We went over this in my 11:45 session, but it's an important enough point that everyone should see it. The issue is, how do you string together an argument with more than just three steps? Here's an example argument.
1. All apples are red.
2. Object x is an apple.
3. If object x is red, then it is sweet.
4. Object x is red.
5. Object x is sweet.
The above argument actually only has three premises-- that is, three independent statements that lead to the conclusion-- because statement 4 is a sub-conclusion. Premises 1 and 2 form a syllogism that gets us to statement 4. Then that statement is used with Premise 3 in a modus ponens argument to get to the final conclusion.
Oftentimes, it is clearer to present the argument as follows.
P1. All apples are red.
P2. Object x is an apple.
C1. Object x is red.
P3. If object x is red, then it is sweet.
C2. Object x is sweet.
In this presentation, the flow among all the statements is a little more obvious. But notice that the same statements are labelled premises-- regardless of where they appear within the argument. This is because the "premise", "sub-conclusion", and "conclusion" labels are attached according to the logical form of the argument.
When providing support for this argument, only the three premises need evidence. The sub-conclusion is arrived at through a valid argument, so its truth is, of course, guaranteed by the truth of its premises.
Monday, September 17, 2007
Wednesday, September 12, 2007
So How Can I Tell If My Premises Are True?
I said below that to check an argument's soundness, you should first check its validity and then check the truth of the premises. You check the validity by looking to see if the argument follows a valid form. But how do you check the truth of the premises?
It might seem like a silly question-- what could be easier than checking to see if our premise matches the world?-- but it's only really obvious for the simplest of premises.
For intance, if our premise is:
The apple is red.
we simply look to see if the apple actually is red. If it is, the premise is true. If it's not, the premise is false. Easy-peasy.
Things get more complicated if we string multiple propositions together into a single premise. For instance, if our premise is:
The apple is red and the elephant is pink.
both parts must be true for the premise to be true. In the above, the premise must be false, even if the apple is red, because the elephant is definitely not pink.
On the other hand, if our premise is:
The apple is red or the elephant is pink.
the premise might be true (as long as the apple actually is red). When two propositions are connected by "or", only one part needs to be true to make the entire premise true.
The situation is most complicated when the two parts are connected in a conditional sentence. For instance, look at this premise:
If the match is struck, then the match will catch fire.
To decide if the premise is true, we again have to examine the world. And again, we have to examine the world in two parts. First, we look to see whether the match actually was struck. And we also look to see if the match actually caught fire.
If both things happen, then we say the conditional premise was true. This makes sense-- remember, the "if" part of the conditional is supposed to give the sufficient condition for the "then" part. In this case, the match actually was struck, and it did indeed catch fire. Everything the premise claimed would happen did in fact happen. In short, the premise was true.
What if neither thing happens? (That is, what if we don't strike the match, and it doesn't catch fire? Or put another way, what if both parts of the conditional are false?) In this case, we still say that the overall conditional was true. Again, think about what the conditional is saying (this time, from the other direction): The "then" part gives a necessary condition for the "if" part. In our example, the match didn't catch fire, meaning that the necessary condition for it being struck was not fulfilled. And, as it turns out, the match wasn't struck. Our conditional held up again. We say it was true.
What about other combinations? Let's say we don't strike the match, but it catches fire anyway. (Maybe we lit it with a magnifying glass aimed at the sun.) In this case, the sufficient condition for the match catching fire wasn't fulfilled-- but that's OK, because there might be other ways to light the match (and in fact there are). The necessary condition for striking the match was fulfilled-- but that's OK, because that alone doesn't guarantee that the match was struck. In fact, our evidence in this case doesn't tell us anything at all about whether the conditional actually held up. In this case, we give the conditional the benefit of the doubt, and say it was true anyway.
Finally, what if we strike the match, but it doesn't catch fire? In this case, the conditional predicts that striking the match must make it catch fire, but it didn't happen. The conditional lied! When the "if" part turns out true, but the "then" part is false, this is the only time we can say that the overall conditional was false.
All of the above is typically organized into a diagram called a truth table. The truth table takes the truth values for the individual parts, and tells us whether the premise that connects them is true or false. See below:
It might seem like a silly question-- what could be easier than checking to see if our premise matches the world?-- but it's only really obvious for the simplest of premises.
For intance, if our premise is:
The apple is red.
we simply look to see if the apple actually is red. If it is, the premise is true. If it's not, the premise is false. Easy-peasy.
Things get more complicated if we string multiple propositions together into a single premise. For instance, if our premise is:
The apple is red and the elephant is pink.
both parts must be true for the premise to be true. In the above, the premise must be false, even if the apple is red, because the elephant is definitely not pink.
On the other hand, if our premise is:
The apple is red or the elephant is pink.
the premise might be true (as long as the apple actually is red). When two propositions are connected by "or", only one part needs to be true to make the entire premise true.
The situation is most complicated when the two parts are connected in a conditional sentence. For instance, look at this premise:
If the match is struck, then the match will catch fire.
To decide if the premise is true, we again have to examine the world. And again, we have to examine the world in two parts. First, we look to see whether the match actually was struck. And we also look to see if the match actually caught fire.
If both things happen, then we say the conditional premise was true. This makes sense-- remember, the "if" part of the conditional is supposed to give the sufficient condition for the "then" part. In this case, the match actually was struck, and it did indeed catch fire. Everything the premise claimed would happen did in fact happen. In short, the premise was true.
What if neither thing happens? (That is, what if we don't strike the match, and it doesn't catch fire? Or put another way, what if both parts of the conditional are false?) In this case, we still say that the overall conditional was true. Again, think about what the conditional is saying (this time, from the other direction): The "then" part gives a necessary condition for the "if" part. In our example, the match didn't catch fire, meaning that the necessary condition for it being struck was not fulfilled. And, as it turns out, the match wasn't struck. Our conditional held up again. We say it was true.
What about other combinations? Let's say we don't strike the match, but it catches fire anyway. (Maybe we lit it with a magnifying glass aimed at the sun.) In this case, the sufficient condition for the match catching fire wasn't fulfilled-- but that's OK, because there might be other ways to light the match (and in fact there are). The necessary condition for striking the match was fulfilled-- but that's OK, because that alone doesn't guarantee that the match was struck. In fact, our evidence in this case doesn't tell us anything at all about whether the conditional actually held up. In this case, we give the conditional the benefit of the doubt, and say it was true anyway.
Finally, what if we strike the match, but it doesn't catch fire? In this case, the conditional predicts that striking the match must make it catch fire, but it didn't happen. The conditional lied! When the "if" part turns out true, but the "then" part is false, this is the only time we can say that the overall conditional was false.
All of the above is typically organized into a diagram called a truth table. The truth table takes the truth values for the individual parts, and tells us whether the premise that connects them is true or false. See below:
| p | q | p and q | p or q | if p, then q |
|---|---|---|---|---|
| T | T | T | T | T |
| T | F | F | T | F |
| F | T | F | T | T |
| F | F | F | F | T |
Saturday, September 08, 2007
Review-- This Stuff Is Important
Since we'll be using these terms and concepts for the remainder of the semester, I figured I'd collect all of the main points in one post. So, a review of yesterday's class:
Truth is a description we give to propositions-- not to arguments. We decide if a proposition is true by looking at the world and seeing if it matches the proposition.
Validity is a description we give to arguments as a whole. Or to be even more specific, we look to see if the argument follows a valid argument form-- a pattern we've already identified as valid. An argument is valid if, when the premises are accepted as true, the conclusion must also be true (regardless of whether the premises actually are true or false).
Soundness is a further description we give to arguments. An argument is sound when it is valid, and the premises are true. (Note: This, of course, guarantees that the conclusion is also true).
So, the usual procedure is that we check for validity by pretending the premises are true, then checking what that does for the conclusion. If we have a valid argument, we look for the actual truth of the premises, by examining the world. If that also checks out, we can say we have a sound argument. And that's the bestest kind of argument in the whole world. (As long as it's not a stupid argument, which I explained in class.)
The three most important valid argument forms are:
Syllogism
All A are B
x is A
x is B
Modus Ponens
If p, then q
p
q
Modus Tollens
If p, then q
~q
~p
One final point. We talked about how to read a conditional proposition-- an if-then statement. For a proposition of the form:
If p, then q
we say that p is a sufficient condition for q, and that q is a necessary condition for p. But like I said yesterday, this is not a causal relationship between the two parts, or a temporal relationship (that is, it's not about which one comes first), but a purely logical relationship. To wit:
If the match is struck, then the match will catch fire.
According to this sentence, striking the match is a sufficient condition for the match to catch fire. Why? Because when the match is struck, nothing else needs to be done. All the conditions for a fire have been met.
But note also that the match catching fire is a necessary condition for the match being struck. This might not sync up well with your immediate intuition, but it is true. Why? Because when the match is struck, there is no way the fire can be avoided. Or, put another way, without a fire we can automatically tell that the match was never struck. The logical conditions for the match being struck were not fulfilled.
Wrap your head around that, and you'll be all set.
Truth is a description we give to propositions-- not to arguments. We decide if a proposition is true by looking at the world and seeing if it matches the proposition.
Validity is a description we give to arguments as a whole. Or to be even more specific, we look to see if the argument follows a valid argument form-- a pattern we've already identified as valid. An argument is valid if, when the premises are accepted as true, the conclusion must also be true (regardless of whether the premises actually are true or false).
Soundness is a further description we give to arguments. An argument is sound when it is valid, and the premises are true. (Note: This, of course, guarantees that the conclusion is also true).
So, the usual procedure is that we check for validity by pretending the premises are true, then checking what that does for the conclusion. If we have a valid argument, we look for the actual truth of the premises, by examining the world. If that also checks out, we can say we have a sound argument. And that's the bestest kind of argument in the whole world. (As long as it's not a stupid argument, which I explained in class.)
The three most important valid argument forms are:
Syllogism
All A are B
x is A
x is B
Modus Ponens
If p, then q
p
q
Modus Tollens
If p, then q
~q
~p
One final point. We talked about how to read a conditional proposition-- an if-then statement. For a proposition of the form:
If p, then q
we say that p is a sufficient condition for q, and that q is a necessary condition for p. But like I said yesterday, this is not a causal relationship between the two parts, or a temporal relationship (that is, it's not about which one comes first), but a purely logical relationship. To wit:
If the match is struck, then the match will catch fire.
According to this sentence, striking the match is a sufficient condition for the match to catch fire. Why? Because when the match is struck, nothing else needs to be done. All the conditions for a fire have been met.
But note also that the match catching fire is a necessary condition for the match being struck. This might not sync up well with your immediate intuition, but it is true. Why? Because when the match is struck, there is no way the fire can be avoided. Or, put another way, without a fire we can automatically tell that the match was never struck. The logical conditions for the match being struck were not fulfilled.
Wrap your head around that, and you'll be all set.
Sunday, September 02, 2007
Saturday, September 01, 2007
Prime Number Proof
Since it came up, I thought I'd show you the quick proof that there is no greatest prime number. This type of proof is called a reductio ad absurdum. That means it works by showing that a denial of the thing to be proved leads to a contradiction.
We assume (for the moment) that there is a greatest prime number-- call it n. This means the list of prime numbers is finite, and looks something like this:
{2,3,5,7,...,n}.
Now, we multiply all of those numbers together, and add 1 to the result. This gives us a new number P.
P= (2*3*5*7*...*n) + 1.
What do we know about P? P is not divisible by any of the prime numbers on our list-- they all give a remainder of 1. There are two possibilities. P might be prime. But in this case, P is a prime greater than n, which contradicts our initial assumption. Or, P isn't prime. In this case, P is factorizable-- and by extention, P is prime-factorizable. But because our list of primes was supposed to be complete (up to n), P's prime factors won't be on the list-- they must be greater than n. Again, this contradicts our assumption. And this exhausts all the possibilities.
Our conclusion? The initial assumption was false. We can't name the greatest prime number-- because there isn't one. That's it.
(No more math. I (almost) promise.)
We assume (for the moment) that there is a greatest prime number-- call it n. This means the list of prime numbers is finite, and looks something like this:
{2,3,5,7,...,n}.
Now, we multiply all of those numbers together, and add 1 to the result. This gives us a new number P.
P= (2*3*5*7*...*n) + 1.
What do we know about P? P is not divisible by any of the prime numbers on our list-- they all give a remainder of 1. There are two possibilities. P might be prime. But in this case, P is a prime greater than n, which contradicts our initial assumption. Or, P isn't prime. In this case, P is factorizable-- and by extention, P is prime-factorizable. But because our list of primes was supposed to be complete (up to n), P's prime factors won't be on the list-- they must be greater than n. Again, this contradicts our assumption. And this exhausts all the possibilities.
Our conclusion? The initial assumption was false. We can't name the greatest prime number-- because there isn't one. That's it.
(No more math. I (almost) promise.)
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