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In my undergraduate logic courses I introduce several types of logics to my students including propositional, first order, second order, intuitionistic and fuzzy logics and it usually happens that somebody asks:

Which logic is better? Which logic is correct? Which one is more useful in mathematics, business, philosophy, etc?

I don't like to tell my students that your question is meaningless because even amongst professional logicians, mathematicians and philosophers, some of them prefer some logics more than others.

Question: What is a nice answer to above question?

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    $\begingroup$ I guess you could tell them that intuitionistic logic isn't not the best kind of logic. $\endgroup$ – Steven Gubkin Nov 16 '14 at 14:48
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    $\begingroup$ Which type of wrench is best? Which type of car is best? Which vitamin is best? Is algebra or geometry better? $\endgroup$ – user2338816 Nov 17 '14 at 0:05
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    $\begingroup$ I think the answer you give to that question is: "yes." $\endgroup$ – James S. Cook Nov 17 '14 at 6:33
  • $\begingroup$ Better compared with what? Your question IS the problem and it must be solved before going ahead. Then which of these "logic" are testable, fruitful, extended, simplest and "conservative" in the meaning of keeping results previously acquired? $\endgroup$ – climenole Dec 3 '14 at 16:01
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The following is written as if I were giving my own best answer to a student.

It's probably an accident of history that Aristotle defined classical logic in a specific way, and that classical logic has been almost exclusively used as the foundation of mathematics ever since. Common sense tells us that Aristotelian logic is an oversimplification of how we actually reason about the real world. For example, Aristotelian logic claims that every proposition is either true or false, but the following are statements that many reasonable people would say are neither true nor false:

  • Green is a nicer color than black.

  • Harry Potter had a great-great-great-great grandfather named Joe.

  • Based on the equal protection clause of the U.S. constitution, it's unconstitutional to deny gay people the right to get married.

  • My terrier is an intuitive Buddhist.

The vast majority of the literature of mathematics assumes classical logic, so for professional mathematicians in most fields, strong knowledge of classical logic is both necessary and sufficient.

But people didn't just invent fuzzy logic or multi-valued logic for no reason. They invented them because those logics were more suitable for what they wanted to do.

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    $\begingroup$ 1- A proposition is not a sentence. A proposition is a result of the analyse of declarative sentences which result into a model called "proposition": an analytic reduction to the most simple characteristics of declaratives sentences, the bivalence. By the way, P V ¬P, the excluded middle theorem, must be applied proposition by proposition and not to many proposition in the same time and , for not to the unformalised sentences given in your answer. $\endgroup$ – climenole Dec 3 '14 at 16:13
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It is meaningful to ask which logic is more useful in mathematics and business. E.g.:

  • Learning classical first-order logic can help you program, most dramatically in SQL.

I'd rather not confuse undergraduates with the other logics you mention. I would advise them:

  • Instead of learning fuzzy logic, learn probability directly.
  • Instead of learning second-order logic, learn multi-sorted first-order logic.
  • Instead of learning intuitionist logic, learn rhetoric, to be alert to how people use dichotomies.

So I would deflect the questions of "better" or "correct", and stick with "useful", which gets clearer answers.

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    $\begingroup$ Fuzzy logic is not probability! Saying that SILK has degree of SOFTness 0.8 is very different from saying that SILK is SOFT with probability 0.8. Speaking of usefulness, intuitionistic logic has at least as much relevance to programming as classical first-order logic, most dramatically in Haskell. $\endgroup$ – fishlips Nov 18 '14 at 1:50
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    $\begingroup$ @fishlips: For "useful in business", check indeed.com: 250 jobs with Haskell, and 100,000 jobs with SQL. $\endgroup$ – user173 Nov 18 '14 at 9:16
  • $\begingroup$ @MattF. For usefulness in mathematics though, its Haskell for the win. SQL isn't even turing complete I don't think. $\endgroup$ – PyRulez Apr 11 '15 at 2:07
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Logic is a (formal) tool for reasoning. There is not better, or worse one. Only more or less suitable for given problem. To ask which one is better is similar to ask which tool is better: hammer or tweezers, or which language is better japan, or french?

Given a particular task, you may compare suitability of logic variations and results obtained using one, or another.

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