Basic Logic 1

[I. Pigott]

Here's our attempt at an online refresher.
 

If the scientific method is the essence of discovery, then logic is the bedrock.

Logic is the system that allows us to begin with a set of statements that represent reality, and through nothing more than rigorous application of definition and elimination, arrive at the synthesis of a new truth.  Those "newly true" statements can then become the premises of a completely new argument.

For the purposes of Shen Lung logic, and for the clearest possible language, we will attempt to stick to a strict set of names.

Statements

A statement is a sentence that attempts to assert a fact.  "Come over here, Belvedere" is a sentence, but cannot function as a statement because there is no inherent "truth value."  The nature of the sentence does not lend itself to a "true" or "false" determination.

Some statements might never be resolved in real life, like "There are exactly 6-trillion molecules in my body."  We may never know for sure, but we do know that the statement is either true or false.  It cannot be both.  Also, a true statement is true forever, and likewise a false one stays false.

Here's an example:  "The lawn is wet."  Well, in real life my lawn can go back and forth from wet to dry and in between... but that's not what logicians refer to as the assertion.  Every statement carries with it an understood value of time.  When I say "The lawn is wet" I am really saying more than just "The lawn is now wet", but rather "The lawn is now, at the moment of this original utterance, at 12:25am on November 8th 2005, wet."  That sentence is either true for all time or false forever.  If a sentence is capable of retaining a permanent "truth value" of either True or False, then it is a statement.

True and False are words that are used to describe statements, and statements alone.  Statements can be either premises or conclusions.

Structures

Structure refers to the order in which you place the terms of the argument within the statements.  "All Shetlands are ponies" is certainly different than "All ponies are Shetlands." Many seemingly miraculous arguments can instantly fall because of the use of an incorrect structure.  Even if all of the premises are true, and the conclusion is true, that is not in and of itself proof that the structure is proper.  All of the relevant terms must link, and in the proper order.

Later on, we'll go deep into the nature of logical structures.  For the purposes of clear language, we will reserve the words Valid and Invalid to refer to the structure of an argument.

Argument

Simply put, an argument is your assertion along with all of the supporting statements and structure.  We refer to arguments as being either Sound or Unsound.

A sound argument is one where the premises are true, and the structure is valid. 

An unsound argument is one where there is:

• one or more false premises
• an invalid structure
• or both of the above

Yes, it can be a royal pain to keep all of these terms properly sorted, and it is quite easy to slip into common language.  But do your best to train your mind to think of statements as true/false, structures as valid/invalid, and arguments as sound/unsound.  The mental and verbal discipline will help you pinpoint the problems that much faster down the road.

More to come...

[James Burroughs]

So as far as valid or invalid goes, the truth of the premises and conclusion are not looked at. Rather are they ordered right(a=b b=c therefore a=c)? The truth of the staments come into play when considering sound or unsound. Am I close?

[Will Hooper]

So as far as valid or invalid goes, the truth of the premises and conclusion are not looked at. Rather are they ordered right(a=b b=c therefore a=c)? The truth of the staments come into play when considering sound or unsound. Am I close?

That's pretty close. Validity is concerned only with whether or not the premises, as stated, necessarily force the conclusion. A common form of this, as you note, is that - given a=b and b=c - a must equal c. Soundness concerns whether an argument, if it is valid, is factually true or not.

An example:

All cats have scales
All scaly things can fly
---------------------------------
Therefore all cats can fly

This is a perfectly valid syllogism, as the conclusion must be true if we assume the premises are true.
However, it's an unsound syllogism, because despite being valid, one or more (in this case, all) of the statements are factually incorrect.

On the other hand, this syllogism is unsound and invalid:

All cats are warm-blooded
All mammals are warm-blooded
---------------------------------------
Therefore all cats are mammals

It's invalid because the conclusion is not forced by the premises - just because two groups (cats and mammals) are both contained within the same larger group (warm blooded creatures) does not mean they overlap perfectly. Because it is invalid, it is also unsound - despite the fact that all three statements are correct and factual, the logical invalidity of the syllogism means the whole thing is unsound (it doesn't matter that the statements are true; since the truth of the conclusion isn't forced by the truth of the premises, the truth of the conclusion is essentially 'accidental' in terms of logic.)

Hope this helps!

- Will

[James Burroughs]

I see says the blind man. Thank you sir for clarity.

[Luke Anthony]

It always sounds weird when we say that validity in deductive logic "ignores" whether or not premises are true, since validity sounds like the final verdict on an argument. Will's right on the money when he says that validity looks at whether premises force the conclusion; soundness, which is an even tougher standard, judges both validity and the truth of the premises. An argument can be valid but not quite sound; it can never be sound and invalid.

To throw another colorful analogy into the mix, I find it helpful to think of validity and soundness in terms of construction. Validity refers to how well a house/argument is built. The truth of the premises is what the house/argument is built out of - the raw materials. And soundness is the final judge, which looks at both of those. The three little pigs and the big bad wolf offer a pretty good example. No, bear with me here...

Let's say you have four houses, that represent three arguments. I won't go into involved examples here, since we're just talking about concepts, and Will's already given some good examples.

Let's say one argument is invalid, and has false premises. This is complete gibberish (but you'll see it more than you think!). This is akin to if the first little pig built his house out of straw, and just piled the straw every which way. No door, no lock, no real walls to speak of. Shoddy craftsmanship, through and through! And to make matters worse, it's made of substandard parts! The wolf won't even bother to huff and puff with this one - he can just walk right on over to where the pig is cowering under a pile of hay, and pick him up. It's not really hunting so much as shopping. That's an invalid argument with false premises - garbage in, garbage out.

Move on to the second argument. It's valid, but has false premises. This is much more common. This is akin to a very, very well-constructed house made of sticks. Gables, a two-story dumbwaiter, the works. It's even got a security system! But it's built out of sticks, through and through. Maybe the pig got a deal on pine bark, who knows? The stairs creak precariously, and even though it can stand under its own weight, and looks quite respectable, it's been ravaged by termites. I mean, it's pine bark! You can only throw on so many layers of stucco. With this one, the big bad wolf actually has to look twice, before he huffs and puffs and blows it all over, revealing one trembling pig amidst the rubble. That's a valid argument with false premises - good construction, shoddy materials.

Next, we have the third argument. This is probably the most common - an invalid argument with true premises. Oftentimes, people will even throw a bunch of contextless facts (we call them "statistics" in the biz!) together, and declare a conclusion at the end. This is similar to a shoddy construction job made out of the best materials. Let's say the richest and most impetuous of the pigs saw what happened to his last friend, and ordered the best, toughest materials to build his house in a day. It's fine Alabama marble; fine brick and steel; the best of everything, thrown together in the worst way. You can even see where the grout is crumbling, the I-beams aren't welded properly, and it's listing to one side where there's a sinkhole of citations from "a book somewhere." This structure will fool most common burglars, and even rascally coyotes; the pig feels quite comfortable eating his fancy double-meat Papa John's pizza (oh, the cannibalism of circular arguments!!), and just turns the music up when the creaking gets too loud. But the big bad wolf isn't fooled. He walks around the structure, sniffs at it a few times - kicks the mailbox for good measure - and when he huffs and puffs, it goes down like a... well, like several tons of bricks. That's an invalid argument with true premises.

But finally, we have the house at the end of the street. The cul de sac. This is where the last pig has been paying special attention to his materials and his structure - he went to night school for a degree in lupine-resistant architecture, you know! And he has sturdy materials, put together well. This is a valid argument with true premises. There are no structural weak points, no flourishes or false attics to weaken the structure. Everything's put together safe and SOUND. Eventually, the big bad wolf tires himself out trying to blow it down, hyperventilates, and passes out in the petunias.

So, to recap, we've got:

Invalid argument with false premises: bad construction of bad materials
Invalid argument with true premises: bad construction, but good materials
Valid argument with false premises: good construction, but bad materials
Valid argument with true premises: good construction, and good materials

I hope that makes sense.

ANALOGIES WOOOO