« first day (369 days earlier)      last day (2432 days later) » 

user400188
user400188
12:13 AM
@user21820 I've been meaning to ask, do you know of any resources I could use to study logics applications to math as opposed to language? Forall x is good but it doesn't cover mathematical expressions very much.

Ive noticed the deficiency quite acutely. The other day I was working through a proof with a friend, and he was able to (using intuition) jump through a series of logical steps which I later was able to verify. The speed at which they accomplished it was a bit of a shock though.

I also noticed it was quite hard for me to deiced which two expressions in normal math where distinct …
 
 
5 hours later…
user21820
user21820
5:00 AM
@user400188 I personally wouldn't recommend trying to learn logic in an informal way. Intuition is a good thing; we need it both to decide on the deductive rules (including axioms) and to guide us to interesting conjectures and proof ideas. However, relying on intuition lets you easily fall into illogical reasoning, and I myself have done that a lot of times. The only way to prevent that is a careful rigorous logical deduction. So there are actually two separate issues that you need to focus on.
The first is logic itself. You need to use formal deduction sufficiently until it becomes natural to you, so much so that you would be able to quickly find any short proof of any theorem regardless of the content matter.
 
user21820
user21820
5:13 AM
The second is topic knowledge. It is important to attempt to understand the structure of objects in each area of mathematics, in a partly intuitive way, as it can often greatly speed up finding a proof. I'll give 2 quick examples from real analysis. This is hard to solve by pure logic alone, and asymptotic notation is often the best way to think about limits. Another is how you need to use the completeness of reals to prove IVT and MVT and so on.
 
user21820
user21820
5:27 AM
So I suggest you work on both separately but in tandem. One nice book I've read before is "How to Prove It" by Daniel Velleman. Basically, you can work through each problem twice, first by attempting to find an intuitive solution and reasoning, and second by trying to capture it formally.
If you can't, then either something is wrong with the intuition or there is a lack in facility with logic, so feel free to ask about it here and either way you will improve.
And to answer the indirect question, you want both accuracy and speed. When I rely more on intuition and less on rigorous deduction, I can indeed think much faster, but I frequently make silly mistakes and rubbish proofs. It's fine if one checks the intuitive reasoning later, which is not much different from what you were doing when verifying your friend's proof.
@user400188 As for your last paragraph, I can't figure out how the two remarks are related, if by "distinct" you mean "inequivalent". In general there is no algorithm that can decide such a thing, and therefore only experience can help you there. It appears unrelated to what you said about using what you were trying to prove as a premise; that should not even be possible if you're using Fitch-style deduction.
Anyway the reason I recommend "How to Prove It" is because the problems are of a reasonable variety and provides ample material for you to practise logical deduction. Writing everything in symbols is far from necessary, but making the conditional and quantified contexts explicit should always be done whether on paper or in the head.
 
 
9 hours later…
user131753
2:09 PM
Recently, while trying to explain the basics of Propositional Calculus to a freshman, I noticed that it is difficult for me to elaborate the difference between axiom schemes and axioms. In Margaris's book (which is the one I suggested to him primarily because I am familiar with the text of the book) we find the following,
 
user131753
 
user131753
However, as the freshman pointed out, the difference between an axiom scheme and an axiom is not clearly stated here. My question is: what exactly is the difference between an axiom scheme and an axiom? Can you help me @user21820?
 
user21820
user21820
2:26 PM
@user170039: Hello!
An axiom scheme is a collection of axioms.
That's all there is to the difference. It may be easier to work with concrete examples. The basic properties of arithmetic are captured by a finite collection of axioms called PA− (which are listed out in this article)‌​, but it turns out that the induction axiom schema can never be captured by a finite collection of axioms.
Of course, we never need an axiom schema when it consists of finitely many axioms, since we can just conjunct them all together into one axiom. The word "schema" can be thought of as meaning "template", namely a prescribed general format for some collection of axioms. The law of excluded middle (LEM) can be understood as either a deductive rule or an axiom schema (one axiom "P∨¬P" for each sentence P).
Same with all those propositional axiom schemas you have in the picture. You can either chuck them into the deductive system as axioms as in a Hilbert-style system (where there is only one deductive rule, modus ponens) or you can capture them via deductive rules instead of axioms as in a natural deduction system. In the former they would be described via schemas, since you cannot write them all down.
Hope that answers your inquiry! =)
@user170039 By the way, the link is sometimes down, and I found an archive of it here.
 
user131753
2:47 PM
That's what I told him. I also told him that if you want to be more precise then you may say that an axiom scheme is a collection of axioms all of which have the same form.
 
user131753
3:05 PM
If an axiom scheme is a collection of axioms all of which have the same form (which you say as "a prescribed general format for some collection of axioms", if I have correctly interpreted it) then his questions are: (1) How is it possible to have "a collection of axioms all of which have the same form"? (2) How are we differentiating between the forms of two axioms? (3) Are we then saying that the form of an axiom and the axiom itself are two different objects?
 
user131753
(4) What does it mean then to say that logic is formal?
 
user21820
user21820
(1) By same form we mean a rule of generating the axioms. (2) One cannot talk about the form of a given axiom. One can only say whether an axiom is of some given form or not. An axiom may very well belong to multiple forms. (3) Yes. The rule is not the result of using the rule. (4) Hmmm...
 
user131753
I also have a question regarding this (I hope that it's not silly): Why does an axiom have "form" (assuming that it is indeed a property of an axiom and not the collection of axioms) and in what sense exactly?
 
user21820
user21820
(4) You can just define "formal system" the way I always do it, namely that a formal system is simply a proof verifier program that always halts on every input string pair (p,x) and says "valid" if p is a valid proof of x and "invalid" otherwise.
@user170039 It's not. Just like "printed" and "written" are properties of texts, but are not intrinsic to any given text.
17 is of the form a^2+b^2 where a,b are integers.
Same usage here.
 
user131753
So, is the concept of "form" empirical?
 
user21820
user21820
3:14 PM
"A or not A" is of the form ( P+" or not "+P ) where P is a string.
No it's up to you to define a string and a format and then the question of whether the string is of that format is a yes/no question.
Using programming terms, a format is simply a program that halts on every input string and says "yes" or "no". You can see that those 6 axiom schemas (which are collections) can be recognized by corresponding formats (programs).
Each format F can be converted to an enumeration (a program that enumerates the schema) as follows:
> for i from 0 upwards { for each string x of length i { if F(x) then output x } }
Anyway I got to go off now, so I'll respond next time. =)
 

« first day (369 days earlier)      last day (2432 days later) »