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Mathematical Logic



Suppose we assume this is true:

If you've won a cooking contest, then you're good at cooking.

Does the original statement imply this is true?

If you're good at cooking, then you've won a cooking contest.

In logic, we're looking at the structure of an argument, and trying to make general statements that we can apply to any argument.

For the last question, the structure was supposing that

If A, then B.

and asking if this guaranteed that

If B, then A.

An "if ... then" sentence in the format above is known as implication. The second statement is the converse of the first. In general, knowing an implication is true is not enough to say whether the converse is true.

There are examples where this is very intuitive:

If you are standing in Paris, you are standing in France.

does not imply

If you are standing in France, you are standing in Paris.

since there are many towns other than Paris in France!

The useful thing about knowing the logical structure is it allows us to handle much less intuitive examples with confidence, like the next question!

In the original TV show Star Trek, crew members wearing a red shirt died more on screen than any other type of crew. (This statement is true: 25 died wearing red shirts, 10 wearing gold, and 8 wearing blue.)

Does this mean that (supposing you are a crew member on Star Trek) that you are more likely to die if you are wearing a red shirt?

The converse can sometimes be true. For which choice is the converse true?

You might not be sure what the words mean here. Try to just look at the structure of the argument. You can also substitute your own argument into the structure and see if it works.


If a bacteria is of the genus Fusobacterium, it is anaerobic.

also imply the following?

If a bacteria is not of the genus Fusobacterium, it is not anaerobic.

Again, you may not know what the words mean below, but just think of the structure of the argument!

In mathematics, if something is a monoid, then it is a semigroup.

Does this imply that:

In mathematics, if something is not a semigroup, then it is not a monoid.

Just to summarize, given the implication

If A, then B.

Then the following (known as the contrapositive) must be true.

If not B, then not A.

The converse may or may not be true.

If B, then A.

The inverse may or may not be true.

If not A, then not B.

By knowing these structures, we can spot bad arguments in real life, no matter what topic! Knowing some implication is true, we can only state for certain the contrapositive is true; the converse and inverse require further argument.

Suppose I make the following claim on TV:

If you buy my SuperDrink 3000, you will be healthy.

Assuming for the moment the statement above is even true, would the statement below also be true?

If you don't buy my SuperDrink 3000, you will not be healthy.


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