Propostional Logic Symposia - [1] - Introduction and Basics

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Reply Mon 2 Jun, 2008 01:38 pm
I have been meaning to do a series of threads on propositional and predicate logic for a while now, so here it goes. The main purpose of this and subsequent threads is to make propositional and predicate logic easy to understand and actually doable which a lot of websites on the internet never seem to get right or make easy enough to understand. It is my hope that as you follow along with the threads, which will be numbered in a specific track, you will gain the ability to formulate logical equations and execute logical proofs and formulas without having to go to university to do it. ANYONE CAN DO IT!!!!

LOGIC IS NOT HARD!!!! It only looks like that superficially because it looks like some sort of complex mathematical process. But it is relatively simple to do when you know the essential fundamentals. When you come to the end of my explanations of propositional logic for example, you will have the power to translate an entire page of written argument into a proof and calculate whether it is logical, illogical, true or not, etc.



This is the method I want to follow;

A)Posting an initial thread on the successive topic in logic (i.e. translation, proofs, etc.) Keep in mind that I fumble up in my logic sometimes, so the initial thread may change a bit as I revise it and make it simpler or correct it. I'll post a few sample proofs as well.

B)Ask any questions if you want, I'm happy to answer them as well as correct or affirm the sample proofs. I can also at your request post additional proofs for you to do.

I'll start with basic things that you probably already know but there good to just get out there. This stuff is important because when you get to translations, you will need to know how to distinguish an arguments constituent parts.


Arguments

I'm sure most of you have been in some sort of an argument or another at some point in your life. When you argue, hopefully, you provide your own beliefs to support a claim that you want to make. So, it seems logical (stay tuned) to define what a logical argument is so it is ridiculously clear to us.


Logical Argument

All dogs are mammals. All mammals are warm blooded. Therefore, all dogs are warm blooded.


What is a logical argument?

First remember the formal definition of a logical argument, which is a set of one or more premises which lead to a conclusion which follow from the premises. This is important and fundamental to know.

[All dogs are mammals.]
[ All mammals are warm blooded.]

Both of these statements are premises, meaning that they support the conclusion.

[Therefore, all dogs are warm blooded.]

This is the conclusion, which follows from the premises.


Think of what has just happened in terms of basic math, 1 + 1 = 2. "All dogs are mammals" is 1 point. "All mammals are warm blooded" is 1 point. "Therefore all dogs are warm blooded" is = 2. All you are doing is adding logical statements together and coming to a conclusion based off of the numbers you just added together.

But notice that the conclusion is preceded by "therefore." This is an indicator which tells you this is the conclusion. Indicator words are important because a conclusion can come at the beginning, middle, or end of a sentence. Words like; thus, therefore, hence, so, it necessarily follows that, etc. are all conclusion indicators.

So basically, think of all of this in terms of that simple mathematical formula.

RECAP!!!

1 + 1 = 2

[Premise] + [Premise] = [Conclusion]

[All dogs are mammals] +[ All mammals are warm blooded]
Therefore(=) [ all dogs are warm blooded.]

There is more to be said on deductive and inductive logic, premise indicators, etc. If you want me to elaborate on any of those, let me know and I'll go into that.

NOTE: to maintain the integrity and usefulness of these symposia threads, posts not pertaining to the subject may be moved or deleted by the staff.
 
de budding
 
Reply Mon 2 Jun, 2008 01:49 pm
@VideCorSpoon,
Thanks! I will be following along for sure. I also have a book with some examples of two premise sylogysms some being true and some being false. I'll post some tomorow if you'd like, to test us Very Happy.

edit: Was just wondewring if... 1 + 1 = 2

[Premise] + [Premise] = [Conclusion]

then 1+1+1 = 3 is premise+premise+premise=conclusion, does the 3 indicate a stronger conclusion?

Dan.
 
VideCorSpoon
 
Reply Mon 2 Jun, 2008 02:06 pm
@de budding,
That's great! Bring 'em along! It's a good thing you said that because I wanted to do the next thread on truth functional logic which is I think what you are talking about and I consider my favorite part of propositional logic.

Also, on the 1 + 1 + 1 = 3, the 3 being a stronger conclusion than a 2. The sum of the mathematical statement (i.e. = 3) is just a conclusion in itself. It shows that all the premises have been incorporated and lend support to the conclusion by analogy. It doesn't make it a stronger conclusion, so long as the conclusion follows from the premises.

Also thinking about it, one of the biggest challenges will be cutting back on the material enough to where its followable and not cutting too much to where more elaboration i needed. We'll find out!
 
VideCorSpoon
 
Reply Mon 2 Jun, 2008 03:11 pm
@VideCorSpoon,
Also, If anybody has any requests on specific areas of propositional logic they want (or need), please tell me what you want and I'll incorporate it into the symposia with more elaboration.
 
Nocturne
 
Reply Mon 2 Jun, 2008 11:11 pm
@VideCorSpoon,
VideCorSpoon,

I think that you have made a mistake with regard to the role of argument. It is important to understand that a logical "proof" does not prove that the conclusion of an argument is true, but only that the conclusion is true when we assume that the premises are true, and it is quite clear that an argument does not prove its premises either. Therefore, the premises of an argument cannot support their conclusion, because the logical content of the conclusion is a subset of the logical content of the premises i.e. every deductive argument begs the question. The situation is even worse with regard to inductive arguments, because to the extent that the conclusion follows it does so circularly, and to the extent that it does not follow it is simply invalid. In other words, the premises of an argument cannot provide any support or good reason to think that their conclusion is true, a trivial consequence of elementary laws.

I also think that you should study the difference between an equation and a deduction. In short, every equation is also a deduction but every deduction is not also an equation i.e. the set of equations is a subset of the set of deductions. The difference is to do with symmetry i.e. the relation of equatibility is always symmetrical whereas deducibility can be asymmetrical. For example,

[indent]1. If A = B then B = A
2. If A |= B then B |= A[/indent]

The first statement is always true whereas the second is not always true. In fact, the second statement is true, if and only if, A is equal to B (remember, every equation is also a deduction). Therefore, while 1 + 1 = 2, the following argument is invalid:

Every dog is a mammal
& Every mammal is warm-blooded
= Every dog is warm-blooded

The invalidity is clear since equative arguments, unlike deductive arguments, must always be symmetrical i.e. the conclusion must follow from the premises and vice versa. However, it is not the case that either of the premises is implied by the conclusion, and so your argument is a nonequative deductive argument. Therefore, the use of the equality relationship is invalid.

Regards,
Lee
 
de budding
 
Reply Tue 3 Jun, 2008 04:01 am
@Nocturne,
[CENTER][CENTER]Syllogymnasium Quiz- Introduction[/CENTER][/CENTER]

Aristotle was the founder of formal logic. Yet he knew just hw weak it could be when it comes to persuading people. Properly reasoned arguments are often hard to follow, while poorly reasoned arguments often work by being emotive and appealing to people's prejudices.



So on with the game but, first, the rules which come from a form of argument know as the syllogism, which Aristotle formalized. All you need to know is one principle, the principle of validity,

An argument is valid if and only if the conclusion necessarily follows from the premises.



All men are mortal.
Socrates is man.
Therefore Socrates is mortal.

As we see the conclusion follows the premises. This is a valid

All cheese is from the moon.
Chocolate is a cheese.
There for chocolate is from the moon.

Cheese isn't from the moon and chocolate isn't cheese, but this argument is still valid. The validity of the argument is still intact even though the argument is false because, if it were true that all cheese is from the moon and at chocolate is a cheese, it would follow as sure as night follows day that chocolate is from the moon.



Vegetarians don't eat pork sausages.
Moby doesn't eat pork sausages.
Therefore Moby is a vegetarian.

True premises this time but, the argument is invalid because the conclusion does not necessarily follow from the premises. Moby doesn't eat sausages because of religious reasons that have nothing to do with vegetarianism.



If today is Tuesday, then I should be at work.
Today is Tuesday.
Therefore I should be at work.



Dan.



 
de budding
 
Reply Tue 3 Jun, 2008 04:15 am
@de budding,


An argument is valid if and only if the conclusion necessarily follows from the premises

If man made global warming is really happening, then the polar ice caps will be melting.
The polar ice caps are melting,
Therefore man made global warming is really happening.

If acupuncture tended to make people ill, then it would be foolish to try it.
Acupuncture does not tend to make people ill.
Therefore it is not foolish to try it.

If I don't get home by six, I'll miss the news.
Therefore, if I get home by six I won't miss the news.

If I work hard, I'll pass my exams.
Therefore if I don't work hard I won't pass.

All men are bastards.
Some bastards are attractive.
Therefore some men are attractive.

All politicians are liars.
No person of integrity is a politician.
Therefore no person of integrity is a liar.

All human life is sacred.
All God's creation is sacred.
Therefore all human life is God's creation.

Every person is a child of the universe.
Every person is a being of light and hope.
Therefore every being of light and hope is a child of the universe.

No vegans are fish eaters.
Some fish eaters are not vegetarians.
Therefore some vegetarians are not vegans.

Today isn't both sunny cold.
Today sunny,
Therefore today is cold.

Answers bellow, highlight to view...
[CENTER]_________________________________________________________
They are all invalid..!

I know, I know but I didn't make the quiz and it's quick and easy to mark youself this way...[/CENTER]
0 mistakes = A
1-2 mistakes = B
3-4 mistakes = C
5-6 mistakes = D
7+ = Fail!
[CENTER]_____________________________________________________________[/CENTER]
 
VideCorSpoon
 
Reply Tue 3 Jun, 2008 07:17 am
@de budding,
Nocturne,

Thank you for your comment,

First, I'm glad that you posted. This is why I wanted to do this thread series. I think everyone knows bits and pieces of what logic is, but no one really has an idea of the structured formal logic system and how those bits and pieces go together.

First, my reference to an argument was meant to be very simple because the premises thus conclusion formation is essential in translation in the proofs and I think that the average reader will not buy into the extended definition easily. This thread series is meant to be a simplified instruction manual. I agree with you that a logical argument does not prove the conclusion true, etc,etc,etc. But that is part of truth functional logic, which is going to be the next thread. You're jumping the gun so to speak, but still it's a valid comment.

Also, a "proof" does not mean "to prove" in logic. A "proof" means a "logical calculation derived from inference and replacement rules." That comes later though.

That I should study the difference between an equation and a deduction, ok. They are both the same thing in logic, but still that needed to be clarified. Propositional logic is built off of equations (proofs) which derive from principles of deductions (inference and replacement rules). The rest of your comment relates somewhat to a tiny bit of one of the replacement rules, namely commutation. That comes later though.

Your comments on symmetricality is jumbled logic, which is again one of the reasons why I want to complete this thread series.

Debudding,


That's awesome, thanks. And I really like what you did with the white lettering for the answer key, can I use that method when I do my next thread?
 
de budding
 
Reply Tue 3 Jun, 2008 07:42 am
@VideCorSpoon,
Of course you can (use white trick) Very Happy, and I'm glad you found the quiz helpful.

Dan.
 
Arjen
 
Reply Tue 3 Jun, 2008 10:41 am
@de budding,
Hold on a second guys, logic is not like math. Trust me on this one. If you guys want I can start a topic on proposition logic to clarify.
 
VideCorSpoon
 
Reply Tue 3 Jun, 2008 02:36 pm
@Arjen,
Wait a tick Arjen! Sorry, I couldn't resist it.

Sucks to say but formal logic is a lot like math. Keep in mind this is an introduction. This is not all propositional logic has to offer. Oh, I trust you... but I think we have drastically different understandings in logic.

But please do start a thread on propositional logic, I'm sure it will be very informative.
 
Arjen
 
Reply Tue 3 Jun, 2008 10:44 pm
@VideCorSpoon,
VideCorSpoon wrote:
Wait a tick Arjen! Sorry, I couldn't resist it.

Sucks to say but formal logic is a lot like math. Keep in mind this is an introduction. This is not all propositional logic has to offer. Oh, I trust you... but I think we have drastically different understandings in logic.

But please do start a thread on propositional logic, I'm sure it will be very informative.

To tell you the truth I had not seen your proader intent with the topic. Especially after the second topic I am inclined to say just a little word on this matter:

Logic is used only to deduce correct reasoning. A correct logical statement has no bearing on what exists in reality. I may be abe to prove given the right logical assumptions that grass is black for instance, but it would no more be true than grass being green.

I hope this is clear to everybody.
 
de budding
 
Reply Wed 4 Jun, 2008 02:19 am
@Arjen,
The difference between validity and truth; I think it is pretty apparent and most of us are familair with people using false logic day to day anyways.
Dan,
 
Arjen
 
Reply Wed 4 Jun, 2008 04:05 am
@de budding,
The thing of it is that the logic can be completely accurate, but the outcome of the reasoning does not have to correspond with reality. This is something that "popular science" exploits with a detrimental effect on the worldview of many.

The distinction I am trying to make is beween the reasoning and the outcome of the reasoning. There is no relation between the two. That is why I can say accurate things, based on an inaccurate reasoning by luck or hearsay for instance.

Hope this helps.
 
VideCorSpoon
 
Reply Wed 4 Jun, 2008 07:26 am
@Arjen,
many subsystems of logic that are incorporated into logic and in a way help determine reality (i.e. truth functional quantification).
 
Arjen
 
Reply Wed 4 Jun, 2008 10:36 am
@VideCorSpoon,
VideCorSpoon wrote:

The problem is not insolveable, even though it may seem to be. There is an ontological problem which can be formalised as falsum. It is, in reality quite easy. Free will presumes deni-all because deni-all allows for fantasies in stead of reality. That is a different topic though.

Quote:

So you are right in a way when you say logic is can be accurate, but not necessarily in line with reality. But propositional logic is the tip of the iceberg. There are many subsystems of logic that are incorporated into logic and in a way help determine reality (i.e. truth functional quantification).

In predicate logic the same closed system exists. It is a formalisation of thought-objects and nothing more. We may think it is "real", but we must seperate what we think and what we think to percieve from the things in-the-selves. If we do not we start to believe in illusions. The only thing logic can do is simplefy our reasonings and make it easier to find fault with them.
 
VideCorSpoon
 
Reply Wed 4 Jun, 2008 12:34 pm
@Arjen,
Arjen wrote:
The problem is not insolveable, even though it may seem to be. There is an ontological problem which can be formalised as falsum. It is, in reality quite easy. Free will presumes deni-all because deni-all allows for fantasies in stead of reality. That is a different topic though.

In predicate logic the same closed system exists. It is a formalisation of thought-objects and nothing more. We may think it is "real", but we must seperate what we think and what we think to percieve from the things in-the-selves. If we do not we start to believe in illusions. The only thing logic can do is simplefy our reasonings and make it easier to find fault with them.


Within the realm of formal quantificational logic... its not. Arjen, I asume from your message that we are both familiar with predicate logic... you don't remember the golden rule about a single proof initially incorporating Existential and Universal quantifiers????

As to your second response. Ok.
 
Arjen
 
Reply Thu 5 Jun, 2008 12:50 pm
@VideCorSpoon,
VideCorSpoon wrote:
Within the realm of formal quantificational logic... its not. Arjen, I asume from your message that we are both familiar with predicate logic... you don't remember the golden rule about a single proof initially incorporating Existential and Universal quantifiers????

I wonder what you are getting at..
 
VideCorSpoon
 
Reply Thu 5 Jun, 2008 01:27 pm
@Arjen,
That the proof is unsolvable because quantificational proof's cannot contain a contradictory and tautological existential and universal quantifier in a disaggregated way. I don't understand what else I could have been getting at?
 
Arjen
 
Reply Thu 5 Jun, 2008 01:54 pm
@VideCorSpoon,
VideCorSpoon wrote:
That the proof is unsolvable because quantificational proof's cannot contain a contradictory and tautological existential and universal quantifier in a disaggregated way. I don't understand what else I could have been getting at?

I think you are misunderstanding this "golden rule". It has no bearing whatsoever on the distinction I was making. The "golden rule" has bearing on incorrect formulations. I was noting that logic is about thought-objects and that correct formulations only correspond to those thought objects. If the formulation is correct it still says nothing about reality.

On top of that I would like to say that predication logic can (and is) used to formulate formula's, but that these still only refer to thought-objects. When anything we think corresponds to reality that is not because the formulation was correct or anything. When what we think corresponds to reality that is because our thoughts are "A" and reality is "A".

According to some leading philosophers a difference would still exist a priori because A(t) refers to a thought object and A(r) refers to reality.

My two cents..

Smile
 
 

 
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