Deductive Logic
Worksheet 1
In our unit on probability and chance, we mentioned how hard it is to prove something. Remember that little ball bearing toy that you flip upside down and watch where all the balls go? Do I think that all the balls will ever end up in the same column? No, I don't believe that will ever happen in my lifetime. I can state that belief with a high degree of confidence. But I can't prove it, because the event could happen.
In this unit on logic, we will be focusing on things that you can prove. In a proof, you start with some initial given information and you arrive at a conclusion that must be true. The given information is some kind of fact or definition that you accept as true. Then you look at the consequences of that given information. We will start our unit on proof by looking at some very simple conclusions and see if we think they are valid conclusions, coming from the given information.
Here's an example:
Given Information: Every time I go to the store, I have fish for dinner that same day.
Conclusion: Since I went to the store today, I will have fish for dinner.
This conclusion is valid from the given information. Remember you have to accept the given as true. You could restate it as a subset problem: every day that I go to the store is a day that I have fish for dinner. Therefore, my store days are a subset of my fish-for-dinner days.
Next look at this example:
Given Information: Every time I go to the store, I have fish for dinner that same day.
Conclusion: I'm having fish for dinner today, so I must have gone to the store today.
This is not a valid conclusion from the given information. All we know is that today is a fish-for-dinner day. The day might be a member of the store day subset, but then again it might not.
For each problem on this sheet, decide if the conclusion is valid or not. Explain on your separate sheet of paper. Remember that you must accept the given information as true.
1) Given Information: I only eat fish on days that I go to the store.
Conclusion: I'm having fish for dinner today, so I must have gone to the store sometime today.
Is this conclusion valid or not? Notice that I changed the given information. Does it matter? Explain on your separate sheet of paper.
2) Given Information: Every time we have a fire drill, the fire department comes to the school.
Conclusion: Since the fire department came to the school today, we must have had a fire drill.
3) Given Information: I always stop at red lights.
Conclusion:If there's not a red light, I don't stop.
4) Given Information: I always stop at red lights.
Conclusion:If I'm not stopped, there's not a red light.
5) Given Information:Every time I get a Yahtzee, I end up with a score over 100 for the game.
Conclusion:Since I scored 150 on this game, I must have gotten a Yahtzee.
6) Given Information:The store will not sell you cigarettes if you do not have proof of age.
Conclusion:Since the store sold you cigarettes, you must have had proof of age.
7) Given Information:Every time there has been a shoplifting incident at the Mini-Mart, you have been seen in the store at that time.
Conclusion:You are the shoplifter.