How can we tell true proofs from false ones? What, exactly, are the rules of a proof? These are the questions which will occupy us.
Proofs are argument by form. We'll illustrate this with three parallel examples of a particular proof form called syllogism.
Example 1.2
1 |
All people are mortal |
Premise |
2 |
Socrates is a person. |
Premise |
3 |
Therefore. Socrates is mortal. |
Syllogism, lines 1,2 |
Example 1.3
1 |
All [substitution ciphers] are [vulnerable to brute-force attacks] |
Premise |
2 |
The [Julius Caesar cipher] is a [substation cipher]. |
Premise |
3 |
Therefore, the [Julius Caesar cipher] is [vulnerable to brute-force attacks]. |
Syllogism, lines 1,2 |
Note that you don't need to know anything about cryptography to know that the conclusion follows from the two premises. (Are the premises indeed true? That's a different question.)
Example 1.4
1 |
All griznoxes chorble happily |
Premise |
2 |
A floober is a type of griznox. |
Premise |
3 |
Therefore, floobers chorble happily. |
Syllogism, lines 1,2 |
You don't need to be a world-class foober expert to evaluate this argument, either.
ASIDE: Lewis Carroll, a logician, has developed many whimsical examplesl0 of syllogisms and simple reasoning. (Relatedly, note how the social context of Carroll's examples demonstrates some feminist issues in teaching logic .)
As you've noticed, the form of the argument is the same in all these. If you are assured that the first two premises are true, then, without any true understanding, you (or a computer) can automatically come up with the conclusion. A syllogism is one example of a inference rule that is, a rule form that a computer can use to deduce new facts from known ones.
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