P is true then well you know what we can use disjunctive syllogism again on lines 1 & 6 we have P as being part of a disjunction with T implies you but we know from line 6 that not P is true so that means that the second part of line 1 must be true which means we get the implication T implies you now we're done essentially with our conditional proof the last thing that we need to do here is just end it and the way we're going to end it is by taking the first line of our conditional proof and taking the last line of our conditional proof and using an implication to separate those two so line 8 says the first line of the conditional proof so line 4s and then the last line of the conditional proof line 7 you'll notice that we have those two lines there and now we just an implication separating them and our justification here is the lines of the conditional proof and we just call it hey this is a conditional proof we're done that's it we have successfully shown that if s is true then T implies you must be true as well so that's a conditional proof for you hope you enjoyed this one I hope to see you next time take care