Proof of logic

drkitten said:
Unfortunately, the second half of your third assumption is simply wrong. No finite amount of experimentation can weed out the infinite number of possible alternative hypotheses. (This particular observation goes all the way back to the logical positivists and before, but it was framed in its most strident form by Popper and Quine.)

That's why "replication" is a key factor in empirical science, but not in "logic"; it would be a very good idea for me (as a bench scientist) to replicate some of the classic findings in my field of study as part of my research program. On the other hand, I have no need to re-prove the classic logical findings, because logical findings are typically "sound" in the sense defined above; there's no possibility that Godel got his Completeness Theorem wrong, but there's a very definite possibility that Milliken got the mass of the electron wrong.
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*Scratching head*

What can we miss with gravity for example?
 
South Sydney are the greatest team in the history of sport.
I predict that meditating on this truth should bring me extreme happiness.
Meditating on this truth brings me extreme happiness.
Meditating on this trutn brings a lot of people extreme happiness.
South Sydney are the greatest team in the history of sport.
 
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JetLeg said:
*
What can we miss with gravity for example?
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Lots. For example, we currently believe that inertial mass and gravitational mass are the same thing (and that no object possesses one but not the other). We believe that mass is a constant barring changes in energy.

For that matter, we believe that an object, when dropped, will fall down. But no one actually knows whether my coffee cup will fall when dropped, because it's never been dropped. Perhaps I have the one magic coffee cup that doesn't fall.

We might be wrong (although that's not the way I would bet). All we have is a confident guess.
 
Complexity said:
I'll use 'iff' as an abbreviation for 'if and only if'

We're working with statements - strings of symbols that are considered to be 'well-formed' if they conform to a specified grammar. One such grammar is that of the first-order predicate calculus; another is that of the second-order predicate calculus. More on these later.

An interpretation of a statement or a set of statements provides meaning to the symbols in the statement(s). For every interpretation of a set of statements, each statement in the system is either true or false under that interpretation.

A statement that is true under every interpretation of the system is called a tautology.

A statement that is false under every interpretation of the system is called a contradiction.

A set of statements is consistent if it they don't imply a contradiction.

Pick a consistent set of well-formed statements and assume that they are true. This set is our set of axioms.

Some statements may exist that are true iff each of the axioms is true (under interpretation). These statements are called consequences of the set of axioms. These statements are also called theorems.

A set of axioms and the set of its theorems are called a theory.

Another important idea, besides consequence, is that of derivability.

A logical calulus is a set of procedures, a logical machine, for taking one or more statements and constructing new statements. It also provides the grammar that statements must conform to to be considered well-formed.

One such logical calculus is called the first-order predicate calculus (FOPC). Another logical calculus, the second-order predicate calculus (SOPC), enhances the first-order predicate calculus by adding in the ideas of 'for every' and 'there exists'.

Whenever we use the procedures of a logical calculus to construct a new statement from one or more existing 'input' statements, we say that we are deriving the new statement from the 'input' statements, and that the new statement is derived from the 'input' statements.

Which statements can we feed into the logical calculus machine? Any of the axioms and any statements which were derived from the axioms, directly or indirectly.

We will say that a statement is derivable iff there is a sequence of derivations that starts with the axioms and results in the construction of that statement.

A logical calculus is sound iff, for every set of axioms that conform to the calculus, every derivable statement is a consequence of that set of axioms.

In other words, given the input of axioms or theorems, a sound logical calculus can only construct theorems.

A logical calculus is complete iff, for every set of axioms that conform to the calculus, every theorem is derivable.

Goedel proved that the first-order predicate calculus is complete; that is, every true statement of a FOPC theory, every theorem, can be derived from its set of axioms. That impressed mathematicians.

Goedel then proved that the second-order predicate calculus is incomplete; that is, there exist SOPC theorems that can not be derived, statements that are true that can not be proved to be true.

Furthermore, Goedel proved that if you try to patch up the calculus so that this true statement becomes derivable, you inevitably result in a calculus that has other true statements that are not derivable. You can't ever, ever fix it. Any logical system of sufficient complexity has theorems that can't be proven.

This shocked the hell out of mathematicians.

I hope that this made some sense. I left out a lot and probably screwed something up.
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I don't know if this question makes any sense, but I do know that you're the right person to ask. First, what exactly is the grammar of first-order predicate calculus; and second, how is one to be certain of its validity without proof of outcome?

(This is the first in a long string of questions I want to ask about your post, but I shall refrain from torturing you with the rest.)
 
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bluharmony said:
I don't know if this question makes any sense, but I do know that you're the right person to ask. First, what exactly is the grammar of first-order predicate calculus; and second, how is one to be certain of its validity without proof of outcome?

(This is the first in a long string of questions I want to ask about your post, but I shall refrain from torturing you with the rest.)
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Hi. By all means, ask what you like.

The three varieties of logic that I'll refer to are propositional (boolean) logic, first-order predicate calculus (FOPC) , and second-order predicate calculus (SOPC).

There are two boolean values: true and false.

A boolean variable can be assigned either boolean value.

Boolean expressions are formed from boolean variables and boolean values by using negators (nots), connectors (junctors; e.g. and, ors, exclusive-or, implies, is-equivalent-to) and parentheses. The first link provides a lot of information on boolean (propositional) logic.

First-order predicate calculus extends propositional logic by adding a universe of objects, variables that can can be assigned one of those objects, functions, predicates, and quantifiers that act over variables.

Consider the mini-world of geneology.

The universe of objects might be all of the people who have ever lived.

A variable named Dufus could be assigned any any of these people (e.g. 'Gerry').

A predicate is a special function of zero or more arguments that returns a boolean value. For example, IsFatherOf( Horatio, Ben ) returns true iff Horatio is the father of Ben.

IsDescendedFrom( Alice, Mary ) returns true iff Alice is descended from Mary.

AreParentsOf( Betty, Barney, BamBam ) returns true iff Betty and Barney are parents of BamBam.

A function that is not a predicate accepts zero or more arguments and returns an object from the universe of objects. For example, FatherOf( Ben ) should return Horatio.

There are two quantifiers, for-every and there-exists. In FOPC, quantifiers can only be applied to variables.

We can say, for example, that there exists an X such that IsFatherOf( X, Ben ) is true. We can be more general and say that, for every Y, there exists an X such that IsFatherOf( X, Y ) is true [[note: here there be dragons]].

In second-order predicate calculus, quantifiers can be applied to predicates and functions as well as variables. We can then say things like "For every function f, ..." and "There exists a predicate P such that, for every function f, ...".

This additional power is needed to express most of mathematics.

I think the first place to start is propositional (Boolean) logic. Here's a link that looks pretty good.

http://www.iep.utm.edu/p/prop-log.htm

The next level up in in logical power is the first-order predicate calculus, also known as first-order logic.

Here's a series of presentation screens:

http://www.cs.utexas.edu/~mooney/cs343/slide-handouts/fopc.4.pdf

The grammar for the first-order predicate calculus is given in screen 3.

The write-up of first-order logic in wikipedia covers a lot but it is a bit dense.

http://en.wikipedia.org/wiki/First-order_logic

There are external links that might be interesting at the end of that page.

This is all I had time for tonight, but it should give you a starting point.

Ask any questions that occur to you.
 
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JetLeg said:
To be completely honest - it seems to me that it is related to the fact, that I would love something not to be disproved in IS IT thread, and would love something to be proved in this thread.

But, nevertheless, my arguments seem to make sense to me in both of the threads, so I will not do an ad-hominem to myself.
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I appreciate the honest answer here, thanks.
 
bluharmony - I'm thinking of writing up some tutorials on logic, starting with propositional (boolean) logic. I've got some time, and it would be fun, so I'll probably start on it in a day or two.

Propositional logic, boolean expressions, and all that sound trivial to many technically oriented people, but they aren't.

I've been working on some problems related to boolean expressions since 1986 - I did my master's and doctoral work on them (satisfiability determination and simplification). Both are problems I could spend the rest of my life on.

During graduate school, I focused on automated reasoning, also known as automated theorem-proving, using both first- and second-order predicate calculus. Northwestern was one of the centers for work in this area at the time.

While I was teaching, at around 1992, I started to focus on some of the challenging problems in graph theory, especially involving cliques. I never lost my interest in boolean expressions, however. It is just a matter of time, and what you work on first.

If you don't mind, I'll start with propositional logic, touch on some of the work that I've done with boolean expressions, and then move to first-order and then second-order predicate calculus. This will probably take many weeks, but I'll do my best to make some sense of things.

Since we're talking about mathematical logic, I'll start threads on the various logics under the Science and Math forum.

Please let me know in this thread of any topics you'd like to have covered or questions you'd like me to try to answer.

Thanks.
 
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Thanks, that sounds wonderful. Your explanation of boolean logic made what I'm doing when searching Westlaw explicitly clear. I also understand the other explanations you provided, but I would love to read more to feel completely comfortable. It's been ages since I've studied formal logic and mathematics, and those have always been interests of mine. I look forward to anything you put together on the topics, especially if your writing is as easy to follow as in the explanations above.
 
Cool. I'm psyched. I'll start outlining some stuff tomorrow morning.

Sometimes not working has its advantages.
 
Complexity said:
Since we're talking about mathematical logic, I'll start threads on the various logics under the Science and Math forum.

Please let me know in this thread of any topics you'd like to have covered or questions you'd like me to try to answer.

Thanks.
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I would really like that. I'm not sure how much I can absorb but your writing style is pretty good for my level.

Thanks.
 
Robin said:
South Sydney are the greatest team in the history of sport.
I predict that meditating on this truth should bring me extreme happiness.
Meditating on this truth brings me extreme happiness.
Meditating on this trutn brings a lot of people extreme happiness.
South Sydney are the greatest team in the history of sport.
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Eh...

Can you give me another example?

I don't get it, the structure of the argument seems perfectly plausible to me.
 
drkitten said:
Lots. For example, we currently believe that inertial mass and gravitational mass are the same thing (and that no object possesses one but not the other). We believe that mass is a constant barring changes in energy.

For that matter, we believe that an object, when dropped, will fall down. But no one actually knows whether my coffee cup will fall when dropped, because it's never been dropped. Perhaps I have the one magic coffee cup that doesn't fall.

We might be wrong (although that's not the way I would bet). All we have is a confident guess.
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We see that

Magnet is put near iron -> iron moves towards iron.
Therefore, magnet attracts iron.
And, we know that iron does not move by itself.

B occurs after A, and B does not occur by itself.


What else do we need to check? Is it enough?
 
JetLeg said:
Eh...

Can you give me another example?

I don't get it, the structure of the argument seems perfectly plausible to me.
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Sure, but if someone else makes the same argument about another team at the same time, it remains plausible, but you are left with a contradiction. You can't have two Greatest Team of All Time winners.
 
Mangafranga said:
You said before that you can see a hammer, but you can't see logic. Well, compare the inferences to the world. Has it ever been that, for instance, the premises of the argument "p or q, ~p, so q" have been true, while the conclusion has been false. The system of proof here is simply looking at the world to see if there has been a case where this argument hasn't held. If not, we accept the argument as valid (and just like in science we accept that it may be falsified in the future, but so what).
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Can you demonstrate in such a way, or in another that an argument from an authority is a fallacy?

For example, I say : everything that god says is right. Or, everything the dalai lama says is right.

How can you demonstrate it's a fallacy?

If you say "And what if god says 1+1=3?", I will reply - then we just have to reinterpret his sentence.
 
JetLeg said:
If you say "And what if god says 1+1=3?", I will reply - then we just have to reinterpret his sentence.
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Any statement can be correct if we're allowed to change the meaning of all the words.

Out of curiosity, JetLeg -- why do you post here? You don't seem to want to hear anything we have to say, and we can't see anything tangible in what you want to say. What is your goal when you come to this forum?
 
Logic is a description of the basic rules of thought. Does it describe thought well? Why do you want it proven?
 
mr. ottle said:
Any statement can be correct if we're allowed to change the meaning of all the words.

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Can you give me an example ?
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Out of curiosity, JetLeg -- why do you post here? You don't seem to want to hear anything we have to say, and we can't see anything tangible in what you want to say. What is your goal when you come to this forum?

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Discussion
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