I discussed a particular valid argument in class:
All humans are mortal
Socrates is a human
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Therefore, Socrates is mortal.
This has the argument form:
All A's are B's
x is an A
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Therefore, x is a B.
We went over what makes this argument valid in class. However, I would like to go over other valid arguments.
A friend tells you “I’ll be either at the movies or home.” Later on you find out she is not home, then the logical thing for you to conclude is “she is at the movies.” We use this kind of reasoning all the time, without realizing it. This kind of argument is valid and its form is:
Either A or B
Not B
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Therefore A
Assuming your friend means to keep her promise, and is not impeded by any other circumstance, and you find out she is not home, then it must be true that she is at the movies. Of course, your friend can lie when she says, “I’ll be either at the movies or home” she could go to a lake, then the first premise would be false. Even then, we would say the conclusion you reached is a valid one, even though your friend lied (and the premise was false). Again, it is valid because if the premises are true then the conclusion must be true. If you accept the premises then you must accept the conclusion, logically speaking.
A friend tells you, “If you see me home tonight I will be studying.” So we know that if she is home, then she will be studying. That’s our first premise. Later on tonight, we find out she is in fact home. Now we know our second premise: “She is home tonight.” Thus, we conclude she must be studying. Assuming all premises are true, then the conclusion must also be true.
If she is home tonight, then she is studying
She is home tonight.
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Therefore she is studying.
This is a very common valid argument form, called “Modus Ponens” and has the form:
If A then B
A
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Therefore B
Take the same example of your friend who says she’ll study tonight. She calls you and lets you know that she is not studying right now. Again, if it’s true that if she is home tonight then she is studying, a valid conclusion you can get is that she is not home. If she were home, she would be studying. If she’s not studying it’s because she is not home.
If she is home tonight, then she is studying.
She is not studying.
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She is not home tonight.
This argument form is called “Modus Tollens” and has the form:
If A then B
Not B
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Therefore not A
Be careful of Modus Ponens and Modus Tollens! There are other similar forms to Modus Ponens and Modus Tollens that are actually invalid, such as:
If she is home tonight, then she is studying
She is studying.
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Therefore she is home tonight.
At first glance, it seems this argument is valid, since it is very similar to Modus Ponens, but do not be fooled by appearances. Although it might be true that our first premise “If she is home tonight, then she is studying” is true, and our second premise“she is studying” is also true, it would be invalid to conclude that “She is home tonight.” She could be at the Java House studying. Keep in mind that although our premises are true, our conclusion need not be true (because she could be at the Java House). This kind of invalid argument form is called “Affirming the Consequent” and it has the form:
If A then B
B
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Therefore A
This looks similar to Modus Ponens, but notice that instead of “A” as our second premise, we have “B” instead (the consequent of “If A then B”).
See if you can figure out an invalid argument form that is similar to Modus Tollens! Again, here is how Modus Tollens looks like:
If A then B
Not B
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Therefore not A
p.s. Here's a link to a webpage that explains the basics of Logic and deductive standards (or deductive arguments) in good detail, please read through it!
http://kristopher.g.phillips.googlepages.com/logicprimer
p.p.s. Your homework for this week includes reading The Republic Books I and II.
Email me with any questions you might have, or see me during office hours!
