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Logic provides a way to talk about truth and correctness in a rigourous way,so that we can prove things, rather than make intelligent guesses and just hope they are correct.

The ancient Greeks loved to hang around on the stoa , sip some wine, and debate.But at the end of the day, they wanted to sit back and decide who had won the argument.When Socrates claims that one statement follows from another, is it actually so?Shouldn't there be some set of rules to officially determine when an argument is correct?Thus began the formal study of logic.

The three fundamental studies were the Trivium grammar (words), logic (reasoning), and rhetoric (effective communication).These allowed study of the Quadrivium arithmetic (patterns in number),geometry (patterns in space), music (patterns in tone), andastronomy (patterns in time). All together, these subjects comprise the seven liberal arts .

These issues are of course still with us today. And while it might be difficult to codify real-worldarguments about (say) gun-control laws, programs can be fully formalized, and correctness can be specified.We'll look at three examples where formal proofs are applicable:

  • playing a simple game, WaterWorld;
  • checking a program for type errors;
  • circuit verification.
Many other areas of computer science routinely involve proofs, although we won't explore them here.Manufacturing robots first prove that they can twist and move to where they need to go before doing so, in order to avoid crashing intowhat they're building. When programming a collection of client and server computers,we usually want to prove that the manner in which they communicate guarantees that no clients are always ignored.Optimizing compilers prove that, within your program, some faster piece of code behaves the same as and can replace what you wrote.With software systems controlling more and more life-critical applications, it's important to be able to prove that a program always does what it claims.

Waterworld

Consider a game called WaterWorld, where each location is either empty sea or contains a pirate.When you enter a location, you must correctly anticipate whether or not it contains pirates.

  • If you correctly anticipate open sea, you are able to enter and determine how many of the (up to 3) adjacent locations containa pirate.
  • If you correctly anticipate a pirate, the location is tagged as dangerous, and you gather no further information.
Furthermore, there are really two types of moves: guesses, and assertions.If you make an assertion, then even if you happen to be correct but it is possible you could have been wrong, then it is an error.Also, it is an error if you make a guess about a location if it is actually possible to assert a location's contents.The interesting fact about these types of games is that while sometimes guesses are necessary (when?),surprisingly often an assertion can be made.

(You can freely download WaterWorld .)

Glimpses of two different WaterWorld boards

Questions & Answers

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Source:  OpenStax, Intro to logic. OpenStax CNX. Jan 29, 2008 Download for free at http://cnx.org/content/col10154/1.20
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