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An Introduction to Logic

Posted Feb 17, '11 at 2:15pm

abcdaa

abcdaa

33 posts

This sounds a lot like C.S. Lewis... One crazy mathematician, I tell you. Logical fallacies? Anyone done TOK in IB? rofl...

 

Posted Feb 27, '11 at 3:53pm

pls

pls

4 posts

Look this one
All parrots are birds v
All birds are parrots x

Ok another one
1+1=2 BUT WHY

 

Posted Mar 5, '11 at 11:51pm

Moegreche

Moegreche

3,315 posts

Moderator

BASIC SYMBOLIZATION: THE OPERATORS AND, OR, and NOT

Sorry, it's been a while - I've been quite busy. We're not that far from constructing arguments using propositional logic; but before we can do that, we have to be able to symbolize these propositions. We can also connect propositions together using logical operators, which we'll see.

In logic, a capital letter, like 'P' would stand for an actual proposition. When you symbolize an argument from natural language to logic, you want to include a dictionary so people will know what you're talking about. Here's the basic idea.

Dictionary:
S: Socrates is a man.
A: All men are mortal.
M: Socrates is mortal.

The argument could then go like this:

1. S
2. A
/ M

It's not a very interesting argument (and, it turns out, it's not one you can prove in this logical system), but it's there. And it's symbolized. Now, I could have used any letters I wanted to symbolize those propositions - I just picked those because they made it easier to remember. Just note that you can use the same letter twice - that would get confusing real quick.

LOGICAL OPERATORS

With our propositions symbolized, we can now connect them using our logical operators. To do this kind of logic, you actually only need 2 logical operators. But most texts use 5; I'm going to cover the first 3.

NOT:

When we symbolize a proposition, we want to make sure it's atomic. That means that it's a basic as it can get - we don't want any logical operators contained in there.
So here's a sentence:
I am not at home.

The way we would symbolize this statement is ~H (read: not H). That little squiggle is called a tilde (TIL-duh) and in logic it just means "not." It says of whatever it's connected to that whatever it says is false.
So if C = The Cubs will win the World Series, then ~C = The Cubs won't win the World Series.

It important to take any negations (nots) out of your sentences and just use the ~ to symbolize the negation.
There are some exceptions to this rule, but they're very minor. Let's take the sentence:
Dan thinks that this won't work.

If we took out the not and symbolized it like ~D, then someone might read it as "Dan doesn't think this will work" or "It's not the case that Dan thinks this will work." But those aren't quite right - if you don't see why, just trust me on this one
But there's no need to worry because we don't care what Dan thinks. These kinds of sentences aren't even dealt with logically. You tell a logician that Dan thinks this won't work and the logician will ask you, so what?

AND

The logical operator and, which we use a ^ for, works just like an 'and' in grammar - it connects two sentences together.
So,
B: Bill went to the park. J: Jim went to the park.

We can put these together like this:
B ^ J : Bill and Jim went to the park.

And if you're trying to symbolize a statement with an 'and' in it, then you're probably going to have to represent it logically. It's important to note here that the logical 'and' just connects two sentences together - it says that they're both true.
So the sentence:
Bill went to the park but Jim didn't.
Would be:
B ^ ~J : Bill went to the park and Jim didn't go to the park.

Notice we represented 'but' just like an 'and'. But if you understand what the sentence is saying, it makes sense why we do this. Logically, they're the same thing. Just like 'Bill went to the park although Jim didn't'. As long as the sentence is saying that both parts are true, you use 'and'.

OR

In logic, the 'or' operator says that either one thing is true or the other is true or that they're both true. So the only time an 'or' statement in logic is false (at least this logic) is when both sides (disjuncts) are false.

So if our statement was B v J then it would be true is Bob or Jim went to the park - even if they both did. It would only be false if neither of them went to the park.


You can do a lot of symbolizing with just these operators. Look around and find stuff to symbolize if you want some practice. And if you guys want me to post a few for you to work out, just let me know. I just figured there are already plenty of sentences out there

 

Posted Mar 7, '11 at 8:10pm

Moegreche

Moegreche

3,315 posts

Moderator

CONTINUED:

I realized after looking at this that I could've been much clearer about what we're doing. So recall that these logical operators combine propositions.
Let's say we have two props: P and Q. Now, P could be true or false - as could Q (in other words, they're contingent statements). But these logical operators are like mathematical operators, like plus, minus, times, etc. But instead of putting in numbers and getting out numbers, we're putting in truth values and getting out truth values.
Let's take the mathematical operator plus '+'. We can plug in two numbers, say 2 and 5, combine them with the + operator and we'll get a new number - 7.
It's the same thing with logical operators. The ~ just negates the truth value put in. So if P is true, then ~P is false. If P is false, then ~P is true. Easy peasy.
The ^ operator will return true if and only if both statements (we call them conjuncts, because it's a conjunction) are true. So P ^ Q is true iff P is true and Q is true. Again, we're plugging in truth values and getting out a new truth value.
With the or operator 'v', it will return true just so long as at least one disjunct is true. It will return false just in case both are false. So P v Q is false is P and Q are both false, but it will be true otherwise.

Hope this helps clear up what we're doing. Not only can we assess individual statements, but we can also determine whether combinations of these statements will be true or not.
I will continue with the other 2 operators so long as there is still interest. So let me know on my profile if you want this to continue, cuz I really don't want to waste my time

Any questions, please post them here.

 

Posted Mar 15, '11 at 3:10am

Moegreche

Moegreche

3,315 posts

Moderator

I think interest in this has fizzled out. Besides, there are only a few things left that I'm sure would easy enough to figure out. I bet there are plenty of website out there with loads of info.
Logic is an amazingly powerful yet elegant tool, but is often undervalued as a resource. Many studies have pointed to the notion that as advanced as we are as a species, we are also quite illogical - perhaps irrational. I think that by being aware of your logical commitments and understanding their entailments, you become a better cognizer. To put it simply: to think logically is to make the best of this thing called "rational thought". On a cognitive level, it's what truly does make us unique as a species.

 

Posted Mar 16, '11 at 1:42am

Sonatavarius

Sonatavarius

1,371 posts

My interest hasn't fizzled out... I just haven't had the time to properly study the information and commit it to memory. It's quite educational, and I'm sure that there are those who are reading that are just not saying anything... and that there will be those who show up eventually that have questions... its just that our population of people old enough to comprehend it is somewhat small. The hard work you've put into organizing and presenting the information merits applause. If this were Exit Path... I would give you lots of Kudos for this (I like random humor...). Hard work is seldom met w/ the proper support and gratification (and appreciation). I'll try and get back to this some day when I have the time to study it, and I will most likely have questions for you.

 

Posted Apr 30, '11 at 12:56pm

molybdenum42

molybdenum42

17 posts

if 2+2=4, then 4+4=10. now that is what we call logic, and logic had to have a creator, without the creator we would just have insanity, and we call this creator God. so because creation had to have a creator, what logicly follows is that source had to have a sorcerour, aka God

 

Posted Apr 30, '11 at 8:12pm

Einfach

Einfach

1,502 posts

if 2+2=4, then 4+4=10. now that is what we call logic, and logic had to have a creator, without the creator we would just have insanity, and we call this creator God. so because creation had to have a creator, what logicly follows is that source had to have a sorcerour, aka God

Logic didn't actually have to have a creator. It exists whether or not God exists - and it is inherent. Because it's not like the letters "p -> q" is inherent in the universe; instead, p and q are variables, and the constant -> is defined.

So logic doesn't need a creator at all - it exists independent of God.
 

Posted Apr 30, '11 at 8:15pm

Einfach

Einfach

1,502 posts

I didn't explain well ^.

What I mean is that, once p -> q is defined, there are certain properties about the constant -> that create universal truths. These are logical tautologies.

 

Posted May 8, '11 at 4:45pm

crazyape

crazyape

1,800 posts

Logic. Very logical. Where's the logic in contradicting every idea having to do with a higher being? Is it psychological? Or illogical? Maybe both? Maybe niether. Maybe the incentive is..... Not wanting to subject your life to dogmatic schedules, habits, or interactions. Maybe dogmatism is a government cover-up. Maybe Darwin made a deal with the goverment. There are many If's here, and no logical answers.

 
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