Logic and Algebra

StephenPrivitera

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?"

Related General Discussion News on Phys.org

Mentat

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.

Staff Emeritus
Gold Member
Dearly Missed
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".

Mentat

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.

Staff Emeritus
Gold Member
Dearly Missed
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.

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving