The entire purpose of the logic we're looking at is assessing arguments. So it'll be helpful to understand exactly what an argument is. Here's a working definition:

ARGUMENT: A series of propositions consisting of premises which are purported to support a conclusion.

Of course, to understand this definition, we need to know what a proposition, premise, and conclusion are. A proposition is a special kind of statement - it's one that can be true or false. Obviously, questions like 'What time is it?' can't be true or false. And neither can commands like 'Shut the door.' A proposition says something about the world. Here are some examples of propositions:

1) It is sunny outside. 2) Mercury is the closest planet to the sun. 3) All mammals lay eggs.

Notice that 3 is false. But that's okay, it's still a proposition. Remember, these are statements that can be true OR false. As it turns out, there are some philosophers who have some strong arguments about what is and isn't a meaningful proposition. But that discussion is for an Analytic Philosophy class. We don't really care about these things in logic. If it's something to make sense to say it's true or false, then it's a proposition.

So an argument consists of premises and a conclusion. The premises are propositions that give you a reason to accept the conclusion, which is also a proposition. The conclusion is what you're supposed to, well, conclude! Here's as example:

1) All men are mortal. 2) Socrates is a man. 3) Therefore, Socrates is mortal.

In this argument, 1 and 2 are the premises which support 3, the conclusion. You can usually tell the conclusion by keywords like 'therefore' 'so' and 'thus.' Look back at the definition of an argument - notice it says that the premises are PURPORTED to support the conclusion. That just means that they are intended to give support - but they may fail miserably. The result would be a bad argument, but it's still an argument. Here's an example:

1) Monkeys like bananas. 2) I like bananas. 3) Therefore, I'm a monkey.

In this argument, the premises lend very little support to the conclusion. You may even have an argument where the premises have nothing at all to do with the conclusion. But these are still arguments - just really bad ones!

So now you know what an argument is. Up next, we'll go over the basics of how to assess an argument. This, remember, is the central goal of logic (at least, the logic we're talking about). It's worth noting here that the kind of logic we'll be talking about is called PROPOSITIONAL LOGIC. This logic deals with, you guessed it, propositions. Overall, it's very weak - there are many arguments it can't assess. More powerful logical systems like predicate logic and modal logic can handle more arguments. But you have to walk before you can run, and this kind of logic is a very good place to start. If you can understand this, you'll have a much easier time learning more powerful logical systems.

These are the basics, so if there are any questions, please post them. It's vital that you understand these definitions so that the next part will make sense.

Bob is a sponge who lives under the sea. My name is Bob. Therefore, I live in a pineapple under the sea and am square.

The argument itself is an invalid deductive argument, am I right?

Yep, that's right. Though we would just call it invalid - including the 'deductive' part is a bit confusing.

The reason it is invalid is because it is a weak analogy - While the premises may be true, the conclusion might be false.

Yep, if it's possible for the premises to be true and the conclusion false, then the argument is invalid. Unfortunately, the reason why this argument is invalid can't be shown with propositional logic - the logic that this thread is based upon. Predicate logic simply cannot represent the proposition: 'Bob is a sponge who lives under the sea'. We have to use predicate logic for sentences like this, which is a bit more complicated (but only slightly). As it turns out, it's questionable whether this argument is even deductive. The three statements, logically, are unrelated. Consider the following argument:

1) Bob is over there. 2) My name is Bob. 3) Therefore, I am over there.

This is structurally similar to the SpongeBob argument, I've just simplified it so we can see where the flaw is. What this argument lacks is a statement like: 'I am Bob' or 'I am identical to Bob'. Here, 'Bob' names something, and so does 'I'. But it never says the two are the same thing.

So we have 2 ways to go:

1st stab:

1) Bob is a sponge that lives under the sea. 2) I am Bob. 3) Therefore, I am a sponge that lives under the sea.

Here, the argument is deductively valid but clearly not sound. It commits the fallacy of equivocation.

2nd stab:

1) All Bobs live under the sea. 2) I am a Bob. 3) Therefore, I live under the sea.

This is also deductive valid but not sound. Premise 1 is false.

The only reason something is illogical is because of logic itself. Once logic contradicts itself the previous logic is no longer valid, in practical terms. So if it is impossible to gain every bit of knowledge, logic itself is illogical, because there is, and I quote "...an infinite amount of logic". So arguing logic is basically arguing something you don't know if either argument is true, or any common hypothesis.

The only reason something is illogical is because of logic itself. Once logic contradicts itself the previous logic is no longer valid, in practical terms. So if it is impossible to gain every bit of knowledge, logic itself is illogical, because there is, and I quote "...an infinite amount of knowledge". So arguing logic is basically arguing something you don't know if either argument is true, or any common hypothesis.

Spongebob lives under the sea There is only one Spongebob Therefore, there are no Spongebobs who live above sea level

You would need the additional premise that nothing that lives under the sea also lives above sea level, but yeah, it's valid. Although you would need to use Predicate Logic to show its validity.

The logic introduced in this thread is Propositional Logic. Predicate Logic is sort of the next step, like learning arithmetic and then learning algebra. Sometimes it's just called Symbolic Logic, but that's a bit misleading. Unfortunately, I don't have the time or resources to do a Predicate Logic introduction. It's doubly challenging because Predicate Logic uses symbols that aren't supported by this site. The two main ones are the existential quantifier (which looks like a backward 'E' and the universal quantifier (which looks like an upside-down 'A'. I'm sure that there are plenty of websites out there that have information on this more advanced, and much more powerful logical system.