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Fax esigning conditional

in this video we're going to investigate something called conditional statements and a conditional statement is one of the form if P then Q you want to say that there is an initial statement I hypothesis P and that if that hypothesis is true then the conclusion the second statement Q must also be true now we've seen some statements like these before but what we want to investigate in this video is how can we deal with things like truth tables how we can we combine this with other statements and so on so for instance let's try to come up with a truth table for the implication for the conditional p.m. flight Q so our first step as always going to be to fill in the variables so I'm going to say that I thought a prove the possibility through the possibility false is a possibility and fall to the possible possibility for my P variable and then for my Q variable we could have it being true we could have it being false we could have it being true we could have it being false now the idea of the implication is that whenever the first thing is true whenever the hypothesis is true then so two is going to be the conclusion so if I read over the first row this is a scenario where the hypothesis P is true and the conclusion also ends up being true so the the implication is true in this scenario this is an arrow where when the first thing is true the second thing is going to be true and so I'm going to say that my implication is going to be true now for the second scenario the hypothesis is still true but the conclusion Q is false so the implication is not true is not the case that when P is true then Q is going to be true so I'm going to go and say that this is going to be false as well and then in the second set of scenarios in a sense to conditional doesn't apply here so what it says is that if P is true then the conclusion is going to be but in these bottom two scenarios here the assumption isn't even true so the condition isn't true to be even considered it's not the case that P is true so I don't have if P is true so for these bottom two rows we are going to say they are vacuously true we are going to give them the truth value of being true and the reason why we're doing this is that there's first of all no reason to think that they're false right it's not a scenario where the first thing was true and the second thing was false so I'm not saying the implication is false and we're going to sort of think that we haven't even had a chance to get off the ground of considering this and so we're going to just label them as being true nonetheless to make a little bit more sense of this I'm going to put up a couple different columns in my truth table so I'm going to put up naught P in the statement not P or Q so let me first just fill in that table and then we're going to try to analyze them so if I have my keys over here then if I want to look at what my not Peas are going to be they're just not whatever the P was so I'm going to put it as a false I input this as P is true not P is false P is false here and therefore not P is true and T is false here and therefore not P is likewise true so that was easy to fill in that table and now what we have is an or statement not P or Q so in this scenario we have true here for my Q false here for a naught T because it's an or only one of them to be true so I'm going to say this is going to be true down here false and false therefore my or statement is going to be false true and true therefore true false and true I only need one of them and so true as well now here's the big takeaway this column that we're interested in this is the P implies Q column and this column that we're interested in this is the not P or Q column these things are both the same in other words the implication which might seem a little bit confusing at first has the same truth table as the statement not P or Q and indeed this gets to this point of how we're going to think about why these bottom roads are going to be true in the case of implication if we think of it as my implication is one of two different things either my conclusion is true or it is the case that my initial assumption my p-value was false if my if my initial assumption my p-value is false then I not worrying about my conditional I haven't even got started my initial assumption if this thing is true is is not even true and so I can think of this as not P or Q and that is the same logical thing as P implies Q so let's look at this statement if I study hard then I will pass so this is the conditional we have two different components to it first of all we have this statement here if I study hard I am going to denote that by the symbol P and then we have the property that if I study hard then I will pass and I will pass I will give the statement Q so I can rewrite we write this as saying if P then Q and I can even combine this into a shorthand by saying the symbol P implies Q so all of these things mean the same thing first I replaced by English phrases with just logical symbols P and Q I've cut the intima then and then my implication arrow was a way that I could sort of hide the English words if P then Q they mean the same thing all right so let's look at the other one we just said that a logical implication or a conditional was the same thing as not P or Q so let me try to write out not P or Q I claim it is going to be this sentence so here we have either I don't study hard I claim that this is the symbol not P right if P is study hard then I don't study hard is not P and then I will ass that was my statement q and so I'm going to go and put that as a Q and then you'll note that this bottom one is phrased as an either/or either it's this first thing that not P or is the second so I can connect them by putting in my disjunctive symbol my or and I think that these two things do logically mean the same thing so if I study hard that I will pass that sets up a couple different possibilities it might be that I pass the course or it might be that I don't study hard so indeed these two things are largely to say