Reddit mentions of An Introduction to Formal Logic

I think you might be looking for formal logic? The reasoning in mathematical proofs is based on systems of logic that can be formalized and treated as mathematical objects in and of themselves so that we can study the properties of the reasoning we employ in mathematics.

Though formal logic is used in mathematics, it is not limited to it. Formal logic can be used in thinking about anything. Philosophers, for instance, use it all the time in trying to think rigorously and carefully about difficult philosophical questions.

For an introductory book on formal logic, I like Peter Smith's Introduction to Formal Logic.

Since you also said that you're looking to understand the world in a probabilistic way, it also might be worth taking a look into Bayesian Epistemology

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

Hey there.

You mention that your brother is bright -- how bright exactly? First of all, Computability and Logic is quite an advanced book that is typically aimed towards 2nd year logic students, and is usually for students who have taken a rigorous discrete mathematics course in their first year.

It delves quite deeply into the theory of logic and the philosophy of mathematics and would not be suited as a light exercise book for someone unless they have taken a math-heavy first-year logic course and are planning on taking up electrical engineering or something of the sort.

As for Hurley's book, a Concise Intro to Logic, well, this is on the other side of the spectrum -- it is very watered down compared to other logic readings, and pales in comparison (to most other introductory logic books) with regard to depth and breadth on formal logic.

It's usually aimed at first-year philosophy students who are taking introductory courses in logic or critical thinking, and most of it is simply rote-learning certain forms of argument as well as a lot of "quick and dirty" techniques which mimic that of a dry maths textbook. If you're looking for an interesting exposition into logic, then this book is certainly not it -- it would serve better as a high-school introduction for logic, and if prescribed to anyone older, would be very lackluster.

Here are some suggestions for you:

• Raymond Smullyan's A Beginner's Guide to Mathematical Logic
  
  
• Nicholas JJ's Logic: The Laws of Truth
  
  
• Peter Smith's An Introduction to Formal Logic
  
  
• RL Simpson's Essentials of Symbolic Logic
  
  If you're looking for mathematically-oriented logic puzzles, then you should check out Smullyan's series of puzzle books. You can find them on Amazon.
  
  
  
  
  

No. That's not quite there. But at this point, I've handed you the keys, you have to take the drivers ed yourself. This is a pretty good book to start with

Yes, it's a very big guide. It's also intended for academics (mostly philosophers) with no training in logic who want to have a solid grasp of the topic for research work, so you definitely don't need to go through the whole thing; if you do, you'll probably be better equipped than most PhD students in logic!

I think sticking to Epstein, or using Peter Smith's introduction (Or Paul Teller's book that Smith recommends and is free) is reasonable for a high-school course; If you go with Smith, you may want to cover a chapter or two from a more advanced book (for example, on the incompleteness theorems, the basics of computability, or the basics of model theory) if you move quickly.

Epstein's and Teller cover more ground, and Epstein presents a more "traditional" syllabus for mathematical logic. On the other hand, if you're interested in either philosophy or computer science (or any of the more exotic modern developments in logic), Smith's (And Teller's) presentation might translate more directly into the ideas you'll see there: the traditional syllabus on mathematical logic tends to better prepare students for model theory and set theory, which have less relevance outside of pure math.

Anyway, best of luck!