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Hello Mathematics community,

As finals are approaching, my professor (regardless of how good is he) decided to just skim through the material that is going to be on the test. However, I wrote down two questions he constantly asked...

1) What are the advantages and disadvantages of Categorical Syllogism?

2) What are some of the advantages and disadvantages of Natural Deduction?

I have searched and read multiple papers and none of them had a clear cut on what these advantages and disadvantages are.

From what I have read thus far from my textbook and papers was the difference between Symbolic logic and Aristotelian logic?

Traditional Aristotelian logic (also called Classical Deductive Logic or Categorical Syllogisms) is an ancient method of deductive reasoning. Historians say that Aristotle, that ancient Greek philosopher guy, was the first to talk about syllogisms. In ancient Greece, men often first presented their arguments in syllogisms, and then they gave their arguments in a more rhetorical form. Men were held to a higher standard back then.

Modern Symbolic Logic is another method of deductive reasoning developed after the seventeenth century. Gottfried Leibniz and other logicians had a vision for a more simple and useful method for translating ordinary reasoning into a universal language of symbols. Among other things, this method can more quickly analyze longer arguments to see if they are valid. This method of deductive reasoning made logic even more "mathematical." It also made it uglier.

This is for two weeks from now so take your time answering and thank you so much for you help and making me try to understand the difference.

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That's a very broad question with a wide variety of possible answers, but here is just my 2 cents:

Aristotelean Categorical Logic

Advantages:

  • Can be analyzed using handy visual Venn diagrams
  • Syllogisms not uncommon in real life, including the fallacious versions
  • You can potentially memorize all the valid syllogisms ... there are not so many of them

Disadvantages:

  • Limited applicability; hard to deal with 'or', for example

Natural Deduction

Advantages:

  • Much more powerful and much wider applicability. Can do propositional logic, predicate logic, and other logics yet

Disadvantages:

  • Trying to fit reasoning into the 'straightjacket' of formal logic can take away from understanding of logic at more intuitive level
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