10.3: Basic Arguments- Using Logic
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An argument requires a number of premises (facts or assumptions) which are followed by a conclusion (point of the argument). The premises are used as justification for a conclusion. A conclusion which is correctly supported by the premises is known as a valid argument, while a fallacy is a deceptive argument that can sound good but is not well supported by the premises.
We will look at examples where the first two statements are the premises, and the third statement is the conclusion.
Determine if the following argument is valid.
All men are mortal.
John Smith is a man.
John Smith is mortal.
There are two premises (the first 2 sentences) and one conclusion (the last sentence). If we think of the premises as a and b, and the conclusion as c, then the argument in symbolic form is: \(a \land b) →c\). In order for the argument to be valid, we need this conditional statement to always be true. If there is ever a time, even just one time, when this conditional statement is false, then it is an invalid argument. Another way to think of this is to say that the conclusion must follow from the premises. If the premises are true, then the conclusion must be true in order for the argument to be valid.
Look at the argument – if we assume that a and b are both true, then does the conclusion have to follow? YES! If all men are mortal, and if John Smith is a man, then John Smith must be mortal. This is valid.
Determine if the following argument is valid.
All dogs are yellow.
Flippy is a dog.
Flippy is yellow.
All dogs are yellow is equivalent to “If it is a dog then it is yellow.” That is equivalent to “If it is not yellow, then it is not a dog” by the contrapositive. Assume the premises are true. Does the conclusion have to follow? YES! So this is valid!
In both of the examples above, the first statement of the premises could be written as an “if-then” statement. “All dogs are yellow” means the same thing as “If it is a dog, it is yellow."
The above examples are examples of Modus Ponens, which is always a valid argument.
Format of Modus Ponens (which is a valid logical argument)
p → q
p
q
Basically Modus Ponens states that if p implies q, and p is true, then q must also be true!
One could create a truth table to show Modus Tollens is true in all cases : [\((p → q) \land p ] → q\)
Determine if the following argument is valid. (Hint: rewrite the “all” as “if-then”, then also write the contrapositive)
All dogs are yellow.
Chipper is yellow.
Chipper is a dog.
“All dogs are yellow” is equivalent to “If it is a dog then it is yellow.” or “If it is not yellow, then it is not a dog” by the contrapositive. Assume the premises are true. Does the conclusion have to follow? It states all dogs are yellow, but doesn’t say anything about yellow things, or that everything yellow is a dog. It is possible to have something yellow (like a lemon) that is not a dog; that means the conclusion isn’t necessarily true. This argument is invalid. Let p stand for “It is a dog.” Let q stand for “It is yellow.” The format of the above argument, shown below, is not Modus Ponens.
It is an example of Fallacy by Converse Error.
| p → q |
| p |
| q |
Remember the example where p is “You live in Vista” and q is “You live in California”? Consider
This is a valid logical statement because it is of the form Modus Ponens. |
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This is an invalid argument, and is an example of Fallacy by Converse Error. |
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This is also an invalid argument, and is an example of Fallacy by Inverse Error. |
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This is a valid argument, and is an example of Modus Tollens. Modus Tollens is based on the contrapositive. Remember that p → q is logically equivalent to (~ q) → (~ p) So the above argument could be written in four steps:
The last three statements LOOKS like Modus Ponens. But the original argument only had three lines. It wasn’t written as the contrapositive. So it’s not called Modus Ponens. One could create a truth table to show Modus Tollens is true in all cases: [(p → q) \(\land ~q] → ~p\) |
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Another reasoning argument is called the Chain Rule (transitivity). Below is an example. The first two sentences are the premises, and the last is the conclusion. If the first two are true, the conclusion is true. If I have a bus pass, I will go to school. If I go to school, I will attend class. If I have a bus pass, I will attend class. So the idea is that if “if p, then q” and “if q, then r” are both true, then “if p, then r” is also true. Symbolically, the chain rule is: [(p → q) \(\land (q → r)] → (p → r)\) One could create a truth table to show the truth table is true in all cases, but it’s more complicated because there are 3 statements, hence 8 rows in the truth table. The format for the Chain Rule where the first two lines are the premises and the third is the conclusion is: q → r p → r |
17. If the two statements below are premises, use the Chain Rule to state the conclusion.
If Mia doesn’t study, then Mia does not pass the final.
If Mia does not pass the final, then Mia does not pass the class.
What about a logic statement where all of the outcomes of a formula are true in every situation? When this happens, it is called a tautology. Modus Ponens, Modus Tollens, and the Chain Rule (transitivity) are tautologies. A truth table will show the statement true in each row of the column for that statement. A fallacy is when all the outcomes of a logic statement are false. An example of a fallacy in words is “I called Jim and I did not call Jim.” If p is “I called Jim,” the logic statement in symbols for this fallacy is \(p \land ~ p\)). A tautology would be “I called Jim or I did not call Jim,” which is written as \(p \lor ~ p\))
You will create your own truth tables for Modus Ponens and Modus Tollens in the next exercises. Create intermediate columns so it is clear how you get the final column, which will show each is a tautology.
18. Make a Truth Table showing Modus Ponens is a valid argument. In other words, create and fill out a truth table where the last column is [(p → q) \(\land p] → q\), and show that in all four situations, it is true, which means it is a tautology
19. Make a Truth Table showing Modus Tollens is a valid argument. In other words, create and fill out a truth table where the last column is [(p → q) \(\land ~ q] → ~ p\), and show that in all four situations, it is true.
SUMMARY of arguments, where the first two statements are premises, and the third is the conclusion.
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Format of Modus Ponens (which is a valid logical argument) p → q p q |
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Format of Modus Tollens (which is a valid logical argument) p → q ~q ~p |
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Format of Fallacy by the Converse Error (an invalid argument) p → q q p |
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Format of Fallacy by the Inverse Error (an invalid argument) p → q ~p ~q |
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Format of Chain Rule (which is a valid logical argument) p → q q → r p → r |
20. Determine if the following arguments are valid or not. If they are valid, write if it is by Modus Ponens, Modus Tollens, or the Chain Rule. If it is not valid, write if it is by Fallacy by Converse Error, or Fallacy by Inverse Error, or neither. If it looks like the chain rule, but has a false conclusion, write the correct conclusion.
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If you are a gambler, then you are not financially stable. Sean isn’t financially stable. Sean is a gambler. |
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If you have a college degree, then you are not lazy. Marsha has a college degree. Marsha is not lazy. |
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If you have a college degree, then you are not lazy. Shannon is lazy. Shannon does not have a college degree. |
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If you have a college degree, then you are not lazy. Beth is not lazy. Beth has a college degree. |
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If you are a kitten, then you are a cat. If you are a cat, then you can purr. If you are a kitten, then you can purr. |
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If you are a comedian, then you are funny. If you are funny, then you are smart. If you are smart, then you are a comedian. |
21. Write a conclusion that would make each argument valid, and state if you used Modus Ponens or Modus Tollens.
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If you are a gambler, then you are not financially stable. Hollis is financially stable. |
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If a defendant is innocent, then he does not go to jail. Podric went to jail |
22. Determine whether there is a problem with the person’s thinking. Explain your reasoning.
A) John’s mom told him “If you get home after 10pm, then you are grounded.” John got home at 9:30pm and was grounded. He was really ticked off because he said that she lied to him. Did she?
B) Marcia told her daughter: “If you get home before 10pm, then I will give back your cell phone.” Her daughter got home at 9:45pm, but her mom didn’t give back the cell phone. Did her mother lie?
23. Create a truth table for \(p \lor (~ p → q)\)