# Forums → WEPR → An Introduction to Logic

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Part 1: ARGUMENTS

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.

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Part 2: ASSESSING ARGUMENTS

So now that we know what an argument is, we can start to figure out if it's a good argument or a bad one. There are two types of arguments: deductive and inductive. You may have heard these terms in a science class, but they have a specific meaning in logic.

Good deductive arguments will have premises that GUARANTEE the truth of the conclusion.

Good inductive arguments will have premises that MAKE PROBABLE the truth of the conclusion.

Let's look at the Socrates argument again:

1 All men are mortal.

2 Socrates is a man.

3 Therefore, Socrates is mortal.

In this argument, if the premises are true, there's no way the conclusion can be false. You don't even need to symbolize this or do any formal proofs to see this.

And here's an inductive argument:

1 Most birds can fly.

2 This animal is a bird.

3 So this animal can probably fly.

Here, the premises - even if they're totally true - don't guarantee the truth of the conclusion. But the do make the conclusion more likely. These are the kind of arguments we're used to seeing in everyday life. Think about the arguments lawyers give during a trial.

With any court case, the arguments are inductive. Suppose an alleged murderer's fingerprints have been found at the crime scene, his DNA was found on the victim, and 2 eye witnesses saw him. It's highly likely that the suspect murdered someone. But we can imagine a crazy world where all this happened and he didn't do it. But that world is so unlikely that this outcome is well supported.

That's why jurors can't have a reasonable doubt about the suspect's guilt. If lawyers used deductive arguments, the jurors would have to be absolutely certain of the suspect's guilt. There could be no possibility of innocence at all.

Our disussion will be limited to deductive arguments. Inductive arguments are intuitive, for the most part, until you start getting deep into the philosophy of them. But these issues are beyond the scope of what we're concerned with.

So, when you assess a deductive argument, there are several 'hases' you go through. Several questions you have to ask. We'll take them each in turn.

IS IT VALID?

This is the first question you need to ask about any argument. A good understanding of validity is crucial for logic. Here's a strict definition:

VALID: an argument is valid if and only if it is impossible for the premises to be true and the conclusion false.

Different logic texts define validity in different ways, but this is the most helpful definition that I've found. Read this definition a few times and make sure it makes sense. Here are some examples to help it sink in:

Some Valid Arguments:

All mammals have hair.

A dog is a mammal.

Therefore, a dog has hair.

Jim is taller than Bob.

Bob is taller than Sam.

So, Jim is taller than Sam.

Mars is closer to the Sun than Earth.

Earth is closer to the sun than Venus.

So Mars is closer to the sun than Venus.

"Wait a minute, stupid," you say. "None of those statements in that last argument are true." Well, you're right - none of them are true. But the argument is still valid. That because the premises, if they're true, will guarantee the conclusion is true. Draw it out if you want as the premises have it set up - there's just no way for the conclusion to be false. It fits our definition of validity.

An invalid one is where the premises can be true and the conlusion false. Here's an example:

If my car starts, then it has fuel.

My car has fuel.

So, my car will start.

In this, we can imagine a scenario where my car has fuel but it fails to start. My car may have fuel in it, but may not start for any number of other reasons. Maybe my spark plugs are bad or maybe my engine is gone completely! These scenarios are completely compatible with the premises. The reason this argument is invalid is because of faulty reasoning. But since we can imagine a world where the premises are true and the conclusion is false, the argument is definitely invalid. We'll see why later.

If an argument is invalid, you can stop assessing it. The fact that it's invalid makes it a bad argument, and it's the most damaging attack on an argument. There's nothing you can do for an invalid argument but to revise it.

So back to the planet argument. It's valid, but it's still a bad argument. Where it fails is with the second question you need to ask:

IS IT SOUND?

A deductive argument is sound if its premises are true. Easy enough. But notice I said a 'deductive argument.' This is because an invalid argument CAN'T be sound. So this is only a question we ask of valid arguments.

The planet argument is valid, but it's unsound. Both the premises are false. So if the premises were true, then the conclusion would be true. But since the premises are, in fact, false, the argument is useless.

An unsound argument is a bad argument simply because it's unsound. When you attack an argument this way, you'll want to identify with premise(s) is(are) false and why.

IS IT CONVINCING?

An argument can be valid and sound, but still not be convincing. Consider the following argument:

All mammals have hair.

Every number divisible by 2 is even.

So, all bachelors are unmarried men.

This argument is valid and sound. It's valid because the premises can't be true and the conlusion false. Now that has nothing to do with the premises - it's just that the conclusion can't be false. But this is still a valid argument! It's also sound because all the premises are actually true. But it's still a bad argument because it's not convining at all. The conclusion is simply a definition of 'bachelor,' so it's not even that interesting of an argument at all.

If you've answered 'yes' to all these questions, then the argument is a good one. Now, soundness can be attacked in the realm of philosophy or science. The same can be said for whether it's convincing. As for logicians, we start off by assessing only if the argument is valid. A surprising number of arguments aren't, but it's hard to tell unless you symbolize it an assess its form. So that's what the major part of logic is. As you progess, you can start to see which premises are key to the argument and worth attacking. In this way, you become a better philosopher, debater, lawyer, whatever.

Next we'll start symbolizing arguments so we can begin to see their form.

A comprehensive list of fallacies:

http://www.don-lindsay-archive.org/skeptic/arguments.html

I find it very useful, and it's got funny quotes, too.

I'd appreciate comments at this point to be limited to questions about logic or any necessary clarification. I hope for this thread to be a resource, but the more it gets cluttered the harder it will be for users to get what they need.

Again, thank you so much Moe! It looks fantastic. I think it will be a great resource. It just seems that sometimes in the WEPR we need to go back to the basics.

1 Most birds can fly.

2 This animal is a bird.

3 So this animal canprobablyfly.

Because of the word probably, doesn't that make the conclusion true? There is a 100% chance that this bird *probably* can fly.

Wait. That sounded better in my head.

1 All men are mortal.

2 Socrates is a man.

3 Therefore, Socrates is mortal.

I thought I may as well point out that while Socrates the man is mortal, his works are immortal. So, if you want to get down to the nitty-gritty, than if:

Socrates is mortal

The works of Socrates are immortal

Therefore, the name of Socrates is immortal.

It contradicts your statement that Socrates is immortal, becuse the name Socrates is immortal. It is valid becsuse the premise is impossible to be false, since Socrates the man died years ago, and the conclusion true. My logic is also sound, because both of my premises are true. Logically it is even convincing, because it makes perfect sense...

Because of the word probably, doesn't that make the conclusion true? There is a 100% chance that this bird *probably* can fly.

It does make the conclusion true, in a sense. We call this kind of argument 'strong' because the conclusion is highly probable based on the premises. Once you try to say something like 'It's true that it's possible,' the kind of statement you make becomes more complex. Unfortunately, this new kind of statement falls beyond the scope of this discussion, but that's a very good point.

It contradicts your statement that Socrates is immortal, becuse the name Socrates is immortal.

That's an interesting response. But I'm not presenting these examples so they can be challenged, I'm presenting them simply as examples to help those reading understand the concepts.

But to answer you're question (if it is a question), your conclusion does not contradict the conclusion from my argument. Notice my conclusion is 'Socrates is mortal' while yours is 'The name Socrates is immortal.' One is talking about the man, the other is talking about the name. So they're not contradictions.

Your argument also commits the fallacy of equivocation. You use the word 'immortal' in an ambiguous way that does not fit the standard definition. The works of Socrates are not immortal in the same way that Socrates could be immortal. If he were, he could not die. But his works were never alive.

Part 3 - SYMBOLIZING ARGUMENTS

We're getting into the meat and potatoes of logic. At this point, we're not concerned with the truth of the premises or even of the conclusion. All we care about is the form of the argument.

Once you can symbolize an argument, you'll better understand how to attack it philosophically.

Here's a quite famous argument for skepticism:

1 If I know that I have hands, then I know that I'm not simply dreaming that I have hands.

2 I don't know that I'm not simply dreaming that I have hands.

3 So, I don't know that I have hands.

This is a valid argument for sure. So if you're going to attack it, you need to show that one of its premises is unsound - that is, false. G.E. Moore famously shifted the statements of the argument to get a different conclusion.

1 If I know that I have hands, then I know that I'm not simply dreaming that I have hands.

2 I know that I have hands.

3 So I know that I'm not simply dreaming that I have hands.

This move, called a Moorean shift, was only available to Moore because he understood logic. He understood what he would have to deny to get the conclusion that he wanted and he also understood how to support this shift.

So let's get cracking.

When you're symbolizing, you'll want to focus on what are called atomic propositions. The word 'atomic' denotes simplicity. An ATOMIC PROPOSITION is one that is at its most basic. It will contain only one statement about the world and will include no logical operators.

Here are some examples:

The sun is shining.

Bob likes to run.

Tim went to the store.

Remember in elementary school when you were learning to put two simple sentences together using words like 'and' 'but' and 'or?' In logic, we're doing that in reverse - taking complex sentences and making them simple.

So:

Bill and Tim went to the store.

Would become:

Bill went to the store. Tim went to the store.

This is because we need atomic propositions in order to use symbolization. The sentence 'Bill and Tim went to the store' is not atomic because it has a logical operator: 'and.'

Logical operators are what bring atomic propositions together. They show how the propositions relate to one another. It will be easier to see where this is going with a list of all the logical operators we will be using.

LOGICAL OPERATORS:

^: and

v: or

~: not

->: if-then

=: identity

iff: if and only if

The first three you are probably quite familiar with. We use these in everyday conversations and it should be clear how these operators work on propositions. The last three may not be as clear, but hopefully they will be clear later.

Here are some examples using each operator. when you symbolize, you'll want to provide a dictionary which will tell your readers what the letters you're using stand for.

B: Bob went to the store.

T: Tim went to the store.

J: Jim went to the store.

(B ^ T) : Bob and Tim went to the store.

(B v T) : Either Bob or Tim went to the store.

~B : It's not the case that Bob went to the store. / Bob didn't go to the store.

(B -> T) : If Bob went to the store, then Tim went to the store.

(B iff T) : Bob went to the store if and only if Tim went to the store.

I didn't do the identy one because it wouldn't really make sense with these propositions. You don't really use identity that much in propositional logic, but it's still a very important concept. Also, you may not find the right arrow '->' in all textbooks and you won't find 'iff' in any of them. Some books may use what's called a 'horseshow' for the right arrow (it looks like a 'U' turned on its side) and the iff is what's called a triple bar. It's an equals sign '=' but with three bars instead of two. AG can't show these kind of symbols, so we're limited to what we can type on a keyboard.

Here are some more complex examples:

(B ^ T) ^ ~J : Bob and Tim went to the store, but Jim didn't.

(B ^ T) -> J : If Bob and Tim went to the store, then Jim went to the store.

(B v T) iff ~J : Bob or Tim went to the store if and only if Jim didn't go to the store.

See if you can put these into normal sentences:

~(B ^ T) v J

(~B v ~T) iff ~J

(B -> T) ^ J

(B -> T) -> J

I'll stop here for questions and continue this part once everyone is on the same page.

What you have said seems very logical, however, it is confusing to getjust by looking at it. I would test it out with my family, but they hate it when I do my deep thinking, logical crap, so I may have to test it at school...

Can you make them even more complex? If you could, than it opens up even more possibilities, like an "and....but" it would allow for contradictions in the same sentence, while still having to use only one symbol. Or did I miss something, can you do that right now?

Can you make them even more complex? If you could, than it opens up even more possibilities, like an "and....but" it would allow for contradictions in the same sentence, while still having to use only one symbol. Or did I miss something, can you do that right now?

Heh, it gets much, much more complex. Unfortunately, so does the symbolism, which can't be represented on this site. Although I guess we could use other symbols in their place. But we will see a greater complexity unfold we we turn to proofs.

As for the "and ... but" possibility, those are actually the same in logic.

So the following two propositions:

Bill went to the store and Tim did not.

Bill went to the store but Tim did not.

Can be represented by:

(B ^ ~T)

As for representing a contradiction, we can already do that:

B ^ ~B : Bill went to the store and Bill did not go to the store.

In logic, contradictions do have a use. If you can show the denial of a premise results in a contradiction, you can show that premise to be necessarily true.

There is also a method of proof called an Indirect Proof that tries to generate a contradiction by denying the argument's conclusion. But we'll get to that.

Good questions, though. I'll have the next part up soon, just want to make sure there are no more questions before proceeding.

Just curious, does there exist an XOR operator in this kind of logic?

Just curious, does there exist an XOR operator in this kind of logic?

Great question. The 'or' operator in this system is not an exclusive one. It can be true if both disjuncts are true.

There are two ways around this. You can still get the logical force of an exclusive or:

(B v T) ^ ~(B ^T)

This would read Bob and Tom went to the store but not both. Thus you can get the same thing as an XOR operator - it's just longer.

Alternatively, we could introduce a logical operator to handle exclusive or stuff. The reason there isn't is because a) it's not necessary and b) we don't do a whole lot of exclusive or stuff.

But as long as you're clear about the truth conditions for the operator (it's false just in case only 1 disjunct is true) then you can certainly introduce it to this logic system.

But as long as you're clear about the truth conditions for the operator (it's false just in case only 1 disjunct is true) then you can certainly introduce it to this logic system.

Sorry, that should've read "it's TRUE just in case only 1 disjunct is true." Sorry about that.

I'm getting ready to post the next part on inferences, but I thought I'd give you guys a few to practice. If any questions come up, just post them here.

Determine if the following are valid or invalid.

*Remember the definition for validity: An argument is valid just in case it's impossible for the premises to be true and the conclusion false. So if you can imagine a world where the premises for the argument are true and the conclusion is false, then it's invalid.

Mars is closer to the sun than Venus.

Earth is closer to the sun than Mars.

So, Earth is closer to the sun than Venus.

Jim is taller than Bob.

Sam is taller than Bob.

So, Jim is taller than Sam.

Monkeys eat bananas.

Old people eat grapefruit.

So, 2+2=4.

The sun is hot.

The sun is not hot.

Therefore, a square is a circle.

If you go outside when it's raining, you'll get wet.

You go outside when it's raining.

So, you'll get wet.

If I get a haircut, chicks will be all over me.

Chicks are all over me.

So I got a haircut.

These are meant to be tricky and a bit weird, but they should adequately test your understanding of validity. If any of them don't make sense or you want to verify your answer, please just post here.

Mars is closer to the sun than Venus.

Earth is closer to the sun than Mars.

So, Earth is closer to the sun than Venus.

Valid

Jim is taller than Bob.

Sam is taller than Bob.

So, Jim is taller than Sam.

Invalid since Sam can be the tallest of the three based on the premises.

Monkeys eat bananas.

Old people eat grapefruit.

So, 2+2=4.

Valid simply because the conclusion can't be false.

The sun is hot.

The sun is not hot.

Therefore, a square is a circle.

Invalid because the premises can't be both true and the conclusion can't be true.

If you go outside when it's raining, you'll get wet.

You go outside when it's raining.

So, you'll get wet.

Valid

If I get a haircut, chicks will be all over me.

Chicks are all over me.

So I got a haircut.

Invalid since it's possible that chicks are all over you for some other reason, like hypnotism.

I hope I did that right...

(~B v ~T) iff ~J

Woo some symbolism, always makes things simpler. Just as a personal test, is the quoted line logically equivalent to;

(B ^ T) iff J

(B ^ T) -> J

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