Logic and Algebra

  • #1
All A's are B's.
can be written as
For all x, if x is A, then x is B.
If F = {x : x in domain, x is A}
and G = {y : y in domain, y is B}
Then I can write, "For all x, if x is A, then x is B" as
F intersect G = F

Similarly, I can write, "Some A's are B's" as
F intersect G [x=] [null]

I can write, "No A's are B's" as
F intersect G = [null]

I can write, "Only A's are B's" as
F intersect G = G

It seems that this approach might bring about considerale results (if only I knew more about the algebra of sets).
Is there some branch of logic that studies logic in this manner? Or is it simply more convenient to study logic conventionally? What is meant by the term "mathematical logic?"
 

Answers and Replies

  • #2
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Originally posted by StephenPrivitera
What is meant by the term "mathematical logic?"
I think that term refers to Symbolic Logic, which uses variables like p and q, and functions like "and", "if-then", and "if and only if", to form logical statements.
 
  • #3
selfAdjoint
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The modern term for symbolic logic is propositional calculus. Sometimes more specifcally second order propositional calculus, the difference from first order being basically the quantifiers "There Exists" and "For All".
 
  • #4
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Originally posted by selfAdjoint
The modern term for symbolic logic is propositional calculus. Sometimes more specifcally second order propositional calculus, the difference from first order being basically the quantifiers "There Exists" and "For All".
Interesting. I wasn't aware of this.
 
  • #5
selfAdjoint
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A good google key is Zermelo-Frankel. This is the name of one of the systems of axioms for set theory, expressed mostly in the language of the propositional calculus. Also look up Foundations of Mathematics.

IIRC we had some threads about all this here in the old days.
 

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