Understanding Modus Tollens: Deductive Reasoning with an Example

[albinoblanke]

TL;DR Summary

Is this example of Modus tollens correct?

My example:
If you have a value of 1 dollar, then you don't have the value of 10 dollars.
You have the value of 10 dollars.

Therefore, you don't have the value of 1 dollar.

Modus tollens is a form of deductive reasoning.
It goes as follows:"If P, then Q. If not Q, then not P."
Example:
If the dog detects an intruder, the dog will bark.
The dog did not bark.
Therefore, no intruder was detected by the dog.

I was reading about deductive reasoning and came across Modus Tollens. My gut felt something was off and I tried to figure my feelings out. So I came up with the example: If you have the value of 1 dollar, then you don't have the value of 10 dollars; If you have the value of 10 dollars, then you don't have the value of 1 dollar.

This was a bit problematic. The value of 10 dollars does contain the value of 1 dollar, it is part of the value. However, according to the deductive reasoning, you shouldn't have 1 dollar.

I didn't know where to ask this question so I decided to drop it here, I bet some of you can probably explain where I went wrong or if this is an actual exception to the rule, it feels like it is. Even if it isn't, I think it comes very close.

 

[Office_Shredder]

If you think that when you have ten dollars you also have one dollar, then why does the statement that you have one dollar mean you cannot have ten dollars?

Either you mean you have exactly 1 dollar, or you mean you have at least 1 dollar. You seem to be confusing the two statements in various places.

 

[Nugatory]

The statement "If A then B" implies the statement "If not B then not A", meaning that if the first statement is true then the second statement is necessarily also true; and also if the first statement is not true then the second statement may or may not be true. But either way Modus Tollens is working.

In your example we have "We have one dollar" for A and "We don't have ten dollars for B". So now we ask ourselves whether it is true that "if A then B", and it is not; if I have one dollar I might also have another nine to go with it. Because "if A then B" is not true here, the contrapositive "if not B then not A" may or may not be true, and that's what's going on your example.

Change proposition A to be "We have exactly one dollar" so that "if A then B" is true and try working through the logic again.

As an aside, there is a formal notation for representing logical propositions that is worth learning; it is more compact and avoids the ambiguities that plague ordinary English and have contributed to your confusion here.

 

[Dale]

The value of 10 dollars does contain the value of 1 dollar, it is part of the value. However, according to the deductive reasoning, you shouldn't have 1 dollar.

If you want to allow the value of 10 dollars to include the value of 1 dollar then the original statement “If you have the value of 1 dollar, then you don't have the value of 10 dollars” doesn’t work.

 

[Mark44]

My example:
If you have a value of 1 dollar, then you don't have the value of 10 dollars.
You have the value of 10 dollars.

Therefore, you don't have the value of 1 dollar.

I think you have the basic idea, but your example is not very convincing, due to the lack of specificity already mentioned.

A better example might be the following:
If your pet is a cat, then that pet is a mammal. ($p \Rightarrow q$)
If your pet is not a mammal, then it is not a cat. ($\neg q \Rightarrow \neg p$)

The term used in mathematical logic for this pair of implications is contrapositive. Courses devoted to logic, sometimes taught as philosophy classes, use the terms modus tollens and modus ponens, along with others.