Logic and Conditional SentencesDate: 10/04/2005 at 03:36:36 From: Cintia Subject: How to determine is a conditional sentence is truth I have a question about conditional statements. I am having a hard time understanding why two false statements in a conditional makes it true. I tried to use different statements to create a truth table but I get stuck on the same concept. I tried a sentence like "If a polygon is a square, then the sides are equal." If I assume a rectangle, then it seems to me that the statement is undefined. Being true or false does not even apply. Date: 10/04/2005 at 15:31:39 From: Doctor Achilles Subject: Re: How to determine is a conditional sentence is truth Hi Cintia, Thanks for writing to Dr. Math. This is the most difficult part of basic logic, and this exact question gave me headaches when I was first learning logic. The first thing to remember is that logic has only two possible answers: true or false. Let's take the following sentences and say that all of them are true: 1) I am hungry 2) I will eat soon 3) I am not thirsty 4) I will not drink soon We can make a few conditional statements out of these (and some related sentences): A) If I am hungry, then I will eat soon. This sentence has the form "if TRUE, then TRUE". I think we can agree that a sentence of this form is TRUE. B) If I am hungry, then I will drink soon. This has the form "if TRUE, then FALSE". The fact that I am hungry does not cause me to drink. Furthermore, I am in fact hungry, and I won't be drinking anything anytime soon. Hopefully we can agree that conditionals like this are FALSE. C) If I am thirsty, then I will eat soon. This has the form "if FALSE, then TRUE". This gets a little difficult. However, the fact of the matter is I will eat soon. And even if I were thirsty (in addition to being hungry), I would still eat soon. Therefore, the statement "if I were thirsty, I would eat soon" is still TRUE. D) If I am thirsty, then I will drink soon. This has the form "if FALSE, then FALSE". Here is where things can get downright difficult. I am not thirsty. However, if I were, then I would go get something to drink. So it is true that if I were thirsty, I would drink soon. So the conditional is TRUE. 3 of the 4 types of condionals came out TRUE: "If TRUE, then TRUE" comes out TRUE "If FALSE, then TRUE" comes out TRUE "If FALSE, then FALSE" comes out TRUE The only conditional that comes out false is: "If TRUE, then FALSE" comes out FALSE The reason that a conditional always comes out true where the first part (the antecedent) is false is because you haven't disproven it. You can think of it sort of like a burden of proof. I could say something like, "If I were 7 feet tall, I would be a professional basketball player". Because I am not 7 feet tall, you cannot prove that that conditional is false. There are some difficulties with this system. For example, the following two conditionals are both true: "If I were 7 feet tall, I would be a professional basketball player" "If I were 7 feet tall, I would not be a professional basketball player" Some logicians have attempted to create different types of logic where the only way a conditional is true is if it is of the form "If TRUE, then TRUE" Still, the only way for it to be false is for it to have the form: "If TRUE, then FALSE" The other two forms (where the antecedent is false) are assigned some other value, such as "untestable", or "N/A". Hope this helps. If you have other questions or you'd like to talk about this some more, please write back. - Doctor Achilles, The Math Forum http://mathforum.org/dr.math/ |
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