Logic: More than just a list of fallacies (Part 3)
April 2, 2013
Welcome back. Last time we looked at how logicians decided they wanted to unlock the secrets of mathematicians: Proving Shit. We saw how natural deduction is one of the easier ways logicians model the rigorous deductive reasoning that goes into a mathematical proof, and we played around with it. Natural deduction allows us to prove that a valid argument is valid, but we were left wondering how we could prove that an invalid argument is invalid. After all, half the fun of arguing with someone is pointing out that their reasoning is flawed. The other half is feeling all serious and philosophical.
So, how do we prove that an argument is invalid?
Pointing at the fallacy files like a bunch of grunting apes won’t do (though this is also fun!).
That doesn’t show why it’s invalid, and it doesn’t show why it’s bad. Not all invalid arguments are bad (induction works well for the most part, but it is not deductively valid), and not all valid arguments are good (circular reasoning is valid, but it is usually considered bad reasoning).
Enter the truth table. A truth table is way to model an argument and check it for validity and invalidity. The cool thing about truth tables is that they always tell you one way or the other, and they are always right.
The most difficult part of working with truth tables has to do with setting them up and memorizing a few simple rules. But don’t worry, the rules are way easier to remember than the ones we used in natural deduction.
Once you’ve got the formalized argument, you make a list of the sentence letters that get used as variables, i.e., the p’s and q’s. Each letter gets its own column of the table.
Then, you put each premise and the conclusion at the top of its own column. Finally, you use the rules we’re about to memorize to fill in the table and check for the Mark of Invalidity.
1. Sasha is a mammal.
2. If Sasha is a dog, then Sasha is a mammal.
3. Therefore, Sasha is a dog.
Intuition check: Do you think this argument is valid?
Step 1: Formalize.
2. p → q
3. Therefore, p
Step 2: List the letters.
“Hey wait a minute, what are those T’s and F’s doing there?
“This wasn’t part of the agreement! You said each letter gets its own column, not that a bunch of T’s and F’s get randomly filled in beneath them! I hate logic! I quit!”
Chilllllll out, dude. Clearly you’re not a golfer. We are just working ahead a little. What makes you think the T’s and F’s are random? Take a closer look at them, noting that T stands for True, and F stands for False…
Remember, logicians don’t really care whether an individual claim is true or false, they only care about the way that claim relates to the other claims in the arguments. So, in our argument, we’ve got two sentence letters, and each of those statements can be true or false. That means there are four possible combinations of the statements, corresponding to each row in the truth table.
Step 3: Give each premise and the conclusion a column.
That was easy enough! Now we need to look at some rules.
“Rules?! I hate memorizing rules!”
This isn’t Post-Modern Literary Criticism: There are rules!
The rules tell us whether the premises and conclusion are true in each row. You could also think of it this way: given the way the world is according to the row, is the premise/conclusion true or false?
It depends on the logical connective used in the premise. We are using the connectives \/ (“or”), & (“and”), → (“if… then…”), and ~ (“not”). Here are the rules for each. For the most part they make sense, but you will have to memorize them.
When is (p & q) true? When p is true, and q is true; otherwise, it is false. That’s what the truth table rule says.
When is (p \/ q) true? When at least one of them is true. You might think, “hey, either p or q… shouldn’t that be false if they are both true?” Strictly speaking, there are two “OR"s: inclusive, and exclusive. An inclusive OR is true if both of its statements are true, while an exclusive OR is not. In logic, we are mostly concerned with the inclusive OR, so that is the one we assume we’re using. If we ever use an exclusive OR, it will be abundantly clear, and specially noted.
When is (p → q) true? This one is a little bit weird, as you might have noticed. It makes sense that it’s true when p and q are both true, but what gives with the other rows? Think of it like this: “if p, then q” statement is basically saying “the truth of p guarantees the truth of q.” That statement is false when p is true, but q is false. Ask, “Do I go from a true p to a false q?” If the answer is yes, then the if/then is false. If the answer is no, then it is true. When p is false, the answer is no, because you don’t go anywhere from a true p.
When is (~p) true? When p is false.
Okay, now that we have our rules, we can apply them to the truth table we’ve been making for the argument.
Step 3: Fill in the table.
In this argument, it’s easy to fill in premise 1 and the conclusion: each row is the same as the sentence letter on the left. If we are in the top row world, then we are in the world where p is true and q is true, so premise 1, q, is true, and the conclusion, p, is true, too.
And we just followed the rules for → to fill in premise 2’s rows. What now?
Step 4: Check for the Mark of Invalidity.
Recall that logic is about jumping through BeliefSpace. We said in Part 1 that logic is like a map through BeliefSpace, saying “if you are on a true position in BeliefSpace, and you follow my rules, you will never jump onto a false position.” False positions have spiked pits, so we want to avoid jumping on them. The Mark of Invalidity tells us how.
The spot we’re jumping to with this argument is the conclusion, p. We want to avoid jumping from a true spot to a false spot, so let’s look at the rows where the conclusion is false: the bottom two rows. Now, we ask, in either of those rows, are ALL of the premises true (REMINDER: the far left columns are not premises, but variables from the argument. The premises are the numbered columns)? Yes, in fact, the penultimate row is such a row: in that row, q is true and p → q is true, too. But p, our conclusion, is false in that row. So, if that’s the way the world is, then our argument would be taking us from a true position in BeliefSpace to a false one. We’d be done for! Skewered!
That’s the Mark of Invalidity: A way the world could be where the premises are ALL true, but the conclusion is false. That is what invalidity is, and that’s why invalid arguments are bad: they just might take you from true positions to false ones. Not only are there spikes at the bottom of false pits, but there are like, these old trolls that smell really bad, and all they want to do is talk to you about their right-wing political views, about how we need to deregulate logic and let the market decide. It is horrible down there. Please learn your logic.
After following these 4 simple steps, if you use the rules correctly, you are guaranteed a correct answer one way or the other. We just proved that the argument is invalid. If the truth table you construct lacks the Mark of Invalidity, then it is Valid. Crucially, there are weird cases. We might have an argument with contradictory premises, so that they can’t ALL be true in the same row: VALID (this property of classical logic is called explosion, for realz). We might have an argument with a conclusion that can’t possibly be false, no matter what the premises are: VALID. The argument is invalid when it has a row that goes from ALL TRUE PREMISES to a FALSE CONCLUSION; otherwise it is valid. Even the row with all false premises and a true conclusion is valid. If you are standing on a false position, logic is like,
“Hey, you’re fucked buddy. Do whatever you want. I only promised to take you from True to True. What is this, some sort of interrogation? Who do you work for?! NSA? CIA? The Vatican? I never promised you anything if you hand me a false position… Get real, man. Get real.”
Logic can be a bit feisty sometimes.
And we’ve just been talking about classical propositional logic, as it’s applied to reasoning about arguments! There are shitloads of other logics, and shitloads of applications!
“There are other logics than these???”
Yes, Dear Reader. And if you come with me, I will show them to you.
Above: Sonja Smets, doing unspeakable things with logic.
Below: Johan van Benthem, liking it.
About the Author: Seth KurtenbachSeth Kurtenbach is pursuing his PhD in computer science at the University of Missouri. His current research focuses on the application of formal logic to questions about knowledge and rationality. He has his Master's degree in philosophy from the University of Missouri, and is growing an epic beard in order to maintain his philosophical powers. You can email Seth at Seth.Kurtenbach@gmail.com or follow him on Twitter: @SJKur.
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