Reddit reviews Mathematical Logic (Oxford Texts in Logic)

What this expressions says

First of all let's specify that the domain over which these statements operate is the set of all people say.
Let us give the two place predicate P(x,y) a concrete meaning. Let us say that P(x,y) signifies the relation x loves y.

This allows us to translate the statement:
∀x∀yP(x,y) -> ∀xP(x,x)

What does ∀x∀yP(x,y) mean?

This is saying that For all x, it is the case that For all y, x loves y.
So you can interpret it as saying something like everyone loves everyone.

What does ∀xP(x,x) mean?

This is saying that For all x it is the case that x loves x. So you can interpret this as saying something like everyone loves themselves.

So the statement is basically saying:
Given that it is the case that Everyone loves Everyone, this implies that everyone loves themselves.
This translation gives us the impression that the statement is true. But how to prove it?

Proof by contradiction

We can prove this statement with a technique called proof by contradiction. That is, let us assume that the conclusion is false, and show that this leads to a contradiction, which implies that the conclusion must be true.

So let's assume:
∀x∀yP(x,y) -> not ∀xP(x,x)

not ∀xP(x,x) is equivalent to ∃x not P(x,x).
In words this means It is not the case that For all x P(x,x) is true, is equivalent to saying there exists x such P(x,x) is false.

So let's instantiate this expression with something from the domain, let's call it a. Basically let's pick a person for whom we are saying a loves a is false.

not P(a,a)

Using the fact that ∀x∀yP(x,y) we can show a contradiction exists.

Let's instantiate the expression with the object a we have used previously (as a For all statement applies to all objects by definition) ∀x∀yP(x,y)

This happens in two stages:

First we instantiate y
∀xP(x,a)

Then we instantiate x
P(a,a)

The statements P(a,a) and not P(a,a) are contradictory, therefore we have shown that the statement:

∀x∀yP(x,y) -> not ∀xP(x,x) leads to a contradiction, which implies that
∀x∀yP(x,y) -> ∀xP(x,x) is true.

Hopefully that makes sense.

Recommended Resources

Wilfred Hodges - Logic

Peter Smith - An Introduction to Formal Logic

Chiswell and Hodges - Mathematical Logic

Velleman - How to Prove It

Solow - How to Read and Do Proofs

Chartand, Polimeni and Zhang - Mathematical Proofs: A Transition to Advanced Mathematics

> Never read it, will google them after this reply.

It's so fucking cool it's unreal. Not up to date with recent developments but wanna check it out again properly soon.

>Mendelson can be useful but, heck, you need some strong background. There's a lot of books mistitled as "introductions", mendelson is one of them.

That'd explain why it was so dense lol - I dived from no mathematical logic (apart from like basic predicate calculus) and using first order symbols sparingly.

>There's actually no perfect book to serve as introduction to mathematical logic, but I highly recommendthat you check out https://www.amazon.com/Mathematical-Logic-Oxford-Texts/dp/0199215626
>
>Also get this little fella here: https://www.amazon.com/Mathematical-Logic-Dover-Books-Mathematics/dp/0486264041 for a nice, short survey.

Thanks :D I'll check it out. Given your breadth of knowledge on it I imagine your background is pure mathematics?

Never read it, will google them after this reply.

Mendelson can be useful but, heck, you need some strong background. There's a lot of books mistitled as "introductions", mendelson is one of them.

There's actually no perfect book to serve as introduction to mathematical logic, but I highly recommend that you check out https://www.amazon.com/Mathematical-Logic-Oxford-Texts/dp/0199215626

The price is outrageous, so get a pdf here if available.

Also get this little fella here: https://www.amazon.com/Mathematical-Logic-Dover-Books-Mathematics/dp/0486264041 for a nice, short survey.

Great to hear! Some good texts to consider, which will take you through the end of the main syllabus in an introduction to first-order logic are Chiswell and Hodges and Smith. If you ever get the chance to look back through this material I'd recommend taking a look at Goldfarb, but I don't think that's a great place to start in your situation.