Predicate Logic and Sets

What is the basis for reason? And mathematics?

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nguyentruongnghia
Posts: 1
Joined: Sun Sep 11, 2022 9:42 pm

Predicate Logic and Sets

Post by nguyentruongnghia »

Hi friends,

I don't seem to be able to formalize a statement about the equivalence of two sets. Specifically, how do I formalize "the number of apples in this box A is the same as the number of pears in that box B". Surely one has to use universal and existential quantifiers in some way, and I suppose Russell must have done it 100 years ago, but I can't do it, and just can't find the formula anywhere.

Thanks
Nghia
wtf
Posts: 1177
Joined: Tue Sep 08, 2015 11:36 pm

Re: Predicate Logic and Sets

Post by wtf »

nguyentruongnghia wrote: Sun Sep 11, 2022 9:56 pm Hi friends,

I don't seem to be able to formalize a statement about the equivalence of two sets. Specifically, how do I formalize "the number of apples in this box A is the same as the number of pears in that box B". Surely one has to use universal and existential quantifiers in some way, and I suppose Russell must have done it 100 years ago, but I can't do it, and just can't find the formula anywhere.

Thanks
Nghia
One way is to say that there's a bijection between the two sets. That means that there is a bijective function between them.

A function is a relation such that every element of A appears exactly once; that is, at least once and no more than once.

A relation is a subset of the Cartesian product A X B.

A bijective function is a function that is injective, meaning that no two elements of A get mapped to the same element of B; and surjective, meaning that every element of B gets hit by some element of A.

So you have to put into symbolic logic the concepts of relation, function, injection, and surjection. That's a bit of work if you have to do it from scratch.

But if you're allowed to assume that you know what a bijective function is, you can say that A and B have the same cardinality if there exists f : A -> B such that f is bijective. It's up to you how much of that you want to drill down to first principles.
alan1000
Posts: 312
Joined: Fri Oct 12, 2012 10:03 am

Re: Predicate Logic and Sets

Post by alan1000 »

Unless I've misunderstood the question (could happen: story of my life, really), the answer is very simple, and Russell gave it. Imagine a society which is perfectly monogamous. In this society, the number of husbands would exactly equal the number of wives, because each wife would have one husband, and each husband would have one wife.

Similarity of sets means that every element of one set can be paired with one element (any element) in the other, in one-one relation, just like husbands and wives, or like children lining up in two's and holding hands. In set theory, it is literally as simple as that. You line them up together, and they match perfectly. A zipper might be another useful analogy.

You seem to have some familiarity with Russell! In which case, your best introduction is probably his "Introduction to Mathematical Philosophy", which explains the whole thing in (reasonably) simple terms. Please read carefully what he has to say, especially the first few chapters, and you will realise that the problem is much simpler than you had thought.

Can you tell us why you are interested to transpose this to the context of predicate logic?
alan1000
Posts: 312
Joined: Fri Oct 12, 2012 10:03 am

Re: Predicate Logic and Sets

Post by alan1000 »

nguyentruongnghia wrote: Sun Sep 11, 2022 9:56 pm Hi friends,

I don't seem to be able to formalize a statement about the equivalence of two sets. Specifically, how do I formalize "the number of apples in this box A is the same as the number of pears in that box B". Surely one has to use universal and existential quantifiers in some way, and I suppose Russell must have done it 100 years ago, but I can't do it, and just can't find the formula anywhere.

Thanks
Nghia
Re-reading my post, I should have emphasised that you don't need to know how to count, you don't need any definition of "number", in fact you don't need ANY mathematics! to establish whether two sets have the same number of elements. You simply line them up, side by side, and if there's nothing left over by itself at the end, the sets are equal. Don't look for anything more profound. That's all there is to it.
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