#$&* course Mth 151 If your solution to stated problem does not match the given solution, you should self-critique per instructions at
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Given Solution: This argument certainly seems valid. We say what will happen if rains, and what will happen is that happens. Then we say that it rains, so the whole chain of happenings, rained then wet grass then smell, should follow. STUDENT QUESTION ??? What if the grass does not smell or there’s a person unable to smell? INSTRUCTOR RESPONSE None of that affects the logical validity of the argument. If the premises are accepted, then the conclusion is valid. Any argument about whether the premises are valid is not relevant to the logical validity of the argument. Of course if the premises aren't so then it's possible the conclusion isn't so either, but that doesn't prevent the argument from being logically valid. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):O K ------------------------------------------------ Self-critique Rating: 2 ********************************************* Question: `q002. Is the following argument valid: 'If it snows, the roads will be slippery. If the roads are slippery they'll be safer to drive on. Yesterday it snowed. Therefore yesterday the roads were safer to drive on.' YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Yes. Even though one of the premises is obviously false. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: The validity of an argument has nothing to do with whether the statements in that argument are true or not. All we are allowed to do is assume that the statements are indeed true, and see if the conclusions of the argument therefore hold. In this case, we might well question the statement 'if the roads are slippery they'll be safer to drive on', which certainly seems untrue. However that has nothing to do with the validity of the argument itself. We can later choose to reject the conclusion because it is based on a faulty assumption, but we cannot say that the argument is invalid because of a faulty assumption. This argument tells us that something will happen if it snows, and then tells us what we can conclude from that. It then tells us that it snows, and everything follows logically along a transitive chain, starting from from the first thing. STUDENT COMMENT: so it does not matter that the roads are not safer when they're slippery, what matters is that the statement said they are when snows and snowed yesterday therefore the roads were safer yesterday INSTRUCTOR RESPONSE: Right. The statements don't have to be true for the argument to be valid. Of course, if the statements aren't true then even though the argument is valid the conclusion might not be true. The old saying is 'garbage in, garbage out'. If you put 'garbage' (i.e., false statements) into a logical argument, that argument can indeed result in 'garbage' (i.e., a false statement as the logical conclusion). STUDENT COMMENT: According to the statement it is true, but I might question this about driving on slippery roads. INSTRUCTOR RESPONSE: That assumption is deliberately absurd, to help make a clear distinction between correct assumptions and correct logic. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: 2 ********************************************* Question: `q003. Is the following argument valid: 'Today it will rain or it will snow. Today it didn't rain. Therefore today it snowed.' YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The “or” means at least one of the things will happen. Since it didn’t rain, then it must snow. It is valid. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: If we accept the fact that it will do one thing or another, then at least one of those things must happen. If it is known that if one of those things fails to happen, then, the other must. Therefore this argument is valid. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: 2 ********************************************* Question: `q004. Is the following argument valid: 'If it doesn't rain we'll have a picnic. We don't have a picnic. Therefore it rained.' YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Not valid I drew a circle for days that it doesn’t rain I drew a circle inside it for a picnic If we don’t have a picnic, an x could be placed inside the outer circle without intersecting the inner circle or it could have been drawn totally outside of the outer circle.
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Given Solution: In this argument where told the something must happen as a result of a certain condition. That thing is not happen, so the condition cannot have been satisfied. The condition was that it doesn't rain; since this condition cannot have been satisfied that it must have rained. The argument is valid. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Not quite understanding this.
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Given Solution: The argument gives three conditions, 'If it rains, the grass gets wet. If the grass gets wet, we'll be able to smell the wet grass. It rained yesterday.', which are symbolized p -> q, q -> r and p. It says that under these three conditions, the statement r, 'we can smell the wet grass', must be true. Therefore the argument can be symbolized by the complex statement [ (p -> q) ^ (q -> r) ^ p] -> r. STUDENT COMMENT: becuase the statment is valid r will be on the outside of the parenthesis INSTRUCTOR RESPONSE: It doesn't matter whether the statement is valid or not. The premises go into the parentheses or brackets, the conclusion follows the -> sign. The form of the argument is [premises] -> conclusion, where the premises inside the brackets are joined by conjunctions. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
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Given Solution: For these truth values p -> q is true since p is false (recall that the only way p -> q can be false is for p to be true and q to be false), q -> r is false since q is false, and p itself is false, therefore [ (p -> q) ^ (q -> r) ^ p] is false. This makes [ (p -> q) ^ (q -> r) ^ p] -> r true, since the statement can only be false if [ (p -> q) ^ (q -> r) ^ p] is true while r is false. ********************************************* Question: `q007. Symbolize the following argument: If it rains, it pours. It doesn't rain. Therefore it doesn't pour. Then set up a truth table to test the validity of the argument. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: p = it rains q = it pours -p→-q p q -p -q -p→-q T T F F T T F F T T F T T F F F F T T T