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Well that's just classical logic, which is pretty much what you expect when you hear the word logic ('False AND True = False', etc) the other ones were develope

Well that's just classical logic, which is pretty much what you expect when you hear the word logic ('False AND True = False', etc) the other ones were developed to address it's shortcomings and paradoxes. http://plato.stanford.edu/search/searcher.py?query=logic http://plato.stanford.edu/search/searcher.py?query=logic https://en.wikipedia.org/wiki/List_of_paradoxes#Logic https://en.wikipedia.org/wiki/List_of_paradoxes#Logic

danharaj 11y ago
Of course, but the most well developed theory pedagogically and theoretically is first order classical logic. So, it's the best place to learn all of the standard concepts like derivability, completeness and soundness theorems, and all of that jazz. That's how I learned it and then my interests soon turned to intuitionistic logics and type theories. Still, the metatheory is very similar and the metatheoretic differences between logics are often framed as deviations from the metatheory of first order classical logic.
- darkmighty 11y ago
  And classical logic is the one used in mathematics (and methatheories) because those are the most useful axioms in that setting. (would you agree that's a fair statement?) For example, it's not useful to think an equality has 30% chance of being true, or not having a truth value, and so on. It allows building structure on solid foundations.
  - danharaj 11y ago
    I think intuitionistic logic is just as valid a foundation for mathematics as classical logic. They both have very attractive features for mathematics. They both appeal to different intuitions about what logic ought to be. When it comes to mathematics, one is more natural than the other in different contexts. The proof theory of a logic is what is useful to mathematics. It's hard to beat either classical or intuitionistic first order logic when it comes to their proof theories. I think that's why nothing else really catches on. You can build other logics on top of one or the other anyway. Another perspective is that you can encode other logics in them.

danharaj 11y ago

Of course, but the most well developed theory pedagogically and theoretically is first order classical logic. So, it's the best place to learn all of the standard concepts like derivability, completeness and soundness theorems, and all of that jazz. That's how I learned it and then my interests soon turned to intuitionistic logics and type theories. Still, the metatheory is very similar and the metatheoretic differences between logics are often framed as deviations from the metatheory of first order classical logic.

darkmighty 11y ago
And classical logic is the one used in mathematics (and methatheories) because those are the most useful axioms in that setting. (would you agree that's a fair statement?) For example, it's not useful to think an equality has 30% chance of being true, or not having a truth value, and so on. It allows building structure on solid foundations.
- danharaj 11y ago
  I think intuitionistic logic is just as valid a foundation for mathematics as classical logic. They both have very attractive features for mathematics. They both appeal to different intuitions about what logic ought to be. When it comes to mathematics, one is more natural than the other in different contexts. The proof theory of a logic is what is useful to mathematics. It's hard to beat either classical or intuitionistic first order logic when it comes to their proof theories. I think that's why nothing else really catches on. You can build other logics on top of one or the other anyway. Another perspective is that you can encode other logics in them.