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# Logic and Proof

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# Detachment and Syllogism

Those words sound awfully fancy, don't they? It should be illegal to say them without a top hat and a monocle. They give two answers to the question, "So, what good are conditional statements, anyway?" They are used as a way of getting new information from information we already have.

The law of **detachment** allows you to "detach" the hypothesis from the conclusion. More precisely, if we know both *p* and *p* → *q* to be true, then we may conclude that *q* is true. When the traffic law says, "If the red light is blinking, then come to a full stop," and you see a blinking red light up ahead, it's clear what you'll do next (hint: resist the temptation to slam the gas pedal).

### Sample Problem

What can you infer from the following two statements: "All chickens hatch from eggs" and "Betsy is a chicken."

We can write the two statements in shorthand as follows: "chicken → hatch from egg" and "Betsy is a chicken." Since Betsy satisfies the hypothesis of chickenhood, she must by detachment meet the conclusion of egginess. In other words, "Betsy hatched from an egg."

The law of **syllogism**, on the other hand, allows us to squeeze together conditional statements. If we know both *p* → *q* and *q* → *r* to be true, we can squeeze them together to get *p* → *r*. After all, going through *q* when we can go straight from *p* to *r* would be just plain silly-gism.

### Sample Problem

"If you make a right turn, you must use a turn signal," and, "There is a right turn on the way to school." What can be inferred from these two statements?

As usual, it's a good idea to write the statements in their shorthand to make the structure of the implications clear. The first becomes "right turn → signal" and the second "school → right turn." (That is, if you go to school, then you must make a right turn.) Syllogism lets us write the chain "school → right turn → signal" and then cut out the middle part, leaving us with "school → signal." Translating out of our shorthand, "If you are driving to school, then you will use a turn signal."

We'll list the two laws one last time:

- Detachment: from
*p*→*q*and*p*you may infer*q*. - Syllogism: from
*p*→*q*and*q*→*r*you may infer*p*→*r*.