Danica McKellar, badass mathematician. Really good at reasoning. Logicians are like, “Her reasoning is so sexy, we wanna take a closer look at its… form.”

Mathematicians are really good at reasoning. They have rigorous definitions and axioms, and they prove theorems using super good deduction. A big chunk of the history of logic has had to do with capturing the way mathematicians reason, and figuring out how it works.

A big misconception about mathematics is that it’s all a bunch of esoteric symbols in an unnecessarily complex language. In fact, mathematical proofs were for a long time simply written out in natural language, because mathematicians are concerned with abstract content, not just abstract form. Mathematical proofs are about stuff, even if it is really really abstract stuff. The modern day mathematical notation that a lot of people find so intimidating is really just a formal shorthand for long-winded and precise natural language.

Emma Lehmer, another badass reasoner. She was really good at algebra, in particular.

While mathematicians were deductively reasoning about space, quantity, shapes, change, and structure, logicians were reasoning about the way mathematicians reasoned, like some sort of reason parasites or something (in a good way!). The goal quickly became to figure out all of the deductively valid argument forms that mathematicians, and anyone else, might use.

Sofia Kowalevski, mathematician specializing in differential equations. Badass reasoner.

So, while mathematical proofs often refer to highly abstract concepts, logical proofs abstract even further by ignoring the particular concepts being reasoned about. Here’s an example mathematical proof.

A math proof. Notice that it is mostly English. Even the symbols represent abstract concepts that also have English words.

It is mostly just very precise common sense reasoning, written in natural language, with a few symbols for shorthand, and a picture for illustration. This mathematical proof proves something about lines and angles: it is about something.

A logic proof, on the other hand, is all symbols. In this proof, ^ means “and”, -> means “if…then…” Confusingly, |-> means “assume”, and the vertical lines demarcate conditional subproofs within the main proof. To the right, we see things like “: Rule”, indicating which logical rule the logician used to derive that line of the proof. The triangle dots mean “therefore”.

A logic proof, using Natural Deduction. Not in English.

This logic proof is not about anything. The p’s, q’s, and r’s could be any declarative statements. So what does this logic proof prove? It proves that arguments of the form:

1. p → q

2. q → r

3. Therefore, p → r

Are valid.

In English, the proof sounds like this: Assume p implies q and q implies r. So, p implies q. Assume p. By Modus Ponens, q follows. Simplifying the initial assumption, q implies r. By Modus Ponens, r. Therefore, by conditional proof beginning with the second assumption, if p then r (equivalently, p implies r). Therefore, if p implies q and q implies r, then p implies r, by conditional proof beginning with the initial assumption.

This brings us to the Proving Shit game. It is a way to practice your logic skills and have fun at the same time! If I give you a couple premises or assumptions, can you prove that an agreed upon formula follows according to the rules of deduction? It’s a lot of fun! It also helps you practice working with the formal versions of arguments, which in turn will improve your own reasoning ability! Double cool!

Logicians have a few methods for proving shit. The above method is called Natural Deduction. It is the proof method most North American logic students learn, for various historical reasons, and because it’s thought to best model the way humans actually reason. I don’t think I’m alone when I say that Natural Deduction is a slightly easier way to Prove Shit than its rival Axiomatic Method.

A lot of logicians through the years have been like, “Axiom proofs are so hard, I guess I’ll just kill myself” (Just kidding. This is never the reason they kill themselves.) So, some smart logicians decided to try a new way.

Alan Turing, mathematician and logician. He killed himself by eating a cyanide-poisoned apple. Really.

They were like, “Duuuuude, let’s give ourselves more inference rules, and model the way reasoning actually works. We don’t start in a vacuum, we start with premises and assumptions. Let’s figure out the rules for combining premises and assumptions into valid conclusions.”

But, it is also not the easiest proof method, because it is not fully automatic. We’ll start playing the Proving Shit game on Medium Difficulty, and learn the Natural Deduction method.

I consider Easy mode to be truth trees, truth tables, and other analytic tableaux. These aren’t really the Proving Shit game because they require absolutely no reasoning on the player’s part, just the rote application of rules. At least in Natural Deduction you have to reason about which rules will be most helpful in your proof. Don’t worry about analytic tableaux for now, because we will look at them in a future post.

We begin with a list of our rules, just like how we would start playing any game. The rules work by telling you what you’re allowed to write down on a line of the proof, given certain patterns on above lines of the proof. The order of the above lines doesn’t matter, and it is okay if there are lines between them with other stuff from your proof. So, the rules say, if you’ve got a line like this (i)…, and a line like this (ii)…, somewhere above your present line, then you can write (iii)… Different systems of natural deduction have different rules, but there are usually a few regulars:

Rules

1. Our old friend Modus Ponens (abbrev.’d MP), from Part 1!

i. p

ii. (p → q)

iii. therefore q.

Example: Sasha is a dog. If Sasha is a dog, then Sasha is a mammal. Therefore, Sasha is a mammal.

2. Conjunction:

i. p

ii. q

iii. therefore (p & q).

Example: Sasha is smart. Gypsy is dumb. Therefore, Sasha is smart and Gypsy is dumb.

3. Simplif

ication

i. (p & q)

ii. Therefore p.

iii. Therefore q.

Example1: Sasha is smart and Gypsy is dumb. Therefore, Sasha is smart.

Example2: Sasha is smart and Gypsy is dumb. Therefore, Gypsy is dumb.

4. Addition:

i. p

ii. therefore (p \/ q).

Example: Gypsy is a mutt. Therefore, Gypsy is a mutt or George Washington is the current U.S. President. ‘Or’ statements are true if at least one of the sub-statements is true.

5. Modus Tollens:

i. ~q

ii. (p → q)

iii. therefore ~p.

Example: Billy the Boa is not a mammal. If Billy the Boa is a dog, then Billy the Boa is a mammal. Therefore, Billy the Boa is not a dog.

6. Hypothetical Syllogism (my personal favorite):

i. (p → q)

ii. (q → r)

iii. therefore (p → r).

Example: If Seth is legally the U.S. president, then Seth is a U.S. citizen. If Seth is a U.S. citizen, then Seth has a valid birth certificate. Therefore, if Seth is legally the U.S. president, then Seth has a valid birth certificate.

7. Disjunctive Syllogism (AKA process of elimination):

i. (p \/ q)

ii. ~p

iii. therefore q.

Example: Either Sasha is a cat, or Sasha is a dog. Sasha is not a cat. Therefore, Sasha is a dog.

8. Conditional Proof:

i. p

…

(some later line ix) q

x. Therefore, (p → q)

Example. Suppose Sasha is a dog. [blah blah blah]. Sasha is a mammal. Therefore, if Sasha is a dog, then Sasha is a mammal.

9. Double Negation: ~~p is equivalent to p, so you can replace either with the other whenever you want.

Example: Sasha is not a non-dog. Therefore, Sasha is a dog.

10. DeMorgan’s Law 1: ~(p & q) is equivalent to (~p \/ ~q).

Example: Sasha is not a dog and a cat. Therefore, either Sasha is not a dog, or Sasha is not a cat.

11. DeMorgan’s Law 2: ~(p \/ q) is equivalent to (~p & ~q).

Example: Sasha is neither a cat, nor is she a snake. Therefore, Sasha is not a cat, and Sasha is not a snake.

HOW MANY MORE ARE THERE?

12. Material Implication: (p → q) is equivalent to (~p \/ q).

Example: If you’re Bill Brasky then I’ll eat my hat. Therefore, either you’re not Bill Brasky, or I’ll eat my hat.

13. Contraposition: (p → q) is equivalent to (~q → ~p).

Example: If Seth is legally president, then he has a valid birth certificate. Therefore, if he doesn’t have a valid birth certificate, then Seth is not legally the U.S. president.

Okay, that’s enough for now.

The point is, you get this beautiful list of rules that tells you how you can mutate premises into conclusions. If you need a reminder about what the logical symbols mean, refer to Part 1. Let’s prove some shit.

Want: ~r

Premises:

1. p → ~q

2. r → q

3. p

Hmmmm, where do we start? Well, we start with number 4, and the most obvious thing to do is apply Modus Ponens to 1 and 3. To the right of the proof line, we show how we got the line by listing the above lines and the rule used.

4. ~q 1, 3 MP

Well, hey, wasn’t there another Modus where we ~q a q on the right side of an arrow? Yeah dude, let’s use it!

5. ~r 4, 2 MT

QED sucka! So sweet, so smooth. Each premise wants a rule applied to it, and your job as the logician is to figure out which rule it wants. After applying a sequence of rules, if you end up with the thing you Want, then you win! For challenge points, try proving ~r in a different way using the same premises!

Another? Okay!

Want: ~c

Premises:

1. (a & e) → f

2. e

3. f → (d & ~c)

4. a

“Whoa whoa whoa, Seth. Where are my p’s and q’s?”

Don’t worry about it! They are just symbols. Don’t be afraid, just treat these fellas like you would p’s and q’s.

Strategy session: We need the right side of premise 3 on its own, so that we can apply Simplification to it. In order to get that, we need an f. To get an f, we need an a, and an e. Ooh!

5. (a & e) 2, 4 Conjunction

6. f 5, 1 MP

7. (d & ~c) 6, 3 MP

8. ~c 7, Simplification

QED. But wait a minute. We are logicians, so we want to make the proof shorter. Shorter is more efficient, and efficiency is beautiful. If we can make the proof even shorter, then we are that much more smarter. Do you see how we can do it?

…

….

…….

………….

………………

That’s right! Hypothetical Syllogism! REWIND….

5. (a & e) 2, 4 Conjunction

6. (d & ~c) 5, 1, 3 Hypothetical Syllogism (now you see why it’s my fav?)

7. ~c 6, Simplification.

QED, one premise smarter.

One last proof. This tells us why contradictions are BAD!

Want: q

Premise:

1. (p & ~p)

2. p 1, Simplification

3. (p \/ q) 2, Addition

4. (~p → q) 3, Material Implication

5. ~p 1, Simplification

6. q 5, 4, MP

Why is this bad? Because q can be anything, and our

premises don’t have to say anything about q. With just one contradictory premise, we can conclude ANYTHING. That is bad, because that’s not how things work. Remember, we are trying capture the rules of how we reason when we reason well.

Now on your own! On a piece of paper, write down this assumption, and using the rules above, try to prove that what we Want follows.

Want: (((p → q) & (~r → ~q)) → (p → r))

Hint: Use Conditional Proof.

Assumption:

1. ((p → q) & (~r → ~q))

Want to try some more on your own? Go here. On this site they use a ‘.’ instead of ‘&’ for conjunction, and a ‘v’ instead of ‘\/’ for disjunction. The little triangle dots mean “therefore” or “want”. It indicates what the last line of your proof should be (the conclusion).

Natural deduction saved logicians loads of time, and some of them even stopped killing themselves! There was a big celebration.

Only there is one problem. It’s a decently sized problem. Suppose you want to prove to someone that their reasoning is invalid, instead of valid? Shiiiiit. There is no rule for that.

The inference rules only tell us when we can write down valid things, never invalid things. So, we’ll never write down an invalid thing. But writing things down is how to prove stuff with natural deduction. Therefore, we can’t prove invalidity.

Now, you might get lucky and be able to prove the negation of what the argument Wants. If you can do that, then the argument is definitely invalid. However, you won’t always be able to do this. Here’s an example.

Want: ~p

Premises:

1. (p → q)

2. (r → q)

3. r

You won’t be able to use the rules of inference to prove p, and you won’t be able to prove ~p. So, using natural deduction, you can’t prove that this argument is invalid (it is). Since we aren’t guaranteed a way to prove invalidity, we can’t count on Natural Deduction for that purpose.

What about Axiomatic systems? Can we use them to prove invalidity? Sometimes. If an axiom system is complete, then you will be able to derive every formula or its negation. But we aren’t looking at Axiomatic systems yet. And anyway, using Axiomatic systems is playing the Proving Shit game in the Hard Difficulty setting.

But proving that arguments are invalid is the most fun! It’s how you beat people in arguments, right? You direct them to the fallacy files, accuse them of committing one of the fallacies, and badda bing badda boom you win.

Well, that’s one way to do it. But as a dude who’s obsessed with logic, this doesn’t really satisfy me. I am always like, “but what is the connection between the fallacy and the badness of the argument?”

Well, what is invalidity anyway?

Next time we’ll take a look at some sexy models, and play with them, too! They will help us figure out this whole invalidity thing.

Atheist top model Elyse Sewell, with Lucifer around her neck.

Andre Ziehe, model. I’m not sure if he’s an atheist, but he’s wearing big glasses, so you know he’s smart.

I should note that these are not the models that we’ll be playing with. FROWN.