Learn about the Law of Syllogism
Now let’s learn about the law of syllogism, which is another way to have conditional statements be validated based on deductive reasoning. We will do two problems and both of which will be used to determine if the conjecture is valid using the law of syllogism. So let’s go over the law of syllogism real quick.
The laws says if P then q is true, and if q then r is true. So we have two related conditionals. If P then q and if q then r, if P then r is also true. If P is true, then q is true, it’s one statement. If q is true, then r is true is another statement, then I can assume that P and r area also are valid true conditional statement. Let’s use two examples to do that.
It says if a figure is a square, then the figure is a rectangle. And it also says if a figure is a square, then it’s a parallelogram. Conjecture is if the figure is a parallelogram, then it’s a rectangle. So here is what we do. Let’s write down all these three in terms of P, q, and. So would P b? P, well it says, the figure is a square, q says, figure is a rectangle, and r says figure is a parallelogram. Okay. So now what I need to do is see what the conjecture told us or what these statements told us. This tells me if P then q, this tells me if P then r. So the conjecture says if the figure is a parallelogram, which is r, then it is a rectangle, which means if r then q are of q then r, I cannot determine this based on what I have been told because what I’ve been told is P implies q, and P implies r, but I’ve not been told that q implies r. So I cannot prove this. It is not valid. I cannot prove it without any doubt, so it is not valid based on what I have been given. I’ve not been given that P implies q and q implies r. I’ve been this, I’ve been given this, but that does not necessarily mean that q implies r or r implies q is also true. Let’s look at the second problem.
It says, in similar ways, if we fly from Texas to California, so P is fly from Texas to California, second part of the statement says, we travel from Central to pacific time zones and r says if you travel from Central to Pacific, then you gain two hours. So what does this tell us? If P is true, if you fly from Texas to California, then q is true. So P implies q is proven. Second, it says, if q is true, which is if you travel from Central to Pacific time zone, then you gain two hours, so q implies r. So based on this, I can deduce that P implies r. If P is true, then r is also true. So then it says, if you fly from Texas to California, which is P, you gain two hours, which is r. So if P then r. So this is a valid conjecture, this is a valid conjecture based on what I’ve been told because of the Law of syllogism, I can prove it. Here however, the Law of syllogism basically told me that if P implies q, then P implies r is true, but I do not know if q implies r, so I could not prove this based on the Law of syllogism.
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